Route Reciprocity and the Logarithmic Yukawa-Curvature Plane of Charged Fermions in the SU(15)_p Composite Theory: Four-Dimensional Composite Dynamics, Five-Dimensional Spectral Reconstruction, and a Prospective Six-Dimensional Completion

1. Introduction: The Physical Question and the Hierarchy of Claims

The charged-fermion spectrum contains large hierarchies, so phenomenological formulas can be made visually attractive by introducing enough adjustable parameters. The purpose of this article is deliberately narrower. A single second-difference relation among the charged-lepton, down-quark, and up-quark Yukawa spectra is taken as a weak empirical clue, and the question is whether the same relation emerges as a sharply defined operator condition in an anomaly-consistent confining preon theory.

The microscopic framework is the chiral $\mathrm{SU}{\left(15\right)}_p$ compositeness model, with the Pati–Salam spectator group

$$G_{\mathrm{PS}}=\mathrm{SU}{\left(4\right)}_{\mathrm{PS}}\times \mathrm{SU}{\left(2\right)}_L\times \mathrm{SU}{\left(2\right)}_R.$$

The central result does not depend on an extra spacetime dimension. Four dimensions contain the microscopic operator problem and the decisive physical correlator. Five dimensions enter only as an effective spectral reconstruction. Six dimensions are studied as a prospective ultraviolet environment for chirality and localization. The tested seven-dimensional branch is retained as a consistency test because its zero-mode multiplicity fails the intended count.

The article separates five logically different levels.

1.

Phenomenological target. The observable ${\Delta }_{\Pi }$ is a fixed combination of three logarithmic Yukawa curvatures. Its current standardized departure from zero is only about $1.56\sigma$.

2.

Operator structure. The Pati–Salam route basis gives the exact identity ${\Delta }_{\Pi }=C_{21}-C_{12}$, and the unchanged field content admits explicit crossed source representatives.

3.

Family alignment. The charged-sector operator must select the basis-covariant median family direction $D_{+}$—that is, distinguish the median eigendirection from the two outer eigendirections in the ordered three-family space. This condition is independent of route reciprocity.

4.

Composite route dynamics. Hidden states, spectral measures, and route-resolved transfer kernels determine whether the physical correlator suppresses the route-breaking component $K_{\perp }$.

5.

Low-energy matching. A nonzero route-even and vanishing route-odd amputated matching vertex must connect the microscopic correlator to the charged-fermion operator, and the regular route-odd remainder must vanish in the same parity-resolved scheme.

No result at one level is used as a substitute for a missing result at another.

Three statements that must not be conflated.The notation permits three superficially similar but logically different statements:

$${\Delta }_{\Pi }\simeq 0,{\Delta }_{\Pi }=C_{21}-C_{12},C_{21}=C_{12}.$$

The first is a weak empirical proximity statement. The second is an exact identity inside the fixed route basis. The third is the dynamical reciprocity condition that the confining theory would have to realize in order to explain an exact plane. Keeping these three statements separate prevents the observed numerical closeness from being mistaken for a derivation of the dynamics.

Logical dependency of the four-dimensional explanation.The empirical clue and the theoretical explanation are separated as

$$\mathrm{mass}\mathrm{data}⟶{\Delta }_{\Pi }\simeq 0,{\Delta }_{\Pi }=C_{21}-C_{12}\mathrm{exactly}.$$

Explaining the exact plane then requires three independent necessary physical conditions (gates),

$$\begin{array}{c}\left[\mathrm{median}\mathrm{family}\mathrm{alignment}{D}_{+}\right]\wedge \left[\mathrm{dynamical}\mathrm{crossed}-\mathrm{route}\mathrm{reciprocity}\right]\\ \wedge \left[F_{+}\left(0\right)\ne 0,F_{-}\left(0\right)=0,C_{-}^{\mathrm{reg}}=0\right].\end{array}$$

All three are necessary, and none follows from either of the other two.

Corrections to Superseded Intermediate Results

... For clarity, three intermediate conclusions that were replaced after internal consistency checks are collected here rather than presented as separate narrative episodes. They concern the statistical weight of the phenomenological clue and two optional higher-dimensional branches; none changes the exact four-dimensional identity ${\Delta }_{\Pi }=C_{21}-C_{12}$ or substitutes for the still-open four-dimensional correlator and matching calculation.

Phenomenological covariance. An earlier $2.23\sigma$ estimate of the standardized displacement from the plane is superseded by $Z_{\Pi }=1.556\simeq 1.56\sigma$. The larger value required a near-maximal residual d–s correlation that was not obtained from a consistent source covariance. The present status is therefore a statistically weak phenomenological clue, not evidence for an exact empirical law; the covariance construction is given in Section 2.5.

Six-dimensional dipole interpretation. The fixed-metric dipole eigenvalue $-0.34796305/R_c^2$ is not a physical tachyon mass. The corresponding eigenfunction lies in the admissible scalar-clock gauge image, and stationary metric completion is Schur-null on that direction. This removes that particular negative fixed-metric mode from the physical spectrum, but it does not prove full dipole stability: the remaining constraint-reduced metric problem is governed by the Birman–Schwinger threshold ${\beta }_0$, which is still uncomputed. The derivation is given in Appendix D.

Seven-dimensional zero-mode multiplicity. The number 405 is the gauge-component multiplicity of the tested all-preon construction, not the final four-dimensional Weyl zero-mode count. A seven-dimensional complex Dirac spinor has eight complex components, and under the tested projection each gauge component yields two selected-chirality four-dimensional Weyl modes. The corresponding count is therefore $2\times 405=810$. The tested seven-dimensional branch consequently fails the intended multiplicity and remains a negative control; no repair has yet passed the representation, anomaly, and Kaluza–Klein checks. The spinor decomposition is given in Appendix E.

This summary records the present scientific status of the three corrected intermediate claims. The detailed sections retain the calculations needed to reproduce the corrected conclusions, but the corrections are not used as independent support for the central four-dimensional theorem.

Several strong operator-algebra questions are already closed. The crossed route tensors can be generated by fixed unchanged-field recoupling words. The effective $-1/2$ color coefficient follows from walled–Brauer projector algebra. The stronger proposal that the two complete kernels are the same closure-preserving microscopic process executed in reverse order is excluded under explicitly stated assumptions. This negative result sharpens rather than weakens the physical hypothesis: the relevant equality, if realized, must be dynamical.

The family direction is logically independent of route reciprocity; the symbols used for family alignment and route breaking are therefore kept distinct throughout. The corrected symmetric spurion defines the median family projector algebraically. Separately, a gauge-neutral ${\overline{6}}_F$ composite potential supplies a global effective-field-theory vacuum-selection branch. A separate four-flavour parent embedding contains a restricted $U{\left(1\right)}_X\times SU{\left(4\right)}_F$ spurion branch. Within that branch, the parent cubic invariant that would induce the light-family determinant operator is forbidden to all polynomial orders. This selection rule is conditional on that embedding; the minimal three-$\psi$ Pati–Salam theory by itself does not supply an exact $SU{\left(4\right)}_F$ symmetry. More general light-family EFT branches admit a finite determinant-tolerant domain. The physical $4D$ theory must still determine which branch is realized and the corresponding renormalized coefficients. The matching condition is likewise independent: route reciprocity alone does not imply a nonzero coupling of the even channel to the charged-fermion operator.

The technical support for these claims is separated by physics rather than by chronology. Appendix B.1 gives the four-route algebra, the 24-field source, the Temperley–Lieb quotient, the walled–Brauer recoupling, and the closure-preserving chronological no-go. Appendix B.2 gives the genuinely confining $1+1$ dimensional $\mathrm{SU}\left(3\right)$ route control; the representation-matched $\mathrm{SU}\left(15\right)$ gap and pole-tracked multi-link calculations are in Appendices B.2.4 and B.2.10. The final finite-moment and matching criteria are in Appendix B.11. Appendix C develops the exact effective five-dimensional quotient, Schur–Feshbach reduction, Jacobi–Robin reconstruction, and finite-time route certificates. Appendix D contains the six-dimensional background and fluctuation analysis, and Appendix E gives the corrected seven-dimensional zero-mode count.

1.1. Core Results and Remaining Physical Inputs

The central claims can be summarized before the technical development. The table deliberately separates exact results from physical quantities that still require the confining theory.

Item	Status	Meaning
${\Delta }_{\Pi }=C_{21}-C_{12}$	established	Exact four-dimensional coordinate/operator identity in the fixed route normalization.
24-field source and crossed recoupling	established	Both crossed routes exist in the unchanged chiral field content; the nontrivial $-1/2$ coefficient follows from fixed recoupling algebra.
Strict closure-preserving chronological $Z_2$	excluded	Under the stated closure assumptions the two complete crossed kernels are not the same microscopic process executed in reverse order. Any physical equality must therefore be dynamical.
Median family vacuum	conditional	The symmetric spurion fixes the median direction basis covariantly. A gauge-neutral light-family ${\overline{\mathbf{6}}}_F$ composite has a global stable domain. In a separate four-flavour parent embedding, a restricted $U{\left(1\right)}_X\times SU{\left(4\right)}_F$ branch conditionally forbids the cubic invariant that would descend to ${\kappa }_{det}$.
Physical four-dimensional propagation and matching	open	${\Delta }_{-}^{\left(15\right)}$, $K_{\perp }$, ${\eta }_{15}$, the physical amputated $F_{\pm }\left(0\right)$, and the regular odd remainder $C_{-}^{\mathrm{reg}}$ require the renormalized route correlator and three-point vertex.
5D / 6D / 7D roles	effective / prospective / negative control	Five dimensions provide spectral reconstruction, six dimensions are an optional ultraviolet completion with one sharply defined metric gate, and the tested seven-dimensional branch fails the zero-mode count.

1.2. What Is Still Missing

No renormalized first-principles ensemble for the explicit point-split route sources has yet been generated in the full four-dimensional chiral $\mathrm{SU}{\left(15\right)}_p$ theory. Consequently the physical four-dimensional values of ${\Delta }_{-}^{\left(15\right)}$, $K_{\perp }$, ${\eta }_{15}$, the source-accessible pole residue, and the absolute matching form factors remain uncomputed.

This statement should not be read as saying that the dynamical mechanism has been left untested. A substantial hierarchy of simpler confining and spectral control problems has already been solved. Their role is to test individual ingredients of the proposed mechanism under conditions in which the gauge constraints, route ordering, hidden-state elimination, or volume dependence can be handled exactly or numerically to high precision. What remains missing is the final transfer of those ingredients into the physical four-dimensional source correlator and matching vertex.

1.3. Dynamical Controls Completed Before the Full Four-Dimensional Calculation

The control calculations deliberately answer different questions. They should therefore be viewed as a sequence of mechanism tests rather than as multiple estimates of the same four-dimensional number.

Control problem	Result	What it establishes
$1+1$D confining $\mathrm{SU}\left(3\right)$ baryon-transfer model	At the first nonzero, third order in hopping, equal route-sensitive excitation costs give $A_{12}=A_{21}$ exactly; for unequal costs ${ϵ}_{\mathrm{rt}}^{\mathrm{SU}\left(3\right)}=({\delta }_1-{\delta }_2)/(2{\Delta }_F+{\delta }_1+{\delta }_2)$.	A genuine non-Abelian confining Hamiltonian can dynamically suppress memory of which constituent crosses first; approximate reciprocity does not require the intermediate configurations to be identical.
Representation-matched $1+1$D open-chain $\mathrm{SU}\left(N\right)$ control	After exact axial-gauge/Gauss-law reduction, ${\Delta }_{-,L}^{\left(N\right)}={\kappa }_{E}N/2+O\left(1\right)$ uniformly in chain length. For the normalized additive deformation, ${\eta }_{N,L}^{\mathrm{ctrl}}\sim 2\left|x\right|/\left({\kappa }_{E}N\right)$.	The confinement-scale odd gap survives dynamical links and large volume in the actual fundamental–fundamental–conjugate-symmetric representation pattern relevant to $\mathrm{SU}{\left(15\right)}_p$.
Point-split, plaquette, and cross-junction controls	The coincident one-link chronology collapses to rank one; finite point splitting restores a non-null odd sector. Minimal plaquette lightening is tiny at large N, while the cross-junction recoupling contains unsuppressed $O\left(1\right)$ color matrix elements.	Distinguishes several logically different effects: source oddness, odd spectral support, a light odd pole, and a route-breaking numerator are not the same quantity. It also excludes a universal claim that every nonlocal recoupling is $1/N$ suppressed.
Pole-tracked Schur–Feshbach open-chain response	Exact sparse differentiation through $L=30$ gives ${\mathcal{T}}_{\delta t,\infty }^{\mathrm{lin},\mathrm{ctrl}}\simeq 0.01279$ and ${\mathcal{T}}_{y,\infty }^{\mathrm{lin},\mathrm{ctrl}}\simeq 0.045586$; the mode-resolved cancellation factor approaches about $1.46\times {10}^{-2}$.	A small route-breaking response can survive volume enlargement with finite conditioning; evaluating the kernel at a frozen rather than tracked pole can give a misleading enhancement.
Local $2+1$D and $3+1$D strong-coupling dressing	The displayed $\mathrm{SU}\left(15\right)$ shared-link controls give ${\rho }_J=1+O\left({10}^{-5}\right)$ for the local parity denominator, but an additive route-odd numerator is not filtered.	Extra transverse plaquettes by themselves do not solve reciprocity. The strong filtering seen in the long-chain control is a spectral/resolvent effect, not a generic dimensional effect.
Dynamical screening / string-breaking control	Exact recoupling weights are $w_{F\leftarrow S}=2/\left[N(N+1)\right]$ and $w_{F\leftarrow A}=2/\left[N(N-1)\right]$. In the common-tower control, screening eventually removes unscreened long-string protection: the corresponding route ratio tends to zero at asymptotically large separation.	Large-N string-energy arguments cannot be extrapolated through string breaking without checking the screened spectrum. This is a control caveat only: the canonical 24-field source has no inter-junction precolor Wilson string.
Canonical source-metric / variational control	The exact source Gram matrix $\widehat{G}=\left(\begin{array}{cc}56& 24\\ 24& 56\end{array}\right)$ has eigenvalues 80 and 32 and condition number $2.5$. In the declared long-chain control, a contaminant-null variational combination increases the odd-pole residue from $5.46\times {10}^{-5}$ to about $0.498$ while suppressing the designated contaminant to numerical zero.	Canonical whitening is well conditioned in the explicit source algebra, and a multi-profile basis can separate a small physical odd component from a large source contaminant. The numerical gain is a control result, not a prediction for the four-dimensional residue.
Finite-moment / block-Krylov and effective Jacobi controls	Positivity, extremal moment bounds, flat-extension criteria, and the common-reflection defect can be evaluated without an uncontrolled $t\to \infty$ extrapolation.	The spectral side of the decision problem has finite-data certificates; exact uniqueness requires the stated closure conditions.

Taken together, these calculations establish that the proposed route-reciprocity mechanism is neither a purely algebraic possibility nor an untested qualitative picture. Confinement-induced suppression, large-N gap scaling, hidden-sector dressing, pole tracking, finite-volume conditioning, and several important failure modes have all been examined in controlled settings. They do not prove that the physical four-dimensional $\mathrm{SU}{\left(15\right)}_p$ vacuum realizes the same numerical suppression. The remaining decisive step is therefore narrower: compute the renormalized four-dimensional route correlator and three-point matching vertex for the explicit source family and test whether the control mechanisms survive in the physical theory.

The corresponding calculations are given in Appendix B.2 for the $\mathrm{SU}\left(3\right)$ Hamiltonian and exact two-site correlator, Appendix B.2.4 for the representation-matched $\mathrm{SU}\left(N\right)$ and $\mathrm{SU}\left(15\right)$ gap control, Appendix B.2.10 for the pole-tracked multi-link analysis, and Appendix G for the supplementary volume, cancellation, family, and bridge scans.

2. Phenomenological Motivation: Logarithmic Curvature of the Mass Spectrum

The present mass-plane discussion has two logically distinct inputs. Historically, the combination later denoted ${\Delta }_{\Pi }=A_{\ell }+2A_d-A_u$ first emerged in the shell-motivated rank-and-scale study of Ref. [10]. The same combination is subsequently re-expressed as a route-sensitive observable and analyzed in the preon-theory setting. The analysis therefore uses the shell-motivated model only as phenomenological motivation. No interpretation as a literal shell spectrum is required for the preon-theory argument itself.

2.1. Exact Change of Coordinates for Three Generations

For a charged sector $s\in \{\ell ,d,u\}$ take, respectively, $(e,\mu ,\tau )$, $(d,s,b)$, and $(u,c,t)$. Define

$$z_{s,g}\left(\mu \right)=ln{y}_{s,g}\left(\mu \right),g=1,2,3.$$

The three numbers $z_{s,1},z_{s,2},z_{s,3}$ can be written exactly as

$$z_{s,g}=I_s+B_s(g-2)+A_s{\xi }_g,({\xi }_1,{\xi }_2,{\xi }_3)=(1,-1,1),$$

where

$$\begin{array}{cc}\hfill I_s& ={\displaystyle \frac{1}{4}}(z_{s,1}+2z_{s,2}+z_{s,3}),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill B_s& ={\displaystyle \frac{1}{2}}(z_{s,3}-z_{s,1}),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill A_s& ={\displaystyle \frac{1}{4}}(z_{s,1}-2z_{s,2}+z_{s,3}).\hfill \end{array}$$

(3)

Hence

$$A_s={\displaystyle \frac{1}{4}}ln{\displaystyle \frac{y_{s,1}{y}_{s,3}}{y_{s,2}^2}}.$$

The transformation is invertible. In matrix form,

$$\left(\begin{array}{c}{I}_s\\ B_s\\ A_s\end{array}\right)=\left(\begin{array}{ccc}\frac{1}{4}& \frac{1}{2}& \frac{1}{4}\\ -\frac{1}{2}& 0& \frac{1}{2}\\ \frac{1}{4}& -\frac{1}{2}& \frac{1}{4}\end{array}\right)\left(\begin{array}{c}{z}_{s,1}\\ z_{s,2}\\ z_{s,3}\end{array}\right),det={\displaystyle \frac{1}{4}}\ne 0,$$

so that

$$z_{s,1}=I_s-B_s+A_s,z_{s,2}=I_s-A_s,z_{s,3}=I_s+B_s+A_s.$$

This inverse formula is useful conceptually: $I_s$ fixes the overall logarithmic level, $B_s$ the endpoint slope, and $A_s$ the genuinely second-difference information. No information is lost by using these coordinates.

If the three Yukawa eigenvalues form a geometric progression, then $A_s=0$. A common multiplicative normalization of the sector changes $I_s$ but leaves $A_s$ unchanged.

Remark.$A_s$ measures the departure of the second generation from the geometric mean of the first and third generations in logarithmic coordinates. It is not an additional fit parameter; it is an exact coordinate transformation in the three-dimensional space of logarithmic Yukawa values.

2.2. Historical Origin of the Plane in Ref.  [10]

The curvature plane was not discovered in a blind test of the present preon theory. It emerged a posteriori from the earlier shell-motivated rank-and-scale analysis of the nine charged-fermion masses in Ref. [10]. That study first organized each charged sector in the same logarithmic basis used here, with an intercept, a first-difference slope, and a discrete curvature. It then compared low-rank charged-sector models by exact rank constraints, conditional displacement, leave-one-out (LOO) prediction, and nested model selection.

The plane itself arose from the complete six-parameter integer-curvature family conventionally denoted $M_{B+\ell T_n}$. Here $M_B$ denotes the shell/rank backbone of Ref. [10], while the $\ell T_n$ label identifies the integer-curvature completion indexed by n. In curvature space the family is

$$(A_{\ell },A_d,A_u)=\lambda (3n+2,-n-1,n),n\in {\mathbb{Z}}_{>0},$$

where $\lambda$ is an overall curvature scale. The first two phenomenologically relevant members are

$$n=2:A_{\ell }:A_d:A_u=8:-3:2,n=3:A_{\ell }:A_d:A_u=11:-4:3.$$

The $n=2$ model is the conservative predictive member: nested selection over $1\le n\le 15$ chose it in eight of nine LOO folds. It has complete $9/9$ LOO coverage, a largest conditional shift of about $1.25\%$, and a worst LOO error of about $7.71\%$. The $n=3$ member is the best central-data minimax ray, with a largest conditional shift of about $2.25\%$ and a smaller worst LOO error of about $4.99\%$, but it is more selection-sensitive. These numbers describe predictive performance of the phenomenological mass model; they are not microscopic predictions of the present $\mathrm{SU}{\left(15\right)}_p$ theory.

The common plane is not an additional fit parameter. It follows algebraically because the entire integer family can be written as

$$(3n+2,-n-1,n)=n(3,-1,1)+(2,-1,0).$$

Both fixed directions are orthogonal to

$${\mathbf{n}}_{\Pi }=(1,2,-1),$$

so every member of the family satisfies

$${\mathbf{n}}_{\Pi }·(A_{\ell },A_d,A_u)=0,A_{\ell }+2A_d-A_u=0.$$

Thus the historical discovery was a data-driven model-selection result: the normal ${\mathbf{n}}_{\Pi }$ was recognized only after the successful curvature family had been identified. The look-elsewhere issue associated with that discovery is therefore real and is not removed retroactively by the present theory.

Figure 1 reproduces the predictive comparison from Ref. [10]. The two integer-ray models $M_{B+\ell T_2}$ and $M_{B+\ell T_3}$ lie in the lower-left part of the plot and therefore combine small conditional displacement with comparatively small worst-case LOO error. The same figure also displays the distinct targeted strange-quark model $M_{B+dLR}^{s14}$, defined by $A_d=B_d/14$. That model has a $1.99\%$ largest conditional shift and reconstructs s with a $0.51\%$ LOO error, but its global worst-state LOO error is $25.06\%$; it is therefore an excellent targeted-s reconstruction, not the origin of the curvature plane and not the globally most stable model. The distinction is important because the plane and the targeted-s rule answer different phenomenological questions.

The logical status of the present result is therefore deliberately two-stage. Historically,

$$\mathrm{charged}-\mathrm{mass}\mathrm{fits}⟶\mathrm{integer}\mathrm{curvature}\mathrm{family}⟶\Pi :A_{\ell }+2A_d-A_u=0.$$

In the present construction, by contrast,

$$\mathrm{fixed}\mathrm{Pati}--\mathrm{Salam}\mathrm{route}\mathrm{algebra}⟶{\Delta }_{\Pi }=C_{21}-C_{12}.$$

The second statement removes any further freedom to scan the plane normal inside the specified route basis, but it does not turn the original observation into a preregistered prediction. The empirical plane remains a weak post-selected clue; the theoretical content is the independent operator meaning assigned to that already identified combination.

2.3. The Plane $\Pi$ and Its Central Displacement

Define

$$\mathbf{A}=(A_{\ell },A_d,A_u),{\mathbf{n}}_{\Pi }=(1,2,-1),$$

$${\Delta }_{\Pi }={\mathbf{n}}_{\Pi }·\mathbf{A}=A_{\ell }+2A_d-A_u.$$

The plane $\Pi$ is the set ${\Delta }_{\Pi }=0$. For the PDG-2024 / Antusch–Hinze–Saad central running inputs at $M_Z$ [11],

$$A_{\ell }=-0.6294621,A_d=+0.2465294,A_u=-0.1553674,$$

$${\Delta }_{\Pi }\left(M_Z\right)=+0.0189642.$$

For transparency, the numerical ingredients are collected before any statistical interpretation:

sector s	$A_s$	$e^{4A_s}=y_{s,1}{y}_{s,3}/y_{s,2}^2$	$y_{s,2}/\sqrt{y_{s,1}{y}_{s,3}}$	contribution to ${\Delta }_{\Pi }$
ℓ	$-0.6294621$	$0.080633$	$3.52163$	$A_{\ell }=-0.6294621$
d	$+0.2465294$	$2.68081$	$0.610755$	$2A_d=+0.4930588$
u	$-0.1553674$	$0.537154$	$1.36443$	$-A_u=+0.1553674$
combined	—	—	—	${\Delta }_{\Pi }=+0.0189641$

The last displayed digit differs from the quoted $0.0189642$ only because the tabulated $A_s$ values have been rounded. The third and fourth columns are two equivalent ways of reading the same curvature. From

$$A_s={\displaystyle \frac{1}{4}}ln\left({\displaystyle \frac{y_{s,1}{y}_{s,3}}{y_{s,2}^2}}\right)$$

one obtains

$$R_s\equiv e^{4A_s}={\displaystyle \frac{y_{s,1}{y}_{s,3}}{y_{s,2}^2}},{\displaystyle \frac{y_{s,2}}{\sqrt{y_{s,1}{y}_{s,3}}}}=R_s^{-1/2}=e^{-2A_s}.$$

Thus $A_s=0$ means that the middle Yukawa eigenvalue is exactly the geometric mean of the first and third. Negative $A_s$ places the middle eigenvalue above that geometric mean, while positive $A_s$ places it below. This table is retained because it makes the phenomenological observable directly reproducible from the displayed numbers.

The intermediate arithmetic is useful because it displays what the curvature measures. From $A_s={\displaystyle \frac{1}{4}}ln(y_{s,1}{y}_{s,3}/y_{s,2}^2)$, the ratio $e^{4A_s}$ is exactly one when the middle Yukawa eigenvalue is the geometric mean of the first and third. The three sectoral ratios are

$${\displaystyle \frac{y_{\ell ,1}{y}_{\ell ,3}}{y_{\ell ,2}^2}}=e^{4A_{\ell }}=0.080633,{\displaystyle \frac{y_{d,1}{y}_{d,3}}{y_{d,2}^2}}=e^{4A_d}=2.68081,{\displaystyle \frac{y_{u,1}{y}_{u,3}}{y_{u,2}^2}}=e^{4A_u}=0.537154.$$

Thus the charged-lepton and up-quark middle eigenvalues lie above their respective geometric means, whereas the down-quark middle eigenvalue lies below it. The plane combines these three departures:

$$-0.6294621+2\left(0.2465294\right)-(-0.1553674)=0.0189641,$$

where the last-digit difference from the quoted $0.0189642$ reflects rounding of the displayed $A_s$ values. Once the plane normal is fixed, no additional fit enters this arithmetic; the historical identification of this particular normal was nevertheless data-driven, as explained above. The geometric angular leakage

$${ϵ}_{\Pi }={\displaystyle \frac{|{\Delta }_{\Pi }|}{\parallel {\mathbf{n}}_{\Pi }\parallel \parallel \mathbf{A}\parallel }}$$

is approximately $1.12\%$. This is a geometric statement, not a statistical significance.

2.4. Specificity of the Curvature Channel: First-Difference and Intercept Controls

The three coordinates $(I_s,B_s,A_s)$ separate the overall logarithmic level, the first generation-to-generation slope, and the second-difference curvature. This makes it possible to ask a limited but useful control question: does the same sector-space normal $n_{\Pi }=(1,2,-1)$ also produce an accidentally small number in the other coordinate channels?

Using the displayed PDG-2024 / Antusch–Hinze–Saad central Yukawa eigenvalues at $M_Z$ in the same $\overline{\mathrm{MS}}$ convention gives

$$\begin{array}{cccccc}\hfill B_{\ell }& =4.0913415,\hfill & \hfill B_d& =3.4822764,\hfill & \hfill B_u& =5.9151728,\hfill \end{array}$$

(4)

$$\begin{array}{cccccc}\hfill I_{\ell }& =-8.0732890,\hfill & \hfill I_d& =-7.8453960,\hfill & \hfill I_u& =-5.7933622.\hfill \end{array}$$

(5)

Therefore

$$\begin{array}{cc}\hfill {\Delta }_B& \equiv B_{\ell }+2B_d-B_u=4.0913415+2\left(3.4822764\right)-5.9151728=5.1407215,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\Delta }_I& \equiv I_{\ell }+2I_d-I_u=-17.9707189.\hfill \end{array}$$

(7)

The first result is the clean control. Because

$$B_s={\displaystyle \frac{1}{2}}ln{\displaystyle \frac{y_{s,3}}{y_{s,1}}},$$

$B_s$ is invariant under an arbitrary sector-wide rescaling $y_{s,g}\mapsto c_s{y}_{s,g}$. The same normal that gives the small curvature residual ${\Delta }_{\Pi }=0.0189642$ thus gives a first-difference combination of order five. The near-plane is not a generic property of applying $(1,2,-1)$ to any of the three logarithmic coordinates.

The intercept combination requires a different interpretation. Under a common rescaling of all Yukawa eigenvalues, $y_{s,g}\mapsto k{y}_{s,g}$,

$$I_s\mapsto I_s+lnk,{\Delta }_I\mapsto {\Delta }_I+2lnk,$$

because the sector coefficients $1+2-1$ do not sum to zero. In particular, with $y_f=\sqrt{2}{m}_f/v$ the conversion between an intercept constructed from Yukawa eigenvalues and one constructed from masses in a fixed reference unit contains

$$2ln{\displaystyle \frac{\sqrt{2}}{v\left(M_Z\right)}}=-10.33694$$

for the quoted $v\left(M_Z\right)=248.401\mathrm{GeV}$. Hence ${\Delta }_I=-17.9707189$ is a convention-fixed descriptive coordinate, not a normalization-invariant control. It is retained here to make this limitation explicit and is not used as evidence for the plane.

Statistical status of the controls.${\Delta }_B$ and ${\Delta }_I$ were examined after the curvature relation had been identified. They are therefore post-hoc diagnostic controls, not independent confirmations and not additional contributions to $Z_{\Pi }$. No “many-sigma” claim is attached to their distance from zero. If a standardized $Z_B$ is quoted in a future update, it must be obtained by propagating the same joint source covariance through the B-coordinate Jacobian; marginal errors must not be treated as independent merely because the central value is large.

2.5. Covariance-Aware Distance from the Exact Plane: $Z_{\Pi }\simeq 1.56$

Let ${\sigma }_{{\Delta }_{\Pi }}$ denote the propagated standard uncertainty and define

$$Z_{\Pi }={\displaystyle \frac{|{\Delta }_{\Pi }|}{{\sigma }_{{\Delta }_{\Pi }}}}.$$

A source-level covariance analysis includes not only common ${\alpha }_s$ running but also publicly traceable correlations among quark-mass inputs. The strongest fully specified public anchor used here is Table IV of Bazavov et al. [12]. That table publishes a $5\times 5$correlation matrix, not a covariance matrix, together with the marginal central values and total uncertainties of $m_c\left(3\mathrm{GeV}\right)$ and four quark-mass ratios. The corresponding covariance subblock is constructed explicitly below.

Covariance treatment	${\sigma }_{{\Delta }_{\Pi }}$	$Z_{\Pi }$
Marginal errors treated independently	$0.0179200$	$1.058$
Earlier two-loop ${\alpha }_s$ covariance control	$0.01403$	$1.35$
Five-loop ${\alpha }_s$ running covariance only	$0.01242$	$1.527$
$+$Bazavov-2018 correlation anchor converted to a covariance subblock	$\simeq 0.01227$	$1.545$
$+$additional publicly traceable source overlaps	$0.01219$	$1.556$

The most defensible public-information diagnostic is therefore

$$Z_{\Pi }=1.556\simeq 1.56\sigma ,p_{\mathrm{two}-\mathrm{sided}}\simeq 0.120.$$

As summarized in the introductory correction block, the earlier $2.23\sigma$ estimate is superseded because it effectively required a near-maximal residual d–s correlation that was not derived from a consistent source covariance. An intentionally aggressive stress test in which residual correlations are made Bazavov-like yields $2.214\sigma$, but that number is a sensitivity test, not central evidence.

Correct phenomenological conclusion.With the best public source-informed covariance proxies, the current central residual is only about $1.5$–$1.6\sigma$ away from the exact plane. It is therefore an interesting clue but statistically weak. Any serious microscopic case for the mechanism must come from independent operator, topology, and dynamics, not from overstating the significance of the mass-plane residual.

2.6. Why the Plane Normal Is Fixed by the Route Construction Rather than Scanned over the Data

The phrase “not a fitted mass formula” can be made algebraically precise. Write a general normal in the four-component bookkeeping space as

$$n=(a,b,c,d),$$

where the fourth entry multiplies the auxiliary neutrino-Dirac curvature. A relation restricted to the charged sectors has $d=0$. Requiring the two uncrossed route images to lie in the candidate plane gives

$$n·r_{22}=2a-b=0,n·r_{11}=3a-b+c=0.$$

Hence $b=2a$ and $c=-a$, so, for $a\ne 0$,

$$n\propto (1,2,-1,0)=n_{\Pi }^{\left(4\right)}.$$

The crossed routes then supply an internal normalization check:

$$n_{\Pi }^{\left(4\right)}·r_{12}=-1,n_{\Pi }^{\left(4\right)}·r_{21}=+1.$$

Thus the route algebra singles out the charged-sector normal uniquely up to an overall nonzero factor. After the conventional integer normalization is chosen, the mass inputs enter only through the one scalar observable ${\Delta }_{\Pi }=n_{\Pi }·\mathbf{A}$.

Pre-specification and its precise scope.Within the stated $\mathrm{SU}{\left(15\right)}_p$ operator basis, $n_{\Pi }$ is a model-fixed direction rather than the maximizer of a scan over coefficients $(a,b,c)$. This removes an internal fitting freedom when the model is tested on updated inputs. It does not erase the historical look-elsewhere question associated with first noticing the relation, and the paper does not claim a blind preregistration. The appropriate interpretation is therefore: one fixed observable with weak present evidence, supported by an independent exact route identity, not a discovery-level statistical regularity.

Four statements that are sometimes compressed into the phrase “the plane exists” must be kept separate:

Statement	Status	Meaning
${\Delta }_{\Pi }=C_{21}-C_{12}$ in the fixed route normalization	exact	A coordinate/operator identity; no statistical significance is attached to it.
The displayed central values give ${\Delta }_{\Pi }\left(M_Z\right)=0.0189642$ and ${ϵ}_{\Pi }\simeq 1.12\%$	descriptive	A numerical distance and an angular leakage, not a confidence level.
The null hypothesis ${\Delta }_{\Pi }=0$ tested with the stated public-information covariance proxy	diagnostic	$Z_{\Pi }=1.556$ and $p_{\mathrm{two}-\mathrm{sided}}\simeq 0.120$; the data neither establish nor significantly disfavor the exact plane.
The confining theory dynamically enforces $C_{12}=C_{21}$ and matches it to the charged-fermion spectrum	open	Requires the renormalized route correlator, pole test, family alignment, and three-point matching vertex.

It is also useful to state precisely why “increasing the sigma” is not the objective. Here $Z_{\Pi }$ measures the standardized departure from the null plane. At the fixed central residual,

$$Z_{\Pi }\ge 3⟺{\sigma }_{{\Delta }_{\Pi }}\le {\displaystyle \frac{0.0189642}{3}}=0.0063214,$$

roughly one half of the present source-informed proxy $0.01219$. If the central value did not move, this would be a three-sigma discrepancy from the exact plane, not three-sigma evidence for it. Compatibility with a point null is not converted into positive evidence merely by obtaining $p>0.05$. A positive equivalence claim would instead require a physically specified tolerance $|{\Delta }_{\Pi }|<\epsilon$ fixed before the test. With the present proxy, the ordinary two-sided $95\%$ interval is approximately

$$-0.00493<{\Delta }_{\Pi }<0.04286,$$

so no equivalence margin tighter than about $0.043$ is supported by that interval. Choosing another normal, dropping source uncertainties, treating the RG crossing as an independent observation, or promoting the aggressive correlation stress test to central evidence cannot alter this conclusion legitimately. The present work therefore aims to sharpen the structural hypothesis and its independent observables, not to inflate a nominal sigma from the same nine inputs.

For a nonspecialist, the logical division is simple: the operator identity answers which combination the theory asks us to test; the covariance analysis answers how accurately present data test its vanishing; and the future nonperturbative correlator must answer why, if at all, the combination should vanish. These are independent questions.

2.7. Why Covariance Matters

Let $x_a$ be the primary input vector and ${\Sigma }_{ab}=cov(x_a,x_b)$ its covariance matrix. Linear error propagation gives

$${\sigma }_{{\Delta }_{\Pi }}^2=J_a{\Sigma }_{ab}{J}_b,J_a={\displaystyle \frac{\partial {\Delta }_{\Pi }}{\partial x_a}}.$$

Off-diagonal entries matter because lattice quark masses may share scale setting, renormalization, gauge ensembles, and ratio information. The published correlations demonstrate that treating all quark masses as statistically independent is not justified.

For reproducibility, order the Bazavov input vector as

$$x_B=(m_c,r_b,r_s,r_d,r_u),(r_b,r_s,r_d,r_u)=\left({\displaystyle \frac{m_b}{m_c}},{\displaystyle \frac{m_s}{m_c}},{\displaystyle \frac{m_d}{m_c}},{\displaystyle \frac{m_u}{m_c}}\right).$$

The published central values and one-standard-deviation marginal uncertainties are

$$\begin{array}{cc}\hfill x_B=& (983.7\mathrm{MeV},4.578,0.08487,0.004291,0.001955),\hfill \\ \hfill {\sigma }_B=& (5.6\mathrm{MeV},0.008,0.00018,0.000039,0.000037).\hfill \end{array}$$

In the same order, Table IV gives the symmetric correlation matrix

$${\rho }_B=\left(\begin{array}{ccccc}1& -0.58607809& -0.11425384& 0.14213251& -0.16516954\\ -0.58607809& 1& 0.45502225& 0.04855992& 0.23627864\\ -0.11425384& 0.45502225& 1& 0.43609054& 0.47252309\\ 0.14213251& 0.04855992& 0.43609054& 1& -0.32724921\\ -0.16516954& 0.23627864& 0.47252309& -0.32724921& 1\end{array}\right).$$

The covariance subblock used in propagation is therefore

$${\Sigma }_B=diag\left({\sigma }_B\right){\rho }_{B}diag\left({\sigma }_B\right),{\left({\Sigma }_B\right)}_{ij}={\rho }_{ij}{\sigma }_i{\sigma }_j.$$

The variance of any derived scalar $f\left(x\right)$ is then propagated linearly as

$${\sigma }_f^2=J\Sigma J^T,J_i={\displaystyle \frac{\partial f}{\partial x_i}}.$$

For $f={\Delta }_{\Pi }$, this is the formula behind every row of the covariance table above: the rows differ only in which source correlations are retained. Positive and negative off-diagonal terms can either increase or decrease the final variance, so the result cannot be inferred from marginal errors alone.

This distinction is important: the correlation coefficients are published, while ${\Sigma }_B$ is the covariance block reconstructed from them and from the published marginal uncertainties.

Equivalently, retaining the dimensionless top–charm ratio explicitly, the dependence on this block can be embedded in the full expression

$${\Delta }_{\Pi }=A_{\ell }+{\displaystyle \frac{1}{4}}ln\left({\displaystyle \frac{m_c}{m_t}}\right)+{\displaystyle \frac{1}{2}}ln{r}_b-ln{r}_s+{\displaystyle \frac{1}{2}}ln{r}_d-{\displaystyle \frac{1}{4}}ln{r}_u.$$

When the charged-lepton and top-quark inputs are held fixed, differentiation with respect to $x_B$ gives the Jacobian

$$J_B=\left({\displaystyle \frac{1}{4m_c}},{\displaystyle \frac{1}{2r_b}},-{\displaystyle \frac{1}{r_s}},{\displaystyle \frac{1}{2r_d}},-{\displaystyle \frac{1}{4r_u}}\right),$$

and direct substitution gives

$$J_B{\Sigma }_B{J}_B^T=6.55160\times {10}^{-5},{\sigma }_{{\Delta }_{\Pi },B}=0.00809420.$$

If the same five marginal uncertainties are artificially treated as independent, the corresponding block value is $0.00709403$. These numbers are an internal reproduction of the Bazavov subblock only; neither is the total uncertainty quoted above, because the complete calculation also contains the charged-lepton, top-quark, running, matching, and source-overlap contributions.

The complete joint covariance matrix of the combined PDG/AHS and lattice running-mass inputs is not public. The quoted $1.556\sigma$ value is therefore a maximum-public-information reconstruction, not an exact confidence level supplied by the collaborations. The displayed ${\rho }_B$, ${\Sigma }_B$, and $J_B$ make the strongest exact public correlation anchor independently reproducible while keeping this limitation explicit.

2.8. RG Trajectory and the Central Crossing near $3.2\mathrm{TeV}$

A published two-loop SM central trajectory evolved with standard two-loop SM RG equations [13] gives approximately

$$\begin{array}{cccc}\hfill {\Delta }_{\Pi }\left(M_Z\right)& =+0.01896,\hfill & \hfill {\Delta }_{\Pi }\left(1\mathrm{TeV}\right)& =+0.00509,\hfill \\ \hfill {\Delta }_{\Pi }\left(3\mathrm{TeV}\right)& =+0.00033,\hfill & \hfill {\Delta }_{\Pi }\left(10\mathrm{TeV}\right)& =-0.00616,\hfill \end{array}$$

with the diagnostic crossing

$${\mu }_{\mathrm{cross}}^{\mathrm{SM}}\simeq 3.2\mathrm{TeV}.$$

The same information is useful in tabular form:

renormalization scale	${\Delta }_{\Pi }\left(\mu \right)$
$M_Z$	$+0.01896$
$1\mathrm{TeV}$	$+0.00509$
$3\mathrm{TeV}$	$+0.00033$
$10\mathrm{TeV}$	$-0.00616$

The sign change is a diagnostic property of this single Standard-Model running trajectory. It does not create a second observation, and it should not be combined statistically with the $M_Z$ value.

This is not a second independent measurement; it is one RG trajectory. Above the Pati–Salam breaking scale, continuing the pure SM equations is only a diagnostic exercise.

If matching shifts the logarithmic Yukawa values by ${\delta }_{s,g}$,

$$\delta A_s={\displaystyle \frac{1}{4}}({\delta }_{s,1}-2{\delta }_{s,2}+{\delta }_{s,3}).$$

A generation-universal shift ${\delta }_{s,g}=c_s$ and any shift linear in the generation label, $a_s+b_{s}g$, cancel exactly. Therefore an ultraviolet threshold can move ${\Delta }_{\Pi }$ only through a genuinely generation-curved correction.

Remark.In simple terms, the proximity to the plane is worth explaining but is not evidence for preons by itself. The independent value of the theory arises only if the microscopic operator space naturally produces the same reciprocity condition $C_{12}=C_{21}$ that corresponds to the plane.

2.9. External Two-Spurion Texture Check: an Exponent-Level Bridge and Its Limits

A recent flavour analysis of a related $\mathrm{SU}{\left(15\right)}_p$ realization uses two independent preon-flavour spurions and a Froggatt–Nielsen-like hierarchy controlled by a small parameter $\kappa \simeq 0.17$ [7]. This construction is not identical to the Pati–Salam branch used for the route operators below, so its fitted coefficients are not imported into the present model. It nevertheless provides a useful algebraic control because logarithmic curvature has a particularly simple expression in any power-counting texture.

Write the positive Yukawa singular values in one charged sector as

$$y_{s,g}=c_{s,g}{\kappa }^{n_{s,g}},c_{s,g}>0,$$

where $n_{s,g}$ is the texture exponent and $c_{s,g}$ contains the order-one coefficient, loop, and nonperturbative information not represented by the integer power. Define the exponent curvature

$$q_s\equiv n_{s,1}-2n_{s,2}+n_{s,3}.$$

Then the definition of $A_s$ gives the exact decomposition

$$A_s={\displaystyle \frac{1}{4}}ln\left({\displaystyle \frac{c_{s,1}{c}_{s,3}}{c_{s,2}^2}}\right)+{\displaystyle \frac{q_s}{4}}ln\kappa .$$

Consequently,

$$\begin{array}{cc}\hfill {\Delta }_{\Pi }& ={\Delta }_{\Pi }^{\left(c\right)}+{\displaystyle \frac{ln\kappa }{4}}\left(q_{\ell }+2q_d-q_u\right),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\Delta }_{\Pi }^{\left(c\right)}& \equiv {\displaystyle \frac{1}{4}}ln\left[{\displaystyle \frac{c_{\ell ,1}{c}_{\ell ,3}}{c_{\ell ,2}^2}}{\left({\displaystyle \frac{c_{d,1}{c}_{d,3}}{c_{d,2}^2}}\right)}^2{\displaystyle \frac{c_{u,2}^2}{c_{u,1}{c}_{u,3}}}\right].\hfill \end{array}$$

(9)

This identity separates an integer-power contribution from the curvature carried by order-one matching coefficients. It is exact once the factorization $y=c{\kappa }^n$ is adopted; it does not assume that the coefficients $c_{s,g}$ are close to one.

The leading texture quoted in Ref. [7] has

$$(n_{u,1},n_{u,2},n_{u,3})\sim (n_{d,1},n_{d,2},n_{d,3})\sim (n_{\ell ,1},n_{\ell ,2},n_{\ell ,3})\sim (4,2,0),$$

so $q_u=q_d=q_{\ell }=0$ at this coarse power-counting level. This produces a trivial curvature-plane compatibility because every sector is individually geometric in the leading exponent approximation; it is not the nontrivial observed pattern by itself.

More interestingly, the same analysis notes that the up-quark mass is numerically closer to ${\kappa }^7$ than to its naive ${\kappa }^4$ estimate, the charm mass is closer to ${\kappa }^3$, and the electron mass is closer to ${\kappa }^7$ than to the naive ${\kappa }^6$ mass scaling; the benchmark also has $v_d/v\sim {\kappa }^2$. Translating these stated mass scalings to Yukawa exponents gives the schematic effective assignments

$$n_u\sim (7,3,0),n_d\sim (4,2,0),n_{\ell }\sim (5,2,0),$$

and therefore

$$q_u=1,q_d=0,q_{\ell }=1,q_{\ell }+2q_d-q_u=0.$$

Thus the integer-exponent part of the published two-spurion texture lies exactly in the same curvature-plane normal direction as the route relation. The remaining physical displacement is entirely in ${\Delta }_{\Pi }^{\left(c\right)}$, i.e., in the order-one coefficients, cancellations, running, and matching corrections.

Scientific status of the texture comparison.This is an algebraic compatibility check, not an independent statistical confirmation and not a derivation of route reciprocity. The texture of Ref. [7] was itself chosen and fitted to the charged-fermion spectrum, and its nonperturbative coefficients are adjustable. That analysis also reports non-negligible cancellations in the light-quark sector, especially for the up quark, so exponent-level agreement must not be promoted to a naturalness claim. The useful result is narrower: the same discrete second-difference combination that defines the present plane is visible in the exponent bookkeeping of an independently developed $\mathrm{SU}{\left(15\right)}_p$ flavour construction.

The identity above also gives a stopping rule for a future microscopic matching. If a derived leading spurion expansion of the Pati–Salam route operators yields

$$q_{\ell }+2q_d-q_u\ne 0,$$

then the exact plane cannot arise at leading texture order without compensation from ${\Delta }_{\Pi }^{\left(c\right)}$. Conversely, $q_{\ell }+2q_d-q_u=0$ removes the leading logarithmic hierarchy from ${\Delta }_{\Pi }$ but still does not enforce the coefficient-level reciprocity $C_{12}=C_{21}$.

2.10. Right-Handed-Isospin Symmetry Is Not the Route Symmetry

Ref. [7] also identifies an approximate right-handed-isospin symmetry $\mathrm{SU}{\left(2\right)}_I$ relating the up- and down-type sectors. In its exact limit the two Yukawa matrices are proportional, so their ordered singular values satisfy

$$y_{u,g}=\rho y_{d,g}$$

with one generation-independent positive factor $\rho$. Because $A_s$ is invariant under such a sector-wide rescaling,

$$A_u=A_d\left(\mathrm{exact}\mathrm{SU}{\left(2\right)}_I\right).$$

If this symmetry alone were identified with the origin of the curvature plane, then

$${\Delta }_{\Pi }=0⟹A_{\ell }+A_d=0.$$

For the central values used in Section 2, however,

$$A_{\ell }+A_d=-0.6294621+0.2465294=-0.3829327,$$

which is not a small residual. Hence exact right-handed isospin is not the symmetry behind the observed near-plane. This agrees with the independent conclusion of Ref. [7] that realistic masses and CKM mixing require additional spontaneous $\mathrm{SU}{\left(2\right)}_I$ breaking.

No-go under exact right-handed isospin.Under the assumptions $Y_u\propto Y_d$ and the present charged-sector definition of $A_s$, exact $\mathrm{SU}{\left(2\right)}_I$ implies $A_u=A_d$ and cannot by itself reproduce the observed curvature pattern. The route involution, if dynamically realized, must therefore be logically distinct from right-handed isospin.

3. Microscopic Basis of $\mathrm{SU}{\left(15\right)}_p$

3.1. Preon Fields

For the Pati–Salam branch the analysis uses the published chiral fermion content of Ref. [6]:

$$\begin{array}{cc}\hfill {\Psi }_W& \sim (\mathbf{15};\mathbf{4},\mathbf{2},\mathbf{1}),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\Psi }_{W^{\prime }}& \sim (\mathbf{15};\overline{\mathbf{4}},\mathbf{1},\mathbf{2}),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\psi }_i& \sim (\mathbf{15};\mathbf{1},\mathbf{1},\mathbf{1}),i=1,2,3,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \Omega & \sim (\overline{\mathbf{120}};\mathbf{1},\mathbf{1},\mathbf{1}).\hfill \end{array}$$

(13)

Here $\mathbf{120}$ is the two-index symmetric representation of $\mathrm{SU}\left(15\right)$, because $15\times 16/2=120$, while $\Omega$ transforms in its conjugate $\overline{\mathbf{120}}$. In two-component Weyl notation one may write

$${\left({\Psi }_W\right)}_{\rho }^{\alpha am},{\left({\Psi }_{W^{\prime }}\right)}^{\alpha }{a}^{\dot{r}}{}_{\rho },{\left({\psi }_i\right)}_{\rho }^{\alpha },{\Omega }_{\alpha \beta ,\rho }={\Omega }_{\beta \alpha ,\rho }.$$

The precolor indices are $\alpha ,\beta =1,\dots ,15$; a is an $\mathrm{SU}{\left(4\right)}_{\mathrm{PS}}$ index; m and $\dot{r}$ are $\mathrm{SU}{\left(2\right)}_L$ and $\mathrm{SU}{\left(2\right)}_R$ indices; and $\rho$ is a left-handed Weyl spinor index.

3.2. Cubic Gauge Anomaly

The fields ${\Psi }_W$ and ${\Psi }_{W^{\prime }}$ each contain $4\times 2=8$ copies of the fundamental $\mathrm{SU}{\left(15\right)}_p$ Weyl representation, while the three ${\psi }_i$ add three more. Thus there are 19 fundamentals. In the normalization $A\left(\mathbf{N}\right)=1$, the cubic anomaly coefficient of the two-index symmetric representation of $\mathrm{SU}\left(N\right)$ is $N+4$ [14]; the conjugate representation has the opposite sign. Therefore

$${\mathcal{A}}_{\mathrm{SU}{\left(15\right)}_p^3}=8+8+3-(15+4)=0.$$

This is a necessary ultraviolet consistency condition. It does not determine which infrared phase is realized after confinement.

3.3. Two Scalar Flavour Spurions and the Corrected Precolor Tensor Assignment

The renormalizable Pati–Salam realization contains elementary precolor scalars that mediate flavour breaking in addition to the chiral fermions listed above. The 2025 Pati–Salam construction wrote two such scalar mediators in the conjugate antisymmetric representation [6]. A later dedicated flavour analysis of the same $\mathrm{SU}{\left(15\right)}_p$ compositeness framework states that the second Yukawa structure used in earlier studies actually requires a conjugate symmetric tensor and corrects that assignment [7]. Applying this representation-theory correction to the analogous Pati–Salam singlet-preon Yukawa contraction gives the consistent two-spurion choice

$$\mathcal{A}\sim (\overline{\mathbf{105}};\mathbf{1},\mathbf{1},\mathbf{1}),{\mathcal{A}}^{\prime }\sim (\overline{\mathbf{120}};\mathbf{1},\mathbf{1},\mathbf{1}),$$

with an antisymmetric and a symmetric preon-flavour spurion, respectively. The detailed four-singlet $\mathrm{SU}{\left(4\right)}_F$ benchmark of Ref. [7] is not part of the minimal Pati–Salam route theorem; only the symmetry of the $\mathrm{SU}{\left(15\right)}_p$ tensor contraction is used in this subsection. A four-flavour parent embedding is invoked later only as an explicitly conditional family-potential selection-rule branch. That contraction is independent of whether the weak spectator group is written directly as the SM group or in Pati–Salam form.

For the three Pati–Salam singlet preons ${\psi }_i$, $i=1,2,3$, write the Lorentz-scalar couplings as

$${\mathcal{L}}_{\mathrm{flav}}={\displaystyle \frac{1}{2}}{\lambda }_{ij}{\mathcal{A}}_{\alpha \beta }{ϵ}^{\rho \sigma }{\psi }_i^{\alpha }{}_{\rho }\psi _j^{\beta }{}_{\sigma }+{\displaystyle \frac{1}{2}}{\lambda }_{ij}^{\prime }{\mathcal{A}}_{\alpha \beta }^{\prime }{ϵ}^{\rho \sigma }{\psi }_i^{\alpha }{}_{\rho }\psi _j^{\beta }{}_{\sigma }+\mathrm{h}.\mathrm{c}.$$

The two-Weyl bilinear is symmetric under the simultaneous interchange $(i,\alpha )\leftrightarrow (j,\beta )$. Therefore contraction with the antisymmetric precolor tensor ${\mathcal{A}}_{\alpha \beta }=-{\mathcal{A}}_{\beta \alpha }$ retains only the antisymmetric flavour part, whereas contraction with the symmetric tensor ${\mathcal{A}}_{\alpha \beta }^{\prime }={\mathcal{A}}_{\beta \alpha }^{\prime }$ retains only the symmetric flavour part:

$${\lambda }^T=-\lambda ,{\left({\lambda }^{\prime }\right)}^T=+{\lambda }^{\prime }.$$

This short derivation is the representation-theory reason for the corrected assignment. It also separates two logically different uses of “field content” in the paper: the route-odd Wilson source below uses only the published chiral fermion content, while the family-alignment discussion may use the scalar spurions already present in the renormalizable flavour completion.

Status of the scalar update.The chiral-fermion anomaly calculation and the point-split route-source construction are unchanged. For the scalar-mediated flavour completion the analysis adopts the representation-theory correction made explicit in Ref. [7]; scalars do not contribute to the cubic chiral gauge anomaly. Any beta-function or detailed scalar-potential analysis of this corrected completion that depends on scalar Dynkin indices must use the $\overline{\mathbf{105}}/\overline{\mathbf{120}}$ assignment rather than the older two-$\overline{\mathbf{105}}$ bookkeeping. This is a correction of the scalar completion, not a modification of the chiral fermion content used in the route-source theorem.

3.4. Unitary Flavour Commutant of the Gauged Spectator Embedding

Before spectator gauging, the nineteen fundamental copies may be organized in a formal precolor flavour space. Gauging $G_{\mathrm{PS}}$ decomposes that space as

$$(\mathbf{4},\mathbf{2},\mathbf{1})\oplus (\overline{\mathbf{4}},\mathbf{1},\mathbf{2})\oplus 3(\mathbf{1},\mathbf{1},\mathbf{1}).$$

The three summands are inequivalent irreducible $G_{\mathrm{PS}}$ modules. More explicitly, the nineteen-dimensional multiplicity space decomposes as

$${\mathcal{F}}_{19}\simeq V_W\otimes \mathbb{C}\oplus V_{W^{\prime }}\otimes \mathbb{C}\oplus \mathbf{1}\otimes {\mathbb{C}}^3,$$

where $V_W=(\mathbf{4},\mathbf{2},\mathbf{1})$ and $V_{W^{\prime }}=(\overline{\mathbf{4}},\mathbf{1},\mathbf{2})$ are inequivalent irreducible spectator modules. Hence

$${End}_{G_{\mathrm{PS}}}\left({\mathcal{F}}_{19}\right)\simeq \mathbb{C}\oplus \mathbb{C}\oplus M_3\left(\mathbb{C}\right).$$

Taking the unitary subgroup gives the full unitary commutant of the gauged kinetic term,

$${\mathcal{C}}_{\mathrm{PS}}^U=U{\left(1\right)}_W\times U{\left(1\right)}_{W^{\prime }}\times U{\left(3\right)}_{\psi }.$$

If one restricts the formal nineteen-copy flavour action to its determinant-one subgroup, the two one-dimensional multiplicity phases $e^{i{\alpha }_W}$, $e^{i{\alpha }_{W^{\prime }}}$ and $U_{\psi }\in U\left(3\right)$ obey

$$e^{8i{\alpha }_W}{e}^{8i{\alpha }_{W^{\prime }}}det{U}_{\psi }=1,$$

and the corresponding subgroup is

$${\mathcal{C}}_{\mathrm{PS}}^{SU}=S\left(U{\left(1\right)}_W\times U{\left(1\right)}_{W^{\prime }}\times U{\left(3\right)}_{\psi }\right).$$

Possible anomalous Abelian combinations or additional interactions may reduce these groups further. The route conclusion is independent of that Abelian refinement: Schur’s lemma shows directly that no continuous off-diagonal intertwiner can exchange $V_W$ with $V_{W^{\prime }}$, because they are inequivalent irreducible $G_{\mathrm{PS}}$ modules. The commutant acts by separate phases on ${\Psi }_W$ and ${\Psi }_{W^{\prime }}$ and by a $U\left(3\right)$ rotation on the singlets ${\psi }_i$; it cannot exchange the two inequivalent spectator blocks.

The scalar Yukawa spurions of Section 3.3 explicitly reduce the $U{\left(3\right)}_{\psi }$ factor to the common stabilizer of $\lambda$ and ${\lambda }^{\prime }$. They cannot enlarge the symmetry or create an intertwiner between $V_W$ and $V_{W^{\prime }}$. Thus the commutant calculation identifies the largest continuous unitary copy-space symmetry of the gauged kinetic term before scalar flavour breaking, up to the Abelian anomaly qualifications stated above; the physical flavour vacuum has a subgroup of it.

Consequently, the route reflection used below is not a Ward identity of the chiral-fermion kinetic theory or of its gauged spectator embedding. An exact route symmetry, if present in the physical correlator, must be emergent in the composite/orbital sector or arise from additional ultraviolet data. This statement does not invalidate the operator sign construction; it fixes the stronger claim that the construction does not by itself select the physical even plane.

3.5. Three-Preon Prebaryons

Convenient interpolating currents have the structure

$$\begin{array}{cc}\hfill {\left(P_4^i\right)}_{\rho }^{am}& =N_4{\Omega }_{\alpha \beta ,\rho }{\left({\Psi }_W\right)}_{\sigma }^{\alpha am}{\left({\psi }_i\right)}_{\tau }^{\beta }{ϵ}^{\sigma \tau },\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \left(P_{\overline{4}}^i\right)a^{\dot{r}}{}_{\rho }& =N_{\overline{4}}{\Omega }_{\alpha \beta ,\rho }\left({\Psi }_{W^{\prime }}\right)\alpha a^{\dot{r}}{}_{\sigma }\left({\psi }_i\right)_{\tau }^{\beta }{ϵ}^{\sigma \tau }.\hfill \end{array}$$

(15)

Both precolor indices are contracted with $\Omega$, so these currents are $\mathrm{SU}{\left(15\right)}_p$ singlets. Their spectator quantum numbers are

$$P_4^i\sim (\mathbf{4},\mathbf{2},\mathbf{1}),P_{\overline{4}}^i\sim (\overline{\mathbf{4}},\mathbf{1},\mathbf{2}).$$

Different Lorentz contractions of three left-handed Weyl spinors must not automatically be counted as independent operators: Schouten identities and Fierz transformations reduce the local basis. The routing analysis below is therefore formulated only after quotienting by these local identities.

3.6. Pati–Salam Breaking and Primitive Tensors

Let

$$\Phi \sim (\mathbf{4},\mathbf{1},\mathbf{2}),{\Phi }^{a\dot{r}},$$

denote an effective composite Pati–Salam-breaking interpolator. It is not introduced as a new fundamental field of the theory. From bilinears $\Phi {\Phi }^{†}$ define

$$\begin{array}{cc}\hfill {\left(P_{\Phi }\right)}^{a}b& ={\Phi }^{a\dot{r}}{\Phi }_{b\dot{r}}^{†},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {{\left(Q_{\Phi }\right)}^{\dot{r}}}_{\dot{s}}& ={\Phi }^{a\dot{r}}{\Phi }_{a\dot{s}}^{†},\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\sigma }_{\Phi }& =Tr{P}_{\Phi }=Tr{Q}_{\Phi },\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {M^{a\dot{r}}}_{b\dot{s}}& ={\Phi }^{a\dot{r}}{\Phi }_{b\dot{s}}^{†}.\hfill \end{array}$$

(19)

The four primitive tensors are

$$B_0={\sigma }_{\Phi }{I}_4\otimes I_2,B_R=I_4\otimes Q_{\Phi },B_L=P_{\Phi }\otimes I_2,B_X=M.$$

They form the spectator-tensor basis on which the crossed-route algebra acts below.

4. Four-Dimensional Construction I: Operator Target and Microscopic Source

4.1. Exact Operator Meaning of the Curvature Plane

The Pati–Salam route construction contains four operator images $r_{11},r_{12},r_{21},r_{22}$. Their charged-sector projections satisfy

$${\mathbf{n}}_{\Pi }·r_{11}=0,{\mathbf{n}}_{\Pi }·r_{12}=-1,{\mathbf{n}}_{\Pi }·r_{21}=+1,{\mathbf{n}}_{\Pi }·r_{22}=0,$$

for ${\mathbf{n}}_{\Pi }=(1,2,-1)$. Therefore, if

$$\mathbf{A}=C_{11}{r}_{11}+C_{12}{r}_{12}+C_{21}{r}_{21}+C_{22}{r}_{22},$$

then

$$\begin{array}{|c|}\hline {\Delta }_{\Pi }=C_{21}-C_{12}.\\ \hline\end{array}$$

In the operator derivation presented here, the plane normal is not obtained by scanning the mass data. Once the stated route basis is fixed, the two uncrossed images determine ${\mathbf{n}}_{\Pi }$ uniquely up to normalization before any numerical mass values are inserted. This is fully consistent with the historical fact that the same normal was first noticed a posteriori in Ref. [10]: the historical discovery was data-driven, whereas the present operator derivation independently fixes the already identified direction. The identity is exact but carries no statistical significance by itself.

4.2. What the Microscopic Source Construction Establishes

The unchanged chiral field content contains a gauge-invariant point-split source with eight local $\Omega FF$ junctions, hence 24 fermionic field insertions. The six internal mixed currents support a Schouten-reduced singlet quotient. Invariant Temperley–Lieb generators define two noncommuting ordered route operations on that quotient, producing a nonzero internal odd direction. The singlet-projected internal current is $\mathrm{SU}{\left(4\right)}_{\mathrm{PS}}$ blind, so the unprojected current with its adjoint component $K_{15}\sim (15,2,2)$ is required for the full crossed color structure.

A single fundamental $\mathrm{SU}\left(4\right)$ crossing is insufficient. In the larger unchanged-field walled–Brauer contraction algebra, fixed pair projectors generate both color characters. In particular, a five-projector word gives

$${\displaystyle \frac{3}{32}}(D-3X)={\displaystyle \frac{3}{32}}{\mathcal{C}}_2^{\left(4\right)},$$

so the required effective direct/crossed ratio $-1/2$ is a consequence of projector coefficients and loop factors. Both crossed tensors therefore have explicit microscopic recoupling representatives.

The stronger chronological interpretation does not survive. The two closed kernels have different singlet support and different rank, whereas literal Hermitian word reversal preserves the fully closed singlet scalar. No closure-preserving reverse-order involution in that class can exchange the complete kernels. Accordingly, the four-dimensional hypothesis is

$$\mathrm{dynamical}\mathrm{crossed}-\mathrm{recoupling}\mathrm{reciprocity},$$

not a microscopic route Ward identity. Detailed projector words, quotient matrices, rank arguments, and the associated scoped no-go theorems are given in Appendix B.1.

4.3. Localization of the Possible Route Defect

Each local $\Omega FF$ junction is already an $\mathrm{SU}{\left(15\right)}_p$ singlet. The inter-junction Wilson network in the explicit source transports spectator $\mathrm{SU}{\left(2\right)}_L\times \mathrm{SU}{\left(2\right)}_R$ indices rather than precolor. Hence the source contains no route-dependent inter-junction $\mathrm{SU}{\left(15\right)}_p$ minimal-area sheet. Any deliberately added precolor dressing that is exactly reflected is route neutral order by order in the character expansion. A separate local $3+1$ dimensional shared-link cluster obeys center/N-ality selection, which removes all odd plaquette orders below N; for $\mathrm{SU}\left(15\right)$ the first center-allowed odd local order is 15. These results eliminate several source-level origins of $K_{\perp }$ without asserting that the physical connected-matter contribution vanishes.

5. Independent Family Alignment: the Median Direction and Its Vacuum Selection

The route mechanism must not silently absorb the separate problem of generation structure. In the Pati–Salam branch used here the three Pati–Salam-singlet but precolor-fundamental preons ${\psi }_i$, $i=1,2,3$, already provide a three-dimensional flavour space

$${\mathcal{F}}_{\psi }\simeq {\mathbb{C}}^3.$$

No auxiliary fourth flavour direction or subsequent heavy-direction projection is required for the algebraic construction in this section. This is an important distinction from the four-singlet $\mathrm{SU}{\left(4\right)}_F$ realization studied in Ref. [7]. That realization is a useful external flavour comparison, but it is not identified with the Pati–Salam flavour space of the present operator proof.

The composite operator requires a family tensor ${D^i}_j$ selecting the discrete curvature direction. The aim is not to assign a preferred generation label by hand, but to reconstruct the required ordered direction from basis-covariant spurion data. For three light families the desired normalized spectrum is

$$D_{+}=diag(1,-2,1)$$

in an ordered eigenbasis. The question is therefore not whether this diagonal matrix can be written down, but whether the corrected scalar-spurion completion defines the ordered eigendirections without inserting the observed generation labels by hand.

5.1. Exact No-Go for a Single Antisymmetric Spurion

With the corrected two-scalar assignment of Section 3.3, the first spurion satisfies

$${\lambda }^T=-\lambda .$$

Every complex $3\times 3$ antisymmetric matrix has vanishing determinant,

$$det\lambda =det{\lambda }^T=det(-\lambda )=-det\lambda ,$$

so $det\lambda =0$. More strongly, the unitary-congruence (Youla) normal form gives, for nonzero generic rank,

$$U^T\lambda U=\left(\begin{array}{ccc}0& \sigma & 0\\ -\sigma & 0& 0\\ 0& 0& 0\end{array}\right),\sigma >0.$$

Therefore the Hermitian matrix

$$H_A\equiv \lambda {\lambda }^{†}$$

has spectrum

$$spec{H}_A=\{{\sigma }^2,{\sigma }^2,0\}.$$

Antisymmetric-spurion alignment no-go.A single nonzero complex antisymmetric spurion in three flavour dimensions cannot define three nondegenerate ordered family directions through $\lambda {\lambda }^{†}$: it necessarily leaves a two-dimensional degenerate singular subspace. Hence it cannot by itself generate the unique median projector required for $D_{+}=diag(1,-2,1)$.

This result is exact and independent of the numerical entries of $\lambda$. It turns the need for additional flavour structure into a representation-theory statement rather than a matter of fit quality.

5.2. Symmetric-Spurion Ordering Theorem

The second spurion satisfies

$${\left({\lambda }^{\prime }\right)}^T={\lambda }^{\prime }.$$

By Takagi factorization there exists a unitary $U_S$ and nonnegative singular values $s_i$ such that

$${\lambda }^{\prime }=U_{S}diag(s_1,s_2,s_3)U_S^T.$$

Define the Hermitian flavour operator

$$H_S\equiv {\lambda }^{\prime }{\lambda }^{\prime †}=U_{S}diag(s_1^2,s_2^2,s_3^2)U_S^{†}.$$

If its eigenvalues are nondegenerate, order them as

$$h_1<h_2<h_3.$$

The spectral projector onto the median eigendirection has the Lagrange–Sylvester form

$$P_2\left(H_S\right)={\displaystyle \frac{(H_S-h_1{I}_3)(H_S-h_3{I}_3)}{(h_2-h_1)(h_2-h_3)}}.$$

Then

$$D_{+}=I_3-3P_2\left(H_S\right)$$

has eigenvalues $(1,-2,1)$ and obeys the basis-covariant polynomial identities

$$D_{+}^2+D_{+}-2I_3=0,Tr{D}_{+}=0,Tr{D}_{+}^2=6.$$

Proved algebraic statement.For a nondegenerate symmetric spurion ${\lambda }^{\prime }$, the ordered family tensor $D_{+}$ is reconstructed exactly and basis covariantly from $H_S={\lambda }^{\prime }{\lambda }^{\prime †}$. The previous auxiliary projector onto a three-dimensional subspace of a four-flavour space is unnecessary in the Pati–Salam branch.

There is also a precise stopping rule. If $H_S$ has a degeneracy involving the median eigenvalue, then $P_2\left(H_S\right)$ is not uniquely defined by $H_S$ alone. In that case this single-operator alignment mechanism fails and additional flavour data are required. Near degeneracy does not invalidate the projector algebraically, but it makes any dynamical implementation sensitive to the small eigenvalue gaps.

5.3. Polynomial Selector Without Inverse Gaps

The median eigendirection can be identified without inverse spectral gaps by the finite polynomial

$$J_{+}\left(H_S\right)={\chi }_S^{\prime }\left(H_S\right).$$

For ordered nondegenerate eigenvalues $h_1<h_2<h_3$, the eigenvalue signs of $J_{+}\left(H_S\right)$ are $(+,-,+)$. Therefore the median family direction is the unique negative eigenspace of this polynomial operator. This form is useful dynamically because it does not require inserting the inverse gaps appearing explicitly in the Lagrange–Sylvester projector. The projector and polynomial selector encode the same ordered eigendirection whenever the spectrum is nondegenerate.

5.4. What the Second Spurion Adds: A Basis-Invariant Misalignment Diagnostic

The two Hermitian spurion combinations

$$H_A=\lambda {\lambda }^{†},H_S={\lambda }^{\prime }{\lambda }^{\prime †}$$

permit the basis-invariant nonnegative diagnostic

$${\mathcal{I}}_{AS}\equiv Tr\left({[H_A,H_S]}^{†}[H_A,H_S]\right)\ge 0.$$

For Hermitian $H_A$ and $H_S$,

$${\mathcal{I}}_{AS}=0⟺[H_A,H_S]=0,$$

so they are simultaneously diagonalizable. A nonzero ${\mathcal{I}}_{AS}$ establishes genuine relative flavour orientation of the two spurion sectors.

This does not by itself predict a nontrivial CKM matrix. The dedicated flavour analysis of Ref. [7] shows that an additional approximate right-handed-isospin symmetry correlates the up- and down-type Yukawas, and realistic CKM mixing requires sector-dependent spontaneous breaking through composite scalar vacuum expectation values. The invariant ${\mathcal{I}}_{AS}$ therefore diagnoses spurion misalignment, not the full quark mixing matrix.

The logical factorization of the present explanation can now be written more sharply as

$$\begin{array}{c}\mathrm{charged}-\mathrm{fermion}\mathrm{plane}⟸\\ \left[\mathrm{dynamical}\mathrm{realization}\mathrm{of}\mathrm{the}\mathrm{algebraically}\mathrm{defined}{D}_{+}\right]\wedge \left[C_{12}=C_{21}\right]\\ \wedge \left[F_{+}\left(0\right)\ne 0,F_{-}\left(0\right)=0,C_{-}^{\mathrm{reg}}=0\right].\end{array}$$

The first factor no longer requires an auxiliary four-flavour projection: the corrected symmetric spurion can define $D_{+}$ directly in the three-family Pati–Salam flavour space. What remains open is the vacuum selection of that direction and its coefficient in the physical matching vertex. Flavour alignment and the route-sign mechanism are therefore still logically independent.

5.5. Composite Median-Family Vacuum: Global Stable Branches

The algebraic selector can be promoted to a constructive composite vacuum without introducing a new fundamental flavon. Let $\Sigma$ be a gauge-neutral complex symmetric composite in ${\overline{\mathbf{6}}}_F$, and let the quadratic operator contain the tensor lift of $J_{+}\left(H_S\right)$. On the symmetric tensor space the six selector eigenvalues are

$$2j_1,j_1+j_2,j_1+j_3,2j_2,j_2+j_3,2j_3,$$

where $j_i={\prod }_{k\ne i}(h_i-h_k)$. For $h_1<h_2<h_3$, the only negative direction is the median rank-one tensor $u_2^{*}\odot u_2^{*}$, because

$$2j_2=-2(h_2-h_1)(h_3-h_2)<0.$$

Consider the determinant-free quartic class

$$V=\langle \Sigma ,Q_2\Sigma \rangle +{\lambda }_1{r}^4+{\lambda }_2{Q}_4,r^2=Tr\left({\Sigma }^{†}\Sigma \right),Q_4=Tr\left[{\left({\Sigma }^{†}\Sigma \right)}^2\right].$$

The inequalities $Q_4\le r^4$ and

$$\langle \Sigma ,Q_2\Sigma \rangle \ge {\mu }_{min}^2{r}^2,{\mu }_{min}^2=m_{\Sigma }^2-2{\eta }_{\Sigma }(h_2-h_1)(h_3-h_2),$$

are saturated simultaneously by the median rank-one tensor. If

$${\eta }_{\Sigma }>0,0<m_{\Sigma }^2<2{\eta }_{\Sigma }(h_2-h_1)(h_3-h_2),$$

$${\lambda }_2\le 0,{\lambda }_1+{\lambda }_2>0,$$

and the determinant term is absent, then

$$V\ge {\mu }_{min}^2{r}^2+({\lambda }_1+{\lambda }_2)r^4$$

has a unique global family direction, namely the median rank-one tensor, up to the overall complex phase left free by the determinant-free potential. Its norm and vacuum energy are

$$r_{\mathrm{vac}}^2={\displaystyle \frac{2{\eta }_{\Sigma }(h_2-h_1)(h_3-h_2)-m_{\Sigma }^2}{2({\lambda }_1+{\lambda }_2)}},$$

$$V_{min}=-{\displaystyle \frac{{[2{\eta }_{\Sigma }(h_2-h_1)(h_3-h_2)-m_{\Sigma }^2]}^2}{4({\lambda }_1+{\lambda }_2)}}.$$

The determinant-free theorem is sufficient but not necessary. A finite determinant-tolerant global domain can also be established. Write the complex symmetric field in the selector eigenbasis as

$$\Sigma =x{E}_{22}+Y,{\langle E_{22},Y\rangle }_F=0,y={\parallel Y\parallel }_F.$$

Here ${\langle A,B\rangle }_F\equiv Tr\left(A^{†}B\right)$ is the Frobenius inner product, ${\parallel A\parallel }_F=\sqrt{{\langle A,A\rangle }_F}$, and $E_{22}$ denotes the normalized rank-one tensor along the selected median family direction. Thus $x{E}_{22}$ is the aligned component and Y collects all transverse family fluctuations. An exact joint maximization of the determinant gives

$$|det\Sigma |\le \left\{\begin{array}{cc}{\displaystyle \frac{\left|x\right|y^2}{2}},\hfill & \left|x\right|/y\ge 1/\sqrt{2},\hfill \\ {\displaystyle \frac{{\left(\right|x|}^2+y^2{)}^{3/2}}{3\sqrt{3}}},\hfill & \left|x\right|/y\le 1/\sqrt{2}.\hfill \end{array}\right.$$

This bound is sharper than adding separate quadratic-minor and cubic estimates because those two pieces cannot generally saturate simultaneously.

For the benchmark coefficient

$$2(h_2-h_1)(h_3-h_2)=0.156599862399$$

and the nearest transverse selector gap

$${\Gamma }_J=0.156998596111,$$

the determinant-allowed potential remains globally median aligned throughout the already stated branch

$${\lambda }_2\le 0,{\lambda }_1+{\lambda }_2>0$$

whenever the additional sufficient condition

$$|{\kappa }_{det}|<1.4178\sqrt{\left(0.156599862399{\eta }_{\Sigma }-m_{\Sigma }^2\right)({\lambda }_1+{\lambda }_2)}$$

is satisfied. The coefficient is obtained by a one-dimensional minimization of the exact determinant bound; the detailed maximization is given in Appendix F.3. Thus exact absence of the determinant operator is not required for the existence of a globally stable median vacuum. The physical renormalized values of $m_{\Sigma }^2$, ${\eta }_{\Sigma }$, ${\lambda }_{1,2}$ and ${\kappa }_{det}$ remain dynamical inputs.

The numerical coefficients above belong to the corrected-spurion benchmark discussed in the technical appendix. They are illustrative benchmark values, not universal predictions of the Pati–Salam branch.

5.6. A Conditional Four-Flavour Parent Branch with an Exact Cubic Selection Rule

The light-family field ${\Sigma }_{ij}\in \overline{\mathbf{6}}$ belongs to residual $SU{\left(3\right)}_F$, for which $det\Sigma$ is an allowed cubic invariant. Therefore no all-orders prohibition of $det\Sigma$ follows from the minimal three-family Pati–Salam field content alone. The stronger statement below is deliberately conditional on an additional four-flavour parent embedding.

Let ${\Sigma }_{\mathrm{par}}$ denote the gauge-neutral parent composite before the heavy flavour direction is selected. Its relevant S-wave precursor lies in ${\mathrm{Sym}}^2{\mathbf{6}}_F={\mathbf{1}}_F\oplus {\mathbf{20}}_F^{\prime }$, and the light $\overline{\mathbf{6}}$ is a descendant after $SU{\left(4\right)}_F\to SU{\left(3\right)}_F$. Denote by $I_3^{\mathrm{par}}$ the parent cubic invariant whose light descendant contains $det\Sigma$. With the anomaly-matched $U{\left(1\right)}_X$ assignment used in this completion, each light descendant carries charge 30. Here “gauge-neutral” refers to the gauged precolor/spectator quantum numbers and does not mean neutral under this conditional global $U{\left(1\right)}_X$. Thus

$$X\left(\Sigma \right)=30,X\left(I_3^{\mathrm{par}}\right)=90.$$

A bare parent cubic is therefore forbidden before spontaneous $U{\left(1\right)}_X$ breaking.

A stronger all-orders statement holds in the restricted flavour-breaking branch containing only $\varphi$, $\lambda$, ${\lambda }^{\prime }$ and their conjugates. The field $\varphi$ carries $X\left(\varphi \right)=30$ and odd $SU{\left(4\right)}_F$ center 4-ality, whereas $\lambda$ and ${\lambda }^{\prime }$ carry zero $U{\left(1\right)}_X$ charge and even 4-ality. A polynomial dressing of $I_3^{\mathrm{par}}$ must carry total $X=-90$. Therefore the net number of $\varphi$ minus ${\varphi }^{†}$ insertions is $-3$, so the total number of odd-4-ality $\varphi$ or ${\varphi }^{†}$ factors is odd. Adding only even-4-ality spurions cannot convert that dressing into an $SU{\left(4\right)}_F$ singlet. Consequently $I_3^{\mathrm{par}}$, and therefore its descendant contribution to the light-family determinant coefficient, is absent to all polynomial orders in this restricted spurion branch. Equivalently, after restriction to the light EFT,

$$\begin{array}{|c|}\hline {\kappa }_{det}=0\\ \hline\end{array}$$

within this conditional parent branch. This conclusion must not be attributed to the minimal three-$\psi$ Pati–Salam field content without the four-flavour embedding.

The obstruction is lifted if the completion contains a flavour-singlet, even-4-ality condensate Y with $X\left(Y\right)=30$. A representative parent source is

$${\displaystyle \frac{c_Y}{{\Lambda }^2}}{\left(Y^{†}\right)}^3{I}_3^{\mathrm{par}}+\mathrm{h}.\mathrm{c}.,$$

so ${\kappa }_{det}\sim c_Y{\langle Y\rangle }^3/{\Lambda }^2$ after condensation. Combining this scaling with the sharp global-vacuum bound gives the sufficient condition

$$|c_Y|{\left({\displaystyle \frac{|\langle Y\rangle |}{\Lambda }}\right)}^3<1.417814059\sqrt{{\displaystyle \frac{0.156599862399{\eta }_{\Sigma }-m_{\Sigma }^2}{{\Lambda }^2}}({\lambda }_1+{\lambda }_2)}.$$

The same Y can generate a lower-spurion linear source for $\Sigma$; that tadpole must therefore be matched before the determinant term is treated as the leading correction. The physical question is reduced to a census of such additional condensates and their matching coefficients, not to an arbitrary choice of ${\kappa }_{det}$.

6. Four-Dimensional Construction II: Renormalized Route Dynamics and the Physical Reciprocity Test

The first four-dimensional construction established that the required operators exist. This section asks the different question that ultimately matters physically: after renormalization and propagation through the confining vacuum, do the two route labels become dynamically reciprocal? The answer is encoded in a two-source correlator rather than in source algebra alone.

6.1. Canonical Route Kernel and Source-Normalization-Robust Symmetry Test

Let $C_{ij}^{AB}(t;a_{\mathrm{src}},\mu )$ be the connected Euclidean correlator of two route labels $A,B\in \{12,21\}$ and a set of smearing/translation profiles $i,j$, all constructed from the same canonical source geometry. Choose $t_0>0$ after contact subtraction and define the positive source metric

$$G=C\left(t_0\right).$$

Whitening gives

$$\widehat{C}\left(t\right)=G^{-1/2}C\left(t\right)G^{-1/2},\widehat{C}\left(t_0\right)=I.$$

Under any nonsingular source recombination the whitened matrix changes by unitary similarity, so the generalized-eigenvalue spectrum and pole energies are source-normalization invariant. The route-exchange operator must be transformed by the same unitary similarity; with this simultaneous transformation, the statement $K_{\perp }=0$ is basis invariant. Keeping $U={\sigma }_x$ fixed after an arbitrary basis rotation would instead define a different exchange test.

In the canonical two-route basis the independent exchange is $U={\sigma }_x$. For a Hermitian kernel

$$K=k_{0}I+k_x{\sigma }_x+k_y{\sigma }_y+k_z{\sigma }_z,$$

route exchange is preserved if and only if $k_y=k_z=0$. The symmetry-breaking component is

$$K_{\perp }={\displaystyle \frac{1}{2}}(K-UK{U}^{†})=k_y{\sigma }_y+k_z{\sigma }_z,$$

with exact norm

$$\parallel K_{\perp }{\parallel }_2=\sqrt{{\displaystyle \frac{{(K_{11}-K_{22})}^2}{4}}+{(\Im K_{12})}^2}.$$

This definition distinguishes route-exchange breaking from route-odd parity. An odd pole proportional to $P_{-}=(I-{\sigma }_x)/2$ is perfectly compatible with exact route-exchange symmetry.

6.2. Schur–Feshbach Positivity and the Route-Breaking Susceptibility

For a self-adjoint hidden composite sector $H_H$ coupled to the visible route doublet by V, choose an energy E below the relevant hidden-sector spectrum. Then ${(H_H-E)}^{-1}$ is positive and the below-threshold susceptibility is

$$M\left(E\right)={\displaystyle \frac{1}{h}}V{(H_H-E)}^{-1}{V}^{†}⪰0,$$

and the effective visible block is $K_{\mathrm{eff}}=K_0-hM$. In a real canonical basis exact diagonal route locking is the single scalar condition

$$M_{11}\left(E_{*}\right)-M_{22}\left(E_{*}\right)=D_{\mathrm{rt}},D_{\mathrm{rt}}=A_{\mathrm{rt}}-B_{\mathrm{rt}}.$$

The positive-semidefinite matrix with minimum trace and Frobenius norm subject to this constraint is obtained exactly; one- and multi-hidden-state Hermitian embeddings show that the required channel is compatible with positivity and can be made stable over a finite energy window. This scalar condition removes the diagonal route mismatch only. If the canonical kernel is not real, exact exchange symmetry also requires the corresponding imaginary off-diagonal route-breaking component to vanish. These are existence and extremal theorems, not replacements for the physical $M\left(E\right)$.

6.3. Physical Normalized Reciprocity Defect

If an isolated route-odd state exists, denote its relevant spectral gap by ${\Delta }_{-}^{\left(15\right)}$. The dimensionless propagation-level defect is

$$\begin{array}{|c|}\hline {\eta }_{15}={\displaystyle \frac{\parallel K_{\perp }\left(E_{*}\right){\parallel }_2}{{\Delta }_{-}^{\left(15\right)}}}.\\ \hline\end{array}$$

Both numerator and denominator are physical four-dimensional observables once the source metric, renormalization prescription, and pole branch have been fixed. Static operator algebra does not determine either number.

6.4. What Lower-Dimensional Confinement Controls Show

The lower-dimensional calculations are mechanism controls rather than substitutes for the four-dimensional correlator. The $\mathrm{SU}\left(3\right)$ confinement calculation and its exact route-defect formula are given in Appendix B.2; the extended volume, mode, rank, cancellation, and bridge scans are in Appendix G.

Control quantity	Large-volume numerical result	Physical lesson
Connector susceptibility	${\mathcal{T}}_{\delta t,\infty }^{\mathrm{lin},\mathrm{ctrl}}\simeq 0.01279$	A nonzero low-energy route perturbation can be strongly filtered by the pole-tracked resolvent response.
Exchanged-pair susceptibility	${\mathcal{T}}_{y,\infty }^{\mathrm{lin},\mathrm{ctrl}}\simeq 0.045586$	Configuration-space “heavy” support is not equivalent to spectral heaviness; the transition measure is the relevant object.
Connector cancellation factor	${\mathcal{C}}_{\mathrm{canc}}^{(\infty ),\mathrm{ctrl}}\simeq 1.46\times {10}^{-2}$	The small connector response is cancellation driven but the available sequence approaches a finite conditioning factor rather than an increasingly singular cancellation.
Local $2+1$/$3+1$D transverse dressing	${\rho }_J=1+O\left({10}^{-5}\right)$ in the displayed $\mathrm{SU}\left(15\right)$ controls	Transverse plaquettes preserve the local parity denominator very efficiently but do not by themselves suppress an additive route-odd numerator.

Three points follow. First, large-N protection is numerator and placement dependent rather than universal. Second, the Feshbach kernel must be evaluated at the tracked physical pole; a frozen matching energy can generate a spurious near-resolvent enhancement. Third, the strong connector filtering is specifically spectral/resolvent physics, not an automatic consequence of confinement or additional transverse dimensions. None of these numerical values is identified with the physical four-dimensional ${\eta }_{15}$.

6.5. Finite-Volume and Cancellation-Stability Contract

A small finite-volume $\eta$ is not accepted as a physical result merely because it is numerically small. Pole tracking must produce a controlled limit of the state energy, the odd gap, and the source-accessible residue. In addition, if

$$K_{\perp }=\sum _i{c}_i,{\mathcal{C}}_{\mathrm{canc}}={\displaystyle \frac{|{\sum }_i{c}_i|}{{\sum }_i\left|c_i\right|}},$$

then a relative mode-level uncertainty $|\delta c_i|\le \delta |c_i|$ implies

$${\displaystyle \frac{|\delta K_{\perp }|}{|K_{\perp }|}}\le {\displaystyle \frac{\delta }{{\mathcal{C}}_{\mathrm{canc}}}}.$$

Thus a cancellation-protected continuum limit is stable only if ${\mathcal{C}}_{\mathrm{canc}}$ tends to a nonzero value or the absolute precision improves fast enough that $\delta /{\mathcal{C}}_{\mathrm{canc}}\to 0$. This criterion is fixed before the physical calculation is performed.

7. Low-Energy Matching, Normalization Robustness, and Phenomenological Closure

Even exact reciprocity of the two-route propagation kernel would not yet explain the charged-fermion plane unless the reciprocal channel couples correctly to the low-energy fermion operator. The matching problem is therefore kept as a separate third gate, with its own renormalization-scheme requirements.

7.1. Even and Odd Matching Amplitudes

The crossed low-energy operators are combined as

$${\mathcal{O}}_{+}={\displaystyle \frac{{\mathcal{O}}_{12}+{\mathcal{O}}_{21}}{\sqrt{2}}},{\mathcal{O}}_{-}={\displaystyle \frac{{\mathcal{O}}_{21}-{\mathcal{O}}_{12}}{\sqrt{2}}}.$$

After pole amputation and operator-mixing subtraction, denote the zero-momentum form factors by $F_{+}\left(0\right)$ and $F_{-}\left(0\right)$. For a pole-resolved route-symmetric matching branch, the adopted stronger gate is

$$\begin{array}{|c|}\hline F_{+}\left(0\right)\ne 0,F_{-}\left(0\right)=0,C_{-}^{\mathrm{reg}}=0.\\ \hline\end{array}$$

The first condition prevents a trivial zero-matching solution. The second removes the odd contribution of the isolated pole, while the third removes route-odd regular spectral/matching weight. An algebraic cancellation between a nonzero pole-odd term and a nonzero regular remainder could reproduce ${\Delta }_{\Pi }=0$, but it would be a different cancellation mechanism and is not called route-symmetric matching here.

Under any finite invertible parity-preserving renormalization or source recombination,

$$F_{+}\mapsto Z_{+}{F}_{+},F_{-}\mapsto Z_{-}{F}_{-},$$

with nonsingular $Z_{\pm }$. Hence the statements $F_{-}=0$ and $F_{+}\ne 0$ are invariant in this class. The regular remainder must be transformed in the same operator scheme; its vanishing is likewise a parity-resolved statement once that decomposition is fixed. If a scheme mixes the parity blocks, $F_{-}$ alone is not an invariant object and the full matching vector must be transformed. This is the scheme stopping rule used throughout the article.

7.2. Canonical Practical Protocol

The unambiguous practical order is:

1.

contact subtraction at a fixed positive $t_0$;

2.

construction of $G=C\left(t_0\right)$ and whitening by $G^{-1/2}$;

3.

a parity-resolved multi-profile GEVP;

4.

pole tracking under changes of volume, regulator, and microscopic deformation;

5.

evaluation of $K_{\perp }$ at the tracked pole rather than at a frozen reference energy;

6.

pole-amputated, mixing-subtracted extraction of $F_{+}\left(0\right)$ and $F_{-}\left(0\right)$ in the same parity-preserving scheme;

7.

controlled finite-volume, regulator, and continuum limits before comparison with the phenomenological tolerance.

Route-resolved step scaling and finite Euclidean moment/Krylov tests provide independent consistency checks and are detailed in Appendices Appendix B.1.13 and Appendix B.11.

7.3. Finite Channel Jacobian for the Remaining Bridge

The remaining microscopic-to-phenomenological bridge can be written as a finite derivative. Let the channel vector be ordered as

$$a=(a_{00},a_{03},a_{150},a_{153}),$$

and let $a_X^{\prime }=da/d{ϵ}_X$ describe the response to a microscopic route deformation ${ϵ}_X$. In the explicitly defined linear flavour-response benchmark, the observable covector is

$${\mathbf{j}}_{\Pi }=(-5.2174599612,-5.2592962711,4.7375121485,5.7392440838),$$

while the scalar slope along the canonical crossed-amplitude coordinate is

$$s_{\Pi B}={\left.{\displaystyle \frac{\partial {\Delta }_{\Pi }}{\partial {ϵ}_B}}\right|}_0=40.947129303807.$$

The observable-equivalent bridge is therefore

$${\chi }_X={\displaystyle \frac{{\mathbf{j}}_{\Pi }·a_X^{\prime }}{s_{\Pi B}}}.$$

The corresponding unit-channel coefficients are approximately

$$(-0.12742,-0.12844,+0.11570,+0.14016).$$

These numbers belong to the stated linear benchmark and are not a first-principles four-dimensional prediction. Their value is conceptual: the formerly abstract matching susceptibility is reduced to a finite four-component channel derivative. In particular, source-level channel support is not by itself a mass-plane response: the observable covector has a nonzero $E_{153}$ component, so the final matching dynamics must be evaluated in the full flavour-response map.

For the present public uncertainty ${\sigma }_{{\Delta }_{\Pi }}=0.01219$, a one-percent microscopic deformation remains below a one-standard-deviation mass-plane shift if

$$|{\chi }_X|<0.02977010.$$

This is a phenomenological tolerance, not a naturalness theorem.

7.4. Why the Physical Channel Jacobian Cannot Yet Be Run from the Published EFT Pieces

The finite bridge above is well defined once a complete renormalized Pati–Salam plus flavour effective theory is fixed. The presently published ingredients do not yet provide such a closed one-loop theory, and this is a concrete rather than merely formal obstruction.

For the Pati–Salam breaking bilinear ${\Phi }^{†}\Phi$ the four irreducible channels are

$$E_{00},E_{03},E_{150},E_{153}.$$

After subtracting the common anomalous dimension, the most general channel-preserving running already contains three independent splittings,

$${\kappa }_{03}={\gamma }_{03}-{\gamma }_{00},{\kappa }_{150}={\gamma }_{150}-{\gamma }_{00},{\kappa }_{153}={\gamma }_{153}-{\gamma }_{00},$$

not the two independent portal rates used in an earlier sensitivity parametrization. Even in this diagonal channel approximation the induced running in the primitive basis $(B_0,B_R,B_L,B_X)$ is non-diagonal.

More importantly, the six saved flavour tensors $F^{\prime },F,G,I,J,K$ do not form a closed one-loop running basis under the allowed preon wave-function directions

$$\delta \lambda =H^T\lambda +\lambda H,H\in \{{\lambda }^{†}\lambda ,{\lambda }^{\prime †}{\lambda }^{\prime },{\lambda }^{†}\lambda +{\lambda }^{\prime †}{\lambda }^{\prime }\}.$$

At the full-precision benchmark the complex span grows from rank 6 to rank 9, while the real-coefficient span grows from rank 6 to rank 12; the largest residuals outside the old span are approximately $0.167$, $0.353$, and $0.369$ for the three displayed directions. The missing directions have nonzero projection on ${\Delta }_{\Pi }$. Therefore a closed $6\times 6$ topology anomalous-dimension matrix is insufficient.

The minimal non-derivative physical class makes the size of the completion explicit. For

$${\mathcal{O}}_{a,ij}={\left(P_R^c{P}_L\right)}_{ij}H{\left({\Phi }^{†}\Phi \right)}_a,a\in \{00,03,150,153\},i,j=1,2,3,$$

there are four Pati–Salam contraction channels and nine independent complex $3\times 3$ flavour directions, hence

$$4\times 9=36\mathrm{complex}\mathrm{operators}$$

or 72 real Wilson components. A nine-direction flavour spanning set can cover the full complex $3\times 3$ matrix space, but the saved topology basis is numerically ill conditioned, with condition number about $3.10\times {10}^4$. In a canonical matrix-unit basis all 72 real components have nonzero projection on ${\Delta }_{\Pi }$ at the stated central benchmark. Thus unknown counterterm directions cannot be discarded as automatically invisible to the observable.

This result explains the remaining status of ${\chi }_X$. The obstruction is not a lack of a general renormalization method; it is the absence of one fully specified combined renormalized Lagrangian, threshold spectrum, counterterm basis, and nonperturbative boundary matching. Consequently numerical values assigned to ${\kappa }_{03},{\kappa }_{150},{\kappa }_{153}$ or to the physical ${\chi }_X$ before those inputs are fixed are completion diagnostics rather than predictions. The correct stopping rule is to construct the complete running basis first and project onto the observable covector ${\mathbf{j}}_{\Pi }$ only at the end.

7.5. Pole-Response-Resolved Route-21 Bridge

For route 21 the source algebra is more specific than a general four-component Jacobian. The two exact crossed source tensors are

$$B_{12}=4(E_{153}-E_{150}),B_{21}=6E_{153}-2E_{03}.$$

The second relation isolates the two irreducible channels entering the route-21 side, so the ratio $-2:6$ is fixed at source level rather than fitted in the low-energy theory. The two irreducible channels can nevertheless acquire different pole residues, amputated vertices, and renormalization factors. Denote the corresponding reduced responses by ${\mathcal{P}}_{03}\left(0\right)$ and ${\mathcal{P}}_{153}\left(0\right)$ and define

$$r_{153/03}\equiv {\displaystyle \frac{{\mathcal{P}}_{153}\left(0\right)}{{\mathcal{P}}_{03}\left(0\right)}}.$$

After factoring out the common $E_{03}$ response and collecting the remaining overall normalization into the route-21 benchmark amplitude ${\alpha }_{21}$,

$$B_{21}^{\mathrm{eff}}\left(r_{153/03}\right)=-2E_{03}+6r_{153/03}{E}_{153}.$$

In the specified linear flavour-response benchmark,

$$\delta {\Delta }_{\Pi }={\alpha }_{21}{D}_{21}\left(r_{153/03}\right)+O\left({\alpha }_{21}^2\right),$$

with

$$D_{21}\left(r\right)=10.518592542157+34.435464502830r.$$

The route-21 contribution to ${\Delta }_{\Pi }$ changes sign only at $r_0=-0.305458128532$. Thus a positive route-21 contribution corresponds to $r_{153/03}>r_0$; equal channel response is not required. This sign statement concerns the route-21 contribution alone and is not itself a statement of full reciprocity.

At $r_{153/03}=1$,

$$D_{21}\left(1\right)=44.954057044987,{\displaystyle \frac{D_{21}\left(r\right)}{D_{21}\left(1\right)}}=0.233985389386+0.766014610614r.$$

The interval preserving the equal-response response coefficient within $10\%$ is $0.869454<r_{153/03}<1.130546$; the corresponding $20\%$ interval is $0.738908<r_{153/03}<1.261092$. If the two reduced responses have the same sign, $r_{153/03}\ge 0$, then $D_{21}>0$ automatically. The overall factor ${\alpha }_{21}$ is a route-21 matching amplitude, not the low-energy coefficient $C_{21}$ itself. Numerical amplitudes for a separate positive benchmark displacement are retained in the supplementary controls, where their reference shift is defined explicitly. The main result here is only that exact source Clebsches reduce the route-21 channel ambiguity to one response ratio and one overall matching amplitude. The fixed coefficients $-2$ and 6 are therefore source-level representation factors; they do not imply equal propagation responses or equal low-energy Wilson coefficients. This is a channel-resolved construction for the route-21 contribution only. It does not by itself establish the full reciprocity condition $C_{21}=C_{12}$, because the renormalized route-12 response and the common physical matching normalization must be determined in the same scheme. The result narrows one side of the bridge without replacing the two-route correlator test.

7.6. Finite Euclidean Information and Exact Decision Rules

The physical pole branch should not be established by an uncontrolled large-time one-exponential fit when only finite Euclidean information is available. Positive moment and block-Krylov constructions give exact extremal statements about subthreshold support. If the matrix moment problem reaches flat extension, the representing spectral measure is unique and route symmetry of the closing moment set implies route symmetry of the reconstructed regular measure. Before closure, symmetric low moments are not sufficient to exclude unresolved asymmetric spectral weight. The finite-moment, block-Krylov, flat-extension, and matching criteria are derived in Appendix B.11.

8. Five-Dimensional Branch: Effective Spectral Reconstruction, Not a Fundamental Assumption

The five-dimensional construction has a sharply limited role. Once the four-dimensional source space already contains the relevant route-odd direction, no extra coordinate is needed to manufacture a parity of the fundamental preons. The useful statement is instead spectral: under the stated cyclicity and moment assumptions, the positive projected measure admits a local Jacobi representation.

For a positive projected measure $d\mu \left(s\right)$,

$$c\left(\zeta \right)=\int {\displaystyle \frac{d\mu \left(s\right)}{\zeta +s}},\zeta >0,$$

Lanczos recursion constructs a semi-infinite Jacobi operator

$$H_J=\left(\begin{array}{cccc}{\alpha }_1& -{\beta }_1& 0& \dots \\ -{\beta }_1& {\alpha }_2& -{\beta }_2& \dots \\ 0& -{\beta }_2& {\alpha }_3& \ddots \\ ⋮& ⋮& \ddots & \ddots \end{array}\right),{\beta }_n>0,$$

such that

$$c\left(\zeta \right)=\langle 1\left|{(\zeta I+H_J)}^{-1}\right|1\rangle .$$

This is an exact spectral representation of the chosen cyclic composite sector. It does not imply that nature contains a literal fifth spacetime dimension.

There is also an exact quotient statement behind this reconstruction. Let $P_{\mathrm{rt}}^{-}=e_{-}{e}_{-}^{†}$ be the rank-one projector onto the selected route-odd line. For any positive covering-space resolvent $C\left(\zeta \right)$,

$$P_{\mathrm{rt}}^{-}C\left(\zeta \right)P_{\mathrm{rt}}^{-}=c_{-}\left(\zeta \right)P_{\mathrm{rt}}^{-},c_{-}\left(\zeta \right)={\int }_0^{\infty }{\displaystyle \frac{d{\mu }_{-}\left(s\right)}{\zeta +s}},d{\mu }_{-}\left(s\right)\ge 0.$$

If the quotient is taken as a low-energy datum, the exact Schur–Feshbach elimination of its orthogonal complement produces a scalar energy-dependent kernel on this one-dimensional physical channel. The route sector is therefore structurally closed without requiring the unprojected covering-space dynamics itself to be exactly route symmetric. In the orbifold/quotient completion this is the mechanism by which the crossed-even relation can be enforced. The stronger statement that the same quotient is generated dynamically by the unchanged four-dimensional theory is not assumed.

In the minimal one-band class with a homogeneous tail and one endpoint defect, the essential band edges $s_{-}$ and $s_{+}$ determine

$${\alpha }_{\infty }={\displaystyle \frac{s_{+}+s_{-}}{2}},{\beta }_{\infty }={\displaystyle \frac{s_{+}-s_{-}}{4}}.$$

Normalize the projected measure when forming moments and write $\langle s\rangle =\int sd\mu \left(s\right)/\int d\mu \left(s\right)$. The first moment determines the endpoint diagonal, so the defect and canonically normalized Robin coefficient are

$$c_J=\langle s\rangle -{\displaystyle \frac{s_{+}+3s_{-}}{4}},$$

$$c_{★}={\displaystyle \frac{\langle s\rangle -(s_{+}+3s_{-})/4}{\sqrt{(s_{+}-s_{-})/4}}}.$$

Higher moments reconstruct ${\beta }_1,{\alpha }_2,{\beta }_2,\dots$ and therefore test rather than assume the one-defect local-tail hypothesis.

The corresponding homogeneous endpoint phase diagram is exact in this class: a negative defect produces one state below the band; a moderate positive defect produces no state outside the band; a sufficiently large positive defect can produce only an above-band ultraviolet-localized state. These statements make the effective fifth coordinate falsifiable.

The common-reflection test can be made quantitative. Let $M_{*}$ be a nondegenerate Hermitian whitened $2\times 2$ correlator matrix with spectral gap

$${\Delta }_{*}=|{\lambda }_{+}\left(M_{*}\right)-{\lambda }_{-}\left(M_{*}\right)|,$$

and let $U_{*}$ be the reflection about its spectral axis. For any other Hermitian member $M\left(t\right)$ of the same whitened family,

$${∥{\displaystyle \frac{M\left(t\right)-U_{*}M\left(t\right)U_{*}}{2}}∥}_2={\displaystyle \frac{\parallel [M_{*},M\left(t\right)]{\parallel }_2}{{\Delta }_{*}}}.$$

If $M\left(t\right)$ is also nondegenerate, with gap $\Delta \left(t\right)$, the angle between the two spectral axes satisfies

$$sin\theta \left(t\right)={\displaystyle \frac{2\parallel [M_{*},M\left(t\right)]{\parallel }_2}{{\Delta }_{*}\Delta \left(t\right)}}.$$

Thus a near-degenerate reference spectrum amplifies a fixed commutator residual by $1/{\Delta }_{*}$. This is the precise form of the finite-time contamination warning: approximate pairwise commutation is physically meaningful only relative to the relevant spectral gap.

The same matrix correlator also admits route-resolved finite-time certificates. For a chosen threshold $E_{\mathrm{th}}$, the positive inertia of the threshold-normalized transfer matrix gives a lower bound on the number of source directions that must have subthreshold support; equivalently, generalized effective energies below $E_{\mathrm{th}}$ certify such support without fitting individual poles. In the route-odd trial direction, three Euclidean time slices give a Cantelli lower bound on the total subthreshold odd weight and, after normalization by the full two-source trace, a rigorous lower bound on the visible odd fidelity of at least one subthreshold state. These tests do not require a unique-pole assumption and do not require an uncontrolled $t\to \infty$ extrapolation.

Two limitations remain central. First, a scalar spectrum does not determine route parity. Second, an exact reflection in a two-state spectrum can be algebraically automatic, so genuine evidence for a physical route reflection requires stability under additional states or continuum weight and an independent matching vertex. Detailed quotient/Feshbach proofs, the Stieltjes–Jacobi transfer theorem, the Robin reconstruction, the finite-time inertia and Cantelli bounds, and the two-state anti-tautology result are given in Appendix C.

9. Six-Dimensional Branch: Prospective Ultraviolet Completion and the Remaining Metric Gate

The six-dimensional branch addresses a different question from the four-dimensional mass-plane mechanism: whether chirality, localization, and a smooth higher-dimensional environment can coexist without introducing an inconsistency. None of the six-dimensional results is used to prove ${\Delta }_{\Pi }=C_{21}-C_{12}$.

Several ingredients are established under their stated assumptions. A smooth gravitating vortex can localize exactly one four-dimensional Weyl zero mode per suitable parent multiplet. The standard graviton mode is normalizable, and several spectator/vector sectors admit nonnegative fluctuation operators. In the nonminimal no-turn branch, the geometry

$$L\left(r\right)=R_{c}tanh(r/R_c),M\left(r\right)={cosh}^{-\kappa }(r/R_c)$$

has a regular axis and no finite-radius turning point. With transverse bulk confinement, both the four-dimensional Planck norm and the appropriately weighted gauge norm can converge.

For the axisymmetric scalar/radion problem of the displayed local model, the quadratic operator has a positive-square structure. In the notation of the detailed derivation,

$$\mathbb{H}={\mathbb{Q}}^{†}\mathbb{Q}+{\mathbb{K}}^{†}\mathbb{K},$$

and smooth-cap regularity removes the boundary term. This excludes normalizable $m_4^2\le 0$ modes in the stated $n_{\theta }=0$ sector.

The first non-axisymmetric sector is subtler. The scalar-clock directions associated with translations are gauge/reparametrization directions of the coupled system. The completed clock sector has an exact Schur-null property, so a negative eigenvalue obtained by freezing the metric cannot be interpreted as a physical tachyon. After the scalar-clock image is removed, the remaining gauge-invariant metric problem can be written as

$$H_{\mathrm{phys}}=H_{+}-G_{-},H_{+}⪰0,G_{-}⪰0.$$

Here $H_{+}$ collects the nonnegative part of the physical quadratic operator, whereas $G_{-}$ contains the potentially destabilizing nonnegative contribution that enters with a minus sign. Under the stated self-adjointness and relative form-compactness assumptions, the existence of a negative mode is equivalent to the positive compact Birman–Schwinger operator

$$\mathcal{B}\left({\zeta }_{\mathrm{BS}}\right)=G_{-}^{1/2}{(H_{+}+{\zeta }_{\mathrm{BS}})}^{-1}{G}_{-}^{1/2}$$

reaching eigenvalue one as ${\zeta }_{\mathrm{BS}}\downarrow 0$. The auxiliary parameter ${\zeta }_{\mathrm{BS}}>0$ is a spectral regulator, not a physical mass parameter. The threshold quantity

$${\beta }_0=\underset{\mu \downarrow 0}{lim}{\lambda }_{max}\mathcal{B}\left(\mu \right)$$

has not been computed. A future finite-core result must therefore be reported together with a certified truncation/discretization error. If the threshold operator $\mathcal{B}\left(0\right)$ exists in the stated operator-norm sense and ${\mathcal{B}}_R$ is a finite-core approximation satisfying

$$\parallel \mathcal{B}\left(0\right)-{\mathcal{B}}_R{\parallel }_2\le {ϵ}_R,{\beta }_R={\lambda }_{max}{\mathcal{B}}_R,$$

then Weyl’s inequality gives

$$|{\beta }_0-{\beta }_R|\le {ϵ}_R.$$

Consequently,

$$\begin{array}{|c|}\hline {\beta }_R+{ϵ}_R<1⟹\mathrm{certified}\mathrm{dipole}-\mathrm{stability}\mathrm{PASS},\\ \hline\end{array}$$

whereas

$${\beta }_R-{ϵ}_R>1⟹\mathrm{certified}\mathrm{tachyonic}\mathrm{FAIL}.$$

Only $|{\beta }_R-1|\le {ϵ}_R$ is unresolved at that finite core. Thus the remaining six-dimensional question is not an open-ended search for another background ansatz; it is the construction of the physical constraint-reduced $U_{\mathrm{phys}}$ followed by one certified Birman–Schwinger eigenvalue calculation.

The six-dimensional branch is therefore prospective rather than claimed as a completed ultraviolet theory. The microscopic origin of the confining transverse phase, the complete matter/Yukawa representation content, interface dynamics when required, higher harmonics, and the full anomaly/inflow completion remain separate gates. The axisymmetric positivity theorem, clock-sector reduction, and Birman–Schwinger formulation are derived in Appendix D.

10. Seven-Dimensional Branch: Zero-Mode Multiplicity as a Negative Control

The seven-dimensional construction is not used as support for the four-dimensional route mechanism. Its present role is a consistency test. The tested gauge content has multiplicity 405. A genuine complex Dirac spinor in seven dimensions has eight complex components, and under the tested compactification and projection each gauge component produces two four-dimensional Weyl modes of the selected chirality. The resulting selected-chirality multiplicity is therefore $2\times 405=810$, rather than the intended one-mode-per-component count.

Current seven-dimensional status.The tested all-preon seven-dimensional branch does not reproduce the intended chiral zero-mode multiplicity. No minimal repair has yet passed the required representation, anomaly, and Kaluza–Klein checks. The branch is therefore retained as a negative control rather than as an active explanation of the charged-fermion plane.

The detailed spinor decomposition and zero-mode counting are given in Appendix E.

11. Discussion: What Is Central, What Is Supportive, and What Remains Decisive

The main scientific content is four-dimensional. Historically the mass-plane clue was identified a posteriori in the shell-motivated model-selection study of Ref. [10]; its present statistical status remains weak. What makes the same combination theoretically useful here is that the fixed route basis independently converts it into the exact operator identity ${\Delta }_{\Pi }=C_{21}-C_{12}$. The operator construction then shows that both crossed channels exist within the unchanged field content and that the nontrivial color coefficient required by the second route is generated by fixed recoupling algebra. This closes the operator-existence problem without turning the crossed pair into a microscopic Ward doublet. The all-length chronological no-go is therefore important precisely because it prevents an overstatement: the physical equality must emerge from dynamics. Family alignment and low-energy matching remain separate four-dimensional gates rather than consequences of route reciprocity.

The second central ingredient is the separation of family alignment from route reciprocity. A single antisymmetric three-family spurion cannot define a unique median direction. The corrected symmetric spurion can do so algebraically, and the gauge-neutral composite ${\overline{6}}_F$ channel provides a global vacuum branch. In the separate four-flavour parent embedding, the restricted $U{\left(1\right)}_X\times SU{\left(4\right)}_F$ spurion branch sets the descendant determinant coefficient to zero by an exact charge-and-center selection rule. This is a conditional completion statement, not an exact symmetry of the minimal three-$\psi$ Pati–Salam theory. More general completions can generate the determinant operator, but the sharp joint determinant bound leaves a finite globally stable domain. This materially improves the status of family selection while keeping the genuinely dynamical question explicit: the confining theory must determine which branch is realized and the corresponding renormalized scalar coefficients.

The third central ingredient is matching. A small or vanishing $K_{\perp }$ does not by itself reproduce the charged-fermion relation. The same physical branch must also have $F_{+}\left(0\right)\ne 0$, $F_{-}\left(0\right)=0$, and $C_{-}^{\mathrm{reg}}=0$ after amputation, mixing subtraction, and the same pole/regular decomposition. These pole zero/nonzero statements are robust under finite invertible parity-preserving scheme changes; the regular odd remainder must be controlled in the same decomposition. The general benchmark channel Jacobian shows that the bridge is finite dimensional. For route 21 the source algebra narrows it further: the local coefficient ratio is fixed by $B_{21}=-2E_{03}+6E_{153}$, leaving one reduced response ratio $r_{153/03}$ and one overall amplitude ${\alpha }_{21}$. This is useful progress but not yet reciprocity, because the route-12 response and the common physical normalization must be obtained in the same scheme.

The lower-dimensional confinement calculations are supportive rather than direct evidence for the four-dimensional theory, but the amount of dynamical work they contain is substantial. The $1+1$ dimensional $\mathrm{SU}\left(3\right)$ calculation is a genuine non-Abelian confinement witness: exact leading reciprocity follows when the two route-sensitive excitation costs are equal, and the departure from reciprocity is suppressed by the flux gap when they are unequal. The representation-matched $\mathrm{SU}\left(N\right)$ open-chain calculation proves that the odd gap remains $O\left(N\right)$ uniformly in chain length in the fundamental–fundamental–conjugate-symmetric singlet sector, while the long-chain $\mathrm{SU}\left(15\right)$ calculation shows that the pole-tracked susceptibility and its cancellation conditioning approach finite large-volume limits. Point splitting, plaquette and cross-junction recoupling, string breaking, transverse $2+1/3+1$ dimensional dressing, and variational source decontamination were then used as deliberate countertests. Together they show why several tempting shortcuts fail: large N does not protect every numerator, configuration-space heaviness is not spectral heaviness, transverse plaquettes can preserve a gap without filtering an additive odd source, and unscreened string protection cannot be extrapolated through string breaking. These results are an extensive nonperturbative/control-theory validation of ingredients and failure modes, but their numerical values cannot be imported into four dimensions.

The five-dimensional branch is also stronger than a heuristic extra-coordinate picture. Once a physical rank-one route quotient is specified, exact Schur–Feshbach reduction closes the selected channel, positivity gives a Stieltjes measure, and Lanczos/Favard reconstruction produces a local Jacobi chain whose endpoint resolvent is exactly the original cyclic correlator. In the one-band class, the spectral band and the first moments reconstruct the asymptotic bulk coefficients and the endpoint Robin defect, while higher moments test the local-tail hypothesis. Matrix correlators then supply finite-time inertia/GEVP and Cantelli-type route-odd certificates and a quantitative common-reflection defect. Thus the 5D branch is a mathematically closed effective spectral reconstruction in its stated class if the route quotient is accepted as a low-energy datum. What remains open is the stronger microscopic claim that unchanged four-dimensional $\mathrm{SU}{\left(15\right)}_p$ dynamics generates that quotient by itself. It is not required for the operator identity and should not be advertised as a discovered fundamental extra dimension. The six-dimensional branch has more ambitious ultraviolet content and correspondingly more open physics. Several stability subproblems are solved, but the physical dipole metric threshold remains uncomputed. The seven-dimensional branch presently fails a basic multiplicity test.

The remaining first-principles four-dimensional calculation is therefore conceptually simple even though technically difficult. A common source geometry, one canonical source metric, one renormalization prescription, and a parity-resolved multi-profile basis should be used to obtain $C_{ij}^{AB}\left(t\right)$. The same calculation must identify the pole branch, threshold, $K_{\perp }$, and three-point form factors. A positive result would be an actual dynamical explanation of the operator reciprocity. A negative result would be equally informative because the decision criteria have been fixed before the physical numbers are known.

12. Conclusions

The article addresses a deliberately narrow question: whether a weak regularity in the nine charged-fermion Yukawa eigenvalues can be translated into a nontrivial and testable microscopic statement in an anomaly-consistent $\mathrm{SU}{\left(15\right)}_p$ preon theory.

The phenomenological relation, first identified a posteriori in Ref. [10], is

$${\Delta }_{\Pi }=A_{\ell }+2A_d-A_u.$$

Current common-scale inputs place it near zero, but only at a standardized departure of about $1.56\sigma$ with the best public covariance reconstruction used here. No discovery-level claim follows from this number.

The strongest result is four-dimensional and exact:

$${\Delta }_{\Pi }=C_{21}-C_{12}.$$

Thus the exact plane is equivalent to reciprocity of two crossed composite coefficients. The corresponding crossed operators are independent, gauge-invariant, and constructible within the unchanged chiral field content. Their nonlocal spectator routing contains a genuine order-sensitive quotient, and the required $\mathrm{SU}{\left(4\right)}_{\mathrm{PS}}$ crossed color character is generated explicitly by a multi-vertex walled–Brauer recoupling. The effective $-1/2$ direct/crossed coefficient is fixed by projector algebra. At the same time, a stronger closure-preserving chronological involution is excluded. The physically viable hypothesis is therefore dynamical crossed-route reciprocity rather than a microscopic Ward identity.

The family direction is no longer an undefined label choice. A nondegenerate symmetric spurion reconstructs the median projector and $D_{+}=diag(1,-2,1)$ basis covariantly. A gauge-neutral light-family ${\overline{\mathbf{6}}}_F$ channel of residual $SU{\left(3\right)}_F$ supplies a global nonlinear vacuum branch. In the determinant-free quartic class, a sufficient domain is

$${\eta }_{\Sigma }>0,0<m_{\Sigma }^2<2{\eta }_{\Sigma }(h_2-h_1)(h_3-h_2),$$

$${\lambda }_2\le 0,{\lambda }_1+{\lambda }_2>0.$$

In the separate four-flavour parent embedding, a restricted $U{\left(1\right)}_X\times SU{\left(4\right)}_F$ spurion branch gives ${\kappa }_{det}=0$ to all polynomial orders after restriction to the light EFT. In more general branches ${\kappa }_{det}$ can be nonzero, but the sharp determinant theorem supplies a finite globally stable domain. The remaining family problem is therefore dynamical rather than algebraic: the confining completion must determine which branch and which renormalized scalar coefficients are realized.

The four-dimensional propagation test is now unambiguous. After canonical source whitening,

$$\parallel K_{\perp }{\parallel }_2=\sqrt{{\displaystyle \frac{{(K_{11}-K_{22})}^2}{4}}+{(\Im K_{12})}^2},{\eta }_{15}={\displaystyle \frac{\parallel K_{\perp }\parallel }{{\Delta }_{-}^{\left(15\right)}}}.$$

The same physical calculation must establish the pole-resolved matching conditions

$$F_{+}\left(0\right)\ne 0,F_{-}\left(0\right)=0,C_{-}^{\mathrm{reg}}=0.$$

These zero/nonzero statements are invariant under finite invertible parity-preserving renormalization and source normalization changes when the pole is amputated and mixing is treated consistently.

For the explicit route-21 source the local tensor ratio is already fixed,

$$B_{21}=-2E_{03}+6E_{153}.$$

Writing $r_{153/03}={\mathcal{P}}_{153}\left(0\right)/{\mathcal{P}}_{03}\left(0\right)$, the specified linear benchmark gives

$$D_{21}\left(r\right)=10.518592542157+34.435464502830r.$$

A positive route-21 contribution requires only $r_{153/03}>-0.305458128532$, not equality of the two reduced responses. This removes a source-level fitting freedom, but it remains a one-sided matching result: full reciprocity still requires the route-12 response and common physical normalization in the same scheme.

A substantial hierarchy of nonperturbative and confining controls has already been completed before the final four-dimensional calculation. In the genuine $1+1$ dimensional $\mathrm{SU}\left(3\right)$ baryon-transfer control, the first nonvanishing crossed amplitudes are exactly equal for equal route-sensitive excitation costs, while unequal costs generate the exact defect

$${ϵ}_{\mathrm{rt}}^{\mathrm{SU}\left(3\right)}={\displaystyle \frac{{\delta }_1-{\delta }_2}{2{\Delta }_F+{\delta }_1+{\delta }_2}},$$

showing explicitly how a confinement flux gap suppresses chronological memory. In the representation-matched $\mathrm{SU}\left(N\right)$ open chain, exact Gauss-law reduction gives ${\Delta }_{-,L}^{\left(N\right)}={\kappa }_{E}N/2+O\left(1\right)$ uniformly in L. For $N=15$, pole-tracked calculations through 30 links yield the large-volume control susceptibilities ${\mathcal{T}}_{\delta t}^{\mathrm{lin},\mathrm{ctrl}}\simeq 0.01279$ and ${\mathcal{T}}_y^{\mathrm{lin},\mathrm{ctrl}}\simeq 0.045586$, while the connector cancellation factor tends to a finite value of order $1.46\times {10}^{-2}$. Point-split, plaquette, cross-junction, screening/string-breaking, transverse $2+1/3+1$ dimensional, and variational source controls were used to test failure modes rather than to generate additional fitted numbers. Collectively, these calculations show that small route breaking can arise dynamically, that it can remain finitely conditioned with increasing volume, and that several naive protection arguments fail outside their domains. They are mechanism-level evidence and stopping rules, not numerical predictions for the physical four-dimensional ${\eta }_{15}$.

The five-dimensional branch supplies an exact effective spectral reconstruction with several nontrivial theorems. If the rank-one physical route quotient is accepted as a low-energy datum, exact Schur–Feshbach reduction closes the route channel; the positive projected resolvent has a Stieltjes measure; and Lanczos/Favard reconstruction produces a local Jacobi chain whose endpoint resolvent reproduces the same correlator. In the one-band Jacobi–Robin class, $s_{\pm }$ and the first moment determine ${\alpha }_{\infty }$, ${\beta }_{\infty }$, and the canonically normalized endpoint defect, while higher moments give direct locality tests. Route-resolved matrix correlators further provide finite-time inertia/GEVP certificates of subthreshold support, Cantelli lower bounds on route-odd spectral weight and fidelity, and an exact commutator formula for the common-reflection defect. The 5D construction is therefore structurally closed in its stated effective class if the route quotient is taken as one low-energy datum; whether unchanged four-dimensional $\mathrm{SU}{\left(15\right)}_p$ dynamics generates that quotient remains open. The six-dimensional branch is a prospective ultraviolet extension: several zero-mode and stability sectors are under analytic control, while the remaining physical dipole metric problem has been reduced to a Birman–Schwinger threshold that is not yet computed. The tested seven-dimensional branch yields twice the intended zero-mode multiplicity and is retained as a negative control.

Accordingly, the present work does not claim a first-principles derivation of the charged-fermion spectrum. It establishes the operator target, explicit crossed recouplings, constructive global family-alignment branches, exact positivity and finite-data criteria, a partially resolved route-21 bridge, scheme-robust matching conditions, an exact effective five-dimensional reconstruction in its stated class, and a substantial set of non-Abelian confining controls that test both the proposed suppression mechanisms and their failure modes. The logical burden is now concentrated in three independent four-dimensional gates: the realized family vacuum, dynamical equality of the two crossed route amplitudes, and the physical pole-resolved matching conditions $F_{+}\left(0\right)\ne 0$, $F_{-}\left(0\right)=0$, and $C_{-}^{\mathrm{reg}}=0$. The decisive unresolved calculation is the renormalized four-dimensional route correlator and three-point vertex of the explicit source family. Those observables will determine whether the mechanisms that survive the control hierarchy are actually realized by the confining $\mathrm{SU}{\left(15\right)}_p$ vacuum.

Appendix B.1.1. Local Operator Envelope

Let $H^m{}_{\dot{s}}\sim (\mathbf{1},\mathbf{2},\mathbf{2})$ be an electroweak bidoublet interpolator. For every primitive tensor $B_I$, define the composite-EFT operator

$${\mathcal{O}}_I\left[D\right]=D^i{}_j{ϵ}^{\rho \sigma }\left(P_{\overline{4}}^j\right)a^{\dot{r}}{{{}_{\rho }\left(B_I\right)^a}_{\dot{r}}}^b{}_{\dot{s}}H^m{}_{\dot{s}}\left(P_4^i\right)_{bm\sigma }.$$

All spectator and Lorentz indices are contracted explicitly. Every operator contains one bilinear Pati–Salam-breaking insertion $\Phi {\Phi }^{†}$. The routes differ only in the spectator/Wilson contraction pattern, not in the number of spurion insertions.

Appendix B.1.2. Explicit Crossed Pair, Trace-Character Structure, and the Nonlocal Six-K Witness

The microscopic analysis must distinguish four different statements: (a) whether the unchanged field content admits an explicit 24-field source skeleton; (b) how the required crossed tensors are represented in the local Pati–Salam tensor space; (c) whether a genuinely order-sensitive nonlocal contraction space exists; and (d) whether one microscopic operation ties the nonlocal order to the exchange $B_{12}\leftrightarrow B_{21}$. The first three questions can be answered constructively. The fourth remains open.

Appendix B.1.1.1. Outer prebaryons and a no-go for an eight-P completion

The two outer three-preon currents are

$$\begin{array}{cc}\hfill \left(P_{\overline{4}}^j\right)a^{\dot{r}}{}_{\rho }\left(x_1\right)& =N_{\overline{4}}{\Omega }_{{\alpha }_1{\beta }_1,\rho }\left(x_1\right){{{\left({\Psi }_{W^{\prime }}\right)}^{{\alpha }_1}}_a}^{\dot{r}}{}_{{\sigma }_1}\left(x_1\right){\left({\psi }_j\right)}^{{\beta }_1}{}_{{\tau }_1}\left(x_1\right){ϵ}^{{\sigma }_1{\tau }_1},\hfill \end{array}$$

(A1)

$$\begin{array}{cc}\hfill {\left(P_4^i\right)}^{bm}{}_{\rho }\left(x_8\right)& =N_4{\Omega }_{{\alpha }_8{\beta }_8,\rho }\left(x_8\right){\left({\Psi }_W\right)}^{{\alpha }_{8}bm}{}_{{\sigma }_8}\left(x_8\right){\left({\psi }_i\right)}^{{\beta }_8}{}_{{\tau }_8}\left(x_8\right){ϵ}^{{\sigma }_8{\tau }_8}.\hfill \end{array}$$

(A2)

Suppose, more strongly than required by the general $\Omega FF$ definition, that all eight junctions were restricted to $P_4\sim (4,2,1)$ and $P_{\overline{4}}\sim (\overline{4},1,2)$. If $n_4$ is the number of $P_4$ currents, the single bidoublet $H\sim (1,2,2)$ requires $n_4+1$ to be even, so $n_4$ is odd. The $\mathrm{SU}{\left(4\right)}_{\mathrm{PS}}$ center charge is $2n_4-8$ modulo four, while $\Phi {\Phi }^{†}$ is center neutral. A singlet therefore requires $2n_4-8\equiv 0(mod4)$, so $n_4$ is even. The assumptions are inconsistent.

Eight-P no-go.With eight local junctions, one H, and one $\Phi {\Phi }^{†}$ insertion, no Pati–Salam singlet exists if every junction is restricted to $P_4$ or $P_{\overline{4}}$. The general $\Omega FF$ source space is not excluded.

Appendix B.1.1.2. Six mixed internal junctions and the correct point-split gauge dressing

Define

$$K^{m\dot{r}}{}_{\rho }\left(x\right)=N_K{\Omega }_{\alpha \beta ,\rho }\left(x\right){\left({\Psi }_W\right)}^{\alpha am}{}_{\sigma }\left(x\right){{{\left({\Psi }_{W^{\prime }}\right)}^{\beta }}_a}^{\dot{r}}{}_{\tau }\left(x\right){ϵ}^{\sigma \tau }.$$

(A3)

The $\mathrm{SU}{\left(4\right)}_{\mathrm{PS}}$ index is contracted in the singlet channel of $4\otimes \overline{4}$, so

$$K\sim (1,2,2).$$

It is therefore an $\mathrm{SU}{\left(15\right)}_p$ and $\mathrm{SU}{\left(4\right)}_{\mathrm{PS}}$ singlet but not a full $G_{\mathrm{PS}}$ singlet. This distinction is essential for the point-split analysis. Choose

$${\mathcal{J}}_1=P_{\overline{4}}^j,{\mathcal{J}}_8=P_4^i,{\mathcal{J}}_v=K_v(v=2,\dots ,7).$$

The exact fermion count is

$$N_{\Omega }=8,N_{{\Psi }_W}=7,N_{{\Psi }_{W^{\prime }}}=7,N_{\psi }=2,$$

so the source contains precisely 24 fermionic field insertions.

For separated $K_v$ and $K_w$, local $\mathrm{SU}{\left(2\right)}_L\times \mathrm{SU}{\left(2\right)}_R$ gauge invariance requires spectator Wilson transport. With fundamental Wilson lines $U_L(v,w)$ and $U_R(v,w)$ from $x_w$ to $x_v$, define

$$\begin{array}{cc}\hfill S_{vw}\left[{\gamma }_{vw}\right]=& {ϵ}^{{\rho }_v{\rho }_w}{K}_v^{m\dot{r}}{}_{{\rho }_v}ϵ_{mn}{\left[U_L(v,w)\right]}^n{}_pϵ_{\dot{r}\dot{t}}{\left[U_R(v,w)\right]}^{\dot{t}}{}_{\dot{s}}K_w^{p\dot{s}}{}_{{\rho }_w}.\hfill \end{array}$$

(A4)

The transformation law $U(v,w)\mapsto g\left(x_v\right)U(v,w)g{\left(x_w\right)}^{-1}$ together with $g^Tϵg=ϵ$ makes this bilocal scalar locally gauge invariant. Edge reversal gives $S_{wv}\left[{\gamma }^{-1}\right]=S_{vw}\left[\gamma \right]$ after the Grassmann exchange sign is combined with the three antisymmetric invariant tensors. Thus the bare $ϵK_v{K}_w$ notation is valid only as shorthand for this Wilson-dressed expression in a fixed gauge or coincident limit.

Appendix B.1.1.3. Exact algebraic representation of the crossed tensors and the local-order no-go

Let

$$M^{a\dot{r}}{}_{b\dot{s}}={\Phi }^{a\dot{r}}{\Phi }_{b\dot{s}}^{†}$$

and define partial-trace superoperators

$$\begin{array}{cccc}\hfill {{\left({\mathcal{T}}_{4}M\right)}^{a\dot{r}}}_{b\dot{s}}& ={\delta }^a{}_{b}M^{c\dot{r}}{}_{c\dot{s}},\hfill & \hfill {{\left({\mathcal{T}}_{2}M\right)}^{a\dot{r}}}_{b\dot{s}}& =M^{a\dot{u}}{}_{b\dot{u}}\delta ^{\dot{r}}{}_{\dot{s}}.\hfill \end{array}$$

(A5)

Then

$${\mathcal{T}}_4{\mathcal{T}}_{2}M=B_0,{\mathcal{T}}_{4}M=B_R,{\mathcal{T}}_{2}M=B_L,M=B_X.$$

For $\mathrm{SU}\left(N\right)$ introduce

$${\mathcal{C}}_1^{\left(N\right)}=N\mathcal{I}-{\mathcal{T}}_N,{\mathcal{C}}_2^{\left(N\right)}={\mathcal{T}}_N-(N-1)\mathcal{I}.$$

The first kernel is fixed directly by fundamental Fierz completeness,

$${\left({\mathcal{C}}_1^{\left(N\right)}\right)}^{i\ell }{}_{jk}=N{\delta }^i{}_k\delta ^{\ell }{}_j-{\delta }^i{}_j\delta ^{\ell }{}_k=2N{\left(T^A\right)}^i{}_j\left(T^A\right)^{\ell }{}_k,$$

while ${\mathcal{C}}_2^{\left(N\right)}$ is the complementary trace-subtraction combination entering the exact product factorization. With the common right-factor sign convention,

$$B_{ij}=-{\mathcal{C}}_i^{\left(4\right)}{\mathcal{C}}_j^{\left(2\right)}M,$$

(A6)

and therefore

$$\begin{array}{cc}\hfill B_{11}& =-B_0+2B_R+4B_L-8B_X,\hfill \end{array}$$

(A7)

$$\begin{array}{cc}\hfill B_{12}& =B_0-B_R-4B_L+4B_X,\hfill \end{array}$$

(A8)

$$\begin{array}{cc}\hfill B_{21}& =B_0-2B_R-3B_L+6B_X,\hfill \end{array}$$

(A12)

$$\begin{array}{cc}\hfill B_{22}& =-B_0+B_R+3B_L-3B_X.\hfill \end{array}$$

(A10)

Equivalently,

$$R=A_4\otimes (-A_2),A_N=\left(\begin{array}{cc}-1& N\\ 1& -(N-1)\end{array}\right).$$

This is an exact algebraic representation of the four rows. It explains why the integers $4,3,2,1$ are tied to the dimensions $4,2$ and their complements. It does not by itself prove that a particular microscopic interaction dynamically selects these four superoperator combinations.

The local tensor space decomposes multiplicity-free,

$$End\left(4\right)\otimes End\left(2\right)=(1,1)\oplus (1,3)\oplus (15,1)\oplus (15,3).$$

Hence, by Schur’s lemma, its linear $G_{\mathrm{PS}}$-equivariant endomorphism algebra is Abelian. In particular, two fixed local equivariant maps $\mathcal{A},\mathcal{B}$ satisfy $\mathcal{A}\mathcal{B}M=\mathcal{B}\mathcal{A}M$ and cannot generate $B_{12}\ne B_{21}$ merely by reversing chronological order.

Local chronological-order no-go.The difference $B_{21}-B_{12}$ cannot be the result of applying the same two local $G_{\mathrm{PS}}$-equivariant operations to one $\Phi {\Phi }^{†}$ tensor in opposite order. Any genuine chronological effect must use a larger multiplicity space or nonlocal dynamics.

Appendix B.1.1.4. Rank-one trace-character theorem

The sectoral evaluation used later corresponds to a rank-one Pati–Salam-breaking direction. Write

$${\Phi }^{a\dot{r}}=\sqrt{\sigma }{u}^a{r}^{\dot{r}},u^{†}u=1,r^{†}r=1,$$

and define the rank-one projectors

$$p=|u\rangle \langle u|,q=|r\rangle \langle r|.$$

Then

$$M=\sigma p\otimes q,B_0=\sigma I_4\otimes I_2,B_L=\sigma p\otimes I_2,B_R=\sigma I_4\otimes q.$$

For a rank-one projector $p_N$ in an N-dimensional space define two canonical trace characters

$$X_N^{\left(0\right)}=I_N-N{p}_N,Tr{X}_N^{\left(0\right)}=0,$$

$$X_N^{\left(1\right)}=I_N-(N-1)p_N,Tr{X}_N^{\left(1\right)}=1.$$

The four route tensors become exactly

$$\begin{array}{cc}\hfill B_{11}& =-\sigma X_4^{\left(0\right)}\otimes X_2^{\left(0\right)},\hfill \end{array}$$

(A11)

$$\begin{array}{cc}\hfill B_{12}& =+\sigma X_4^{\left(0\right)}\otimes X_2^{\left(1\right)},\hfill \end{array}$$

(A12)

$$\begin{array}{cc}\hfill B_{21}& =+\sigma X_4^{\left(1\right)}\otimes X_2^{\left(0\right)},\hfill \end{array}$$

(A13)

$$\begin{array}{cc}\hfill B_{22}& =-\sigma X_4^{\left(1\right)}\otimes X_2^{\left(1\right)}.\hfill \end{array}$$

(A14)

Thus the crossed pair is the exchange of the trace labels $(0,1)$ and $(1,0)$. The coefficients N and $N-1$ are uniquely fixed by the trace conditions 0 and 1 and do not use the charged-fermion masses.

Trace-character theorem.Under the rank-one breaking assumption, the complete route matrix is equivalent to the tensor product of the two canonical trace characters of the 4 and 2 Pati–Salam factors. The exact mass-plane target can therefore be called crossed trace-character reciprocity. This is an algebraic identification, not yet a Ward symmetry of the confining theory.

Appendix B.1.1.5. Irreducible-channel decomposition and a stronger local symmetry obstruction

Define the superoperator projectors

$${\mathcal{P}}_1^{\left(4\right)}={\displaystyle \frac{{\mathcal{T}}_4}{4}},{\mathcal{P}}_{15}^{\left(4\right)}=\mathcal{I}-{\displaystyle \frac{{\mathcal{T}}_4}{4}},{\mathcal{P}}_1^{\left(2\right)}={\displaystyle \frac{{\mathcal{T}}_2}{2}},{\mathcal{P}}_3^{\left(2\right)}=\mathcal{I}-{\displaystyle \frac{{\mathcal{T}}_2}{2}},$$

and $M_{R_4,R_2}={\mathcal{P}}_{R_4}^{\left(4\right)}{\mathcal{P}}_{R_2}^{\left(2\right)}M$. Direct substitution gives

$$\begin{array}{cc}\hfill B_{12}& =4\left(M_{15,3}-M_{15,1}\right),\hfill \end{array}$$

(A15)

$$\begin{array}{cc}\hfill B_{21}& =2\left(3M_{15,3}-M_{1,3}\right).\hfill \end{array}$$

(A16)

The two tensors share the $(15,3)$ channel but have different additional support: $(15,1)$ versus $(1,3)$. Because these are inequivalent irreducible $G_{\mathrm{PS}}$ modules, a local symmetry commuting with $G_{\mathrm{PS}}$ cannot exchange the crossed tensors.

$G_{\mathrm{PS}}$-equivariant exchange no-go.Within the local multiplicity-free $\Phi {\Phi }^{†}$ tensor space, no linear symmetry commuting with $G_{\mathrm{PS}}$ can implement $B_{12}\leftrightarrow B_{21}$. A physical exchange, if it exists, must involve nonlocal/composite multiplicity, broken-phase information, or additional dynamical structure.

Appendix B.1.1.6. Quotient decomposition of the fixed-K 1 spectator space

The six internal currents in this source class are projected locally to the Pati–Salam singlet channel of $4\otimes \overline{4}$,

$$K_1\equiv K=\Omega {\Psi }_W{\Psi }_{W^{\prime }}\sim (1,2,2).$$

For six two-dimensional doublets the exact spin decomposition is

$${\left(\mathbf{2}\right)}^{\otimes 6}=5V_0\oplus 9V_1\oplus 5V_2\oplus V_3,$$

(A17)

where $V_J$ denotes the spin-J representation. Equivalently,

$$m_J=\left({\displaystyle \genfrac{}{}{0pt}{}{6}{3-J}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{6}{2-J}}\right),$$

with the endpoint convention that binomial coefficients outside their natural range vanish.

For the Lorentz and $\mathrm{SU}{\left(2\right)}_L$ factors there are two outer doublets. Since

$${\mathbf{2}}^{\otimes 2}=V_0\oplus V_1,$$

the total singlet multiplicity is

$$14=5·1+9·1=5+9.$$

(A18)

The 5 is the K-singlet/outer-singlet factorized sector; the 9 is the K-triplet/outer-triplet mixed sector.

For $\mathrm{SU}{\left(2\right)}_R$ there are four outer doublets,

$${\mathbf{2}}^{\otimes 4}=2V_0\oplus 3V_1\oplus V_2,$$

so

$$42=5·2+9·3+5·1=10+27+5.$$

(A19)

The two outer $\mathrm{SU}{\left(4\right)}_{PS}$ singlet tensors remain independent in the fixed-$K_1$ source class. Therefore the spectator invariant space has

$$2\times 14\times 14\times 42=16464$$

channels. The representation count requires the following qualification: the numbers 14 and 42 are already dimensions of the $\mathrm{SU}\left(2\right)$ invariant spaces after the Schouten relations have been imposed. Thus 16464 is not a pre-Schouten pairing count.

The completely factorized sector in which the six $K_1$ currents are singlets separately in the Lorentz, L, and R factors has dimension

$$2\times 5\times 5\times 10=500=125\times 4.$$

(A20)

Hence

$$16464=500+15964.$$

(A21)

The 15964-dimensional complement contains at least one nonzero total K spin coupled to the corresponding outer non-singlet tensor. This is an exact spectator representation count; it is not a count of physical states or of independent renormalized operators after the common Grassmann, locality, and source restrictions are imposed.

Appendix B.1.1.7. Why a set-theoretic reconnection does not descend to the quotient

The set-theoretic graph argument must be replaced by a quotient-compatible linear operation. Let $\left(ab\right)\left(cd\right)\left(ef\right)$ denote the product of three antisymmetric $ϵ$ contractions for six doublets. Schouten gives, for example,

$$\left(23\right)\left(45\right)\left(67\right)-\left(24\right)\left(35\right)\left(67\right)+\left(25\right)\left(34\right)\left(67\right)=0.$$

(A22)

A set-theoretic rule $E_{23}$ that simply reconnects the two pairs containing 2 and 3 sends all three terms on the left of Equation (A8) to the same matching $\left(23\right)\left(45\right)\left(67\right)$. It would therefore send the zero vector to a nonzero vector. Consequently the naive rewiring map is not a well-defined linear operator on the singlet quotient.

This does not destroy the positive result. It identifies the correct algebraic replacement.

Appendix B.1.1.8. Temperley–Lieb repair: a genuine quotient-compatible ordered route

For adjacent doublets define

$${\left(e_i\right)}^{a_i{a}_{i+1}}{}_{b_i{b}_{i+1}}={ϵ}^{a_i{a}_{i+1}}{ϵ}_{b_i{b}_{i+1}}.$$

(A23)

This is an ordinary invariant linear operator on the tensor product and therefore descends automatically to the singlet quotient. With the convention in Equation (A9),

$$e_i^2=2e_i,e_i{e}_{i+1}{e}_i=e_i,[e_i,e_j]=0\left(\right|i-j|>1),$$

(A24)

which is the Temperley–Lieb algebra with loop weight 2.

Choose the noncrossing basis of the six-doublet singlet quotient

$$\begin{array}{cccc}\hfill m_1& =\left(23\right)\left(45\right)\left(67\right),\hfill & \hfill m_2& =\left(23\right)\left(47\right)\left(56\right),\hfill \\ \hfill m_3& =\left(25\right)\left(34\right)\left(67\right),\hfill & \hfill m_4& =\left(27\right)\left(34\right)\left(56\right),\hfill \\ \hfill m_5& =\left(27\right)\left(36\right)\left(45\right).\hfill & \hfill \end{array}$$

The reference state is $m_5=M_0$, while the two ordered six-K matchings are $m_3=M_{12}^K$ and $m_2=M_{21}^K$. In this basis the exact quotient matrices are

$$e_{23}=\left(\begin{array}{ccccc}2& 0& -1& 0& -1\\ 0& 2& 0& -1& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right),e_{34}=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ -1& 0& 2& 0& 0\\ 0& -1& 0& 2& -1\\ 0& 0& 0& 0& 0\end{array}\right).$$

(A25)

Direct multiplication gives the exact ordered outputs

$$e_{34}{e}_{23}{m}_5=m_3,e_{23}{e}_{34}{m}_5=m_2,m_3\ne m_2.$$

(A26)

Thus the order-sensitive construction survives the Schouten quotient, but the correct statement is about the noncommuting Temperley–Lieb operators rather than an arbitrary set map on pairing diagrams.

The permutation

$${\tau }_K=\left(24\right)\left(57\right)$$

acts in the basis above as

$$T_{\tau }=\left(\begin{array}{ccccc}0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1\end{array}\right).$$

(A27)

One checks exactly that

$$T_{\tau }^2=I,T_{\tau }{e}_{23}{T}_{\tau }=e_{34},T_{\tau }{m}_5=m_5,T_{\tau }{m}_3=m_2.$$

The one-factor singlet quotient has a 3-dimensional even and a 2-dimensional odd subspace. Applying the same geometric permutation to the Lorentz, $\mathrm{SU}{\left(2\right)}_L$, and $\mathrm{SU}{\left(2\right)}_R$ six-K singlet factors gives, on the factorized 125-dimensional internal space,

$$dim{\mathcal{H}}_{K,+}=63,dim{\mathcal{H}}_{K,-}=62.$$

(A28)

At trivial spectator holonomy the single-factor pairing tensors obey

$$\langle m_2,m_2\rangle =\langle m_3,m_3\rangle =8,\langle m_2,m_3\rangle =2.$$

For the three-factor product route tensors this gives the exact index-theoretic overlap

$${\displaystyle \frac{\langle K_{12},K_{21}\rangle }{\sqrt{\langle K_{12},K_{12}\rangle \langle K_{21},K_{21}\rangle }}}={\displaystyle \frac{1}{64}}.$$

(A29)

The corresponding normalized Gram matrix has eigenvalues $65/64$ and $63/64$. Hence the even and odd source directions are both nonzero and the two route tensors are not close to a singular pair in this purely index-tensor benchmark. Equation (A15) is not a physical renormalized correlator and is not transported to nontrivial Wilson holonomy without a separate calculation.

Quotient-compatible nonlocal route theorem.The fixed-$K_1$ 24-field skeleton contains a genuine order-sensitive internal route on the Schouten-reduced singlet quotient. The correct route generators are the invariant Temperley–Lieb operators $e_{23}$ and $e_{34}$; their two ordered products acting on $M_0$ reproduce the two ordered route states and are exchanged by the independently specified ${\tau }_K$. This establishes a nonzero internal odd channel, but it does not yet establish $B_{12}\leftrightarrow B_{21}$.

Appendix B.1.1.9. A new exact no-go: the singlet-K 1 route is SU (4) PS blind

The quotient-compatible route operators above act on the six singlet-projected currents $K_1\sim (1,2,2)$. Since each $K_1$ carries no $\mathrm{SU}{\left(4\right)}_{PS}$ index, any operation generated solely by their Lorentz/L/R contractions and the corresponding Wilson transport acts as the identity on the outer $\mathrm{SU}{\left(4\right)}_{PS}$ tensor factor.

The two crossed tensors, however, have different $\mathrm{SU}{\left(4\right)}_{PS}$ trace-character structure. Using the irreducible superoperator projectors,

$${\mathcal{C}}_1^{\left(4\right)}=4{\mathcal{P}}_{15}^{\left(4\right)},{\mathcal{C}}_2^{\left(4\right)}={\mathcal{P}}_1^{\left(4\right)}-3{\mathcal{P}}_{15}^{\left(4\right)},$$

while

$$B_{12}=-{\mathcal{C}}_1^{\left(4\right)}{\mathcal{C}}_2^{\left(2\right)}M,B_{21}=-{\mathcal{C}}_2^{\left(4\right)}{\mathcal{C}}_1^{\left(2\right)}M.$$

Thus the required exchange changes the $\mathrm{SU}{\left(4\right)}_{PS}$ channel itself.

Singlet-$K_1$ route-selection no-go.No internal route operation acting only in the locally singlet-projected $K_1\sim (1,2,2)$ sector can by itself map $B_{12}$ to $B_{21}$. The positive Temperley–Lieb route is therefore a genuine internal ordering effect, but not a complete microscopic selector of the crossed primitive tensors.

This theorem is stronger and cleaner than the earlier factorized single-outer-graph no-go, but its assumption is also explicit: the internal ${\Psi }_W{\Psi }_{W^{\prime }}$ pair has already been projected locally to the $\mathrm{SU}{\left(4\right)}_{PS}$ singlet. A full preon-level recoupling need not make that projection before the route operation.

Appendix B.1.1.10. Minimal unchanged-field adjoint lift

Before the local $\mathrm{SU}{\left(4\right)}_{PS}$ projection define the unprojected three-preon current

$${{{\mathbb{K}}^a}_b}^{m\dot{r}}{}_{\rho }=N_{\mathbb{K}}{\Omega }_{\alpha \beta ,\rho }{\left({\Psi }_W\right)}^{\alpha am}{{{}_{\sigma }\left({\Psi }_{W^{\prime }}\right)^{\beta }}_b}^{\dot{r}}{}_{\tau }ϵ^{\sigma \tau }.$$

(A30)

The spectator product $4\otimes \overline{4}=1\oplus 15$ yields two currents made from exactly the same three fermion fields,

$$\begin{array}{cc}\hfill {K_1^{m\dot{r}}}_{\rho }& ={{{\mathbb{K}}^a}_a}^{m\dot{r}}{}_{\rho },\hfill \end{array}$$

(A31)

$$\begin{array}{cc}\hfill {K_{15}^{A,m\dot{r}}}_{\rho }& ={\left(T^A\right)}^b{{{}_a\mathbb{K}^a}_b}^{m\dot{r}}{}_{\rho }\sim (15,2,2).\hfill \end{array}$$

(A32)

With $Tr\left(T^A{T}^B\right)={\delta }^{AB}/2$, the fundamental completeness relation gives the exact inversion

$${\mathbb{K}}^a{}_b={\displaystyle \frac{1}{4}}{\delta }^a{}_{b}K_1+2{\left(T^A\right)}^a{}_{b}K_{15}^A.$$

(A33)

No new fundamental field, family index, or fermion insertion has been introduced. The $\mathrm{SU}{\left(15\right)}_p$ singlet contraction and the Weyl structure are the same as in $K_1$; only the spectator $4\otimes \overline{4}$ channel is left unprojected. The adjoint current is therefore the minimal unchanged-field internal mediator capable of carrying the missing $\mathrm{SU}{\left(4\right)}_{PS}$ recoupling information.

For separated vertices the adjoint transport is

$${\left[U_{15}(v,w)\right]}^{AB}=2Tr\left[T^A{U}_4(v,w)T^B{U}_4{(v,w)}^{†}\right],$$

(A34)

combined with the same fundamental $U_L$ and $U_R$ transports. This provides a gauge-covariant point-split realization without changing the 24-fermion census.

Appendix B.1.1.11. The full preon-level space is larger than the fixed-K 1 count

Equation (A7) is the exact spectator count only after each of the six internal $4\otimes \overline{4}$ pairs has been projected to $K_1$. If those $\mathrm{SU}{\left(4\right)}_{PS}$ indices are retained before recoupling, there are eight fundamentals and eight antifundamentals in the full source, including the outer ${\Psi }_W/{\Psi }_{W^{\prime }}$ and the $\Phi /{\Phi }^{†}$ pair. Schur–Weyl duality gives

$$dim{End}_{\mathrm{SU}\left(4\right)}\left(4^{\otimes 8}\right)=\sum _{\lambda ⊢8,\ell \left(\lambda \right)\le 4}{\left(f^{\lambda }\right)}^2=33324.$$

(A35)

Combining this raw $\mathrm{SU}\left(4\right)$ commutant count with the already reduced Lorentz/L/R invariant dimensions gives the ambient tensor-product number

$$33324\times 14\times 14\times 42=274323168.$$

This very large number is not a physical operator count: the local $\Omega FF$ structure, common field species, Grassmann statistics, point-splitting restrictions, and route definition impose strong correlations. Its purpose is to show why a literal unstructured enumeration of “all 24-field contractions” is the wrong computational object. The representation-channel reduction in Equation (A17) is the useful minimal description.

Appendix B.1.1.12. The remaining coherent channel-amplitude gate

The exact kernels can be written for either group factor as

$${\mathcal{C}}_1^{\left(N\right)}=N{\mathcal{P}}_{\mathrm{adj}}^{\left(N\right)},{\mathcal{C}}_2^{\left(N\right)}={\mathcal{P}}_1^{\left(N\right)}-(N-1){\mathcal{P}}_{\mathrm{adj}}^{\left(N\right)}.$$

(A36)

Therefore route 12 requires a pure $\mathrm{SU}{\left(4\right)}_{PS}$ adjoint selector together with the $\mathrm{SU}{\left(2\right)}_R$ singlet–triplet difference, whereas route 21 requires a pure $\mathrm{SU}{\left(2\right)}_R$ triplet selector together with the $\mathrm{SU}{\left(4\right)}_{PS}$ coherent singlet–adjoint combination of relative weights $1:-3$. The new $K_{15}$ current supplies the required adjoint carrier. Group theory by itself does not yet fix the coherent $1:-3$ amplitude.

Appendix B.1.1.13. Canonical crossing test: exact SU (2) exception and SU (4) no-go

In the unnormalized channel basis $({\mathcal{P}}_1,{\mathcal{P}}_{\mathrm{adj}})$, write

$$c_1^{\left(N\right)}=(0,N),c_2^{\left(N\right)}=(1,-(N-1)).$$

The fundamental four-point crossing matrix is

$$F_N={\displaystyle \frac{1}{N}}\left(\begin{array}{cc}1& N^2-1\\ 1& -1\end{array}\right),F_N^2=I.$$

(A37)

A direct multiplication gives $c_1^{\left(N\right)}{F}_N=(1,-1)$ and therefore

$$c_1^{\left(N\right)}{F}_N=c_2^{\left(N\right)}⟺N=2.$$

(A38)

For $N=2$ the reverse equality also holds,

$$c_1^{\left(2\right)}{F}_2=c_2^{\left(2\right)},c_2^{\left(2\right)}{F}_2=c_1^{\left(2\right)}.$$

(A39)

so the $\mathrm{SU}{\left(2\right)}_R$ part of the crossed pair is an ordinary canonical crossing. For $\mathrm{SU}{\left(4\right)}_{PS}$,

$$F_4={\displaystyle \frac{1}{4}}\left(\begin{array}{cc}1& 15\\ 1& -1\end{array}\right),(0,4)F_4=(1,-1)\ne (1,-3).$$

(A40)

The reverse image $(1,-3)F_4=(-1/2,9/2)$ is not proportional to $(0,4)$.

Both $K_1$ and $K_{15}$ are spectator projections of the same Grassmann monomial $\Omega {\Psi }_W{\Psi }_{W^{\prime }}$. A fixed permutation of elementary fermions therefore contributes a channel-independent parity sign. Bosonic Wilson transport does not alter that parity. At trivial holonomy it is the identity; at nontrivial holonomy it rotates adjoint indices rather than generating a universal scalar factor three. Multiplying $F_4$ or $F_4^T$ on either side by arbitrary diagonal sign matrices changes signs but not the $1:1$ magnitude ratio obtained from a pure adjoint input, and cannot produce the target $1:3$ magnitude ratio.

The same obstruction has a metric form. With projector Gram matrix

$$G_N=diag(1,N^2-1),F_N{G}_N^{-1}{F}_N^T=G_N^{-1},$$

the dual squared norms are

$$\parallel c_1^{\left(N\right)}{\parallel }_{*}^2={\displaystyle \frac{N^2}{N^2-1}},{\parallel c_2^{\left(N\right)}\parallel }_{*}^2={\displaystyle \frac{2N}{N+1}},$$

whose ratio is $2(N-1)/N$. It equals one only for $N=2$ and is $3/2$ at $N=4$.

Canonical single-crossing no-go.Within the minimal $K_1\oplus K_{15}$ channel space, one standard four-point $\mathrm{SU}{\left(4\right)}_{PS}$ Fierz crossing cannot exchange ${\mathcal{C}}_1^{\left(4\right)}$ and ${\mathcal{C}}_2^{\left(4\right)}$. The failure survives channel-independent Grassmann/Wilson signs, transposition of the crossing convention, and independent sign conventions for the singlet and adjoint basis vectors. The $\mathrm{SU}{\left(2\right)}_R$ factor is the exact exception: $F_2$ does exchange its two required kernels.

Appendix B.1.1.14. The minimal missing color amplitude

The unique element of $span\{I,F_N\}$ that maps the pure-adjoint covector to the second required covector is

$$Q_N=F_N-{\displaystyle \frac{N-2}{N}}I,c_1^{\left(N\right)}{Q}_N=c_2^{\left(N\right)}.$$

(A41)

For $N=4$,

$$Q_4=F_4-{\displaystyle \frac{1}{2}}I={\displaystyle \frac{1}{4}}\left(\begin{array}{cc}-1& 15\\ 1& -3\end{array}\right),(0,4)Q_4=(1,-3).$$

(A42)

Thus the previously isolated $1:-3$ singlet/adjoint coefficient is equivalent to a sharper two-diagram condition: ordinary crossed color recoupling plus a direct-color contribution of relative weight $-1/2$. In the full factorized channel space,

$$(c_1^{\left(4\right)}\otimes c_2^{\left(2\right)})(Q_4\otimes F_2)=c_2^{\left(4\right)}\otimes c_1^{\left(2\right)}.$$

(A43)

The entire extra forward bare requirement is therefore localized in the $-\frac{1}{2}I$ term of the $\mathrm{SU}{\left(4\right)}_{PS}$ factor.

A stronger two-way statement is more restrictive. If one asks for one operator $T=aI+b{F}_N$ satisfying both $c_{1}T=c_2$ and $c_{2}T=c_1$, the only solution is $N=2$, $T=F_2$. For general N the unique abstract channel involution is

$$T_N={\displaystyle \frac{1}{N}}\left(\begin{array}{cc}N-1& 2N-1\\ 1& 1-N\end{array}\right),T_N^2=I,$$

(A44)

so $T_4={\displaystyle \frac{1}{4}}\begin{array}{cc}\hfill 3& 7\\ \hfill 1& -3\end{array}$. This $T_4$ is not contained in $span\{I,F_4\}$. Hence a genuine color-channel ${\mathbb{Z}}_2$ needs at least one further independent recoupling operation beyond direct propagation and one standard crossing. The larger multi-vertex/walled-Brauer contraction algebra of the six unprojected K currents is not excluded.

Microscopic reduction theorem.The positive Temperley–Lieb six-K ordering theorem and the minimal $K_{15}$ carrier survive. The canonical $\mathrm{SU}{\left(2\right)}_R$ crossing already performs the required exchange, but the analogous $\mathrm{SU}{\left(4\right)}_{PS}$ single crossing fails. A forward $B_{12}\to B_{21}$ channel conversion in the minimal direct-plus-crossed class is possible if and only if the $\mathrm{SU}\left(4\right)$ color factor contains the additional direct amplitude $-1/2$, equivalently $Q_4=F_4-\frac{1}{2}I$. A full two-way color involution requires still more structure, represented abstractly by $T_4$. The next microscopic calculation is therefore the derivation or exclusion of the $-1/2$ direct/crossed amplitude from the actual multi-vertex 24-field contraction with canonical Grassmann and Wilson data.

Appendix B.1.1.15. Explicit four-pair walled–Brauer realization of the missing color character

The single-crossing result isolates a precise forward color gate: one ordinary fundamental crossing is insufficient in the $\mathrm{SU}{\left(4\right)}_{PS}$ channel, and the required second channel is equivalently

$${\mathcal{C}}_2^{\left(4\right)}=F_4\left[{\mathcal{C}}_1^{\left(4\right)}\right]-{\displaystyle \frac{1}{2}}{\mathcal{C}}_1^{\left(4\right)}.$$

This subsection evaluates that gate in a larger but still unchanged-field multi-vertex contraction algebra.

Let

$$D={\delta }^{a_0}{}_{b_0}\delta ^{a_1}{}_{b_1},X={\delta }^{a_0}{{}_{b_1}\delta ^{a_1}}_{b_0}$$

be the direct and crossed four-leg tensors in $End\left({\mathbb{C}}^N\right)$. The singlet and adjoint projectors are

$${\mathcal{P}}_1={\displaystyle \frac{1}{N}}D,{\mathcal{P}}_{\mathrm{adj}}=X-{\displaystyle \frac{1}{N}}D,$$

so the two required trace-character kernels are

$$\begin{array}{cc}\hfill {\mathcal{C}}_1^{\left(N\right)}& =N{\mathcal{P}}_{\mathrm{adj}}=NX-D,\hfill \end{array}$$

(A45)

$$\begin{array}{cc}\hfill {\mathcal{C}}_2^{\left(N\right)}& ={\mathcal{P}}_1-(N-1){\mathcal{P}}_{\mathrm{adj}}=D-(N-1)X.\hfill \end{array}$$

(A46)

Choose four unprojected internal color pairs from the existing currents ${\mathbb{K}}_v$, transport their $\mathrm{SU}{\left(4\right)}_{PS}$ fibers to one reference point $x_{★}$, and label the upper fundamental and lower antifundamental indices by $(a_i,b_i)$, $i=0,1,2,3$. The direct seed is

$$|{\mathbf{1}}_4\rangle ={\displaystyle \prod _{i=0}^3}{\delta }^{a_i}{}_{b_i}.$$

Let $s_{ij}^L$ exchange the upper fundamental lines $a_i\leftrightarrow a_j$ and $s_{ij}^R$ exchange the lower antifundamental lines $b_i\leftrightarrow b_j$. Define the canonical group-algebra idempotents

$$A_{ij}^{L,R}={\displaystyle \frac{1-s_{ij}^{L,R}}{2}},S_{ij}^{L,R}={\displaystyle \frac{1+s_{ij}^{L,R}}{2}}.$$

The coefficients $1/2$ are fixed projector coefficients. They are not fitted amplitudes. Close pairs 2 and 3 by the partial trace $b_2=a_2$, $b_3=a_3$, with repeated indices summed; denote this closure by ${cl}_{23}$.

The first projector word is

$${\mathcal{W}}_1=S_{02}^R{A}_{02}^L{A}_{01}^L,$$

(A47)

with the rightmost factor acting first. Exact permutation contraction gives

$${cl}_{23}{\mathcal{W}}_1|{\mathbf{1}}_4\rangle ={\displaystyle \frac{N}{8}}D-{\displaystyle \frac{N^2}{8}}X=-{\displaystyle \frac{N}{8}}{\mathcal{C}}_1^{\left(N\right)}.$$

(A48)

Thus a fixed three-projector word produces the pure-adjoint trace character for every N.

The second projector word is

$${\mathcal{W}}_2=A_{03}^R{A}_{23}^L{A}_{13}^L{A}_{02}^L{A}_{01}^L.$$

(A49)

Its exact expansion has 18 nonzero permutation diagrams. After grouping only by surviving boundary topology and the number of closed color loops,

$${cl}_{23}{\mathcal{W}}_2|{\mathbf{1}}_4\rangle ={\displaystyle \frac{1}{32}}\left[(N^2-5N+7)D-(2N^2-8N+9)X\right].$$

(A50)

The condition for this tensor to be proportional to $D-(N-1)X$ factorizes exactly:

$$B_N+(N-1)A_N={\displaystyle \frac{(N-4){(N-2)}^2}{32}},A_N={\displaystyle \frac{N^2-5N+7}{32}},B_N=-{\displaystyle \frac{2N^2-8N+9}{32}}.$$

(A51)

Therefore this canonical word selects the required second trace character precisely at $N=2$ and $N=4$. In particular,

$${cl}_{23}{\mathcal{W}}_2|{\mathbf{1}}_4\rangle {|}_{N=4}={\displaystyle \frac{3}{32}}(D-3X)={\displaystyle \frac{3}{32}}{\mathcal{C}}_2^{\left(4\right)},$$

(A52)

and at $N=2$ it gives ${\displaystyle \frac{1}{32}}(D-X)={\displaystyle \frac{1}{32}}{\mathcal{C}}_2^{\left(2\right)}$.

This closes the forward-amplitude gate identified by the single-crossing analysis. Since

$$F_N\left[{\mathcal{C}}_1^{\left(N\right)}\right]={\displaystyle \frac{2}{N}}D-X,$$

for $N=4$ one has

$${\mathcal{C}}_2^{\left(4\right)}=F_4\left[{\mathcal{C}}_1^{\left(4\right)}\right]-{\displaystyle \frac{1}{2}}{\mathcal{C}}_1^{\left(4\right)}.$$

(A53)

Equation (A34) realizes this same relative coefficient from fixed multi-vertex projector algebra; the overall factor $3/32$ is merely source normalization. Thus the required effective direct/crossed amplitude $-1/2$ is not fitted.

Normalize the word outputs by

$${\widehat{\mathcal{W}}}_1^{\left(N\right)}=-{\displaystyle \frac{8}{N}}{\mathcal{W}}_1,{\widehat{\mathcal{W}}}_2^{\left(4\right)}={\displaystyle \frac{32}{3}}{\mathcal{W}}_2,{\widehat{\mathcal{W}}}_2^{\left(2\right)}=32{\mathcal{W}}_2.$$

Then the already established factorized local tensors acquire explicit recoupling representatives,

$$\begin{array}{cc}\hfill B_{12}& =-cl\left[{\widehat{\mathcal{W}}}_1^{\left(4\right)}\otimes {\widehat{\mathcal{W}}}_2^{\left(2\right)}\right]M,\hfill \end{array}$$

(A54)

$$\begin{array}{cc}\hfill B_{21}& =-cl\left[{\widehat{\mathcal{W}}}_2^{\left(4\right)}\otimes {\widehat{\mathcal{W}}}_1^{\left(2\right)}\right]M.\hfill \end{array}$$

(A55)

Consequently the primitive rows $(1,-1,-4,4)$ and $(1,-2,-3,6)$ are no longer inserted as unexplained coefficients: they arise from fixed pair projectors and group-dimension loop factors.

At field level, four of the six unprojected currents ${\mathbb{K}}_v$ supply the two boundary and two ancilla color pairs. The remaining two internal currents may remain in the singlet $K_1$ channel and close in the already established Lorentz/L/R network. No new fundamental field, family index, or contact limit is introduced, and the fermionic census remains 24 insertions. At separated points the color fibers are Wilson-transported to $x_{★}$ before the pair projectors act.

Constructive forward multi-vertex recoupling.Within the unchanged published field content, an explicit four-pair/two-ancilla walled-Brauer subnetwork generates both required $\mathrm{SU}{\left(4\right)}_{PS}$ trace-character kernels. The five-projector word ${\mathcal{W}}_2$ yields ${\mathcal{C}}_2^{\left(4\right)}=D-3X$ with no fitted internal coefficient, thereby realizing the effective direct/crossed amplitude $-1/2$ isolated by the single-crossing analysis. Together with the $N=2$ word, both crossed tensors $B_{12}$ and $B_{21}$ have explicit recoupling representatives. This is a forward operator-existence positive result, not yet a proof of a chronological ${\mathbb{Z}}_2$ or of dynamical equality of their renormalized amplitudes.

Appendix B.1.1.16. Finite-class minimality and symmetry search.

An exact exhaustive search in the anti-projector class finds no word shorter than five that produces $D-3X$ at $N=4$. At length five, exactly 1280 labeled anti-projector words do so; after grouping by boundary topology and loop number all reduce to the same normalized polynomial

$$(D_0,D_1,D_2,X_0,X_1,X_2)=(7,-5,1,-9,8,-2),$$

and all 1280 also produce the $N=2$ target. A separate exhaustive length-five search allowing both A and S projectors finds 489728 labeled ${\mathcal{C}}_1^{\left(4\right)}$ words and 1280 labeled ${\mathcal{C}}_2^{\left(4\right)}$ words, but no pair related by a permutation of the four pair labels, an $L\leftrightarrow R$ swap, or literal reversal of the projector order. This is a finite-class negative result only; it is not a no-go for arbitrary longer words or the complete six-current algebra.

Appendix B.1.1.17. All-length strict-chronology stopping rule.

The finite search above can be upgraded to an all-length theorem. For a closed boundary kernel $Y=AD+BX$ on $End\left({\mathbb{C}}^N\right)$, the singlet eigenvalue is ${\lambda }_1\left(Y\right)=NA+B$. Hence

$${\lambda }_1\left({\mathcal{C}}_1^{\left(N\right)}\right)=0,{\lambda }_1\left({\mathcal{C}}_2^{\left(N\right)}\right)=1.$$

(A56)

For any finite word $W=P_m\dots P_1$ built from the Hermitian pair projectors, transpositions and Hermitian cap–cup contractions used in the transported walled–Brauer network, literal chronological reversal is $W_{\mathrm{rev}}=W^{†}$. If $|\Omega \rangle$ denotes the completely direct singlet closure of all color pairs, then in the real projector basis

$${\Lambda }_1\left(W_{\mathrm{rev}}\right)=\langle \Omega |W^{†}|\Omega \rangle ={\Lambda }_1\left(W\right).$$

(A57)

Up to a fixed positive normalization this closed scalar is the singlet eigenvalue of the boundary kernel. Therefore a word closing to a nonzero multiple of ${\mathcal{C}}_1^{\left(4\right)}$ can never be converted by literal reverse order into a nonzero multiple of ${\mathcal{C}}_2^{\left(4\right)}$, at any word length or with any number of closed ancilla pairs.

The same obstruction is visible from rank:

$$rank{\mathcal{C}}_1^{\left(4\right)}=15,rank{\mathcal{C}}_2^{\left(4\right)}=16.$$

(A58)

Any invertible closure-preserving boundary similarity preserves rank and hence cannot exchange the two kernels. On the full Pati–Salam tensor space the additional supports $(15,1)$ and $(1,3)$ are inequivalent irreducible modules, so no transformation commuting with unbroken $G_{\mathrm{PS}}$ can exchange the complete crossed tensors.

Strict microscopic chronological involution excluded under the closure-preserving assumptions.The explicit crossed recoupling representatives remain established, but they are not the same closure-preserving Hermitian projector process executed in opposite chronological order. Longer projector words cannot repair this obstruction. The exact identity ${\Delta }_{\Pi }=C_{21}-C_{12}$ and the existence of both microscopic crossed channels are unchanged; the surviving physical question is whether confinement and renormalization dynamically make their amplitudes reciprocal.

Appendix B.1.1.18. A reduced correlator target from the irreducible channels

The irreducible decomposition also suggests a shorter nonperturbative diagnostic. Let the source components associated with

$$A\equiv M_{15,3},B\equiv M_{15,1},C\equiv M_{1,3}$$

be normalized in one common source scheme. In an unbroken $G_{\mathrm{PS}}$ regime, inequivalent irreducible channels have vanishing cross correlators. With

$$B_{12}=4(A-B),B_{21}=6A-2C,$$

the equality of the two diagonal crossed-source correlators becomes

$$4G_B\left(t\right)-G_C\left(t\right)-5G_A\left(t\right)=0,$$

(A59)

where $G_X\left(t\right)=\langle J_X\left(t\right)J_X^{†}\left(0\right)\rangle$. This is a correlator-level route diagnostic, not yet the low-energy coefficient equality itself. In the broken phase the corresponding identity contains the allowed cross terms,

$$-5G_A+4G_B-G_C-8\Re G_{AB}+6\Re G_{AC}=0,$$

(A60)

with all quantities evaluated in the same canonical source normalization.

Appendix B.1.1.19. Microscopic conclusion after the all-length stopping rule

The forward multi-vertex construction is complete in the following sense: both crossed kernels are generated by explicit unchanged-field recoupling words and the effective $-1/2$ color coefficient is fixed by projector algebra. What is excluded is the stronger identification of those two words as one closure-preserving microscopic process in opposite chronological order. This distinction removes the need for further searches over longer Hermitian projector words in the same class.

The operator problem therefore hands the decision to dynamics. In a canonical route-label basis define

$$K_{\perp }={\displaystyle \frac{1}{2}}(K-{\sigma }_{x}K{\sigma }_x).$$

(A61)

The physically meaningful question is whether the renormalized theory suppresses this component relative to the route-odd spectral gap, rather than whether a microscopic Ward identity sets it identically to zero.

Appendix B.1.3. Sectoral Route Images

For bookkeeping it is useful to temporarily enlarge the charged-sector vector to

$${\mathbf{A}}^{\left(4\right)}=(A_{\ell },A_d,A_u,A_{{\nu }_D}),$$

where $A_{{\nu }_D}$ is the curvature of the singular values of the neutrino Dirac Yukawa matrix; it is not the curvature of the light Majorana-neutrino masses.

The route-image entries can be derived directly rather than taken as a table. For a primitive coefficient vector

$$c=(c_0,c_R,c_L,c_X),$$

the rank-one Pati–Salam-breaking background introduced below gives the sector eigenvalue

$$r_c(x,y)=c_0+c_{R}y+c_{L}x+c_{X}xy,x,y\in \{0,1\}.$$

With the ordered sector assignment

$$\ell :(x,y)=(1,0),d:(0,0),u:(0,1),{\nu }_D:(1,1),$$

the evaluation map is

$$c⟼(c_0+c_L,c_0,c_0+c_R,c_0+c_R+c_L+c_X).$$

Applying this map to the four rows of the route-to-primitive matrix R of Appendix B.1.6,

$$(-1,2,4,-8),(1,-1,-4,4),(1,-2,-3,6),(-1,1,3,-3),$$

gives, row by row,

$$\begin{array}{cc}\hfill (-1,2,4,-8)& \mapsto (3,-1,1,-3),\hfill \\ \hfill (1,-1,-4,4)& \mapsto (-3,1,0,0),\hfill \\ \hfill (1,-2,-3,6)& \mapsto (-2,1,-1,2),\hfill \\ \hfill (-1,1,3,-3)& \mapsto (2,-1,0,0).\hfill \end{array}$$

Thus the four route images are not independent numerical input; they are the explicit evaluation of the primitive tensors in the four bookkeeping sectors:

$$\begin{array}{cccc}\hfill r_{11}& =(3,-1,1,-3),\hfill & \hfill r_{12}& =(-3,1,0,0),\hfill \end{array}$$

(A62)

$$\begin{array}{cccc}\hfill r_{21}& =(-2,1,-1,2),\hfill & \hfill r_{22}& =(2,-1,0,0).\hfill \end{array}$$

(A63)

Extend the charged-sector plane normal to

$$n_{\Pi }^{\left(4\right)}=(1,2,-1,0).$$

Then

$$n_{\Pi }^{\left(4\right)}·r_{11}=0,n_{\Pi }^{\left(4\right)}·r_{12}=-1,n_{\Pi }^{\left(4\right)}·r_{21}=+1,n_{\Pi }^{\left(4\right)}·r_{22}=0.$$

Written without compressed dot-product notation, the four evaluations are

$$\begin{array}{cc}\hfill \left(1\right)\left(3\right)+\left(2\right)(-1)+(-1)\left(1\right)& =0,\hfill \\ \hfill \left(1\right)(-3)+\left(2\right)\left(1\right)+(-1)\left(0\right)& =-1,\hfill \\ \hfill \left(1\right)(-2)+\left(2\right)\left(1\right)+(-1)(-1)& =+1,\hfill \\ \hfill \left(1\right)\left(2\right)+\left(2\right)(-1)+(-1)\left(0\right)& =0.\hfill \end{array}$$

The auxiliary fourth component never contributes because the extended normal has last entry zero. Consequently the two uncrossed routes lie in the plane, while the crossed routes pierce it with equal and opposite unit projections.

Figure A1. The exact operator identity in the crossed-coefficient plane. The phenomenological condition is the diagonal line of equal crossed routes; displacement is measured along the normal $(-1,+1)$. This drawing represents an exact coordinate statement, not a statistical confidence region.

Figure A1. The exact operator identity in the crossed-coefficient plane. The phenomenological condition is the diagonal line of equal crossed routes; displacement is measured along the normal $(-1,+1)$. This drawing represents an exact coordinate statement, not a statistical confidence region.

If

$${\mathbf{A}}^{\left(4\right)}=C_{11}{r}_{11}+C_{12}{r}_{12}+C_{21}{r}_{21}+C_{22}{r}_{22},$$

then

$${\Delta }_{\Pi }=C_{21}-C_{12}.$$

This is an exact linear identity in the fixed operator normalization. The equality $C_{12}=C_{21}$ is therefore the central microscopic target.

Appendix B.1.4. Even-Route Reconstruction and Why There Is No Second Charged-Sector Plane

The reciprocity condition removes one crossed-odd coordinate but does not determine the full charged spectrum. Set

$$C_{12}=C_{21}\equiv C_{\times }.$$

Direct substitution of the route images gives

$$\begin{array}{cc}\hfill A_{\ell }& =3C_{11}-5C_{\times }+2C_{22},\hfill \end{array}$$

(A64)

$$\begin{array}{cc}\hfill A_d& =-C_{11}+2C_{\times }-C_{22},\hfill \end{array}$$

(A65)

$$\begin{array}{cc}\hfill A_u& =C_{11}-C_{\times }.\hfill \end{array}$$

(A66)

These equations have rank two. Indeed, the simultaneous additive shift

$$(C_{11},C_{\times },C_{22})⟼(C_{11}+t,C_{\times }+t,C_{22}+t)$$

leaves all three charged curvatures unchanged. Equivalently,

$$r_{11}+(r_{12}+r_{21})+r_{22}=0$$

after restriction to the charged components. Therefore the charged data reconstruct only route-coefficient differences, not the three coefficients separately and not their ratios. A convenient invariant pair is

$$\begin{array}{cc}\hfill C_{11}-C_{\times }& =A_u,\hfill \end{array}$$

(A67)

$$\begin{array}{cc}\hfill C_{\times }-C_{22}& =A_d+A_u.\hfill \end{array}$$

(A68)

On the exact plane the second difference can equivalently be written as

$$C_{\times }-C_{22}=A_{\ell }+3A_d,$$

because

$$(A_{\ell }+3A_d)-(A_d+A_u)={\Delta }_{\Pi }.$$

For the displayed central values, Equation (86) gives

$$C_{11}-C_{\times }=-0.1553674.$$

The two data-side estimators of the other difference are

$$A_d+A_u=0.0911620,A_{\ell }+3A_d=0.1101262,$$

and their separation is exactly the observed residual $0.0189642$. A constrained numerical reconstruction should therefore project the data onto ${\Delta }_{\Pi }=0$ with the full covariance matrix. It should not select one estimator, average them without their covariance, or report a ratio $C_{11}:C_{\times }:C_{22}$ that the charged-sector map cannot determine.

This rank count also proves that reciprocity supplies exactly one charged-sector relation. After $C_{12}=C_{21}$ is imposed, the image is the two-dimensional plane ${\Delta }_{\Pi }=0$ in the three-dimensional space $(A_{\ell },A_d,A_u)$. There is no second independent linear relation among these three curvatures without an additional dynamical condition on the even routes.

The auxiliary Dirac-neutrino slot does not change this conclusion. From the fourth components of the route vectors,

$$A_{{\nu }_D}=-3C_{11}+2C_{\times }.$$

This quantity changes under the additive null direction above and is therefore not fixed by the charged curvatures. Any prediction for $A_{{\nu }_D}$ requires an additional even-route condition; light-neutrino masses require, in addition, the Majorana sector.

Appendix B.1.5. The Crossed-Only Truncation as a Covariance-Dependent Diagnostic

The stronger ansatz $C_{11}=C_{22}=0$, together with reciprocity, gives

$$(A_{\ell },A_d,A_u)=C_{\times }(-5,2,-1)$$

and hence

$$A_{\ell }=5A_u,A_d=-2A_u.$$

Using the measured central value of $A_u$ predicts

$$A_{\ell }^{\mathrm{cross}}=5A_u=-0.7768371,A_d^{\mathrm{cross}}=-2A_u=0.3107349,$$

to be compared with $A_{\ell }=-0.6294621$ and $A_d=0.2465294$. The corresponding central residuals are

$$R_{\ell }\equiv A_{\ell }-5A_u=0.1473751,R_d\equiv A_d+2A_u=-0.0642054.$$

Thus the displayed central spectrum is not compatible with the exact crossed-only truncation; nontrivial even-route information is required if the route representation is to reproduce it. This does not disfavor route reciprocity itself, which never required $C_{11}=C_{22}=0$.

No exclusion significance is quoted here. $R_{\ell }$ and $R_d$ share the same Yukawa inputs and must be tested with their full $2\times 2$ covariance, including correlations with $A_u$. Moreover, setting both even coefficients to zero is an operator-level truncation stronger than anything reconstructible from charged curvatures, because the common additive coefficient direction is invisible in that sector. The correct present description is therefore “incompatible with the central values”, not “excluded”.

Appendix B.1.6. Transformation to the Primitive Basis

Let

$$\mathcal{O}={({\mathcal{O}}_{11},{\mathcal{O}}_{12},{\mathcal{O}}_{21},{\mathcal{O}}_{22})}^T,{\mathcal{O}}_{\mathrm{prim}}={({\mathcal{O}}_0,{\mathcal{O}}_R,{\mathcal{O}}_L,{\mathcal{O}}_X)}^T.$$

They are related by

$$\mathcal{O}=R{\mathcal{O}}_{\mathrm{prim}},$$

with

$$R=\left(\begin{array}{cccc}-1& 2& 4& -8\\ 1& -1& -4& 4\\ 1& -2& -3& 6\\ -1& 1& 3& -3\end{array}\right),detR=1.$$

Thus the route basis and primitive basis both have rank four.

The crossed exchange $P_{\times }$ interchanges ${\mathcal{O}}_{12}\leftrightarrow {\mathcal{O}}_{21}$ while leaving 11 and 22 fixed. In the primitive basis the same operation is

$$S_B=R^{-1}{P}_{\times }R=\left(\begin{array}{cccc}1& 2& -2& -4\\ 0& 0& 1& 2\\ 0& 1& 0& -2\\ 0& 0& 0& 1\end{array}\right),S_B^2=I_4.$$

Its characteristic polynomial is

$$det(\lambda I_4-S_B)={(\lambda -1)}^3(\lambda +1),$$

so there is exactly one odd primitive direction. A convenient choice is

$$v_{-}={(2,-1,1,0)}^T,{\ell }_{-}={(0,-1,1,2)}^T,$$

for which

$$S_B{v}_{-}=-v_{-},S_B^T{\ell }_{-}=-{\ell }_{-}.$$

In operator notation,

$${\mathcal{O}}_{21}-{\mathcal{O}}_{12}=-{\mathcal{O}}_R+{\mathcal{O}}_L+2{\mathcal{O}}_X\equiv \mathcal{O}\left[B_{-}\right].$$

Appendix B.1.7. Projected Quotient Cross-Check: Why O 12 and O 21 Can Be Independent

This subsection is the sectoral cross-check of the full stopping rule in Appendix B.1.2; it is not a substitute for enumerating the six internal junctions and their complete Lorentz/spectator contraction.

For a rank-one Pati–Salam-breaking background introduce binary labels $x,y\in \{0,1\}$ that select, respectively, the leptonic $\mathrm{SU}{\left(4\right)}_{\mathrm{PS}}$ direction and the upper $\mathrm{SU}{\left(2\right)}_R$ component. For a generic primitive coefficient vector $c=(c_0,c_R,c_L,c_X)$ the eigenvalue is

$$r_c(x,y)=c_0+c_{R}y+c_{L}x+c_{X}xy.$$

The 12 and 21 rows give, in the order $(\ell ,d,u,{\nu }_D)$,

$$R_{12}=(-3,1,0,0),R_{21}=(-2,1,-1,2).$$

Suppose that $a{\mathcal{O}}_{12}+b{\mathcal{O}}_{21}=0$ after all local Schouten identities, Fierz identities, and fermionic-statistics relations have been imposed, with a common nonzero microscopic kernel. Projection onto the u component gives $-b=0$; with $b=0$, projection onto the d component gives $a=0$. Therefore

$$dimspan\{{\mathcal{O}}_{12},{\mathcal{O}}_{21}\}=2$$

at the source/operator level.

Remark.Rank two here does not mean two physical particles. Two linearly independent interpolating operators may couple to the same bound state. Physical spectral rank is determined by correlator residues, not by the number of operator formulas that can be written.

Appendix B.1.8. Large-N Route-Count Theorem: the Reflected Crossed Pair Has Identical Interchange Order

The flavour analysis of Ref. [7] formulates preon-to-prebaryon matching amplitudes as

$$\mathcal{M}\sim {\displaystyle \frac{C}{{\Lambda }_{\mathrm{pre}}^D}}{\left({\displaystyle \frac{1}{N}}\right)}^k,N=15,$$

where k counts preon-line interchanges between different prebaryons and C contains the remaining nonperturbative information. A preon-line interchange is a constituent transfer from one bound state to another accompanied by precolor reconnection; each additional interchange costs one power of $1/N$.

In the earlier large-N analysis the equality of the two crossed exponents was left conditional on an explicit graph-to-large-N identification. For the reflected point-split lift used in the present manuscript, the exponent difference can in fact be fixed without knowing the absolute common exponent.

Let ${\Gamma }_{12}$ be a microscopic point-split representative of the first crossed route and define the second by the independently specified route reflection,

$${\Gamma }_{21}=\tau {\Gamma }_{12},{\tau }^2=1.$$

The source record of Appendix B.4 defines $\tau$ as a bijection of junctions and routed lines that preserves the field species and gauge representation of every mapped line. Let $\mathfrak{I}\left(\Gamma \right)$ be the set of constituent-transfer events entering the large-N count, so that

$$k\left(\Gamma \right)=\left|\mathfrak{I}\right(\Gamma \left)\right|.$$

Every transfer event between distinct junctions,

$$(v_a,e,v_b),v_a\ne v_b,$$

is mapped to

$$(\tau v_a,\tau e,\tau v_b),\tau v_a\ne \tau v_b.$$

Because $\tau$ is its own inverse, this map is a bijection

$${\tau }_{*}:\mathfrak{I}\left({\Gamma }_{12}\right)⟶\mathfrak{I}\left({\Gamma }_{21}\right).$$

Therefore

$$k_{12}=k_{21}\equiv k,\Delta k\equiv k_{21}-k_{12}=0.$$

(A69)

Large-N route-count theorem.For any instantiated point-split route pair related by the declared microscopic reflection ${\Gamma }_{21}=\tau {\Gamma }_{12}$, the Assi–Plestid–Sengupta–Zupan interchange count is identical for the two crossed routes. Hence the two coefficients cannot differ by a parametric power of $1/N$:

$$C_{12}=N^{-k}{\widehat{C}}_{12},C_{21}=N^{-k}{\widehat{C}}_{21}.$$

This is an exponent-level positive result for route reciprocity. It does not imply ${\widehat{C}}_{12}={\widehat{C}}_{21}$.

Appendix B.1.1.1. Relation to the six-K route.

The large-N theorem is conditional on a separately instantiated pair of precolor constituent-transfer diagrams related by a species-preserving involution. The quotient-compatible Temperley–Lieb route of the quotient-compatible six-K analysis acts on spectator/Lorentz contraction data of the internal $K_1$ sector; it does not by itself supply the required precolor constituent-transfer pairing and, by the $\mathrm{SU}{\left(4\right)}_{PS}$-blindness theorem, cannot by itself derive $B_{12}\leftrightarrow B_{21}$. The equality $k_{12}=k_{21}$ is therefore retained as a separate conditional theorem for an explicitly tagged constituent-transfer doublet. The earlier microscopic construction unoriented ten-edge permutation that exchanged inequivalent outer currents is not used as such an involution.

Appendix B.1.1.2. Why the absolute common k is not yet a theorem.

A Wilson edge is parallel transport and should not be counted as a constituent transfer merely because it appears in the point-split network. The current source record fixes connectivity, paths, representations, spectator contractions, and Grassmann ordering, but it does not uniquely tag which routed fundamental lines change prebaryon ownership in a particular preon-to-prebaryon matching diagram. Therefore Equation (A42) fixes the difference $\Delta k$ exactly, while the common absolute value k still requires an explicit constituent-transfer tagging.

A useful minimal witness exists for the separate precolor constituent-transfer bookkeeping. If two tagged internal binding/reconnection events are each instantiated as one elementary constituent transfer between distinct prebaryons, then

$$k_{12}=k_{21}=2,N^{-k}={15}^{-2}={\displaystyle \frac{1}{225}}\simeq 4.44\times {10}^{-3}.$$

This $k=2$ value is a constructive benchmark, not a unique first-principles prediction. It must not be identified with the quotient-compatible Temperley–Lieb generators $e_{23}$ and $e_{34}$ of the quotient-compatible $K_1$ space: those are spectator/spin contraction operators, whereas k counts precolor constituent transfers. A more detailed microscopic matching may assign a different common k without changing the theorem $\Delta k=0$.

The phenomenologically relevant consequence is that the common large-N factor cancels from the normalized reciprocity defect,

$${\delta }_{\mathrm{rt}}\equiv {\displaystyle \frac{C_{21}-C_{12}}{C_{21}+C_{12}}}={\displaystyle \frac{{\widehat{C}}_{21}-{\widehat{C}}_{12}}{{\widehat{C}}_{21}+{\widehat{C}}_{12}}},$$

when the denominator is nonzero. Large-N power counting therefore removes a possible parametric obstruction but does not explain the equality itself. The remaining dynamical target is sharpened to

$${\widehat{C}}_{12}\stackrel{?}{=}{\widehat{C}}_{21}.$$

If

$${\widehat{C}}_{ij}=c_{ij}^{\left(0\right)}+{\displaystyle \frac{c_{ij}^{\left(1\right)}}{N}}+O\left(N^{-2}\right),$$

then no large-N argument presented here forces $c_{12}^{\left(0\right)}=c_{21}^{\left(0\right)}$. An independently derived emergent route involution that enforces this leading equality would push the first reciprocity violation to subleading order; without such a result the equality remains a nonperturbative correlator/matching question.

Sharpened stopping rule.For a valid reflected lift of the two crossed point-split sources, a future calculation that reports $k_{12}\ne k_{21}$ would indicate that the compared diagrams are not actually the same $\tau$-paired microscopic route doublet, or that the proposed interchange-count prescription is not invariant under the declared reflection. The physical decision gate is no longer the large-N exponent difference; it is the order-one reduced-amplitude equality and, ultimately, the renormalized route-resolved correlator and matching vertex.

Appendix B.1.9. Order-One Strong-Dynamics Structure: A Common Local Core and a Single Route-Odd Remainder

After Equation (A42), write

$$C_{12}=N^{-k}{\widehat{C}}_{12},C_{21}=N^{-k}{\widehat{C}}_{21},$$

and introduce the even/odd reduced coefficients

$${\widehat{C}}_{+}={\displaystyle \frac{{\widehat{C}}_{12}+{\widehat{C}}_{21}}{\sqrt{2}}},{\widehat{C}}_{-}={\displaystyle \frac{{\widehat{C}}_{21}-{\widehat{C}}_{12}}{\sqrt{2}}}.$$

The exact plane requires ${\widehat{C}}_{-}=0$. There are two logically different questions: whether the known microscopic algebra already forces this equality, and which parts of the strong amplitude can actually generate a nonzero ${\widehat{C}}_{-}$.

Appendix B.1.1.1. No free ultraviolet equality.

The local quotient established above gives two independent same-order gauge-invariant operators ${\mathcal{O}}_{12}$ and ${\mathcal{O}}_{21}$. They are not Hermitian conjugates; each has its own conjugate term in the action. The unitary Pati–Salam flavour commutant of the gauged kinetic term derived in Section 3.4 acts separately on the inequivalent spectator blocks and does not exchange these two operators. Therefore a generic Wilsonian action consistent with the already established microscopic symmetries permits independent coefficients in the crossed sector. Equality of the full generic order-one coefficients is consequently not protected by the known ultraviolet Ward algebra.

Order-one UV-protection no-go.Within the unchanged published field content and the already identified exact Ward algebra, ${\widehat{C}}_{12}={\widehat{C}}_{21}$ is not a free consequence of gauge invariance, Fierz identities, or large-N power counting. The equality must arise from additional dynamical information: an emergent route involution, a nonanalytic confinement projection, a special infrared spectral structure, or a directly verified correlator/matching relation.

Appendix B.1.1.2. Where an order-one splitting cannot originate.

The local $\Omega FF$ prebaryon building block has a unique precolor singlet contraction: the singlet multiplicity in $\overline{\mathbf{120}}\otimes \mathbf{15}\otimes \mathbf{15}$ is one. Hence there is no hidden “direct versus crossed” local SU(15)_p Clebsch coefficient inside one prebaryon. The two crossed route operators use the same chosen local Weyl current and differ only in spectator/Wilson contractions, so any Schouten/Fierz coefficient multiplying that common local current is shared rather than independently route dependent. Together with $k_{12}=k_{21}$, these facts allow the canonical decomposition

$${\widehat{C}}_{12}={\widehat{C}}_{\mathrm{core}}+{\delta }_{12},{\widehat{C}}_{21}={\widehat{C}}_{\mathrm{core}}+{\delta }_{21},$$

(A70)

where ${\widehat{C}}_{\mathrm{core}}$ denotes the common local precolor–Lorentz contribution and ${\delta }_{ij}$ contains route-distinguishing spectator contractions, nonlocal Wilson-path response, operator renormalization/mixing, and other connected information not fixed by the local singlet.

Universal local-core theorem.For the reflected route pair constructed from the same local $\Omega FF$ junctions, neither the local SU(15)_p singlet multiplicity, nor the local Lorentz quotient, nor the large-N interchange exponent can distinguish routes 12 and 21. Therefore

$${\widehat{C}}_{21}-{\widehat{C}}_{12}={\delta }_{21}-{\delta }_{12}.$$

Every physical order-one reciprocity defect is localized in route-distinguishing nonlocal/spectator/path/renormalization data. This theorem does not assert that the remainder is small.

The spectral analysis of Appendix B.9 gives a second, independent localization: an isolated exactly route-odd pole with a route-even local matching vertex contributes zero to ${\widehat{C}}_{-}$ exactly. After that pole is separated, the entire route defect is carried by the regular remainder.

Appendix B.1.10. Four-dimensional Strong-Coupling Surface Analysis: the Explicit Route Graph Is Spectator, Not Precolor

The lower-dimensional controls motivate a direct question in four dimensions: can the explicit point-split 24-field source be assigned two different $\mathrm{SU}{\left(15\right)}_p$ minimal surfaces whose character expansions first agree at leading order and separate at some higher plaquette order? For the actual source definition the premise must be corrected before such an expansion is attempted.

Appendix B.1.1.1. Exact source-level precolor analysis.

The six internal mixed currents have the local form

$$K_{\rho }^{m\dot{r}}\left(x\right)=N_K{\Omega }_{\alpha \beta ,\rho }\left(x\right){\left({\Psi }_W\right)}_{\sigma }^{\alpha am}\left(x\right){{\left({\Psi }_{W^{\prime }}\right)}^{\beta }}_a{}_{\tau }{}^{\dot{r}}\left(x\right){ϵ}^{\sigma \tau }\sim (1,2,2).$$

The two $\mathrm{SU}{\left(15\right)}_p$ fundamental indices are saturated by ${\Omega }_{\alpha \beta }$ at the same junction. Consequently every local $\Omega FF$ current is already a precolor singlet. The Wilson lines required between separated K currents transport the spectator $\mathrm{SU}{\left(2\right)}_L\times \mathrm{SU}{\left(2\right)}_R$ indices, not precolor. Thus the displayed eight-junction graph

$$E_{\Gamma }=\{12,23,34,45,56,67,78,81,27,36\},\tau =\left(18\right)\left(27\right)\left(36\right)\left(45\right),$$

although it has cycle rank $10-8+1=3$ and is reflection invariant as an unoriented graph, is not an inter-junction $\mathrm{SU}{\left(15\right)}_p$ Wilson loop. In the precise source-level sense relevant to gauge completion,

$$A_{min,precolor}^{\left(\mathrm{source}\right)}=0,$$

(A71)

because no inter-junction precolor plaquette sheet is required to make the source gauge invariant.

Both crossed representatives use the same eight local $\Omega FF$ singlet contractions. If the remaining spectator, Lorentz, route and matching data are denoted by ${\mathcal{S}}_{ij}$, then schematically

$${\mathcal{A}}_{12}^{\mathrm{bare}}={\mathcal{A}}_{\mathrm{pc},\mathrm{loc}}{\mathcal{S}}_{12},{\mathcal{A}}_{21}^{\mathrm{bare}}={\mathcal{A}}_{\mathrm{pc},\mathrm{loc}}{\mathcal{S}}_{21},$$

and therefore

$${\mathcal{A}}_{12,\mathrm{pc}}^{\left(0\right)}={\mathcal{A}}_{21,\mathrm{pc}}^{\left(0\right)}={\mathcal{A}}_{\mathrm{pc},\mathrm{loc}}.$$

(A72)

Equation (A45) is the correct source-level replacement for the phrase “equal leading precolor minimal surfaces.” It does not imply equality of the complete physical kernels: connected matter propagation, singlet interactions, spectator transport, Pati–Salam channel response, renormalization/mixing and the final matching functional remain dynamical.

Source-level strong-coupling stopping theorem.For the explicit six-K 24-field representative, the ten-edge route graph is spectator transport rather than a confining $\mathrm{SU}{\left(15\right)}_p$ Wilson surface. Therefore a route-odd “minimal precolor area” on that graph is not a physical contribution present in the source definition. The bare local precolor factor is exactly common to routes 12 and 21; the first physical nonzero $K_{\perp }$ must be sought in connected correlator dynamics or in the spectator/channel/renormalization/matching data that distinguish the two routes.

Appendix B.1.1.2. Conditional all-order result for an added reflected precolor dressing.

Suppose, only as an auxiliary construction, that both sources are augmented by representation-preserving precolor Wilson networks $W_{\Gamma }$ and $W_{\tau \Gamma }$ related by an exact lattice reflection. In a reflection-invariant Wilson action the plaquette factor has a character expansion

$$e^{-S_g\left[U_p\right]}=c_0\left(\beta \right)\left[1+\sum _{R\ne \mathbf{1}}{d}_R{u}_R\left(\beta \right){\chi }_R\left(U_p\right)\right],$$

and a Wilson-network expectation is a sum over colored surfaces,

$$\langle W_{\Gamma }\rangle =\sum _{\mathcal{F}\in \mathfrak{F}\left(\Gamma \right)}A\left(\mathcal{F}\right)\prod _{p,R}{u}_R^{m_{p,R}\left(\mathcal{F}\right)}.$$

If the reflection maps every edge representation and local intertwiner to its partner, then $\mathcal{F}\mapsto \tau \mathcal{F}$ is a weight-preserving bijection. Hence

$${\langle W_{\Gamma }\rangle }^{\left(n\right)}={\langle W_{\tau \Gamma }\rangle }^{\left(n\right)}\mathrm{at}\mathrm{every}\mathrm{character}\mathrm{order}n.$$

(A73)

Thus an exactly reflected pure-precolor dressing cannot itself generate $K_{\perp }$ at any order. This theorem is conditional on the deliberately added reflected dressing and does not provide the forbidden microscopic exchange of the complete $B_{12}$ and $B_{21}$ Pati–Salam kernels.

Appendix B.1.1.3. A genuine local 3+1D precolor selection rule.

There is nevertheless a direct four-dimensional plaquette statement that does not rely on the spectator ladder. On a cubic spatial lattice one link touches four elementary spatial plaquettes. For

$$B_p={\displaystyle \frac{1}{2N}}\left(Tr{U}_p+Tr{U}_p^{†}\right)$$

let $n_{+}\left(p\right)$ and $n_{-}\left(p\right)$ count the two orientations at total perturbative order k. Each touching plaquette possesses outer links not shared by the other three. Center invariance of the Haar integral over any such outer link gives the necessary condition

$$n_{+}\left(p\right)-n_{-}\left(p\right)\equiv 0(modN)\mathrm{for}\mathrm{every}\mathrm{touching}\mathrm{plaquette}p.$$

(A74)

For $k<N$, the absolute value of the difference is smaller than N, so Equation (A47) forces $n_{+}\left(p\right)=n_{-}\left(p\right)$ separately for every plaquette. Each plaquette then contributes an even number of insertions and therefore

$$\mathrm{all}\mathrm{odd}\mathrm{coefficients}\mathrm{with}k<N\mathrm{vanish}\mathrm{exactly}\mathrm{in}\mathrm{this}\mathrm{shared}-\mathrm{link}\mathrm{cluster}.$$

(A75)

For odd N the first center-allowed odd order is $k=N$; for even N every allowed plaquette insertion count is even and odd total orders are forbidden in this particular cluster. In particular,

$$a_3=a_5=a_7=a_9=a_{11}=a_{13}=0,k_{\mathrm{odd},\mathrm{first}}=15\mathrm{for}\mathrm{SU}\left(15\right).$$

(A76)

At $k=15$ there are eight center-allowed insertion-count patterns (four plaquettes times two orientations). For the auxiliary normalized cluster partition function $Z\left(r\right)={\langle exp\left[r{\sum }_{p=1}^4{B}_p\right]\rangle }_{\beta =0}$, the simplest single-plaquette 15-fold invariant carries the bare expansion coefficient

$${\displaystyle \frac{8}{15!{30}^{15}}}=4.263551910578\times {10}^{-34}.$$

This tiny number is only a normalization illustration for that auxiliary $Z\left(r\right)$ and is not a prediction for the physical $K_{\perp }$.

What the four-dimensional surface analysis decides.The requested first nonzero physical $K_{\perp }^{\left(n\right)}$ is not determined by a minimal-surface expansion on the ten-edge source graph, because that graph carries spectator rather than precolor Wilson transport. If an exactly reflected pure-precolor dressing is added, its character coefficients are route neutral order by order. If instead a candidate local route-breaking mechanism belongs to the shared-link plaquette class above, center/N-ality excludes its odd orders $3,5,\dots ,13$ in $\mathrm{SU}\left(15\right)$ and makes 15 only the first allowed odd order, not a proof of a nonzero physical coefficient. The legitimate next four-dimensional calculation is therefore a connected strong-coupling matter expansion or, ultimately, the canonically normalized renormalized two-route correlator.

Appendix B.1.11. Closing the Regular Route-Exchange Remainder: RG Sign No-Go and Spectral Purification

The order-one analysis above localizes every remaining failure of reciprocity in the regular route-exchange-breaking response. It is important to distinguish this object from a physical route-odd state. In the canonically normalized route-source space let

$$U\equiv {\sigma }_x,U^2=I_2,$$

and decompose a Hermitian two-route matrix as

$$X=x_0{I}_2+x_x{\sigma }_x+x_y{\sigma }_y+x_z{\sigma }_z.$$

(A77)

Then

$$UX{U}^{†}=x_0{I}_2+x_x{\sigma }_x-x_y{\sigma }_y-x_z{\sigma }_z,$$

so exact route-exchange symmetry is

$$[X,U]=0⟺x_y=x_z=0.$$

(A78)

By contrast, both parity projectors

$$P_{\pm }={\displaystyle \frac{I_2\pm {\sigma }_x}{2}}$$

commute with U. Thus a physical route-odd pole proportional to $P_{-}$ is fully compatible with exact route-exchange symmetry. The symmetry-breaking object is the ${\sigma }_y/{\sigma }_z$ part of the regular response, not odd route parity itself.

Appendix B.1.1.1. The transverse RG exponent is a genuine dynamical eigenvalue.

Let $g_{+}$ denote coordinates tangent to a route-symmetric RG manifold and $g_{-}$ a canonically renormalized transverse coordinate for the leading regular route-breaking deformation. If

$$\mu {\displaystyle \frac{d{g}_{-}}{d\mu }}={\beta }_{-}(g_{+},g_{-}),{\beta }_{-}(g_{+},0)=0,$$

then at a symmetric fixed point the stability matrix

$$B_{ab}={\left.{\displaystyle \frac{\partial {\beta }_a}{\partial g_b}}\right|}_{*}$$

satisfies

$$B_{-+}=0,$$

(A79)

because differentiating ${\beta }_{-}(g_{+},0)=0$ along the symmetric manifold gives zero. The transverse linearized equation is therefore

$$\mu {\displaystyle \frac{d\delta g_{-}}{d\mu }}={\theta }_{-}\delta g_{-}+O\left(\delta g_{-}^2\right),{\theta }_{-}=B_{--}.$$

(A80)

For flow from $\Lambda$ to $\mu <\Lambda$,

$$\delta g_{-}\left(\mu \right)=\delta g_{-}\left(\Lambda \right){\left({\displaystyle \frac{\mu }{\Lambda }}\right)}^{{\theta }_{-}}+\dots .$$

Hence ${\theta }_{-}>0$ is the precise smooth-flow condition for infrared suppression of the transverse route-breaking deformation. At an actual four-dimensional conformal fixed point where $g_{-}$ couples to a scaling operator ${\mathcal{O}}_{-}$, one may identify ${\theta }_{-}={\Delta }_{-}-4$; away from such a fixed point, the stability eigenvalue rather than a guessed operator dimension is the correct quantity.

Invariant-manifold linearization theorem.For a differentiable Wilsonian flow with invariant route-symmetric manifold $g_{-}=0$, the transverse row of the stability matrix obeys $B_{-+}=0$. The sign relevant for smooth infrared attraction is the single transverse eigenvalue ${\theta }_{-}=B_{--}$. The existence of the symmetric manifold does not determine this sign.

Appendix B.1.1.2. Reflection positivity does not determine the sign.

Consider a single positive spectral atom with residue

$$R_{\epsilon }=a{I}_2+b{\sigma }_x+\epsilon c{\sigma }_z,a>\left|b\right|.$$

(A81)

Its eigenvalues are

$$a\pm \sqrt{b^2+{\epsilon }^2{c}^2},$$

so $R_{\epsilon }\succ 0$ whenever

$$\left|\epsilon c\right|<\sqrt{a^2-b^2}.$$

(A82)

The ${\sigma }_z$ component is odd under conjugation by U. Over any finite scale interval one may choose the initial amplitude small enough and take

$$\epsilon \left(\mu \right)=\epsilon \left(\Lambda \right){\left({\displaystyle \frac{\mu }{\Lambda }}\right)}^{\theta }$$

with arbitrary real $\theta$ while preserving positivity throughout the interval.

Positivity sign-underdetermination theorem.Reflection positivity and positive-semidefinite matrix spectral weights do not fix the sign of the transverse route exponent. Positive route-resolved spectral families exist for either sign of $\theta$. Therefore ${\theta }_{-}$ must be obtained from the renormalized anomalous-dimension/stability matrix or from direct step scaling; positivity alone cannot establish the IR-attractor rescue.

Appendix B.1.1.3. Why the 24-preon engineering dimension does not solve the relative problem.

In the coincident local limit, a source containing twenty-four four-dimensional Weyl fields and no derivatives has engineering dimension

$${\Delta }_{24}^{\mathrm{cl}}=24\times {\displaystyle \frac{3}{2}}=36.$$

A local Wilsonian coupling to such an operator carries the common engineering exponent $36-4=32$. However the even and route-breaking crossed combinations have the same constituent number and the same engineering dimension. For the local crossed doublet,

$$\mu {\displaystyle \frac{d\mathbf{g}}{d\mu }}=\left[32I_2+{\gamma }^T\left(g_p\right)\right]\mathbf{g}+\dots ,$$

(A83)

with $\gamma$ the operator-mixing anomalous-dimension matrix. The common $32I_2$ suppresses both source deformations equally and does not align their ratio. Since no established microscopic Ward symmetry forces $\gamma$ to be diagonal in the route $(+,-)$ basis, relative suppression is anomalous-dimension and confinement-matching data.

Equal-engineering-dimension no-go.The large canonical dimension of the 24-preon source does not by itself drive the route defect to zero relative to the even amplitude. The tree-level engineering term is common to the crossed doublet and cancels from the relative reciprocity problem.

Appendix B.1.1.4. Ordinary Gauss-law projection is not the missing nonanalytic mechanism.

Let ${\mathcal{P}}_G$ denote gauge-group averaging onto $\mathrm{SU}{\left(15\right)}_p$ singlets,

$${\mathcal{P}}_G\left[A\right]={\int }_{\mathrm{SU}\left(15\right)}dGU\left(G\right)AU{\left(G\right)}^{-1}.$$

The point-split routed sources $J_1$ and $J_2$ are already separately gauge invariant by construction. Hence

$${\mathcal{P}}_G\left[J_1\right]=J_1,{\mathcal{P}}_G\left[J_2\right]=J_2,$$

and therefore

$${\mathcal{P}}_G\left[J_{-}\right]=J_{-},J_{-}\equiv {\displaystyle \frac{J_1-J_2}{\sqrt{2}}}.$$

(A84)

Gauss-projection no-go.Ordinary Gauss-law projection cannot annihilate the regular route-breaking channel, because the route sources are already gauge singlets. Any exact confinement-induced closure of the route defect must contain dynamical information beyond ordinary gauge-singlet projection.

A different naive possibility also fails. If a physical quotient literally identifies the two source operators, $\pi \left(J_1\right)=\pi \left(J_2\right)$, then $\pi \left(J_{-}\right)=0$. The entire odd projected correlator vanishes and no isolated route-odd pole with nonzero residue can survive.

Full-route-quotient incompatibility theorem.A quotient that identifies $J_1$ and $J_2$ in the physical operator algebra removes the odd source itself and is therefore incompatible with the pole-mediated branch studied here. Exact route symmetry is allowed; a full source-identifying quotient is not.

Appendix B.1.1.5. Selective spectral purification is the viable exact closure.

After isolating the physical odd pole, write the canonical matrix spectral measure as

$$d\mu \left(E\right)=Z_{-}{P}_{-}\delta (E-E_{-})dE+d{\mu }_{\mathrm{reg}}\left(E\right),Z_{-}>0,$$

(A85)

and define

$$d{\nu }_{\mathrm{reg}}\left(E\right)=d{\mu }_{\mathrm{reg}}\left(E\right)-Ud{\mu }_{\mathrm{reg}}\left(E\right)U^{†}.$$

(A86)

Then

$$C_{\mathrm{reg}}\left(t\right)-U{C}_{\mathrm{reg}}\left(t\right)U^{†}={\int }_0^{\infty }{e}^{-Et}d{\nu }_{\mathrm{reg}}\left(E\right).$$

(A87)

Regular spectral-purification theorem. For finite matrix spectral measures the following are equivalent:

(i)

$C_{\mathrm{reg}}\left(t\right)=U{C}_{\mathrm{reg}}\left(t\right)U^{†}$ for every $t>0$;

(ii)

$d{\mu }_{\mathrm{reg}}\left(E\right)=Ud{\mu }_{\mathrm{reg}}\left(E\right)U^{†}$ as a matrix measure;

(iii)

the signed matrix measure $d{\nu }_{\mathrm{reg}}$ vanishes.

Thus exact regular reciprocity is equivalent to the statement that the entire regular spectral measure lies in the commutant of the independently defined route involution U.

The nontrivial implication follows componentwise from uniqueness of the Laplace transform of finite signed measures. This criterion preserves the required odd pole because $U{P}_{-}{U}^{†}=P_{-}$. It removes only route-exchange breaking; it does not remove route-odd parity states. Both $P_{+}$ and $P_{-}$ spectral sectors are permitted in the regular continuum or discrete background.

Appendix B.1.1.6. Finite-data exact certificate when the moment problem closes.

Let

$$M_n^{\mathrm{reg}}=\int y^{n}d{\mu }_{\mathrm{reg}}\left(y\right)$$

be the regular threshold-normalized matrix moments. Suppose the block-Hankel hierarchy reaches exact flat closure, so the positive representing matrix measure is unique. If every moment entering the flat reconstruction satisfies

$$U{M}_n^{\mathrm{reg}}{U}^{†}=M_n^{\mathrm{reg}},$$

(A88)

then the transformed measure $Ud{\mu }_{\mathrm{reg}}{U}^{†}$ is a second positive representing measure with the same closing moments. Uniqueness forces the two measures to coincide.

Flat-extension route-symmetry certificate.Flat closure plus route symmetry of the complete closing matrix-moment set implies exact U-invariance of the reconstructed regular spectral measure. Therefore the regular route-exchange-breaking remainder vanishes for all Euclidean times and for every route-equivariant linear matching functional. Conversely, any closing moment that fails (A61) is an immediate exclusion for exact regular spectral purification.

Without flat closure, finitely many symmetric moments do not generically prove symmetry of the full measure because the positive moment problem remains nonunique; weak additional asymmetric spectral weight may remain compatible with the known moments. This is the same finite-information limitation already established for weak additional states, now applied to route symmetry.

Appendix B.1.1.7. A positivity bound useful before closure.

For the regular Hermitian matrix (A50), positivity gives

$$x_0\ge \sqrt{x_x^2+x_y^2+x_z^2}.$$

Hence

$$x_y^2+x_z^2\le x_0^2-x_x^2.$$

(A89)

In a real canonical route basis $x_y=0$, so

$$|x_z|\le \sqrt{x_0^2-x_x^2}.$$

(A120)

This does not prove equality but converts measured symmetric components into a rigorous upper bound on the regular exchange-breaking component.

Decision rule.The problem now has two nonredundant solution paths. The smooth path is to compute the transverse step-scaling eigenvalue ${\theta }_{-}$; ${\theta }_{-}>0$ permits infrared suppression, whereas ${\theta }_{-}\le 0$ closes that rescue branch. The exact nonanalytic path is not ordinary gauge projection and not a full route quotient. It is selective spectral purification: retain the isolated $P_{-}$ pole and require the regular matrix spectral measure to commute with U. If the regular moment problem becomes flat, this exact condition is decidable from finitely many route-resolved Euclidean moments.

Appendix B.1.12. First-principles Route Step Scaling and the Sharp Finite-Moment Cost

The previous subsection identified the two possible closure mechanisms but left the smooth exponent as an abstract observable and did not determine the minimum finite-data cost of the exact spectral route. Both questions can be sharpened without introducing new microscopic assumptions.

Appendix B.1.1.1. Crossed deformation doublet and finite-volume response matrix

Define

$${\mathcal{Q}}_{+}={\displaystyle \frac{{\mathcal{O}}_{21}+{\mathcal{O}}_{12}}{\sqrt{2}}},{\mathcal{Q}}_{-}={\displaystyle \frac{{\mathcal{O}}_{21}-{\mathcal{O}}_{12}}{\sqrt{2}}}.$$

(A121)

Here ${\mathcal{Q}}_{+}$ is route even and ${\mathcal{Q}}_{-}$ is the route-exchange-breaking deformation. Perturb the regulated microscopic action by the same pair of bare coefficients on every matched volume,

$$\delta S_L=h_{+}^{\left(0\right)}{\mathcal{J}}_{+}\left(L\right)+h_{-}^{\left(0\right)}{\mathcal{J}}_{-}\left(L\right),{\mathcal{J}}_a\left(L\right)={\int }_{V_L}{d}^{4}x{\mathcal{Q}}_a^{\mathrm{bare}}\left(x\right).$$

(A122)

The coefficients $h_{+}^{\left(0\right)}$ and $h_{-}^{\left(0\right)}$ have the same engineering dimension because the two crossed operators contain the same fields and derivatives. They are held fixed when L is changed. Any conventional powers of L needed to define dimensionless renormalized coordinates are included in the response normalization below, not in the bare perturbation itself.

Choose two dimensionless renormalized probes ${\Phi }_a\left(L\right)$ with nonsingular linear response and define

$${\mathcal{R}}_{ab}\left(L\right)={\left.{\displaystyle \frac{\partial \langle {\Phi }_a\left(L\right)\rangle }{\partial h_b^{\left(0\right)}}}\right|}_{h^{\left(0\right)}=0},$$

(A123)

with any common engineering normalization required by the chosen finite-volume scheme understood as part of $\mathcal{R}$. Linear response gives

$${\mathcal{R}}_{ab}\left(L\right)=-{\int }_{V_L}{d}^{4}x{\langle {\Phi }_a\left(L\right){\mathcal{Q}}_b^{\mathrm{bare}}\left(x\right)\rangle }_c,$$

(A124)

up to the conventional sign in $e^{-S}$. Thus every entry is a finite-volume connected correlation observable. In practice ${\Phi }_{+}$ and ${\Phi }_{-}$ may be chosen as even and exchange-breaking projections of the pole-subtracted route correlator at fixed $t/L$; the construction itself is probe independent provided $\mathcal{R}$ is nonsingular.

For the same regulator-level perturbation ${\mathbf{h}}^{\left(0\right)}={(h_{+}^{\left(0\right)},h_{-}^{\left(0\right)})}^T$, define local renormalized coordinates

$$\mathbf{u}\left(L\right)=\mathcal{R}\left(L\right){\mathbf{h}}^{\left(0\right)}+O\left({\left(h^{\left(0\right)}\right)}^2\right).$$

Comparison of boxes L and $sL$ therefore gives the exact linear step map

$${\Sigma }_{\mathrm{rt}}(s,L)=\mathcal{R}\left(sL\right)\mathcal{R}{\left(L\right)}^{-1},\mathbf{u}\left(sL\right)={\Sigma }_{\mathrm{rt}}(s,L)\mathbf{u}\left(L\right)+O\left(u^2\right).$$

(A125)

First-principles route step-scaling theorem.For any finite-volume renormalization prescription with nonsingular response matrix $\mathcal{R}$, the route-resolved coupling step-scaling matrix is exactly (A68). It is constructed from connected finite-volume correlators and requires neither a pole fit nor an assumed anomalous dimension.

Appendix B.1.1.2. Invariant-manifold test.

If the route-symmetric manifold is locally $u_{-}=0$, invariance requires that a tangent perturbation does not generate a transverse one. Hence

$${\left({\Sigma }_{\mathrm{rt}}\right)}_{-+}=0.$$

(A126)

A controlled continuum result violating (A69) is therefore an immediate exclusion for the one-transverse-coordinate smooth-restoration ansatz; the full coupled flow must then be used.

When the manifold is invariant and the transverse sector is one real dimension,

$${\Sigma }_{\mathrm{rt}}=\left(\begin{array}{cc}{\Sigma }_{++}& {\Sigma }_{+-}\\ 0& {\Sigma }_{--}\end{array}\right).$$

At a fixed point, $\mu d{u}_{-}/d\mu ={\theta }_{-}{u}_{-}+\dots$ and $\mu =1/L$ give

$${\Sigma }_{--}\left(s\right)=s^{-{\theta }_{-}},{\theta }_{-}^{\mathrm{eff}}(s,L)=-{\displaystyle \frac{ln|{\Sigma }_{--}(s,L)|}{lns}}.$$

(A127)

Thus ${\theta }_{-}>0$ is equivalent to $|{\Sigma }_{--}|<1$ in the fixed-point scaling regime and measures absolute suppression of the transverse deformation. Because the even and odd 24-preon operators have the same engineering dimension, relative route alignment additionally requires comparison with the tangent/even scaling. The ratio-normalized exponent is derived explicitly in Appendix B.1.13.

Under an analytic finite-volume scheme change ${\mathbf{u}}^{\prime }=f\left(\mathbf{u}\right)$ with fixed-point Jacobian J, the linear step map transforms by similarity,

$${\Sigma }_{\mathrm{rt}}^{\prime }=J{\Sigma }_{\mathrm{rt}}{J}^{-1}.$$

(A128)

Hence the fixed-point eigenvalues, and in particular the one-dimensional transverse exponent, are scheme invariant. Without a reality restriction the exchange-breaking sector is two real dimensional; for its transverse block ${\Sigma }_{\perp }$, smooth fixed-point IR attraction requires

$$\rho \left({\Sigma }_{\perp }\right)<1,$$

(A129)

while ${\sigma }_{max}\left({\Sigma }_{\perp }\right)<1$ is a stronger finite-step sufficient contraction criterion.

Step-scaling decision criterion.The smooth branch requires two logically separate tests: first ${\left({\Sigma }_{\mathrm{rt}}\right)}_{-+}=0$ for invariance of the route-symmetric manifold; second control of the transverse scaling. Equation (A70) gives the absolute transverse exponent. For the physical relative reciprocity defect, the common scaling of the even channel must also be divided out; the resulting exponent is ${\Theta }_{\mathrm{rel}}={\theta }_{-}-{\theta }_{+}$ in the parity-diagonal regime, as derived in Appendix B.1.13. A nonzero tangent-to-transverse entry still rejects the one-parameter invariant-manifold mechanism before either exponent is interpreted.

Appendix B.1.1.3. Flat closure at order N: only the lower 2N moments need an explicit symmetry check

After exact subtraction of the isolated physical odd pole, define regular moments

$$M_n^{\mathrm{reg}}=\int y^{n}d{\mu }_{\mathrm{reg}}\left(y\right).$$

At block order N let

$${\mathbb{H}}_0^{\left(N\right)}={\left[M_{i+j}^{\mathrm{reg}}\right]}_{i,j=0}^{N-1},{\mathsf{B}}_N=\left(\begin{array}{c}{M}_N^{\mathrm{reg}}\\ M_{N+1}^{\mathrm{reg}}\\ ⋮\\ M_{2N-1}^{\mathrm{reg}}\end{array}\right),$$

(A100)

and

$${\mathsf{S}}_N=M_{2N}^{\mathrm{reg}}-{\mathsf{B}}_N^{†}{\left({\mathbb{H}}_0^{\left(N\right)}\right)}^{+}{\mathsf{B}}_N⪰0.$$

(A101)

The earlier flat-extension theorem gives ${\mathsf{S}}_N=0$ if and only if the block-Krylov space has become transfer invariant.

Assume

$$U{M}_n^{\mathrm{reg}}{U}^{†}=M_n^{\mathrm{reg}},n=0,\dots ,2N-1.$$

(A102)

With ${\mathbb{U}}_N=I_N\otimes U$, this implies

$$[{\mathbb{H}}_0^{\left(N\right)},{\mathbb{U}}_N]=0,{\mathbb{U}}_N{\mathsf{B}}_N={\mathsf{B}}_{N}U.$$

Since the Moore–Penrose inverse commutes with every unitary commuting with its Hermitian argument, $[{\left({\mathbb{H}}_0^{\left(N\right)}\right)}^{+},{\mathbb{U}}_N]=0$. If ${\mathsf{S}}_N=0$, then

$$\begin{array}{cc}\hfill U{M}_{2N}^{\mathrm{reg}}{U}^{†}& =U{\mathsf{B}}_N^{†}{\left({\mathbb{H}}_0^{\left(N\right)}\right)}^{+}{\mathsf{B}}_N{U}^{†}\hfill \end{array}$$

(A103)

$$\begin{array}{cc}\hfill & ={\mathsf{B}}_N^{†}{\mathbb{U}}_N^{†}{\left({\mathbb{H}}_0^{\left(N\right)}\right)}^{+}{\mathbb{U}}_N{\mathsf{B}}_N\hfill \end{array}$$

(A104)

$$\begin{array}{cc}\hfill & =M_{2N}^{\mathrm{reg}}.\hfill \end{array}$$

(A105)

Thus route symmetry of the final closing moment follows automatically.

Reduced symmetry-check theorem.At a candidate flat-closure order N, exact regular reciprocity is certified by $2N+1$ consecutive matrix moments $M_0^{\mathrm{reg}},\dots ,M_{2N}^{\mathrm{reg}}$. It is sufficient to check U-commutation explicitly only for $M_0^{\mathrm{reg}},\dots ,M_{2N-1}^{\mathrm{reg}}$ and to verify ${\mathsf{S}}_N=0$. The closing moment then commutes automatically, and uniqueness of the flat representing measure forces $d{\mu }_{\mathrm{reg}}=Ud{\mu }_{\mathrm{reg}}{U}^{†}$.

Appendix B.1.1.4. Sharpness: why the final moment cannot be omitted

The count $2N+1$ is not merely convenient. Choose $2N+1$ positive nodes

$$y_j=m+(j-N)d,m>Nd>0,j=0,\dots ,2N,$$

and coefficients

$$c_j={(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{2N}{j}}\right).$$

Finite-difference identities give

$${\displaystyle \sum _{j=0}^{2N}}{c}_j{y}_j^n=0,n=0,\dots ,2N-1,{\displaystyle \sum _{j=0}^{2N}}{c}_j{y}_j^{2N}=\left(2N\right)!d^{2N}\ne 0.$$

(A106)

Let $C_N={\sum }_j\left|c_j\right|=2^{2N}$, $w_j=\left|c_j\right|/C_N$, and choose $0<\left|\epsilon \right|<1/C_N$. The residues

$$R_j=w_j{I}_2+\epsilon c_j{\sigma }_z$$

(A107)

are positive, but the measure ${\sum }_j{R}_j\delta (y-y_j)dy$ is not U invariant because $U{\sigma }_z{U}^{†}=-{\sigma }_z$. Nevertheless every moment through $2N-1$ is exactly route symmetric; the first breaking term is

$$M_{2N}^{\mathrm{br}}=\epsilon \left(2N\right)!d^{2N}{\sigma }_z.$$

(A108)

Sharp $2N+1$ moment theorem.Absent additional spectral assumptions, no universal exact route-symmetry certificate using only $M_0,\dots ,M_{2N-1}$ exists: a positive route-asymmetric measure can hide its breaking component from every one of those moments. Therefore the $2N+1$ moment matrices required by the order-N flat-extension test are information-theoretically sharp.

Appendix B.1.1.5. Earliest closure: three matrices are exactly sufficient and minimal

For $N=1$ and after dropping any null source combination, assume $M_0^{\mathrm{reg}}\succ 0$. The flat residual is

$${\mathsf{S}}_1=M_2^{\mathrm{reg}}-M_1^{\mathrm{reg}}{\left(M_0^{\mathrm{reg}}\right)}^{-1}{M}_1^{\mathrm{reg}}.$$

(A109)

If ${\mathsf{S}}_1=0$, define

$$T_1={\left(M_0^{\mathrm{reg}}\right)}^{-1/2}{M}_1^{\mathrm{reg}}{\left(M_0^{\mathrm{reg}}\right)}^{-1/2}.$$

Then the complete regular sequence is

$$M_n^{\mathrm{reg}}={\left(M_0^{\mathrm{reg}}\right)}^{1/2}{T}_1^n{\left(M_0^{\mathrm{reg}}\right)}^{1/2},n\ge 0.$$

(A110)

Consequently

$$[M_0^{\mathrm{reg}},U]=[M_1^{\mathrm{reg}},U]=0,{\mathsf{S}}_1=0$$

(A101)

forces every higher moment and the unique regular spectral measure to commute with U.

Two moments are not enough. Let $m>d>0$, take nodes $m-d,m,m+d$, and choose positive residues for $0<\left|\epsilon \right|<1/4$,

$$R_0={\displaystyle \frac{1}{4}}{I}_2+\epsilon {\sigma }_z,R_1={\displaystyle \frac{1}{2}}{I}_2-2\epsilon {\sigma }_z,R_2={\displaystyle \frac{1}{4}}{I}_2+\epsilon {\sigma }_z.$$

Then

$$M_0=I_2,M_1=m{I}_2,$$

(A112)

are route symmetric, but

$$M_2=\left(m^2+{\displaystyle \frac{d^2}{2}}\right)I_2+2\epsilon d^2{\sigma }_z$$

(A113)

is not. The same $M_0,M_1$ are compatible with the symmetric one-atom measure $I_2\delta (y-m)dy$.

Minimal three-moment exact reciprocity certificate.In the earliest closure case the three regular $2\times 2$ matrices $M_0^{\mathrm{reg}},M_1^{\mathrm{reg}},M_2^{\mathrm{reg}}$ are sufficient for exact regular reciprocity through (A81). Without additional spectral information, two moments are insufficient. Thus three is the exact minimum data count for an $N=1$ flat-extension route-symmetry positive result.

In Euclidean time,

$$M_n^{\mathrm{reg}}\left(t_0\right)=e^{n{a}_E{E}_{\mathrm{th}}}{C}_{\mathrm{reg}}(t_0+n{a}_E),$$

(A114)

so an $N=1$ closure can be decided from the three consecutive pole-subtracted matrices

$$C_{\mathrm{reg}}\left(t_0\right),C_{\mathrm{reg}}(t_0+a_E),C_{\mathrm{reg}}(t_0+2a_E).$$

(A115)

At higher closure order N, the sharp adaptive cost is $2N+1$ consecutive matrices through $t_0+2N{a}_E$. Failure of ${\mathsf{S}}_1=0$ is not a physical exclusion; it only means that the regular cyclic space has not closed at the earliest order and the hierarchy must be extended.

Decision rule.The regular route problem now has two explicitly measurable paths. For smooth emergence, compute $\mathcal{R}\left(L\right)$ and $\mathcal{R}\left(sL\right)$, test ${\left({\Sigma }_{\mathrm{rt}}\right)}_{-+}=0$, and then extract the transverse fixed-point exponent from the normal block. For exact spectral purification, subtract the resolved odd pole and increase the regular moment order until flat closure; exact route symmetry at closure order N costs sharply $2N+1$ consecutive route matrices, with three matrices in the earliest possible case. No numerical sign of ${\theta }_{-}$ and no finite closure order are claimed before the corresponding renormalized correlators are generated.

Appendix B.1.13. Weak-Coupling UV Asymptotics of the 24-Preon Route Doublet

The response-matrix construction above defines what must be measured nonperturbatively. It is nevertheless useful to ask how much of the route-mixing matrix is already fixed in the asymptotically free ultraviolet regime. The answer separates a common ultraviolet renormalization from the genuinely route-sensitive part and also sharpens which exponent tests relative reciprocity.

Appendix B.1.1.1. Asymptotic freedom of the published chiral precolor theory

For the published chiral fermion content, the precolor sector contains nineteen left-handed Weyl fundamentals and one left-handed Weyl fermion in the conjugate two-index symmetric representation. With

$$T\left(\mathbf{15}\right)={\displaystyle \frac{1}{2}},T\left(\mathbf{120}\right)={\displaystyle \frac{N+2}{2}}={\displaystyle \frac{17}{2}},C_A=N=15,$$

one has

$$\sum _{\mathrm{Weyl}}T\left(R\right)=19{\displaystyle \frac{1}{2}}+{\displaystyle \frac{17}{2}}=18.$$

Therefore, in the convention

$$\beta \left(g_p\right)=-{\displaystyle \frac{b_0}{16{\pi }^2}}{g}_p^3+O\left(g_p^5\right),$$

$$b_0={\displaystyle \frac{11}{3}}\left(15\right)-{\displaystyle \frac{2}{3}}\left(18\right)=43>0.$$

(A116)

The unchanged chiral fermion theory is thus asymptotically free. If the optional corrected two-spurion completion is also treated as containing propagating complex $\overline{\mathbf{105}}$ and $\overline{\mathbf{120}}$ scalars at the same ultraviolet scale, then

$$T\left(\mathbf{105}\right)={\displaystyle \frac{13}{2}},T\left(\mathbf{120}\right)={\displaystyle \frac{17}{2}},$$

and the corresponding conditional coefficient is

$$b_0=43-{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{13}{2}}+{\displaystyle \frac{17}{2}}\right)=38>0.$$

No scalar completion is required for the route theorem below.

At leading order,

$$g_p^2\left(\mu \right)={\displaystyle \frac{8{\pi }^2}{b_{0}ln(\mu /{\Lambda }_p)}}\left[1+O\left({\displaystyle \frac{lnln\mu }{ln\mu }}\right)\right].$$

(A117)

Appendix B.1.1.2. Renormalization matrix and reflection covariance

Let

$$\mathbf{J}={(J_{12},J_{21})}^T$$

denote one instantiated reflected pair of point-split 24-preon sources. Write

$${\mathbf{J}}^{\mathrm{bare}}=Z_{\mathrm{rt}}{\mathbf{J}}^{\mathrm{ren}},{\gamma }_{\mathrm{rt}}=Z_{\mathrm{rt}}^{-1}\mu {\displaystyle \frac{d{Z}_{\mathrm{rt}}}{d\mu }}.$$

A regulator/subtraction prescription is called $\tau$-covariant when reflected local ultraviolet subgraphs receive reflected counterterms with the same coefficient. This is a property of the counterterm map on the explicitly paired source family and is weaker than postulating a new fundamental Ward symmetry of the microscopic field content.

If the complete operator basis retained for renormalization is closed under $\tau$, then the counterterm map commutes with the representation of route reflection. In a closed two-source subspace, with $U={\sigma }_x$,

$$[Z_{\mathrm{rt}},U]=0,[{\gamma }_{\mathrm{rt}},U]=0,$$

(A118)

so

$${\gamma }_{\mathrm{rt}}={\gamma }_0{I}_2+{\gamma }_{\times }{\sigma }_x.$$

(A119)

In the route-parity basis this is diagonal,

$${\gamma }_{+}={\gamma }_0+{\gamma }_{\times },{\gamma }_{-}={\gamma }_0-{\gamma }_{\times }.$$

In a larger renormalization basis the corresponding exact statement is block diagonalization into $\tau$-even and $\tau$-odd sectors; a literal $2\times 2$ matrix is justified only after closure against other operators with the same quantum numbers.

Reflection-covariant UV structure theorem.For a $\tau$-closed renormalization basis and a $\tau$-covariant regulator/subtraction scheme, ultraviolet counterterms do not mix route parity. A closed two-route subblock has the form (A89). This result constrains the anomalous-dimension matrix without assuming equality of the finite physical matching coefficients.

Appendix B.1.1.3. Exact local precolor factors at every ΩFF junction

For $F\sim \mathbf{15}$ and $\Omega \sim \overline{\mathbf{120}}$, with the two fundamentals coupled in the symmetric $\mathbf{120}$ channel,

$$C_F={\displaystyle \frac{N^2-1}{2N}}={\displaystyle \frac{112}{15}},C_S={\displaystyle \frac{(N-1)(N+2)}{N}}={\displaystyle \frac{238}{15}}.$$

Hence

$$T_{F_1}·T_{F_2}={\displaystyle \frac{1}{2}}(C_S-2C_F)={\displaystyle \frac{7}{15}}.$$

(A120)

On the local total singlet,

$$(T_{\Omega }+T_{F_1}+T_{F_2})|1\rangle =0,$$

and symmetry of the two fundamental lines gives

$$T_{\Omega }·T_{F_1}=T_{\Omega }·T_{F_2}=-{\displaystyle \frac{C_S}{2}}=-{\displaystyle \frac{119}{15}}.$$

(A121)

The sum is

$${\displaystyle \frac{7}{15}}-2{\displaystyle \frac{119}{15}}=-{\displaystyle \frac{77}{5}}=-{\displaystyle \frac{1}{2}}(C_S+2C_F),$$

as required for the singlet. These color factors are identical at all eight local junctions and in both reflected routes. The remaining common one-loop scalar coefficient depends on the Lorentz kernel and local contour geometry and is not inferred from group theory alone.

Local color-factor theorem.All $\mathrm{SU}{\left(15\right)}_p$ pairwise color factors needed for a one-loop calculation local to one $\Omega FF$ junction are fixed by Eqs. (A90)–(A91) and are common to the two reflected routes. No second local precolor Clebsch coefficient can generate a route splitting.

Appendix B.1.1.4. Route-neutral one-loop theorem for a genuinely point-split network

The ultraviolet renormalization of Wilson operators is local on their support: smooth non-self-intersecting contour pieces renormalize multiplicatively, cusps and endpoints add local factors, while genuine intersections can produce mixing among different routings [37,38]. The present source definition leaves path shapes among the source data, so a generic Euclidean instantiation can be chosen without unintended self-intersections.

Consider a reflected source pair for which the eight local junctions are separated, Wilson segments have no unintended intersections away from declared junctions, and $\tau$ maps every local geometry, field species, representation, spectator contraction and smearing datum to its reflected partner. A one-loop ultraviolet divergent subgraph is then supported locally on a junction, endpoint/cusp, smooth segment, or other declared local support element. Every such subgraph has a reflected partner with the same Feynman-rule coefficient. Because changing the global route connectivity would require a nonlocal counterterm connecting separated support regions, a one-loop UV pole cannot convert one member of the doublet into the other.

Therefore

$${\gamma }_{\mathrm{rt}}^{\left(1\right)}={\gamma }_{\mathrm{com}}^{\left(1\right)}{I}_2,{\gamma }_{\times }^{\left(1\right)}=0,$$

(A122)

and

$${\gamma }_{+}^{\left(1\right)}={\gamma }_{-}^{\left(1\right)}.$$

The unspecified common coefficient contains wave-function, local-junction, Wilson-line, endpoint and cusp contributions. Equation (A92) fixes the relative one-loop matrix without assigning a number that the present source record does not determine.

Route-neutral one-loop UV theorem.For the stated nonintersecting point-split reflected source class, local one-loop ultraviolet logarithms do not distinguish route order. Weak-coupling one-loop running is therefore neutral with respect to reciprocity: it neither creates a relative route splitting nor attracts a generic nonzero route defect toward zero.

Appendix B.1.1.5. The first route-sensitive perturbative coefficient

If the source is deliberately collapsed so that route-changing contractions become local, or if its Wilson contours contain genuine intersections, ultraviolet mixing between routings becomes possible. Reflection covariance still restricts the closed two-route subblock to Equation (A89), but now ${\gamma }_{\times }^{\left(1\right)}$ need not vanish. The parity-sector one-loop difference is

$$\Delta {\gamma }_{\mathrm{rel}}^{\left(1\right)}\equiv {\gamma }_{-}^{\left(1\right)}-{\gamma }_{+}^{\left(1\right)}=-2{\gamma }_{\times }^{\left(1\right)}.$$

(A123)

For a route-changing exchange of two fundamental precolor lines, the elementary color identity

$${\left(T_F^A\right)}^i{}_j\left(T_F^A\right)^k{}_l={\displaystyle \frac{1}{2}}\left({\delta }_l^i{\delta }_j^k-{\displaystyle \frac{1}{N}}{\delta }_j^i{\delta }_l^k\right)$$

contains both a pairing-changing and a pairing-preserving term. However, the actual coefficient multiplying this color operation depends on which fundamental lines participate, the contact/intersection geometry, Lorentz contractions, and the full counterterm basis. Those data are not fixed by the current generic point-split source record, so neither a number nor a sign for ${\gamma }_{\times }^{\left(1\right)}$ is presently justified.

Single-coefficient perturbative gate.In a reflected $2\times 2$ route subspace closed under one-loop contact/intersection mixing, the entire relative ultraviolet effect is carried by one coefficient ${\gamma }_{\times }^{\left(1\right)}$, equivalently by $\Delta {\gamma }_{\mathrm{rel}}^{\left(1\right)}=-2{\gamma }_{\times }^{\left(1\right)}$. In a larger basis the corresponding object is the difference between the leading even- and odd-sector anomalous-dimension eigenvalues. Determining its sign requires an explicit route-changing contact/intersection calculation; group theory alone does not fix it.

Appendix B.1.1.6. The relevant exponent is relative, not merely absolute

Let $g_{+}$ and $g_{-}$ be parity-basis renormalized couplings in a regime in which their linearized flow is diagonal,

$$\mu {\displaystyle \frac{d{g}_{\pm }}{d\mu }}={\theta }_{\pm }{g}_{\pm }+\dots .$$

Because both 24-preon operators have the same engineering dimension, the relative defect

$$r={\displaystyle \frac{g_{-}}{g_{+}}}$$

obeys

$$\mu {\displaystyle \frac{dlnr}{d\mu }}={\Theta }_{\mathrm{rel}},{\Theta }_{\mathrm{rel}}={\theta }_{-}-{\theta }_{+}.$$

(A124)

The common engineering exponent 32 cancels exactly. For a parity-diagonal finite-volume step map,

$${\Theta }_{\mathrm{rel}}^{\mathrm{eff}}(s,L)=-{\displaystyle \frac{ln|{\Sigma }_{--}(s,L)/{\Sigma }_{++}(s,L)|}{lns}}.$$

(A125)

Relative IR attraction toward reciprocity requires ${\Theta }_{\mathrm{rel}}>0$. The finite-moment quantity $-ln|{\Sigma }_{--}|/lns$ remains the absolute transverse exponent and agrees with the relative criterion only if the even/tangent normalization is fixed so that ${\theta }_{+}=0$ or an equivalent normalization removes the common scaling.

Relative-exponent refinement.The response-matrix construction of the finite-moment analysis remains valid, but the physically relevant smooth-restoration test for the coefficient ratio is the difference between odd and even stability exponents. This removes the common 24-preon engineering dimension and any other common multiplicative UV factor from the reciprocity diagnostic.

Appendix B.1.1.7. Weak-coupling asymptotics of the route ratio

With $a_p=g_p^2/\left(16{\pi }^2\right)$, the relative running has the perturbative form

$$\mu {\displaystyle \frac{dlnr}{d\mu }}=a_p\Delta {\gamma }_{\mathrm{rel}}^{\left(1\right)}+a_p^2\Delta {\gamma }_{\mathrm{rel}}^{\left(2\right)}+\dots .$$

If $\Delta {\gamma }_{\mathrm{rel}}^{\left(1\right)}\ne 0$, asymptotic freedom gives

$$r\left(\mu \right)\propto {\left[ln{\displaystyle \frac{\mu }{{\Lambda }_p}}\right]}^{\Delta {\gamma }_{\mathrm{rel}}^{\left(1\right)}/\left(2b_0\right)}$$

(A126)

in the ultraviolet perturbative regime. For the admissible nonintersecting point-split pair, Equation (A92) instead gives

$$\Delta {\gamma }_{\mathrm{rel}}^{\left(1\right)}=0.$$

If a relative difference first appears at two loops, its ultraviolet integration produces only a finite $O(1/ln\mu )$ correction,

$$r\left(\mu \right)=r_{\infty }\left[1+O\left({\displaystyle \frac{1}{ln(\mu /{\Lambda }_p)}}\right)\right](\mu \to \infty ),$$

rather than a universal one-loop logarithmic attraction to zero.

Weak-coupling UV neutrality theorem.For the clean nonintersecting point-split 24-preon source doublet, the one-loop relative anomalous exponent vanishes. Perturbative ultraviolet evolution therefore preserves a reciprocity equality if it is imposed, but it does not dynamically produce equality from a generic unequal pair. Any substantial alignment must come from a specified route-changing contact/intersection kernel, finite matching, or the nonperturbative infrared/confinement dynamics.

Stopping rule.If the source remains genuinely point split and nonintersecting, there is no missing route-changing one-loop UV coefficient: ${\gamma }_{\times }^{\left(1\right)}=0$ is already fixed by locality and reflected source data. A nonzero one-loop route-sensitive coefficient is meaningful only after a definite collapsed/intersection prescription and a complete $\tau$-closed mixing basis are supplied. The next nonperturbative observable remains the pole-subtracted route correlator/step-scaling matrix, now interpreted through the relative exponent (A95).

Appendix B.1.14. Finite-Separation Physical Matching: Why a Contact Limit Is Not Required

The ultraviolet discussion above identified the only way a route-sensitive one-loop counterterm can arise in the clean two-route subspace: one must deliberately introduce a local contact/intersection or collapse the network so that topology-changing contractions become ultraviolet-local. It remains to ask whether the physical low-energy matching itself forces that operation. It does not.

Appendix B.1.1.1. Three independent limits

Introduce three variables that must be kept distinct:

• y, a signed coordinate on a reflected family of routed source configurations, with $\tau \Gamma \left(y\right)=\Gamma (-y)$;
• $d_{\mathrm{ps}}>0$, a physical point-splitting scale, e.g., the minimum separation between distinct local junctions and nonincident contour pieces;
• q, the momentum transfer through the low-energy operator insertion.

The route fixed-point operation is $y\to 0$. A coincident source would require the separate limit $d_{\mathrm{ps}}\to 0$. The physical zero-transfer form factor uses $q\to 0$ after state isolation and amputation. There is no identity relating these limits.

Three-limit separation theorem.The route/configuration fixed point, spacetime collapse of the point-split source, and zero-momentum low-energy matching are independent operations. Neither $y=0$ nor $q=0$ implies $d_{\mathrm{ps}}=0$. Therefore the notation “fixed-point vertex” in the route-coordinate completion must not be read as a coincident 24-preon contact vertex.

Appendix B.1.1.2. A noncollapsed reflection-compatible source exists

The underlying connectivity of the displayed eight-junction representative is

$${\underline{E}}_{\Gamma }=\{12,23,34,45,56,67,78,81,27,36\},\tau =\left(18\right)\left(27\right)\left(36\right)\left(45\right).$$

A constructive Euclidean embedding is

$$\begin{array}{cccccccc}\hfill x_1& =(-3a,+b),\hfill & \hfill x_2& =(-a,+b),\hfill & \hfill x_3& =(a,+b),\hfill & \hfill x_4& =(3a,+b),\hfill \\ \hfill x_8& =(-3a,-b),\hfill & \hfill x_7& =(-a,-b),\hfill & \hfill x_6& =(a,-b),\hfill & \hfill x_5& =(3a,-b),\hfill \end{array}$$

with $a,b>0$, embedded in a two-plane of Euclidean four-space. Straight representatives of the listed edges have no intersections between nonincident edges and the minimum vertex separation is

$$d_V=2min(a,b)>0.$$

For $a=b=1$ the minimum nonincident-segment separation is also 2 in these units. Hence there is an open neighborhood of source-data space in which all unintended contacts remain absent.

Noncollapse source-geometry theorem.The route connectivity admits reflection-compatible point-split representatives with a strictly positive separation gap. A route fixed configuration is therefore not synonymous with spacetime coincidence of its local three-preon junctions.

Appendix B.1.1.3. The route-normal derivative remains an extended gauge-invariant operator

Let $\Gamma \left(y\right)$ be a $C^1$ reflected source family satisfying

$$\tau \Gamma \left(y\right)=\Gamma (-y),\underset{\left|y\right|<y_0}{inf}{d}_{\mathrm{ps}}\left(y\right)\ge d_{*}>0,$$

and define the derivative after renormalization,

$${\mathcal{D}}_{\mathrm{rt}}{J}_{-}\equiv {\left.{\partial }_y{J}_{-}^{\mathrm{ren}}\left[\Gamma \left(y\right)\right]\right|}_{y=0}.$$

(A127)

Every member of the family is gauge invariant, so differentiation with respect to the external source-geometry parameter preserves gauge invariance. Since $J_{-}(-y)=-J_{-}\left(y\right)$,

$$\tau \left({\mathcal{D}}_{\mathrm{rt}}{J}_{-}\right)=+{\mathcal{D}}_{\mathrm{rt}}{J}_{-}.$$

For a fixed-endpoint deformation of one Wilson segment, direct differentiation of the path-ordered exponential gives the gauge-covariant field-strength insertion

$${\left.{\partial }_y{W}_{{\gamma }_y}\right|}_0=i{g}_p{\int }_0^{1}dsW(1,s)F_{\mu \nu }\left(x\left(s\right)\right)W(s,0){\eta }^{\mu }\left(s\right){\dot{x}}^{\nu }\left(s\right),$$

(A128)

where ${\eta }^{\mu }={\partial }_y{x}^{\mu }{|}_0$; endpoint covariant-derivative terms are added if endpoints move. Equation (A98) inserts a field strength on a finite Wilson path rather than collapsing all junctions.

Finite-separation route-derivative theorem.For a smooth reflected source family with a uniform positive separation gap, ${\mathcal{D}}_{\mathrm{rt}}{J}_{-}$ is a gauge-invariant route-even derivative of a still point-split operator family. The completion-level coupling ${\mathcal{O}}_{+}^{†}{\mathcal{D}}_{\mathrm{rt}}{J}_{-}$ therefore does not require a coincident 24-preon source.

Appendix B.1.1.4. Pole amputation removes the finite-source overlap

Let $J_s$ be any renormalized finite-separation interpolating source with nonzero overlap

$$z_s=\langle 0|J_s|X_{-}\rangle \ne 0$$

onto the same isolated physical state. Near the pole,

$$G_s^{\left(2\right)}\left(p\right)={\displaystyle \frac{|z_s{|}^2}{p^2+M_{-}^2}}+\mathrm{regular},$$

while a three-point function with a renormalized local low-energy operator ${\mathcal{O}}_{\alpha }$ factorizes as

$$G_{s,\alpha }^{\left(3\right)}(p^{\prime },p)={\displaystyle \frac{z_s{z}_s^{*}}{(p^{\prime 2}+M_{-}^2)(p^2+M_{-}^2)}}{\mathcal{V}}_{\alpha }\left(q^2\right)+\mathrm{less}\sin \mathrm{gular}\mathrm{terms}.$$

(A129)

The corresponding product of two external overlaps appears for a transition matrix element. Dividing by the external pole residues and propagators leaves ${\mathcal{V}}_{\alpha }\left(q^2\right)$. Thus finite path geometry or smearing changes the interpolation overlap $z_s$ but not the pole-normalized physical matrix element once the same state has been isolated and the operator renormalization scheme is held fixed.

For

$${\mathcal{O}}_{+}={\displaystyle \frac{{\mathcal{O}}_{12}+{\mathcal{O}}_{21}}{\sqrt{2}}},{\mathcal{O}}_{-}={\displaystyle \frac{{\mathcal{O}}_{21}-{\mathcal{O}}_{12}}{\sqrt{2}}},$$

define $F_{\pm }\left(0\right)={\mathcal{V}}_{\pm }\left(0\right)$. No $d_{\mathrm{ps}}\to 0$ limit occurs in this definition.

Finite-separation pole-amputation theorem.The physical zero-transfer matching form factors can be extracted from an isolated pole using any admissible renormalized finite-separation source with nonzero overlap. External-leg amputation removes the arbitrary source-overlap normalization. A collapsed source is therefore not a prerequisite for $F_{\pm }\left(0\right)$.

With an independently established exact route involution, the odd operator has vanishing diagonal matrix element in an odd state,

$$\langle X_{-}\left|{\mathcal{O}}_{-}\right|X_{-}\rangle =0,$$

whereas the even matrix element is allowed. Hence the physical target remains

$$F_{-}\left(0\right)=0,F_{+}\left(0\right)\ne 0\mathrm{must}\mathrm{be}\mathrm{computed},$$

independently of finite source geometry.

Appendix B.1.1.5. The contact coefficient becomes optional scheme data

A collapsed/OPE source prescription may still be introduced deliberately. Then route-changing local counterterms can occur and the contact/intersection coefficient ${\gamma }_{\times }^{\left(1\right)}$ of Equation (A93) must be embedded in the complete local mixing basis. But this is an alternative matching scheme, not a compulsory physical step. Individual off-diagonal anomalous-dimension entries depend on the chosen operator basis and subtraction convention; physical eigenvalue differences at a fixed point and fully matched amplitudes are the invariant quantities.

Contact-branch demotion theorem.Physical low-energy matching does not require the collapsed/contact limit. Therefore a nonzero ${\gamma }_{\times }^{\left(1\right)}$ is not a missing ingredient of the main clean point-split branch. In that branch the one-loop ultraviolet result is already complete at relative order: ${\gamma }_{\times }^{\left(1\right)}=0$. Contact/intersection mixing is relevant only if a coincident/OPE prescription is adopted deliberately.

Appendix B.1.1.6. Finite-difference implementation.

The completion-level route derivative can be estimated without changing the physical point-splitting gap:

$${\mathcal{D}}_{\mathrm{rt}}^{\left(\delta y\right)}{J}_{-}={\displaystyle \frac{J_{-}^{\mathrm{ren}}\left[\Gamma (+\delta y)\right]-J_{-}^{\mathrm{ren}}\left[\Gamma (-\delta y)\right]}{2\delta y}}={\mathcal{D}}_{\mathrm{rt}}{J}_{-}+O\left(\delta y^2\right).$$

(A130)

This is a continuum in source-geometry space, distinct from both lattice-spacing removal and spacetime collapse.

Stopping rule.The perturbative-contact branch is no longer the next compulsory calculation. The minimal first-principles program is to keep the source point split, compute the renormalized $2\times 2$ route correlator, isolate/subtract the odd pole, determine the regular route-exchange remainder or relative step scaling, and then extract $F_{\pm }\left(0\right)$ from pole-amputated finite-separation three-point functions. A contact calculation is needed only if one deliberately chooses a collapsed/OPE source scheme.

Appendix B.2. A First Genuinely Confining Route-Control Calculation in 1+1 Dimensional SU (3)

The preceding sections reduce the unresolved four-dimensional mechanism to a genuinely nonperturbative question. Before attempting the full chiral $\mathrm{SU}{\left(15\right)}_p$ theory, it is useful to ask whether a simpler but authentic non-Abelian confining Hamiltonian can make different constituent-transfer orders reciprocal. The following calculation provides such a control test. It is not a derivation of any $\mathrm{SU}{\left(15\right)}_p$ coefficient; its role is to isolate a confinement mechanism that can later be tested in a multi-link gauge-invariant simulation.

A convenient control theory is Hamiltonian $1+1$ dimensional $\mathrm{SU}\left(3\right)$ lattice gauge theory with dynamical fundamental fermions. A gauge-invariant loop–string–hadron formulation of this theory is known, with non-Abelian Gauss constraints solved locally and open-boundary numerical equivalence to the completely gauge-fixed Hamiltonian. The analysis uses only the elementary one-link strong-coupling sector, for which the route amplitudes can be derived analytically.

Appendix B.2.1. Gauge-invariant One-Link Baryon Transfer

Take neighboring sites $L,R$ joined by $U\in \mathrm{SU}\left(3\right)$ and three distinguishable fermion species $f=1,2,3$. Define

$$B_L^{†}={\displaystyle \frac{1}{\sqrt{6}}}{ϵ}_{abc}{\psi }_{L1}^{a†}{\psi }_{L2}^{b†}{\psi }_{L3}^{c†},B_R^{†}={\displaystyle \frac{1}{\sqrt{6}}}{ϵ}_{abc}{\psi }_{R1}^{a†}{\psi }_{R2}^{b†}{\psi }_{R3}^{c†},$$

(A131)

and the gauge-invariant constituent hop

$$Q_f={\psi }_{Rf}^{a†}{\left(U^{†}\right)}^a{}_b\psi _{Lf}^b,V=-{\displaystyle \sum _{f=1}^3}{t}_f(Q_f+Q_f^{†}).$$

(A132)

Distinct $Q_f$ are even in fermion number and act on distinct species. Moreover,

$${ϵ}_{abc}{\left(U^{†}\right)}^a{}_{a^{\prime }}\left(U^{†}\right)^b{}_{b^{\prime }}\left(U^{†}\right)^c{}_{c^{\prime }}=(det{U}^{†}){ϵ}_{a^{\prime }{b}^{\prime }{c}^{\prime }}={ϵ}_{a^{\prime }{b}^{\prime }{c}^{\prime }},$$

(A133)

so, after a common baryon phase convention,

$$\langle B_R|Q_{{\pi }_3}{Q}_{{\pi }_2}{Q}_{{\pi }_1}|B_L\rangle =1,\pi \in S_3.$$

(A134)

Thus the route ordering can enter the leading strong-coupling amplitude only through intermediate-state energy denominators.

Let the electric Hamiltonian assign a flux energy

$${\Delta }_R={\kappa }_E{C}_2\left(R\right),{\kappa }_E>0.$$

(A135)

After one constituent transfer the link carries fundamental flux. After two transfers the two transported fundamentals are in the antisymmetric $\overline{\mathbf{3}}$ channel, so the intermediate flux is antifundamental. Since

$$C_2\left(\mathbf{3}\right)=C_2\left(\overline{\mathbf{3}}\right)=C_F={\displaystyle \frac{4}{3}},$$

(A136)

both steps pay the same confinement contribution

$${\Delta }_F={\displaystyle \frac{4}{3}}{\kappa }_E.$$

(A137)

This equality is dynamical rather than purely combinatorial: the two distinct partially separated color configurations are degenerate in electric-flux Casimir energy.

Allow an additional one-particle excitation shift ${\delta }_f$ when species f moves from L to R. For the chronological route $\pi =({\pi }_1,{\pi }_2,{\pi }_3)$, third-order degenerate perturbation theory gives

$$A_{\pi }=-{\displaystyle \frac{t_1{t}_2{t}_3}{[{\Delta }_F+{\delta }_{{\pi }_1}][{\Delta }_F+{\delta }_{{\pi }_1}+{\delta }_{{\pi }_2}]}},$$

(A138)

up to the common sign convention of the hopping Hamiltonian.

Exact leading crossed-route reciprocity in the confining $\mathrm{SU}\left(3\right)$ control model.For the crossed routes $12=(1,2,3)$ and $21=(2,1,3)$,

$$\begin{array}{cc}\hfill A_{12}& =-{\displaystyle \frac{t_1{t}_2{t}_3}{({\Delta }_F+{\delta }_1)({\Delta }_F+{\delta }_1+{\delta }_2)}},\hfill \end{array}$$

(A139)

$$\begin{array}{cc}\hfill A_{21}& =-{\displaystyle \frac{t_1{t}_2{t}_3}{({\Delta }_F+{\delta }_2)({\Delta }_F+{\delta }_1+{\delta }_2)}}.\hfill \end{array}$$

(A140)

Hence ${\delta }_1={\delta }_2$ implies

$$A_{12}=A_{21}$$

(A141)

exactly at the first nonvanishing, third order in hopping. In the completely degenerate case all six permutations are equal and

$$A_B^{\left(3\right)}=-{\displaystyle \frac{6t_1{t}_2{t}_3}{{\Delta }_F^2}}.$$

(A142)

The factor six is the coherent sum of the six gauge-invariant constituent-transfer orders.

Appendix B.2.2. Confinement Suppresses Route Splitting Away from Exact Degeneracy

More importantly, the crossed-route defect can be evaluated exactly:

$${ϵ}_{\mathrm{rt}}^{\mathrm{SU}\left(3\right)}\equiv {\displaystyle \frac{A_{21}-A_{12}}{A_{21}+A_{12}}}={\displaystyle \frac{{\delta }_1-{\delta }_2}{2{\Delta }_F+{\delta }_1+{\delta }_2}}.$$

(A143)

Therefore, in the confinement-dominated regime ${\Delta }_F\gg \left|{\delta }_i\right|$,

$$|{ϵ}_{\mathrm{rt}}^{\mathrm{SU}\left(3\right)}|={\displaystyle \frac{|{\delta }_1-{\delta }_2|}{2{\Delta }_F}}+O\left({\displaystyle \frac{{\delta }^2}{{\Delta }_F^2}}\right).$$

(A144)

The strong electric-flux gap therefore suppresses the memory of which constituent crossed first. Exact flavour degeneracy in this control is sufficient for exact leading reciprocity, but a hierarchy between the common confinement gap and the route-sensitive excitation splitting is enough for approximate reciprocity.

For a target $|{ϵ}_{\mathrm{rt}}^{\mathrm{SU}\left(3\right)}|\le {ϵ}_{*}$, the exact one-link condition is

$$|{\delta }_1-{\delta }_2|\le {ϵ}_{*}|2{\Delta }_F+{\delta }_1+{\delta }_2|,$$

(A145)

or, at strong confinement,

$$|{\delta }_1-{\delta }_2|\lesssim 2{ϵ}_{*}{\Delta }_F.$$

(A146)

A differential route cost of $0.02{\Delta }_F$, with negligible average shift, therefore produces approximately a one-percent route defect. This number is only a calibration of the control model and is not fitted to the charged-fermion plane.

Confinement-dominance mechanism-level result.The $1+1$ dimensional $\mathrm{SU}\left(3\right)$ strong-coupling control theory supplies a genuine dynamical witness for route reciprocity: the first nonzero baryon-transfer amplitude is exactly route reciprocal for equal route-sensitive excitation costs, while small unequal costs are suppressed by the confinement flux gap according to Equation (A111). Thus confinement can make route ordering dynamically weak without requiring the two intermediate configurations to be literally identical.

Appendix B.2.3. Exact Eight-State Route-Source Correlator in the Two-Site Control

The leading transfer amplitudes above test chronological order through perturbative denominators. A stronger control is available without adding any new field or phenomenological parameter: keep the complete two-site, three-species gauge-invariant occupancy sector. Label a basis state by the subset $S\subseteq \{1,2,3\}$ of species transported from L to R. There are exactly $2^3=8$ such Gauss-law sectors. The empty and fully transported subsets carry no link flux, whereas every one- or two-particle split carries a fundamental or antifundamental flux and hence the same electric energy ${\Delta }_F$.

After a common phase choice, a constituent hop toggles one species. In the equal-hopping, equal-route-cost control the eight-state Hamiltonian is therefore

$$H_8={\Delta }_F{P}_{\mathrm{split}}-tA\left(Q_3\right),$$

(A115)

where $P_{\mathrm{split}}$ projects onto Hamming weights one and two and $A\left(Q_3\right)$ is the adjacency matrix of the three-dimensional cube. This is an exact representation of the color-diagonal one-link strong-coupling sector; it is not yet a multi-link LSH Hamiltonian with the complete dynamical link Hilbert space.

Let ${\mathsf{P}}_{12}$ exchange species 1 and 2, and define the two split sources

$$|J_1\rangle =|\left\{1\right\}\rangle ,|J_2\rangle =|\left\{2\right\}\rangle .$$

(A148)

For equal hopping parameters and equal route-sensitive one-particle costs,

$$[H_8,{\mathsf{P}}_{12}]=0,{\mathsf{P}}_{12}|J_1\rangle =|J_2\rangle .$$

(A149)

Hence the Euclidean source matrix

$$G_{AB}\left(\tau \right)=\langle J_A|e^{-\tau H_8}|J_B\rangle$$

(A150)

obeys the exact all-time identities

$$G_{11}\left(\tau \right)=G_{22}\left(\tau \right),G_{12}\left(\tau \right)=G_{21}\left(\tau \right)$$

(A151)

for every $\tau \ge 0$. The equality of the diagonal entries is the nontrivial consequence of $[H_8,{\mathsf{P}}_{12}]=0$ together with ${\mathsf{P}}_{12}|J_1\rangle =|J_2\rangle$.

Off-diagonal equality is not an independent reciprocity test.For any Hermitian transfer Hamiltonian, $e^{-\tau H}$ is Hermitian and therefore $G_{21}\left(\tau \right)=G_{12}{\left(\tau \right)}^{*}$. In the present control $H_8$ and the sources are real, so $e^{-\tau H_8}$ is real symmetric and

$$G_{12}\left(\tau \right)=G_{21}\left(\tau \right)$$

holds kinematically even without using the species-exchange symmetry. Consequently $G_{12}=G_{21}$ must not be counted as independent evidence for route reciprocity in a real symmetric control. The nontrivial exchange diagnostic is $G_{11}=G_{22}$ (or, equivalently, vanishing of the route-odd diagonal response) after the two sources have been defined independently.

The odd sector is also exactly soluble. With

$$|J_{-}\rangle ={\displaystyle \frac{\left|\right\{1\}\rangle -|\left\{2\right\}\rangle }{\sqrt{2}}},|K_{-}\rangle ={\displaystyle \frac{\left|\right\{1,3\}\rangle -|\{2,3\}\rangle }{\sqrt{2}}},$$

(A152)

the Hamiltonian restricts to

$$H_{-}=\left(\begin{array}{cc}{\Delta }_F& -t\\ -t& {\Delta }_F\end{array}\right)$$

(A153)

for the equal-cost control. Therefore

$$E_{-}^{\left(1\right)}={\Delta }_F-t,E_{-}^{\left(2\right)}={\Delta }_F+t$$

(A154)

and the normalized odd correlator is

$$G_{-}\left(\tau \right)=\langle J_{-}|e^{-\tau H_8}|J_{-}\rangle =e^{-{\Delta }_F\tau }cosh\left(t\tau \right)={\displaystyle \frac{1}{2}}\left(e^{-({\Delta }_F-t)\tau }+e^{-({\Delta }_F+t)\tau }\right).$$

(A155)

Thus the finite control does contain nonzero route-odd spectral support. What it does not contain, for $t>0$, is an odd ground state: the connected matrix $H_8$ has strictly non-positive hopping off-diagonal entries, so the Perron–Frobenius ground state can be chosen unique and strictly positive and is therefore route even. The earlier shorthand statement that the odd channel “dies” at large Euclidean time must consequently be interpreted only as dominance of the even ground state in the unprojected matrix, not as absence of odd poles from the parity-projected correlator.

Appendix B.2.2.1. Independent three-site occupancy cross-check.

As a volume-extension check distinct from the representation-matched $\Omega F_1{F}_2$ control below, place the same three distinguishable fundamental species on an open three-site chain. A basis state is $(r_1,r_2,r_3)$ with $r_f\in \{0,1,2\}$, giving $3^3=27$ states. Each of the two cuts carries electric cost ${\Delta }_F$ when one or two fundamentals lie to its right, and nearest-neighbour hopping has amplitude $-t$. For ${\Delta }_F=1$ and $t=0.30$, exact parity-resolved diagonalization gives

$$E_0=-0.537493519248,E_{-}^{\mathrm{abs}}=0.513420263749,$$

so the route-odd gap, which must be measured relative to the even ground state, is

$${\Delta }_{-}=E_{-}^{\mathrm{abs}}-E_0=1.050913782996.$$

Thus the value $0.5134$ is an absolute odd-sector energy, not the odd gap. With route-sensitive costs ${\delta }_1=+0.05$, ${\delta }_2=-0.05$, ${\delta }_3=0$, paid when a species is displaced from the left endpoint, and at $\tau =2$, define

$${\epsilon }_{\mathrm{diag}}={\displaystyle \frac{G_{11}-G_{22}}{G_{11}+G_{22}}}.$$

The exact two-site and three-site values are respectively $-0.0825331$ and $-0.0830717$. The second link therefore changes this diagonal defect only mildly in this benchmark, while $G_{12}-G_{21}$ remains zero to numerical precision for the purely kinematic real-symmetry reason stated above. This is an independent occupancy control, not a derivation of the physical $\mathrm{SU}{\left(15\right)}_p$ correlator.

As a combinatorial cross-check, let the two right-oriented split sources be the complements of the two left-oriented sources. Then exact cube path counting gives

$$\langle J^R\left|V\right|J^L\rangle =\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),\langle J^R|V^3|J^L\rangle =\left(\begin{array}{cc}6& 7\\ 7& 6\end{array}\right),$$

(A156)

with eigenvalues 13 and $-1$ for the cubic transfer block. The integers 6 and 7 are path counts in this finite hopping graph; they are not masses, gaps, or predictions for $\mathrm{SU}{\left(15\right)}_p$.

Perturbative domain of the unequal-cost amplitudes.The normalized defect in Equation (A111) is independent of the overall hopping normalization, but Eqs. (A108) are third-order strong-coupling amplitudes. Numerical tables obtained by setting $t=1$ should therefore be read as values of the normalized coefficient $A/t^3$ unless the denominators are parametrically larger than the hopping. The perturbative expansion fails when any displayed intermediate denominator approaches zero; the exact defect identity should not be extrapolated through such a degeneracy as though the third-order expansion remained controlled.

Appendix B.2.2.2. What is and is not learned for SU (15) p .

The result does not prove $C_{12}=C_{21}$ in the 24-preon theory. The $\mathrm{SU}{\left(15\right)}_p$ regular remainder contains multi-junction spectator, path, mixing, and continuum information absent from the one-link problem. What is established is narrower and useful: a real confining non-Abelian Hamiltonian can generate a parametrically small route defect when the common flux gap dominates route-sensitive excitation splittings. This suggests the following new microscopic diagnostic for the main theory. Let ${\Delta }_{\mathrm{conf}}$ be the common energy penalty of the relevant routed color-flux sectors and ${\Delta }_{\mathrm{odd}}$ their route-sensitive nonconfining splitting. A simple confinement-dominance explanation of a target route tolerance ${ϵ}_{*}$ requires, parametrically,

$${\displaystyle \frac{|{\Delta }_{\mathrm{odd}}|}{{\Delta }_{\mathrm{conf}}}}\lesssim 2{ϵ}_{*}$$

(A157)

up to multi-state and multi-link corrections. A first-principles result $|{\Delta }_{\mathrm{odd}}|\sim {\Delta }_{\mathrm{conf}}$ would close this simple mechanism; a hierarchy $|{\Delta }_{\mathrm{odd}}|\ll {\Delta }_{\mathrm{conf}}$ would give a concrete dynamical route toward approximate reciprocity.

Appendix B.2.2.3. Next controlled extension.

For the representation-matched $\Omega F_1{F}_2$ sector considered below, the open-chain dynamical-link problem can in fact be reduced exactly by axial gauge and Gauss’ law, so a separate local link truncation is unnecessary in $1+1$ dimensions with zero boundary flux. The useful remaining extensions are therefore different: enlarge the matter sector to allow pair creation or additional preon channels, introduce nonzero boundary/global-flux sectors, or move to higher spatial dimension where magnetic plaquette dynamics is genuinely independent. The decisive four-dimensional test remains the renormalized route-resolved $\mathrm{SU}{\left(15\right)}_p$ correlator and matching vertex.

Appendix B.2.4. Precolor-specific SU (15) p One-Link Gap Control

The preceding $\mathrm{SU}\left(3\right)$ calculation isolates a generic confinement mechanism. A closer representation-level control can be constructed without changing the published $\mathrm{SU}{\left(15\right)}_p$ field content. Consider the local precolor singlet

$$B^{†}\sim {\Omega }_{ab}{F}_1^a{F}_2^b,$$

(A158)

where the two fundamentals are coupled to the symmetric color channel selected by the conjugate-symmetric $\Omega$ tensor. For $\mathrm{SU}\left(N\right)$,

$$C_F={\displaystyle \frac{N^2-1}{2N}},C_S={\displaystyle \frac{(N-1)(N+2)}{N}}.$$

(A159)

With electric energy ${\Delta }_R={\kappa }_E{C}_2\left(R\right)$, the actual $N=15$ representation content therefore gives

$${\Delta }_F={\displaystyle \frac{112}{15}}{\kappa }_E,{\Delta }_S={\displaystyle \frac{238}{15}}{\kappa }_E.$$

(A160)

Moving one fundamental leaves a $\overline{\mathbf{15}}$ color source opposite a $\mathbf{15}$ and hence fundamental link flux. Moving both fundamentals leaves the conjugate-symmetric representation opposite the symmetric one and hence the gap ${\Delta }_S$. Moving $\Omega$ together with one fundamental returns to fundamental flux. These representation assignments follow from the same local singlet and introduce no new field.

Let t be the hopping amplitude of either fundamental and $s=t_{\Omega }$ that of $\Omega$. The complete one-link occupancy problem has eight sectors labeled by the subset of $\{F_1,F_2,\Omega \}$ transported across the link. Under the exact precolor exchange $F_1\leftrightarrow F_2$, the route-odd basis

$$|a\rangle ={\displaystyle \frac{|F_1\rangle -|F_2\rangle }{\sqrt{2}}},|b\rangle ={\displaystyle \frac{|F_1\Omega \rangle -|F_2\Omega \rangle }{\sqrt{2}}}$$

(A161)

closes exactly. Fundamental hopping cancels from this odd subspace and the $\Omega$ hop gives

$$H_{-}^{(15,\mathrm{ctrl})}=\left(\begin{array}{cc}{\Delta }_F& -s\\ -s& {\Delta }_F\end{array}\right),E_{-}^{(1,2)}={\Delta }_F\mp s.$$

(A162)

The unique positive ground state is route even. Complement symmetry reduces it to the lowest eigenvalue $E_0$ of

$$H_{++}^{(15,\mathrm{ctrl})}=\left(\begin{array}{ccc}0& -\sqrt{2}t& -s\\ -\sqrt{2}t& {\Delta }_F-s& -\sqrt{2}t\\ -s& -\sqrt{2}t& {\Delta }_S\end{array}\right).$$

(A163)

Hence the exact route-odd gap of this control problem is

$${\Delta }_{-,\mathrm{ctrl}}^{\left(15\right)}={\Delta }_F-s-E_0.$$

(A164)

At strong coupling it behaves as

$${\Delta }_{-,\mathrm{ctrl}}^{\left(15\right)}={\Delta }_F-s+{\displaystyle \frac{2t^2}{{\Delta }_F}}+{\displaystyle \frac{s^2}{{\Delta }_S}}+O({(t,s)}^3/{\Delta }^2).$$

(A165)

An antisymmetric route-sensitive cost

$$V_{\perp }=x(n_1-n_2)$$

(A166)

has exact operator norm $\parallel V_{\perp }{\parallel }_2=\left|x\right|$. If $z_1,z_2$ are the ground-state overlaps with the one-fundamental split sources, then

$${\displaystyle \frac{z_1-z_2}{z_1+z_2}}=-{\displaystyle \frac{x}{{\Delta }_{-,\mathrm{ctrl}}^{\left(15\right)}}}+O\left(x^3\right).$$

(A167)

Thus the gap-protected suppression theorem is realized by the actual $\mathrm{SU}\left(15\right)$ precolor representations, not only by the $\mathrm{SU}\left(3\right)$ baryon analog.

The chronological transfer amplitudes provide an independent check. The routes $F_1$ then $F_2$ and $F_2$ then $F_1$ have the common representation sequence $F\to S\to 1$ and satisfy

$$\begin{array}{cc}\hfill A_{12}& =-{\displaystyle \frac{t_1{t}_2{t}_{\Omega }}{({\Delta }_F+{\delta }_1)({\Delta }_S+{\delta }_1+{\delta }_2)}},\hfill \end{array}$$

(A168)

$$\begin{array}{cc}\hfill A_{21}& =-{\displaystyle \frac{t_1{t}_2{t}_{\Omega }}{({\Delta }_F+{\delta }_2)({\Delta }_S+{\delta }_1+{\delta }_2)}},\hfill \end{array}$$

(A169)

so

$${ϵ}_{\mathrm{rt}}^{(15,\mathrm{ctrl})}={\displaystyle \frac{{\delta }_1-{\delta }_2}{2{\Delta }_F+{\delta }_1+{\delta }_2}}.$$

(A170)

The symmetric-channel denominator cancels from the normalized defect.

For a controlled numerical illustration in units ${\kappa }_E=1$, choose $t=s=0.1$ and $x=0.01$. Then

$${\Delta }_F=7.466666666667,{\Delta }_S=15.866666666667,$$

(A171)

$$E_0=-0.003378582053,{\Delta }_{-,\mathrm{ctrl}}^{\left(15\right)}=7.370045248720,$$

(A172)

and

$${\eta }_{15}^{\mathrm{ctrl}}\equiv {\displaystyle \frac{\left|x\right|}{{\Delta }_{-,\mathrm{ctrl}}^{\left(15\right)}}}=1.356843773\times {10}^{-3}.$$

(A173)

Exact diagonalization gives $(z_1-z_2)/(z_1+z_2)=-0.001356843772$, agreeing with the linear prediction to better than ${10}^{-12}$ at this point. These numbers validate the control calculation only; $t,s,x,$ and ${\kappa }_E$ are not physical parameters extracted from the four-dimensional theory.

Precolor-specific gap-suppression control.The unchanged $\mathrm{SU}\left(15\right)$ representation content admits an explicit strong-coupling regime in which the route-breaking norm is parametrically small compared with a calculable odd spectral gap. This is a constructive realization of the emergent-reciprocity mechanism at the precolor representation level. It is not the physical four-dimensional value of ${\Delta }_{-}^{\left(15\right)}$ or $\parallel K_{\perp }\parallel$.

Appendix B.2.5. Exact Open-Chain Dynamical-Link Reduction and Uniform Casimir Suppression

The one-link control admits a stronger multi-link interpretation. Consider the $1+1$ dimensional Kogut–Susskind Hamiltonian on an open chain with sites $j=0,\dots ,L$, zero external electric flux, one fundamental $F_1$, one fundamental $F_2$, and one conjugate-symmetric constituent $\Omega \sim \overline{S}$. Restrict to the multiplicity-one global singlet selected by

$$F_1\otimes F_2\to S,S\otimes \overline{S}\to \mathbf{1}.$$

(A174)

Before gauge fixing the hopping operators contain the dynamical link matrices $U_{\ell }$ in the appropriate representations. On an open one-dimensional chain an axial gauge sets all $U_{\ell }=I$ without truncating the physical Hilbert space. Gauss’ law then fixes the electric representation on each cut from the cumulative matter charge. In the selected singlet channel the result is unique: the empty and complete subsets carry zero flux; $\left\{F_1\right\}$, $\left\{F_2\right\}$, $\{F_1,\Omega \}$ and $\{F_2,\Omega \}$ carry the fundamental Casimir gap ${\Delta }_F$; and $\left\{\Omega \right\}$ or $\{F_1,F_2\}$ carry the symmetric gap ${\Delta }_S$. Thus the coordinate Hamiltonian

$$H=H_E\left(\mathbf{r}\right)-t(A_1+A_2)-s{A}_{\Omega },\mathbf{r}=(r_1,r_2,r_{\Omega }),$$

(A175)

is the exact gauge-reduced Kogut–Susskind Hamiltonian in this fixed three-particle sector. There is no magnetic plaquette term in one spatial dimension. Additional homogeneous electric-flux sectors would require different boundary data, while pair creation or extra matter channels enlarge the sector and are not included here.

The route exchange $U:(r_1,r_2,r_{\Omega })\mapsto (r_2,r_1,r_{\Omega })$ commutes with the equal-hopping Hamiltonian. Every route-odd basis component has $r_1\ne r_2$, so at least one cut between the two fundamentals carries fundamental flux. Consequently

$$minspec\left(H_E{|}_{{\mathcal{H}}_{-}}\right)={\Delta }_F.$$

(A176)

For the path graph on $L+1$ sites,

$$h_L\equiv \parallel H_{\mathrm{hop}}\parallel =2cos\left({\displaystyle \frac{\pi }{L+2}}\right)\left(2\right|t|+\left|s\right|).$$

(A177)

The unique Perron–Frobenius ground state is route even. Positivity of $H_E$ gives $-h_L\le E_0\le 0$. The odd electric minimum and the hopping norm give $E_{-}\ge {\Delta }_F-h_L$, while the explicit odd trial state $\left(\right|1,0,0\rangle -|0,1,0\rangle )/\sqrt{2}$ has expectation value exactly ${\Delta }_F$, hence $E_{-}\le {\Delta }_F$. Therefore

$${\Delta }_F-h_L\le {\Delta }_{-,L}^{\left(N\right)}\equiv E_{-}-E_0\le {\Delta }_F+h_L.$$

(A178)

This sharpens the earlier finite-link estimate.

Since

$$h_L<h_{\infty }\equiv 2\left(2\right|t|+\left|s\right|)$$

(A179)

independently of L and

$${\Delta }_F={\kappa }_E{\displaystyle \frac{N^2-1}{2N}},$$

(A180)

Equation (A144) yields the uniform-in-length theorem

$${\Delta }_{-,L}^{\left(N\right)}={\displaystyle \frac{{\kappa }_{E}N}{2}}+O\left(1\right),L\ge 1,$$

(A181)

where the $O\left(1\right)$ remainder is bounded independently of L. For the chain-normalized route-breaking operator

$$V_{\perp }^{\left(L\right)}={\displaystyle \frac{x}{L}}(r_1-r_2)$$

(A182)

one has exactly

$$\parallel V_{\perp }^{\left(L\right)}{\parallel }_2=\left|x\right|=O\left(1\right)$$

(A183)

for every N and L. If ${\Delta }_F>h_{\infty }$,

$${\displaystyle \frac{\left|x\right|}{{\Delta }_F+h_{\infty }}}\le {\eta }_{N,L}^{\mathrm{ctrl}}\le {\displaystyle \frac{\left|x\right|}{{\Delta }_F-h_{\infty }}},$$

(A184)

and hence

$${\eta }_{N,L}^{\mathrm{ctrl}}={\displaystyle \frac{2\left|x\right|}{{\kappa }_{E}N}}\left[1+O\left(N^{-1}\right)\right]\mathrm{uniformly}\mathrm{in}L.$$

(A185)

Equivalently,

$$\underset{N\to \infty }{lim}\underset{L\ge 1}{sup}\left|N{\eta }_{N,L}^{\mathrm{ctrl}}-{\displaystyle \frac{2\left|x\right|}{{\kappa }_E}}\right|=0.$$

(A186)

Thus the $O\left(N\right)$ odd gap and $O\left(1\right)$ route-breaking norm survive the full open-chain dynamical-link reduction in this $1+1$ dimensional fixed sector.

At ${\kappa }_E=1$, $t=s=0.1$, $x=0.01$, $h_{\infty }=0.6$ and already ${\Delta }_F(N=2)=0.75>h_{\infty }$, so the odd gap is uniformly positive for every chain length and every $N\ge 2$. Exact sparse diagonalization through $L=25$ gives

$$\begin{array}{cc}\hfill {\eta }_{3,25}^{\mathrm{ctrl}}& =8.019695849\times {10}^{-3},\hfill \end{array}$$

(A187)

$$\begin{array}{cc}\hfill {\eta }_{15,25}^{\mathrm{ctrl}}& =1.357081341\times {10}^{-3}.\hfill \end{array}$$

(A188)

For $\mathrm{SU}\left(15\right)$ the 25-link value differs from the one-link result by only

$${\eta }_{15,25}/{\eta }_{15,1}-1=1.75\times {10}^{-4},$$

(A189)

about $0.0175\%$. A long-chain scan at $L=25$ gives $N{\eta }_{N,25}=0.0203562$ at $N=15$, $0.0200879$ at $N=50$, and $0.0200420$ at $N=100$, approaching $2\left|x\right|/{\kappa }_E=0.02$ exactly as required by Equation (A151).

Dynamical-link scaling gate in the control theory.Within the stated open-boundary, zero-boundary-flux, fixed-particle-number, multiplicity-one singlet sector, the multi-link coordinate problem is the exact gauge-reduced $1+1$ dimensional Kogut–Susskind theory. Its odd gap scales as $O\left(N\right)$ uniformly in chain length. For the explicitly normalized additive deformation with fixed x, the route-breaking operator is exactly $O\left(1\right)$ and the Casimir-protected law $\eta \sim 1/N$ survives the dynamical-link gate. This statement does not assert that an arbitrary physical route mismatch must be $O\left(1\right)$; the scaling of the numerator is an independent dynamical question, tested further below. The control does not determine the physical four-dimensional ${\Delta }_{-}^{\left(15\right)}$ or $\parallel K_{\perp }\parallel$.

Appendix B.2.6. Complementary Point-Split Controls and Cross-Junction Precolor Recoupling

The open-chain result above controls the confinement-scale odd gap, but several independent calculations clarify which microscopic effects can and cannot be identified with route breaking. First consider the extreme one-link equal-time limit. For distinct constituent species the elementary gauge-invariant hopping bilinears $Q_f$ are even in fermion number and commute when they act on disjoint species. Hence, if two proposed routes differ only by chronology on the same one-link geometry,

$$Q_2{Q}_1=Q_1{Q}_2,$$

(A190)

and the two route sources coincide. Their two-source Gram/correlator block is then rank one and the odd combination is null. Genuine point splitting or a larger route geometry is therefore necessary for a non-null route-odd source. This scoped result does not apply to the present 24-field Wilson-network pair, whose routes are distinct multi-vertex recouplings.

Finite point splitting removes that collapse. In plaquette controls an independent odd sector reappears, but odd spectral support must not be confused with a light odd pole. In the simplest projected non-Abelian three-plaquette strong-coupling channel, the relative odd lowering for $N>2$ is

$${\displaystyle \frac{{ϵ}_N-E_{-}^{\left(P\right)}}{{ϵ}_N}}={\displaystyle \frac{\alpha }{4(N^2-1)}}{\displaystyle \frac{{\kappa }_B}{{\kappa }_E}},\alpha ={\displaystyle \frac{\sqrt{5}-1}{2}}.$$

(A191)

At $N=15$ and ${\kappa }_B/{\kappa }_E=0.2$ this equals $1.37954\times {10}^{-4}$, or $0.013795\%$. Thus the minimal plaquette mechanism is parametrically too weak to produce a substantial large-N lightening of the odd state in controlled strong coupling. This is a stopping rule only for that stronger light-pole mechanism; it does not obstruct gap-protected emergent reciprocity.

A complementary compact-$U\left(1\right)$ three-plaquette control resolves the hidden odd sector explicitly. At the benchmark ${\kappa }_B/{\kappa }_E=0.2$, exact diagonalization gives the lowest visible odd level $E_{-}=3.929926021592{\kappa }_E$ and the lowest hidden odd level $E_{H,0}=4.090557677761{\kappa }_E$, with

$${\Delta }_0=0.090557677761{\kappa }_E,g_0=0.094891169239{\kappa }_E.$$

(A192)

The isolated two-level channel accounts for $85.4263\%$ of the exact odd Schur lowering in that finite control. This shows that a hidden susceptibility can be dominated by one identifiable odd channel, but it is not a prediction for the four-dimensional precolor theory.

There is also an exact source-level result in the already constructed $\mathrm{SU}{\left(4\right)}_{PS}$ channel module. With the normalized route overlap $s=1/64$, the relevant singlet-projector expectation is

$$p_4={\displaystyle \frac{130}{193}},$$

(A193)

and the normalized orthogonal visible–hidden projector factor is

$$q_4=\sqrt{p_4(1-p_4)}={\displaystyle \frac{\sqrt{8190}}{193}}=0.468904760487.$$

(A194)

This number belongs to the declared $\mathrm{SU}{\left(4\right)}_{PS}$ index-tensor metric and carries no forced $1/N_p$ factor. Wilson-path, Lorentz, radial, bound-state and renormalization factors are not contained in $q_4$, so it is not a physical coupling.

The actual precolor tensor algebra supplies a second order-one result. Consider two normalized local singlets of type

$$\overline{{\mathrm{Sym}}^2\mathbf{N}}\otimes \mathbf{N}\otimes \mathbf{N}⟶\mathbf{1}$$

(A195)

and let $|D\rangle$ and $|X\rangle$ denote the normalized direct and crossed contractions of the two junctions. Symmetric-projector completeness gives

$$\langle D|X\rangle ={\displaystyle \frac{1}{N}}.$$

(A196)

For $Tr\left(T^a{T}^b\right)={\delta }^{ab}/2$, the fundamental Fierz identity reads

$$T_2·T_3={\displaystyle \frac{1}{2}}\left(P_{23}-{\displaystyle \frac{1}{N}}I\right),$$

(A197)

with $P_{23}|D\rangle =|X\rangle$. Therefore

$$\langle X|T_2·T_3|D\rangle ={\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{1}{N^2}}\right)={\displaystyle \frac{N^2-1}{2N^2}}.$$

(A198)

At $N=15$ the factor is exactly

$${\displaystyle \frac{112}{225}}=0.4977777778,$$

(A199)

and it approaches $1/2$ rather than zero at large N. After canonical normalization of the nonorthogonal direct/crossed pair, the parity-channel matrix elements are

$$\langle +|T_2·T_3|+\rangle ={\displaystyle \frac{N-1}{2N}},\langle -|T_2·T_3|-\rangle =-{\displaystyle \frac{N+1}{2N}}.$$

(A200)

Equivalently, on a shared electric-flux segment the symmetric and antisymmetric two-index Casimirs differ by an $O\left(1\right)$ amount, $C_A-C_S=-2$, even though both absolute flux energies are $O\left(N\right)$. There is no conflict with Equation (A147): the latter concerns the absolute odd excitation above the full even ground state, whereas Equation (A164) compares two recoupling channels inside an $O\left(N\right)$ excitation manifold.

A crucial distinction follows. Let $U_{\mathrm{lab}}$ be the canonical route-label exchange. In the parity basis $U_{\mathrm{lab}}=diag(1,-1)$. An exchange-symmetric interaction may have

$$K_{\mathrm{sym}}=diag(E_{+},E_{-}),E_{+}\ne E_{-},$$

(A201)

while still satisfying

$$K_{\perp }={\displaystyle \frac{1}{2}}(K_{\mathrm{sym}}-U_{\mathrm{lab}}{K}_{\mathrm{sym}}{U}_{\mathrm{lab}})=0.$$

(A202)

Thus an even–odd spectral splitting, an odd hidden-state lowering, the projector factor $q_4$, and the cross-junction color factor in Equation (A162) are not measurements of the physical route-breaking numerator $\parallel K_{\perp }\parallel$. They organize the spectrum and source couplings and can be nonzero even when exact route reciprocity is intact.

The combined scaling statement is therefore conditional but sharper. The open-chain theorem establishes an $O\left(N\right)$ confinement-scale odd gap uniformly in length, while Eqs. (A158) and (A162) show that the nonlocal source algebra itself contains $O\left(1\right)$ matrix elements; there is therefore no universal group-theory rule forcing every nonlocal recoupling amplitude to be $1/N$ suppressed. The exchange-symmetric parts of those interactions contribute nothing to $K_{\perp }$. A physical numerator can arise only from a route-distinguishing mismatch between renormalized route-related structures. If that mismatch is $O\left(1\right)$ rather than $O\left(N\right)$, the $O\left(N\right)$ gap is compatible with ${\eta }_N=O(1/N)$. The sign and magnitude of that four-dimensional mismatch remain uncomputed.

Appendix B.2.7. Dynamical Screening and String Breaking: an Asymptotic Caveat on Flux-Gap Protection

The preceding unscreened electric-energy estimates must be separated from the problem of dynamical string breaking. In a control where a two-index symmetric or antisymmetric source is screened by a dynamical fundamental pair, the exact recoupling weights into the screened fundamental channel are

$$w_{F\leftarrow S}={\displaystyle \frac{2}{N(N+1)}},w_{F\leftarrow A}={\displaystyle \frac{2}{N(N-1)}}.$$

For $N=15$ the corresponding equal-strength reduced couplings are

$$g_S={\displaystyle \frac{\lambda }{\sqrt{120}}},g_A={\displaystyle \frac{\lambda }{\sqrt{105}}}.$$

In the common-tower control the min–max ordering preserves $E_S^{-}\ge E_A^{-}$. The unscreened string-breaking lengths are

$$L_{b,S}={\displaystyle \frac{5M}{42{\kappa }_E}},L_{b,A}={\displaystyle \frac{5M}{32{\kappa }_E}},$$

where M is the pair-creation scale of that control. Beyond string breaking the screened channel dominates, and the unscreened route-protection ratio tends to zero at asymptotically large separation. Thus a linear string-energy argument cannot be extrapolated indefinitely in the presence of dynamical screening.

The result is a stopping rule for the interpretation of long-string controls, not a new correction to the canonical 24-field source. In that source each local $\Omega FF$ junction is already a precolor singlet and the inter-junction Wilson network transports spectator rather than precolor indices. Consequently no physical inter-junction precolor string length should be inserted into the screening formulas above. Screening matters only if a proposed alternative realization introduces an actual extended precolor flux tube.

Appendix B.2.8. Feshbach-Dressed Route Breaking in Lower-Dimensional Controls: When 1/N Protection Survives and When It Fails

The preceding controls establish separately that an absolute route-odd excitation can carry an $O\left(N\right)$ electric cost and that nonlocal recoupling matrix elements can remain $O\left(1\right)$. They do not by themselves determine how a route-distinguishing perturbation is dressed after the remaining physical states are eliminated. This can be calculated exactly in the one-link representation-matched control and compared with a qualitatively different $2+1$ dimensional static-string route manifold.

Appendix B.2.2.1. Exact 1+1 dimensional Schur–Feshbach route kernel.

In the eight-state $\Omega F_1{F}_2$ control, take the two visible route sources to be the states in which respectively $F_1$ or $F_2$ has crossed the link, and denote the other six physical states by H. Introduce the route-odd detuning

$$V_x=x(n_1-n_2),U_{\mathrm{lab}}{V}_x{U}_{\mathrm{lab}}=-V_x,$$

(A203)

where $U_{\mathrm{lab}}$ exchanges $F_1\leftrightarrow F_2$. The exact finite-dimensional visible kernel at real energy E is

$$K_F(E;x)=H_{VV}\left(x\right)-H_{VH}{[H_{HH}\left(x\right)-E]}^{-1}{H}_{HV}.$$

(A204)

No hidden-state truncation is made in Equation (A168). It is a Hamiltonian control kernel and must not be identified with the renormalized four-dimensional 1PI kernel. In the real visible basis $U_{\mathrm{lab}}={\sigma }_x$,

$$K_{F,\perp }(E;x)={\displaystyle \frac{1}{2}}\left[K_F(E;x)-{\sigma }_x{K}_F(E;x){\sigma }_x\right],\parallel K_{F,\perp }{\parallel }_2={\displaystyle \frac{1}{2}}|{\left[K_F\right]}_{11}-{\left[K_F\right]}_{22}|.$$

(A205)

Route covariance gives $U_{\mathrm{lab}}{K}_F(E;x)U_{\mathrm{lab}}=K_F(E;-x)$, so at a matching energy fixed in the symmetric theory,

$$\parallel K_{F,\perp }{(E;x)\parallel }_2=\left|x\right|Z_K(E,N)+{O\left(\right|x|}^3).$$

(A206)

Choose $E=E_0\left(N\right)$, the ground-state energy at $x=0$, before switching on the detuning. The lowest odd level is $E_{-}^{\mathrm{low}}={\Delta }_F-s$, hence

$${\Delta }_{-}^{\left(N\right)}={\Delta }_F-s-E_0\left(N\right).$$

(A207)

For ${\kappa }_E=1$, $t=s=0.1$, $x=0.01$ and $N=15$, exact diagonalization and exact Schur complementation give

$${\Delta }_{-}^{\left(15\right),\mathrm{ctrl}}=7.370045248720,$$

(A208)

$$\parallel K_{F,\perp }(E_0;0.01){\parallel }_2=0.010133877346,$$

(A209)

$${\eta }_{15}^{\mathrm{Fesh},\mathrm{ctrl}}=1.375008837\times {10}^{-3}.$$

(A210)

Thus integrating out all six hidden physical states changes the earlier bare-norm value by only about $1.34\%$ at this point. The suppression is therefore not an artifact of ignoring the hidden part of this finite physical Hilbert space.

The large-N expansion of the exact Schur kernel is more informative. At fixed t, s and additive x,

$$Z_K(E_0,N)=1+{\displaystyle \frac{2s}{{\kappa }_{E}N}}+O\left(N^{-2}\right),{\Delta }_{-}^{\left(N\right)}={\displaystyle \frac{{\kappa }_{E}N}{2}}-s+O\left(N^{-1}\right),$$

(A211)

so

$${\eta }_{\mathrm{add}}^{\left(N\right)}={\displaystyle \frac{2\left|x\right|}{{\kappa }_{E}N}}\left[1+{\displaystyle \frac{4s}{{\kappa }_{E}N}}+O\left(N^{-2}\right)\right].$$

(A212)

A direct scan over $20\le N\le 1000$ gives a log–log slope $-1.004686$ and $N{\eta }_{\mathrm{add}}=0.0200080$ at $N=1000$ for $x=0.01$, approaching $2\left|x\right|/{\kappa }_E=0.02$.

Appendix B.2.2.2. A numerator-scaling stopping rule.

The same exact calculation exposes a failure mode that is invisible if x is held fixed by definition. If the route-distinguishing mismatch is instead a fixed fraction of the electric string scale,

$$x_N=\epsilon {\Delta }_F,\epsilon =\mathrm{fixed},$$

(A213)

then

$${\eta }_{\mathrm{frac}}^{\left(N\right)}⟶\left|\epsilon \right|,$$

(A214)

not zero. For $\epsilon =0.01$, the $N=15$ control gives ${\eta }_{\mathrm{frac}}=0.0102673$ and the large-N sequence tends to $0.01$. Thus an $O\left(N\right)$ odd gap protects reciprocity only if the route-distinguishing numerator remains $O\left(1\right)$ in absolute confinement units. A fixed percentage asymmetry of the confining electric scale is itself $O\left(N\right)$ and is not removed by the Casimir growth.

Appendix B.2.2.3. Contrasting projected 2+1 dimensional static-string route manifold.

A complementary strong-coupling control is supplied by the four shortest Wilson routes on the projected three-plaquette strip. At leading order all four routes already share the same electric string energy ${ϵ}_N=4{\kappa }_E{C}_F$. Adjacent routes mix by $-v_N$ with

$$v_N={\kappa }_B{\mu }_N,{\mu }_N={\displaystyle \frac{1+{\delta }_{N2}}{2N}}.$$

(A215)

The lowest route-parity energies are $E_{+}={ϵ}_N-\varphi v_N$ and $E_{-}={ϵ}_N-\alpha v_N$, where $\varphi =(1+\sqrt{5})/2$ and $\alpha =(\sqrt{5}-1)/2$, so their internal splitting is exactly

$${\Delta }_{\mathrm{parity}}=E_{-}-E_{+}=v_N.$$

(A216)

Put an outer-route detuning $+x$ on one visible route and $-x$ on its reflected partner. Exact elimination of the two inner routes adds only route-even I and ${\sigma }_x$ self-energy terms, hence

$$\parallel K_{\perp }\parallel =\left|x\right|$$

(A217)

for the projected visible kernel. Therefore, for $N>2$,

$${\eta }_{\mathrm{string}}={\displaystyle \frac{\left|x\right|}{v_N}}={\displaystyle \frac{2N\left|x\right|}{{\kappa }_B}}.$$

(A218)

At ${\kappa }_B/{\kappa }_E=0.2$ and $x=0.01{\kappa }_E$, this gives ${\eta }_{\mathrm{string}}=0.3$ for $N=3$, $0.4$ for $N=4$, $0.5$ for $N=5$, and $1.5$ for $N=15$. The result does not contradict the open-chain $O\left(N\right)$ gap theorem: in the mobile three-particle singlet sector the odd excitation must pay an absolute fundamental-flux cost relative to the full even ground state, whereas in the projected plaquette manifold all static routes already carry the common $O\left(N\right)$ string energy and only their residual internal parity splitting, $O(1/N)$, remains.

Lower-dimensional route-kernel scaling dichotomy.The exact $1+1$ dimensional Schur–Feshbach calculation directly computes a dressed route-breaking kernel after all hidden physical states of the one-link sector are eliminated. An additive $O\left(1\right)$ mismatch gives $\parallel K_{\perp }\parallel =O\left(1\right)$ and $\eta =O(1/N)$, but a fixed fractional electric-scale mismatch gives $\parallel K_{\perp }\parallel =O\left(N\right)$ and $\eta \to \left|\epsilon \right|$. The projected $2+1$ dimensional static-string manifold supplies a distinct failure class in which the relevant internal parity gap is only $O(1/N)$ and a fixed additive detuning gives $\eta =O\left(N\right)$. Hence confinement and large N alone do not guarantee emergent reciprocity: the physical four-dimensional calculation must establish both the scaling of the relevant odd gap and the scaling of the independently defined route-breaking numerator.

Appendix B.2.2.4. Connected singlet-matter placement theorem.

The four-dimensional source analysis localizes any genuine route defect away from a separate source-level precolor area term, but it does not by itself determine how a connected matter correction is filtered before it reaches the two-route kernel. That issue can be computed exactly in the same representation-matched one-link $1+1$ dimensional control used above. Take the visible states to be the two one-fundamental-transfer configurations $|1\rangle =|100\rangle$ and $|2\rangle =|010\rangle$, and eliminate the other six physical flux states. At real energy E away from hidden poles,

$$K_F\left(E\right)=H_{VV}-H_{VH}{(H_{HH}-E)}^{-1}{H}_{HV}.$$

(A219)

Let $U=U_V\oplus U_H$ exchange $F_1\leftrightarrow F_2$, with $U_V={\sigma }_x$ on the visible doublet. If the complete connected-matter Hamiltonian is route even,

$$UH{U}^{†}=H,$$

(A220)

then the hidden resolvent transforms covariantly and Equation (A183) gives

$$U_V{K}_F\left(E\right)U_V^{†}=K_F\left(E\right),K_{F,\perp }\left(E\right)=0.$$

(A221)

Thus arbitrarily strong route-even connected singlet dynamics cannot generate the route-breaking numerator by itself.

The location of a route-odd perturbation matters qualitatively. First keep all visible diagonal energies equal and perturb only the constituent propagation,

$$t_1=t+\delta t,t_2=t-\delta t,$$

(A222)

while the $\Omega$ hopping s remains common. Exact elimination of the six hidden states and a large-N resolvent expansion at the symmetric-theory ground energy $E_0$ give

$$\parallel K_{F,\perp }\left(E_0\right){\parallel }_2={\displaystyle \frac{{\kappa }_{E}N}{2\left|t\right|}}\left|\delta t\right|+O\left(1\right),{\eta }_{\mathrm{low}\mathrm{conn}}^{\left(N\right)}⟶\left|{\displaystyle \frac{\delta t}{t}}\right|.$$

(A223)

The route asymmetry couples directly to the near-ground singlet sector, so the corresponding resolvent cancels the apparent Casimir protection. At ${\kappa }_E=1$, $t=s=0.1$ and $\delta t=0.001$, the $N=15$ control gives

$${\Delta }_{-}^{\left(15\right)}=7.370045248720,\parallel K_{F,\perp }\parallel =0.071774701562,\eta =9.738705685\times {10}^{-3}.$$

(A224)

The large-N numerical slope of $\eta$ over $50\le N\le 1000$ is $+0.00231$, consistent with the constant limit in Equation (A187).

Now leave all low-energy couplings route symmetric and put the route-odd perturbation only into the exchanged heavy configuration-space pair,

$$H_{55}\to H_{55}+y,H_{66}\to H_{66}-y.$$

(A225)

The same exact Feshbach elimination yields

$$\parallel K_{F,\perp }\left(E_0\right){\parallel }_2={\displaystyle \frac{2\left|sy\right|}{{\kappa }_{E}N}}+O\left(N^{-2}\right),{\eta }_{\mathrm{heavy}\mathrm{conn}}^{\left(N\right)}={\displaystyle \frac{4\left|sy\right|}{{\kappa }_E^2{N}^2}}+O\left(N^{-3}\right).$$

(A226)

For ${\kappa }_E=1$, $t=s=0.1$ and $y=0.01$, the $N=15$ control gives

$$\parallel K_{F,\perp }\parallel =1.338773464\times {10}^{-4},\eta =1.816506437\times {10}^{-5}.$$

(A227)

The fitted large-N slopes are $-1.00010$ for $\parallel K_{\perp }\parallel$ and $-2.00136$ for $\eta$. The additional inverse power arises because the hidden route mismatch must first be transmitted to the visible doublet through a connected hop and then through a confinement-scale resolvent.

Connected-matter placement rule.In the exact gauge-reduced control, route-even connected singlet dynamics gives $K_{\perp }=0$ identically. Route breaking placed directly in a low-energy connector can be infrared amplified and can leave $\eta =O\left(1\right)$ even though the absolute odd gap is $O\left(N\right)$. A route-odd deformation inserted on a heavy configuration pair can instead be Feshbach-filtered more strongly; in the explicit one-link control $K_{\perp }=O\left(N^{-1}\right)$ and $\eta =O\left(N^{-2}\right)$. The multi-link extension below shows that this extra inverse power is not volume-stable and must not be promoted to a general theorem. However, the eigenmode decomposition below shows that “heavy in configuration space” is not equivalent to “spectrally heavy”: the response is dominated by a mixed low-even/high-odd bridge. Hence the physical four-dimensional question is not merely how large the connected correction is, but where its route-odd component resides in the source-accessible transition spectrum.

Appendix B.2.9. Spectral Placement of the Connected Route-Odd Response

The preceding result classifies where the perturbation is inserted in the finite occupancy basis. That is not yet the same as locating the dressed response in the eigenenergy spectrum. The distinction can be made exact. Work at a route-even reference Hamiltonian and transform the visible doublet to the parity basis $|+\rangle ,|-\rangle$, so that $U_V=diag(1,-1)$. Let the hidden Hamiltonian have parity eigenstates $|\alpha ,+\rangle$, $|\beta ,-\rangle$ with gaps

$$d_{\alpha +}={ϵ}_{\alpha +}-E,d_{\beta -}={ϵ}_{\beta -}-E>0,$$

(A228)

and define parity-preserving baseline couplings

$$g_{\alpha +}=\langle +|H_{VH}|\alpha ,+\rangle ,h_{\beta -}=\langle -|H_{VH}|\beta ,-\rangle .$$

(A229)

For an infinitesimal route-odd deformation $\lambda W_{-}$, direct differentiation of the exact Feshbach map gives

$$\begin{array}{cc}\hfill {\left.{\displaystyle \frac{\partial {\left(K_F\right)}_{+-}}{\partial \lambda }}\right|}_0=& w_{+-}-\sum _{\alpha +}{\displaystyle \frac{g_{\alpha +}\delta v_{\alpha +,-}}{d_{\alpha +}}}-\sum _{\beta -}{\displaystyle \frac{\delta v_{+,\beta -}{h}_{\beta -}}{d_{\beta -}}}\hfill \\ \hfill & +\sum _{\alpha +,\beta -}{\displaystyle \frac{g_{\alpha +}{w}_{\alpha +\beta -}{h}_{\beta -}}{d_{\alpha +}{d}_{\beta -}}}.\hfill \end{array}$$

(A230)

The first term is a direct visible insertion, the next two are one-resolvent connector terms, and the last is the two-resolvent hidden-transition term. In a real canonical route basis, $\parallel K_{\perp }\parallel =\left|\lambda \right||{\partial }_{\lambda }{\left(K_F\right)}_{+-}{|}_0+O\left({\lambda }^3\right)$. For a continuum hidden spectrum the last sum becomes a signed transition Stieltjes integral,

$$\delta K_{+-}^{\left(HH\right)}\left(E\right)=\int \int {\displaystyle \frac{d{\nu }_{+-}({\omega }_{+},{\omega }_{-})}{({\omega }_{+}-E)({\omega }_{-}-E)}},$$

(A231)

where $d{\nu }_{+-}$ is generally signed or phase-sensitive and must not be confused with a positive diagonal spectral measure.

The exact one-link $\Omega F_1{F}_2$ control resolves Equation (249) mode by mode. At $N=15$, ${\kappa }_E=1$, $t=s=0.1$, the six hidden gaps above the symmetric full ground energy are

$$\begin{array}{cc}\hfill & 7.051800326\times {10}^{-5},2.748354985\times {10}^{-3},7.470045249,7.470342629,\hfill \\ \hfill & 15.870675476,15.873055932.\hfill \end{array}$$

(A232)

with parities $+,+,-,+,+,+$. For the connector deformation $t_{1,2}=t\pm \delta t$, a single near-threshold even mode contributes $98.6461\%$ of the total absolute linear response. Its large-N data are

$$d_{e_1}={\displaystyle \frac{4t^2}{{\kappa }_{E}N}}+O\left(N^{-2}\right),g_{+,e_1}=-\sqrt{2}t+O\left(N^{-1}\right),{\partial }_{\delta t}{v}_{e_1,-}=-\sqrt{2}+O\left(N^{-1}\right),$$

(A233)

so Equation (249) directly reproduces

$${\displaystyle \frac{\partial {\left(K_F\right)}_{+-}}{\partial \delta t}}=-{\displaystyle \frac{{\kappa }_{E}N}{2t}}+O\left(1\right).$$

(A234)

Thus the $\eta =O\left(1\right)$ failure class is almost a one-mode near-threshold phenomenon rather than a diffuse many-state effect.

For the deformation $H_{55,66}\to H_{55,66}\pm y$, the configuration-space support is heavy, but the spectral response is not. At $N=15$, one mixed transition between the lowest hidden even mode and the confinement-scale odd mode supplies $98.6627\%$ of the total absolute transition weight. Its leading asymptotics are

$$\begin{array}{cccc}\hfill d_{e_0}& ={\displaystyle \frac{16t^{2}s}{{\kappa }_E^2{N}^2}}+O\left(N^{-3}\right),\hfill & \hfill d_{o_0}& ={\displaystyle \frac{{\kappa }_{E}N}{2}}+O\left(1\right),\hfill \\ \hfill g_{+,e_0}& ={\displaystyle \frac{4\sqrt{2}ts}{{\kappa }_{E}N}}+O\left(N^{-2}\right),\hfill & \hfill w_{e_0{o}_0}& ={\displaystyle \frac{2\sqrt{2}t}{{\kappa }_{E}N}}+O\left(N^{-2}\right),\hfill & \hfill h_{o_0}& =s.\hfill \end{array}$$

(A235)

The two-resolvent term therefore gives

$${\displaystyle \frac{\partial {\left(K_F\right)}_{+-}}{\partial y}}={\displaystyle \frac{2s}{{\kappa }_{E}N}}+O\left(N^{-2}\right).$$

(A236)

The $N^{-2}$ suppression of $\eta$ survives despite the ultra-low even denominator because the source coupling and the route-odd transition matrix element into that mode are themselves $O\left(N^{-1}\right)$. This corrects the interpretation of Equation (A190): the PASS is not a consequence of an exclusively heavy eigenenergy sector, but of a suppressed resolvent-weighted mixed bridge.

Spectral-placement refinement.Configuration-space support is not a reliable proxy for spectral placement. In the exact control the low-connector FAIL is $98.6461\%$ dominated by one near-threshold hidden even mode, whereas the nominally heavy-pair PASS is $98.6627\%$ dominated by one ultra-low-even/confinement-odd transition. What distinguishes the two classes is the scaling of the resolvent-weighted transition matrix elements, not the word “heavy” attached to the original basis configuration. The four-dimensional target is therefore the route-odd transition spectral response, Equation (A194), or its Euclidean response moments.

Appendix B.2.2.1. Exact identifiability boundary and physical extraction.

For a canonical Hermitian route kernel

$$K=\left(\begin{array}{cc}a& c\\ c^{*}& b\end{array}\right),U={\sigma }_x,$$

(A237)

the route-breaking part satisfies

$$\parallel K_{\perp }{\parallel }_2=\sqrt{{\displaystyle \frac{{(a-b)}^2}{4}}+{(\Im c)}^2}.$$

(A238)

Thus in a real canonical route basis it is simply $|K_{11}-K_{22}|/2$. The physical odd gap is independently obtained from the parity-projected correlators by

$${\Delta }_{-}^{\left(15\right)}=\underset{t\to \infty }{lim}{\displaystyle \frac{1}{a}}ln\left[{\displaystyle \frac{C_{-}\left(t\right)C_{+}(t+a)}{C_{-}(t+a)C_{+}\left(t\right)}}\right]$$

(A239)

when the lowest states dominate the two channels. Neither quantity follows from the static operator algebra alone: for any positive on-shell Schur–Feshbach susceptibility $M_{*}$ and any chosen hidden gap $g>0$, the choice $H_H-E_{*}=gI$ and $V=\sqrt{hg}{M}_{*}^{1/2}$ reproduces the same $M_{*}$. Consequently the physical gap is an independent spectral datum.

The four-dimensional emergent-reciprocity test is therefore the dimensionless ratio

$${\eta }_{15}={\displaystyle \frac{\parallel K_{\perp }{\parallel }_2}{{\Delta }_{-}^{\left(15\right)}}}.$$

(A240)

A controlled continuum result with ${\eta }_{15}\ll 1$ and decreasing toward the infrared supports gap-protected emergent reciprocity. A result ${\eta }_{15}=O\left(1\right)$, together with failure of regular spectral purification, closes this dynamical branch.

Appendix B.2.10. Multi-link Spectral Stability and Pole-Tracked Feshbach Response

The spectral-placement result must itself be tested against enlargement of the exact gauge-reduced open chain. The calculation therefore keeps the visible route pair $|1,0,0\rangle$, $|0,1,0\rangle$ and integrate all remaining ${(L+1)}^3-2$ physical states through the same energy-dependent Feshbach map. Two route-odd deformations are followed: the constituent-hopping mismatch $t_{1,2}=t\pm \delta t$ and the exchanged configuration-pair insertion $H_{(1,0,1)}\to H_{(1,0,1)}+y$, $H_{(0,1,1)}\to H_{(0,1,1)}-y$.

There is an exact distinction between the infinitesimal response and a finite deformation at fixed energy. Let $E_{*}\left(\lambda \right)$ denote the physical pole continuously connected to the nondegenerate route-even ground state of $H_0$ under a route-odd perturbation $H\left(\lambda \right)=H_0+\lambda W_{-}$. Parity gives

$$E_{*}^{\prime }\left(0\right)=\langle 0,+|W_{-}|0,+\rangle =0,$$

(A241)

so the pole-tracked linear response is exactly the fixed-$E_0$ derivative used in Equation (249):

$${\left.{\displaystyle \frac{d{K}_{\perp }(E_{*}\left(\lambda \right),\lambda )}{d\lambda }}\right|}_0={\left.{\partial }_{\lambda }{K}_{\perp }(E,\lambda )\right|}_{E=E_0,\lambda =0}.$$

(A242)

At finite $\lambda$, however, the resolvent identity

$$R(E_0,\lambda )-R(E_{*},\lambda )=(E_0-E_{*})R(E_0,\lambda )R(E_{*},\lambda ),R(E,\lambda )={[H_{HH}\left(\lambda \right)-E]}^{-1},$$

(A243)

shows that a frozen-energy Schur complement becomes ill-conditioned when the $O\left({\lambda }^2\right)$ pole displacement is comparable to the smallest hidden detuning. A finite route-breaking kernel must therefore be quoted at a controlled matching energy that follows the physical pole, or through an equivalent pole-amputated quantity.

The $N=15$ mode decomposition changes qualitatively once a second link is opened. For the connector response define the cancellation factor $\mathcal{C}=|{\sum }_i{c}_i|/{\sum }_i\left|c_i\right|$, where $c_i$ are the mode-resolved linear contributions. At one link the response is $98.6461\%$ dominated by a single mode and $\mathcal{C}=0.9729$. At two links the dominant absolute fraction falls to $0.5188$ and $\mathcal{C}=0.0381$; at six links they are $0.5087$ and $0.0178$. Thus the one-link rank-one connector picture is not volume-stable: for $L\ge 2$ the small net linear response is a cancellation between two large near-threshold terms. The exchanged-pair response behaves differently: its dominant bridge fraction remains $0.9944$ at $L=2$ and $0.9723$ at $L=6$, with only weak cancellation.

The finite pole-tracking test is numerically decisive. For ${\kappa }_E=1$, $t=s=0.1$, $N=15$, and $\delta t=0.001$, the ratio of the pole displacement to the smallest hidden detuning grows from $3.70\times {10}^{-3}$ at $L=1$ to $2.63$ at $L=6$. At $L=6$ the same finite deformation gives

$$\parallel K_{\perp }(E_0;\delta t)\parallel =0.05095195,\parallel K_{\perp }(E_{*}\left(\delta t\right);\delta t)\parallel =0.00113635.$$

(A244)

The pole-tracked value agrees with the linear prediction $0.00114295$ at the sub-percent level; the frozen-energy value is larger by a factor $44.8$ because it probes a shifted near-pole resolvent. This distinction is negligible in the one-link example but essential as the volume grows.

The long-chain control remains quantitatively small at the stated benchmark. Selected pole-tracked values are

L	${\Delta }_{-}$	${\eta }_{\delta t}$, $\delta t=0.001$	${\eta }_y$, $y=0.01$
1	7.37004525	$9.7383\times {10}^{-3}$	$1.8164\times {10}^{-5}$
2	7.37094276	$3.3187\times {10}^{-4}$	$4.4640\times {10}^{-4}$
6	7.36893679	$1.5421\times {10}^{-4}$	$4.5461\times {10}^{-4}$
10	7.36881065	$1.3698\times {10}^{-4}$	$4.5540\times {10}^{-4}$
25	7.36875506	$1.2864\times {10}^{-4}$	$4.5578\times {10}^{-4}$

Opening the chain from one to 25 links therefore reduces the pole-tracked normalized response to a one-percent hopping mismatch by a factor $75.7$. The y response grows relative to its exceptional one-link value but saturates near $4.56\times {10}^{-4}$.

The large-N scaling also corrects the one-link “super-suppression” interpretation. High-precision $L=2$ data at $t=s=0.1$ give $\left({\partial }_{\delta t}{K}_{+-}\right)/N=-0.1612986391$ and ${\partial }_y{K}_{+-}=0.3225908420$ at $N={10}^4$, numerically consistent with limiting constants $-5/31$ and $10/31$, respectively; the rational identification is not used as a theorem. Fixed-L scans for $L=2,6,10$ show ${\partial }_{\delta t}{K}_{\perp }=O\left(N\right)$ and ${\partial }_y{K}_{\perp }=O\left(1\right)$, so the corresponding normalized classes are ${\eta }_{\delta t}=O\left(1\right)$ and ${\eta }_y=O\left(N^{-1}\right)$. The $N^{-2}$ behavior of the one-link y response is therefore a one-link special case, not a generic multi-link prediction.

Multi-link spectral and matching refinement.The exact open-chain control retains a small pole-tracked route-breaking numerator at $N=15$, but the mechanism by which it is small changes with volume. A single-mode one-link response can become a cancellation of several near-threshold contributions, and an apparent finite mismatch can be strongly exaggerated if the Feshbach kernel is evaluated at a frozen energy near a shifted hidden pole. For a physical four-dimensional test one must therefore establish not only a small $K_{\perp }$ and a finite odd gap, but also stability under volume enlargement and a controlled matching prescription $E_{*}\left(\lambda \right)$. The one-link $\eta \sim N^{-2}$ hidden-pair scaling is not promoted beyond that truncation.

Appendix B.3. Why Ordinary Route Reflection Is Insufficient, and the Scope of the Single-Graph No-Go Result

Reflected microscopic Wilson networks have the natural operation

$$\Gamma \leftrightarrow \tau \Gamma ,{\tau }^2=1.$$

On the route doublet it is represented by

$$P_{\mathrm{rt}}={\sigma }_x.$$

However, ordinary Wilson recoupling in the published field content does not change the primitive label I. On the product space it therefore acts as

$$U_{\mathrm{rec}}=P_{\mathrm{rt}}\otimes I_4.$$

A literal exchange of the complete primitive structure would instead require

$$U_{\mathrm{tw}}=P_{\mathrm{rt}}\otimes S_B.$$

Since

$$spec\left(I_4\right)=(+,+,+,+),spec\left(S_B\right)=(+,+,+,-),$$

these representations are inequivalent. Their traces and determinants differ. Hence an ordinary basis change, Grassmann reordering, an $\mathrm{SU}\left(15\right)$ center phase, or ordinary Pati–Salam transport cannot manufacture the full $S_B$ holonomy.

Local no-go theorem: scope.Full $4\times 4$ holonomy. For the published multiplicity-one Wilson transport, the primitive-fiber action is $U_{\tau }=I_4$. A literal microscopic holonomy $S_B$ with spectrum $(+,+,+,-)$ therefore cannot be derived within that transport. This no-go statement remains correct.

What survives of the rank-one idea.A primitive odd line exists, but the strict complete-source exchange is excluded in the closure-preserving class. The mass-plane channel is rank one in the primitive image space, so a full four-dimensional $S_B$ holonomy would be stronger than necessary. The six-$K_1$ quotient carries an exact odd representation under its internal pairing exchange, while the explicit walled–Brauer words separately realize the required $B_{12}$ and $B_{21}$ kernels. The later all-length singlet-support and rank obstruction shows that no closure-preserving chronological reversal or gauge-compatible common-closure involution can exchange those complete kernels under the stated assumptions. Thus the rank-one observation remains useful as a decomposition of the physical response, but it is not a microscopic route-${\mathbb{Z}}_2$ theorem; the surviving question is whether the renormalized confining dynamics suppresses the corresponding route-odd response.

Appendix B.3.1. Basis-Invariant Metric Criterion and the Anti-Tautology Test

The existence of a linear exchange in an operator formula is weaker than the existence of a physical isometry of the renormalized source space. Under a nonsingular real basis change ${\mathcal{O}}^{\prime }=T\mathcal{O}$, the two-point metric and coefficient covector transform as

$$Z^{\prime }=TZ{T}^T,C^{\prime }=T^{-T}C.$$

Were an exact source-space involution proposed as a protection mechanism, it would have to satisfy simultaneously

$$UZ{U}^T=Z,U^2=I,U^T{c}_{\Pi }=c_{\Pi },c_{\Pi }={(1,1)}^T.$$

These conditions are basis covariant, but after the closure-preserving microscopic no-go they are retained only as an anti-tautology test for any genuinely new symmetry completion, not as a gate that the present four-dimensional mechanism still assumes. They expose a useful theorem: for every positive-definite Z and nonzero covector c, the matrix

$$U_c={\displaystyle \frac{2Zc{c}^T}{c^{T}Zc}}-I$$

obeys $U_{c}Z{U}_c^T=Z$, $U_c^2=I$, and $U_c^{T}c=c$. Hence a reflection constructed only after the desired plane has been chosen cannot explain that plane; its action must be derived independently from the microscopic dynamics.

A concrete $2\times 2$ example makes the logical point transparent. Take $Z=I_2$ and choose the desired fixed covector only after looking at the target, $c={(1,2)}^T$. The formula manufactures

$$U_c={\displaystyle \frac{2}{5}}\left(\begin{array}{cc}1& 2\\ 2& 4\end{array}\right)-I_2=\left(\begin{array}{cc}-3/5& 4/5\\ 4/5& 3/5\end{array}\right).$$

Direct calculation gives $U_c^T{U}_c=I_2$, $U_c^2=I_2$, and $U_c^{T}c=c$. Nothing dynamical has been learned: a different post-selected c would manufacture a different reflection. The non-tautological question is whether the same U follows from the source transformation law before $c_{\Pi }$ is supplied.

For illustration, the component-counting Gram proxy

$$Z_{\mathrm{PS}}=\left(\begin{array}{cc}12& 9\\ 9& 14\end{array}\right)$$

is not invariant under the literal route swap ${\sigma }_x$. The metric-corrected exchange

$$U_{\mathrm{rt}}=\left(\begin{array}{cc}0& \sqrt{6/7}\\ \sqrt{7/6}& 0\end{array}\right)$$

is an exact isometry of this proxy. The coefficient $\sqrt{6/7}$ is fixed, rather than guessed. Indeed, writing

$$U\left(a\right)=\left(\begin{array}{cc}0& a\\ a^{-1}& 0\end{array}\right),$$

direct multiplication gives

$$U\left(a\right)Z_{\mathrm{PS}}U{\left(a\right)}^T=\left(\begin{array}{cc}14a^2& 9\\ 9& 12a^{-2}\end{array}\right).$$

Therefore $U\left(a\right)Z_{\mathrm{PS}}U{\left(a\right)}^T=Z_{\mathrm{PS}}$ requires

$$14a^2=12,12a^{-2}=14,a=\sqrt{{\displaystyle \frac{6}{7}}},$$

where the positive root is chosen for the exchange continuously connected to positive route normalizations. If $C={(C_{12},C_{21})}^T$, the fixed-covector equation $U{\left(a\right)}^{T}C=C$ then gives

$${\displaystyle \frac{C_{21}}{C_{12}}}=a=\sqrt{{\displaystyle \frac{6}{7}}},$$

rather than unity. Conversely, an isometry that fixes $c_{\Pi }$ can always be manufactured from the theorem above. Neither outcome is a microscopic derivation. Moreover, $Z_{\mathrm{PS}}$ counts spectator components; it is not the renormalized Bethe–Salpeter or Euclidean source metric. The physical criterion is therefore to compute Z and the ultraviolet action independently and then test the three displayed equations without retuning the protected covector.

Appendix B.4. First Main Result: A Constructive Gauge-Invariant Point-Split 24-Preon Sign Line

Appendix B.4.1. Point-Split Sources

Denote by

$${\mathcal{Q}}_I[D;\Xi ,\Gamma ]$$

a fully gauge-singlet point-split source. The index $I\in \{0,R,L,X\}$ chooses the primitive tensor; D is the family insertion; $\Xi$ contains continuous spacetime coordinates, Wilson-line lengths, Lorentz routing, and smearing data; and $\Gamma$ specifies the discrete connectivity of the Wilson network. Schematically,

$${\mathcal{Q}}_I[D;\Xi ,\Gamma ]=D^i{}_j{ϵ}^{\rho \sigma }{{\left(P_{\overline{4}}^j\right)}_a}^{\dot{r}}{}_{\rho }\left(B_I\right)^a{\dot{r}}^b{{}_{\dot{s}}\mathcal{X}_b^i}^{\dot{s}}{}_{\sigma }[\Xi ,\Gamma ].$$

In the full nested expansion this source contains eight three-preon junctions, i.e., 24 preon-field insertions. The term “24-preon” refers to the operator content and is not a proven count of constituents in an isolated physical pole.

Here $\mathcal{X}[\Xi ,\Gamma ]$ abbreviates the remaining gauge-singlet nested junctions and their Wilson transport. It is important to state exactly what is and is not fixed: the paper defines a family of point-split sources indexed by the graph data $(\Xi ,\Gamma )$, not one privileged lattice discretization. A concrete calculation must instantiate the following complete record:

$${\mathfrak{D}}_{\Gamma }=\left(V_{\Gamma },E_{\Gamma },o_{\Gamma };{\{x_v,{\mathcal{J}}_v\}}_{v\in V_{\Gamma }};{\{R_e,{\gamma }_e\}}_{e\in E_{\Gamma }};{\mathcal{C}}_{\mathrm{PS}},{\mathcal{C}}_{\mathrm{spin}},\mathcal{S}\right).$$

Here $V_{\Gamma }=\{v_1,\dots ,v_8\}$ labels the eight local three-preon junctions. In the explicit six-K source skeleton the outer vertices are fixed as ${\mathcal{J}}_1=P_{\overline{4}}^j$ and ${\mathcal{J}}_8=P_4^i$, while ${\mathcal{J}}_2,\dots ,{\mathcal{J}}_7$ are the mixed currents $K=\Omega {\Psi }_W{\Psi }_{W^{\prime }}\sim (1,2,2)$ of Appendix B.1.2. Because separated K currents carry local $\mathrm{SU}{\left(2\right)}_L\times \mathrm{SU}{\left(2\right)}_R$ indices, their pairings are dressed by fundamental spectator Wilson lines. Every local $\Omega FF$ current is already an $\mathrm{SU}{\left(15\right)}_p$ singlet, so no inter-junction precolor Wilson line is required for gauge invariance of this representative. The symbol $o_{\Gamma }$ is the canonical ordering of the 24 Grassmann insertions, and ${\mathcal{C}}_{\mathrm{PS}}$, ${\mathcal{C}}_{\mathrm{spin}}$, and $\mathcal{S}$ store the spectator contractions, spinor contractions, and smearing data.

The explicit six-K source record contains a genuine internal $K_1$-space involution ${\tau }_K=\left(24\right)\left(57\right)$ and quotient-compatible Temperley–Lieb generators whose two ordered products are exchanged by ${\tau }_K$. No full microscopic source involution exchanging ${\mathcal{O}}_{12}^{\left(24\right)}$ and ${\mathcal{O}}_{21}^{\left(24\right)}$ is claimed, because the singlet-$K_1$ route is $\mathrm{SU}{\left(4\right)}_{PS}$ blind. A successful full lift must use the unprojected $1\oplus 15$ current space, minimally including $K_{15}$, and derive the local channel exchange rather than append it as a formal record swap.

For clarity, four distinct statements are separated. First, every fully specified point-split contraction record defines an ordinary gauge-invariant composite source. Second, the fixed-$K_1$ subspace contains an independently defined quotient-compatible order-exchange structure and hence nonzero even/odd combinations within that internal subspace. Third, the primitive crossed coefficient space has an exact algebraic odd line under label exchange $12\leftrightarrow 21$. Fourth, no microscopic operation has yet been shown to identify these two exchanges as one physical source symmetry. Renormalized non-nullness and mixing are additional dynamical questions. Thus “constructive 24-field skeleton” and “order-sensitive K-space witness” are the established terms; “microscopic route-${\mathbb{Z}}_2$” is reserved for a future calculation that closes the missing map.

Conditionally, if a future full contraction pair $\Gamma ,\tau \Gamma$ is obtained with $\tau \Gamma \ne \Gamma$ and ${\tau }^2\Gamma =\Gamma$, one may define

$${\mathcal{Q}}_{I,\pm }={\displaystyle \frac{{\mathcal{Q}}_I[D;\Xi ,\Gamma ]\pm {\eta }_{\Gamma }{\mathcal{Q}}_I[D;\tau \Xi ,\tau \Gamma ]}{\sqrt{2}}},{\eta }_{\Gamma }=\pm 1.$$

The sign ${\eta }_{\Gamma }$ fixes the Grassmann-ordering convention; it is not a new coupling constant. By construction,

$$\tau {\mathcal{Q}}_{I,+}=+{\mathcal{Q}}_{I,+},\tau {\mathcal{Q}}_{I,-}=-{\mathcal{Q}}_{I,-}.$$

Appendix B.4.2. Projection onto the Unique Primitive Odd Covector

Define

$${\mathcal{L}}_{-}[D;\Xi ,\Gamma ]=\sqrt{{\displaystyle \frac{3}{2}}}\sum _I{\ell }_{-}^I\left[{\mathcal{Q}}_I[D;\Xi ,\Gamma ]-{\eta }_{\Gamma }{\mathcal{Q}}_I[D;\tau \Xi ,\tau \Gamma ]\right].$$

The normalization is conventional; the physical residue is determined by the correlator. The primitive odd right-vector line and odd coefficient-covector line are dual but must not be identified without an independently specified metric. In route space,

$$P_{\mathrm{orb}}^{-}={\displaystyle \frac{I_2-{\sigma }_x}{2}}.$$

On primitive vectors,

$$P_{B,\mathrm{vec}}^{-}={\displaystyle \frac{I_4-S_B}{2}},im{P}_{B,\mathrm{vec}}^{-}=span\left\{v_{-}\right\},$$

whereas on coefficient covectors,

$$P_{B,\mathrm{cov}}^{-}={\displaystyle \frac{I_4-S_B^T}{2}},im{P}_{B,\mathrm{cov}}^{-}=span\left\{{\ell }_{-}\right\}.$$

Both primitive projectors have rank one. Since ${\mathcal{L}}_{-}$ is selected by the coefficient covector ${\ell }_{-}$, its coefficient-space projector is

$$P_{{\mathcal{L}}_{-}}^{\mathrm{coeff}}=P_{\mathrm{orb}}^{-}\otimes P_{B,\mathrm{cov}}^{-},rank{P}_{{\mathcal{L}}_{-}}^{\mathrm{coeff}}=1.$$

The right-vector projector $P_{B,\mathrm{vec}}^{-}$ is used separately when discussing the vector representative of the primitive sign line.

For the normalized route vector

$$e_{-}={(1,-1)}^T/\sqrt{2}$$

and primitive vector $v_{-}$, define ${\chi }_{-}=e_{-}\otimes v_{-}$. The primitive crossed-exchange algebra has an exact one-dimensional odd line, and the formal operator difference

$${\mathcal{O}}_{-}^{\left(24\right)}={\displaystyle \frac{{\mathcal{O}}_{21}^{\left(24\right)}-{\mathcal{O}}_{12}^{\left(24\right)}}{\sqrt{2}}}$$

is algebraically odd under exchange of the two coefficient labels. What the quotient-compatible six-K analysis does not claim is that this label exchange has already been derived as a symmetry or involution of one microscopic 24-field contraction. The quotient-compatible $K_1$ subspace supplies a noncommuting order witness, while the minimal $K_{15}$ lift identifies the missing $\mathrm{SU}{\left(4\right)}_{PS}$ carrier; the coherent channel amplitude tying the two structures remains open.

Proved at the stated level.The eight-P no-go, the six explicit mixed K junctions, their spectator Wilson dressing, the 24-field count, the exact trace/Fierz factorization of the crossed tensors, the trace-character theorem, the local $G_{\mathrm{PS}}$-equivariant exchange no-go, and the noncommuting six-K contraction-space witness are established. A full microscopic involution that ties the K order to $B_{12}\leftrightarrow B_{21}$ is not established; nor are the renormalized source metric, operator mixing, physical correlator, pole, or matching vertex.

Appendix B.4.3. The Orbital-Even B - Companion Is a Separate Spectral Obstruction

The explicit rank-one odd line does not by itself make the low-energy $B_{-}$ sector one dimensional. The same factorized route–spectator Hilbert space also contains

$$P_{\mathrm{orb}}^{+}\otimes P_B^{-},$$

which is a second rank-one line: it is orbital even but carries the same primitive $B_{-}$ spectator label. The desired source line is

$$P_{\mathrm{orb}}^{-}\otimes P_B^{-}.$$

Thus the operator construction proves that the nontrivial odd $B_{-}$ sign line exists with no new fundamental gauge content, but it does not prove that this line is the unique physical low-energy realization of $B_{-}$. A route-symmetric kernel of the form $K=aI+b{\sigma }_x$ fixes parity but does not force the odd eigenvalue to zero or below the even companion.

The remaining requirement is dynamical: the physical Bethe–Salpeter/transfer kernel must either lift the orbital-even $B_{-}$ companion above the relevant low-energy window or give it zero overlap with the renormalized local matching source. This companion problem is one reason why the route-resolved GEVP and the physical sign-inheritance test cannot be replaced by source algebra alone.

Appendix B.4.4. Node at a Reflection-Fixed Configuration

Introduce a signed local relative coordinate y such that

$$\Gamma (-y)=\tau \Gamma \left(y\right).$$

Then

$${\mathcal{L}}_{-}(-y)=-{\mathcal{L}}_{-}\left(y\right),$$

so at a fixed configuration

$${\mathcal{L}}_{-}\left(0\right)=0.$$

For a smooth source,

$${\mathcal{L}}_{-}\left(y\right)=y{\partial }_y{\mathcal{L}}_{-}\left(0\right)+O\left(y^3\right).$$

This is an operator node. It does not imply zero mass, zero energy, or zero overlap with physical states. On the contrary, the odd normal derivative ${\partial }_y{\mathcal{L}}_{-}\left(0\right)$ is route-even and later becomes the natural local matching datum.

Appendix B.4.5. What the Existence Theorem Does Not Prove

The operator Hilbert space may contain several physical states with the same global quantum numbers. Therefore the exact sign source does not by itself determine

• whether the lowest spectral support contains an isolated pole;
• whether that pole lies below the multiparticle threshold;
• the absolute mass $E_{-}$;
• the absolute pole residue $|z_{-}{|}^2$;
• the absolute matching coefficient to ${\mathcal{O}}_{+}$.

These are dynamical observables.

Appendix B.5. Supporting Result: The Natural Fierz 2-Complex of Routes

Appendix B.5.1. Why the One-Skeleton Graph Is Not Enough

The discrete routing skeleton considered in this supporting combinatorial calculation acts on six undaggered $\Omega FF$ junctions, hence on twelve labeled fundamental precolor lines. A graph vertex is a perfect matching of these twelve labels, i.e., a partition into six unordered pairs. The number of vertices is

$$N_0=11!!=10395.$$

An elementary Fierz flip selects two of the six matched pairs and reparents their four endpoints in either of the two alternative ways. The degree of every vertex is

$$d_{\mathrm{flip}}=2\left({\displaystyle \genfrac{}{}{0pt}{}{6}{2}}\right)=30,$$

so the number of undirected edges is

$$N_1={\displaystyle \frac{N_0{d}_{\mathrm{flip}}}{2}}=155925.$$

The one-skeleton contains many cycles. A graph cycle, however, is not automatically a physical topological cycle: when two sequences of local Fierz identities are algebraically equivalent, the corresponding loop is filled by a natural 2-cell.

Scope of the reduced complex.The finite complex $X_6$ below is a reduced routing subcomplex for six undaggered junctions/twelve fundamental lines. The full point-split source of Appendix B.4 contains eight three-preon junctions. Simple connectedness of $X_6$ is therefore a theorem about this reduced Fierz sector, not a theorem that the complete eight-junction Wilson-network configuration space is simply connected. No step in the exact 4D identity ${\Delta }_{\Pi }=C_{21}-C_{12}$ or in the construction of the route-odd source requires that stronger statement. Extending the combinatorial reduction to all junctions of a chosen full source is an additional bookkeeping problem.

Appendix B.5.2. Natural Two-Dimensional Cells and Exact Counting

Let $X_6$ be the two-dimensional cellular complex whose vertices are all perfect matchings of the twelve labeled lines, whose one-cells are the elementary two-pair Fierz reconnections, and whose two-cells encode the local relations among such reconnections.

Appendix B.5.5.1. Fierz triangles.

For any two matched pairs there are exactly three pairings of their four endpoints. These three matchings form a triangle and are filled by one 2-cell. At each matching there are $\left({\displaystyle \genfrac{}{}{0pt}{}{6}{2}}\right)=15$ such local supports, and each triangle has three vertices. Hence

$$N_{▵}={\displaystyle \frac{N_0\left(\genfrac{}{}{0pt}{}{6}{2}\right)}{3}}=10395\times 5=51975.$$

Appendix B.5.5.2. Critical diamonds.

Consider an unordered pair of distinct elementary flips leaving one matching. If their two-pair supports are disjoint, choose four of the six matched pairs, partition them into two unordered supports in three ways, and choose one of the two reconnections for each support. This gives

$$\left({\displaystyle \genfrac{}{}{0pt}{}{6}{4}}\right)32^2=180$$

disjoint-support flip pairs through each vertex, each with one opposite endpoint and hence one square incidence. If the supports overlap in exactly one matched pair, choose the shared pair, choose the other two pairs, and choose one of two reconnections for each flip:

$$6\left({\displaystyle \genfrac{}{}{0pt}{}{5}{2}}\right)2^2=240.$$

For each such overlapping pair there are two admissible opposite endpoint matchings, hence two square incidences. Therefore the number of square incidences at each vertex is

$$180+2\left(240\right)=660.$$

Each simple square has four vertices, so

$$N_{\square }={\displaystyle \frac{N_{0}660}{4}}=1715175.$$

These counts have also been independently reproduced by exhaustive enumeration of all 10395 perfect matchings and all elementary flips.

Appendix B.5.3. Proof of Canonical Contraction

Fix a reference matching $M_{★}$ and define the integer disorder

$$d\left(M\right)=6-|M\cap M_{★}|.$$

If $M\ne M_{★}$, choose the first reference pair $\{a,b\}\in M_{★}$ absent from M. In M, the labels a and b belong to two pairs $\{a,c\}$ and $\{b,d\}$. The canonical Fierz move replaces these by $\{a,b\}$ and $\{c,d\}$. It restores one reference pair, destroys no already present reference pair, and therefore strictly decreases $d\left(M\right)$. The reduction terminates after finitely many steps at the unique normal form $M_{★}$.

It remains to show that competing first moves are locally confluent modulo the filled 2-cells. Compare any elementary flip with the canonical first step. Their supports have exactly three possibilities:

1.

the same two matched pairs, in which case the two alternatives are related by the filled Fierz triangle;

2.

disjoint supports, in which case the moves commute and bound a filled square;

3.

one common matched pair, in which case the two reduction orders are related by one of the filled overlapping diamonds.

Thus every critical pair of one-step reductions is joined by the stated local 2-cells. Because the disorder provides a terminating reduction, local confluence propagates inductively to confluence: any two reduction paths from a matching to $M_{★}$ are homotopic relative to their endpoints through the filled triangles and diamonds. Given an arbitrary closed edge path, reduce every vertex path to the common normal form and use these homotopies edge by edge. The loop becomes a canonical reduction path followed by its reverse and therefore contracts. Hence

$${\pi }_1\left(X_6\right)=0.$$

Abelianization gives

$$H_1(X_6;\mathbb{Z})=0,H^1(X_6;\mathbb{Z})=0.$$

Proved for the reduced six-junction Fierz complex.The large cycle degeneracy of the $X_6$ one-skeleton is not fundamental topology. After the stated local triangle and diamond 2-cells are included, $X_6$ is simply connected and its ordinary first homology vanishes. The statement is deliberately not extrapolated to the entire eight-junction point-split Wilson-network space without additional bookkeeping.

Appendix B.5.4. Why a Z 2 Structure Can Nevertheless Remain

The global route reflection $\tau$ acts on the reduced complex and may possess fixed configurations. For a simply connected cover, the ordinary topological quotient can lose isotropy information, while the action groupoid/orbifold retains the stabilizer in the standard groupoid formulation [15]. If the relevant physical configuration space is independently shown to be the quotient by this involution, then

$${\pi }_1^{\mathrm{orb}}\left([X_6/{\mathbb{Z}}_2]\right)\simeq {\mathbb{Z}}_2.$$

There are two real one-dimensional representations, the trivial and sign representations. The source ${\mathcal{L}}_{-}$ realizes the sign representation at the operator level; the finite-complex result by itself does not prove that the full continuum source space has this orbifold completion.

Assumption.Extending the finite-complex result to the continuum space of Wilson paths requires an explicit definition of which path collisions, intersections, junction moves, and smooth homotopies are allowed. Thus simple connectedness of finite $X_6$ is proved, whereas a literal continuum orbifold interpretation remains conditional.

Appendix B.6. Synthesis of the Operator and Topological Parts

The operator and reduced-topology statements now have a deliberately asymmetric status. The exact operator chain is

$$\begin{array}{c}\mathrm{published}\mathrm{SU}{\left(15\right)}_p\mathrm{fields}\\ \Downarrow \\ \mathrm{gauge}-\mathrm{invariant}\mathrm{point}-\mathrm{split}\mathrm{source}\mathrm{family}\mathrm{with}\mathrm{a}\mathrm{fixed}\mathrm{eight}-\mathrm{junction}\mathrm{representative}\\ \Downarrow \\ \mathrm{two}\mathrm{reflected}\mathrm{routes}\Gamma ,\tau \Gamma \\ \Downarrow \\ {\mathcal{L}}_{-}^{\mathrm{bare}}\neg \equiv 0\mathrm{in}\mathrm{the}\mathrm{displayed}\mathrm{local}\mathrm{operator}\mathrm{quotient},\tau {\mathcal{L}}_{-}=-{\mathcal{L}}_{-}\\ \Downarrow \\ \mathrm{rank}-\mathrm{one}\mathrm{microscopic}\mathrm{sign}\mathrm{representation}.\end{array}$$

Here “${\mathcal{L}}_{-}^{\mathrm{bare}}\neg \equiv 0$” refers only to the explicitly constructed operator class after the stated local algebraic quotient; it does not assert nonzero renormalized norm or pole residue. Those are correlator questions.

Independently, the reduced six-junction Fierz complex satisfies

$${\pi }_1\left(X_6\right)=0,$$

and therefore shows that ordinary one-skeleton cycles of that reduced routing sector are filled by local Fierz relations. If a physical continuum quotient by the route involution is separately established, an orbifold ${\mathbb{Z}}_2$ sign representation may survive. The finite $X_6$ theorem is consequently supporting topology, not a premise of the exact 4D mass-plane identity and not a proof of the topology of the full eight-junction source.

Remark.Topology does not “generate the mass”. It explains only why a discrete sign sector can naturally exist in the composite configuration space. The mass and the ordering of energy levels are determined by the Hamiltonian or, equivalently in a relativistic formulation, by the Bethe–Salpeter kernel. The next section is therefore conceptually distinct from the topological construction: it moves from exact operator geometry to dynamics.

Appendix B.7. Four-dimensional Dynamics: from a Local No-Go Theorem to a One-Dimensional Schur–Feshbach Reduction

Reader checkpoint: what changes at this section.Everything up to this point was operator algebra and source construction. From here on, a self-adjoint Hamiltonian/transfer completion is used to prove spectral compatibility and positivity statements. The physical renormalized 1PI kernel remains a separate object until a common spectral matching is established; the final correlator test is therefore not bypassed by the completion theorem.

Appendix B.7.1. Why the Local Negative Conclusion Is Too Strong

In the simplest calculable approximation, the primitive kernel was built from two known group-response matrices, $G_4$ and $G_R$, together with the identity $I_4$. In the ordered four-dimensional primitive basis,

$$G_4=\left(\begin{array}{cccc}15/8& 0& 1/2& 0\\ 0& 15/8& 0& 1/2\\ 0& 0& -1/8& 0\\ 0& 0& 0& -1/8\end{array}\right),G_R=\left(\begin{array}{cccc}3/4& 1/2& 0& 0\\ 0& -1/4& 0& 0\\ 0& 0& 3/4& 1/2\\ 0& 0& 0& -1/4\end{array}\right).$$

Here $G_4$ encodes the factorized $\mathrm{SU}{\left(4\right)}_{\mathrm{PS}}$ contribution, $G_R$ the corresponding $\mathrm{SU}{\left(2\right)}_R$ contribution, and $I_4$ the identity in primitive space. Let a, b, and s be real coefficients of these structures. Then

$$K_{\mathrm{calc}}=a{G}_4+b{G}_R+s{I}_4.$$

Exact closure of the odd primitive direction would require

$$P_{B,\mathrm{cov}}^{+}{K}_{\mathrm{calc}}{\ell }_{-}=0,P_{B,\mathrm{cov}}^{+}={\displaystyle \frac{I_4+S_B^T}{2}}.$$

Direct multiplication gives

$$P_{B,\mathrm{cov}}^{+}{K}_{\mathrm{calc}}{\ell }_{-}=\left(\begin{array}{c}(a-b)/2\\ b/2\\ -a+3b/2\\ -2a-b\end{array}\right).$$

The only exact solution within $span\{G_4,G_R,I_4\}$ has $a=b=0$; the scalar identity term $s{I}_4$ does not mix parity. This is an exact local no-go theorem for that factorized subspace.

It does not follow that the full four-dimensional theory cannot close the $B_{-}$ line. Connected Wilson-line reconnections, composite poles, and Schur–Feshbach reduction can generate matrix directions absent from the factorized Casimir ansatz. This is the possibility studied below.

Local no-go theorem: scope.Factorized Casimir result. A nonzero combination of only the known matrices $G_4$, $G_R$, and $I_4$ cannot exactly close the primitive odd direction. The result excludes only the subspace $span\{G_4,G_R,I_4\}$.

How the obstruction is bypassed.Four-dimensional solution through a connected hidden sector. Wilson-line reconnection or a hidden composite sector adds a positive-semidefinite susceptibility $M\left(E\right)$, i.e., a negative-semidefinite correction $-M\left(E\right)$ to the visible two-route kernel. That matrix direction is absent from the factorized Casimir subspace. Exact route reciprocity in a real basis requires only the single sum rule $M_{11}-M_{22}=D_{\mathrm{rt}}$, and the globally minimal correction has a natural size of order $1/15$. Independently, a rank-one solution exists for the missing compensation vector in the full primitive space. Together these results remove the matrix-space and coefficient-size obstruction. They do not yet prove that the same physical hidden state of the full confining theory simultaneously realizes the route, primitive, and pole conditions.

Appendix B.7.2. Projected Two-Route Kernel and the Exact Content of Route Symmetry

Appendix B.7.7.1. Provenance and status of the numerical visible block

The route maps $r_{ij}$, the change-of-basis matrix R, and the identity ${\Delta }_{\Pi }=C_{21}-C_{12}$ above are exact consequences of the fixed operator normalization. By contrast, the following matrix is the canonical visible-block benchmark adopted for the reduced two-route analysis:

$${\displaystyle \frac{K_0^{\mathrm{bench}}}{h}}=\left(\begin{array}{cc}{A}_{\mathrm{rt}}& C_{\mathrm{rt}}\\ C_{\mathrm{rt}}& B_{\mathrm{rt}}\end{array}\right)=\left(\begin{array}{cc}0.297066666666667& 0.011618237144396\\ 0.011618237144396& 0.247660606060606\end{array}\right).$$

It encodes the adopted canonical projection of the factorized visible operator envelope onto the two route states. Its displayed entries fix a reproducible benchmark convention but are not derived in this paper from a renormalized 24-field correlator. Accordingly, no physical conclusion rests on treating the last displayed digits as measured precision. The general results below are written for arbitrary Hermitian entries $A_{\mathrm{rt}}$, $B_{\mathrm{rt}}$, and $C_{\mathrm{rt}}$; the benchmark is used only to quantify the size of one representative compensation.

The distinction can be made exact. Let ${\Gamma }_{\mathrm{prim}}^{\left(2\right)}(E;\mu )$ denote the renormalized one-particle-irreducible quadratic kernel in the primitive source space, and let ${\mathsf{G}}_{\mathrm{prim}}\left(\mu \right)\succ 0$ be the corresponding positive source metric in the same scheme. Let

$$V_{\mathrm{rt}}=\left(v_{12}{v}_{21}\right):{\mathbb{C}}^2⟶{\mathbb{C}}^4$$

embed the two crossed route coefficient vectors into the primitive four-dimensional space; the columns are fixed by the 12 and 21 rows of the displayed route-to-primitive matrix R. Define

$$Z_{\mathrm{rt}}=V_{\mathrm{rt}}^{†}{\mathsf{G}}_{\mathrm{prim}}{V}_{\mathrm{rt}},{\widehat{V}}_{\mathrm{rt}}=V_{\mathrm{rt}}{Z}_{\mathrm{rt}}^{-1/2}.$$

Whenever $Z_{\mathrm{rt}}$ is positive definite, the canonically normalized physical visible block is

$$K_0^{\mathrm{ren}}(E;\mu )={\widehat{V}}_{\mathrm{rt}}^{†}{\Gamma }_{\mathrm{prim}}^{\left(2\right)}(E;\mu ){\widehat{V}}_{\mathrm{rt}}=Z_{\mathrm{rt}}^{-1/2}{V}_{\mathrm{rt}}^{†}{\Gamma }_{\mathrm{prim}}^{\left(2\right)}(E;\mu )V_{\mathrm{rt}}{Z}_{\mathrm{rt}}^{-1/2}.$$

This equation is the reproducible derivation contract for $K_0$: the primitive kernel, source metric, route embedding, renormalization scale, and energy must all come from one calculation. Component-counting matrices such as $G_4$, $G_R$, or $Z_{\mathrm{PS}}$ can help define algebraic benchmark directions, but they do not determine ${\Gamma }_{\mathrm{prim}}^{\left(2\right)}$ and therefore cannot by themselves produce a physical $K_0^{\mathrm{ren}}$. The numerical matrix below has not been obtained through this full contract. Its honest status remains a benchmark, whereas the projection formula itself is exact.

Here $h>0$ is the common positive normalization scale. The subscript “rt” emphasizes that these are entries of the route kernel and must not be confused with the fermion-curvature coordinates $A_s$. For the benchmark, the diagonal difference is

$$D_{\mathrm{rt}}\equiv A_{\mathrm{rt}}-B_{\mathrm{rt}}=0.049406060606061,$$

and the Pauli decomposition is

$$\begin{array}{cc}\hfill k_0& ={\displaystyle \frac{A_{\mathrm{rt}}+B_{\mathrm{rt}}}{2}}=0.272363636363636,\hfill \\ \hfill k_x& =C_{\mathrm{rt}}=0.011618237144396,\hfill \\ \hfill k_z& ={\displaystyle \frac{D_{\mathrm{rt}}}{2}}=0.0247030303030305.\hfill \end{array}$$

0.94

Verification details. The displayed diagonal entries possess the exact rational representatives

$$A_{\mathrm{rt}}={\displaystyle \frac{557}{1875}},B_{\mathrm{rt}}={\displaystyle \frac{5108}{20625}},D_{\mathrm{rt}}={\displaystyle \frac{1019}{20625}},$$

within the adopted benchmark convention. The off-diagonal entry is retained as the benchmark decimal $C_{\mathrm{rt}}=0.011618237144396$ because no unique simpler exact form has been established from the source construction. These representations reproduce the printed matrix; they do not elevate it to a physical $\mathrm{SU}{\left(15\right)}_p$ calculation. A true derivation must provide ${\Gamma }_{\mathrm{prim}}^{\left(2\right)}$, ${\mathsf{G}}_{\mathrm{prim}}$, $V_{\mathrm{rt}}$, E, and $\mu$ in the projection contract above.

The extra digits in this display are retained only so that the later algebraic and numerical checks can be reproduced without round-off ambiguity. At the level of physical interpretation the benchmark compensation is quoted as $D_{\mathrm{rt}}\simeq 0.0494$. Replacing $K_0^{\mathrm{bench}}$ by a future renormalized visible block simply replaces $D_{\mathrm{rt}}$ and $C_{\mathrm{rt}}$ in the formulas; it does not change the one-scalar-condition proposition, the positivity problem, or the finite-data criteria.

Several dimensionless diagnostics make the numerical status of this benchmark more transparent:

$$\begin{array}{cccc}\hfill {\lambda }_{min}& =0.245064860924,\hfill & \hfill {\lambda }_{max}& =0.299662411803,\hfill \\ \hfill det(K_0^{\mathrm{bench}}/h)& =0.073436727273,\hfill & \hfill {\kappa }_2(K_0^{\mathrm{bench}}/h)& =1.22278817,\hfill \\ \hfill {\displaystyle \frac{A_{\mathrm{rt}}-B_{\mathrm{rt}}}{A_{\mathrm{rt}}+B_{\mathrm{rt}}}}& =0.09069871,\hfill & \hfill {\displaystyle \frac{2C_{\mathrm{rt}}}{A_{\mathrm{rt}}+B_{\mathrm{rt}}}}& =0.04265708.\hfill \end{array}$$

Thus the benchmark matrix is well conditioned, while its diagonal mismatch is a $9.07\%$ contrast relative to the diagonal sum. For example, under the purely algebraic antisymmetric deformation $A_{\mathrm{rt}}\mapsto A_{\mathrm{rt}}(1+\epsilon )$, $B_{\mathrm{rt}}\mapsto B_{\mathrm{rt}}(1-\epsilon )$, the sign of the mismatch changes at $\epsilon =-D_{\mathrm{rt}}/(A_{\mathrm{rt}}+B_{\mathrm{rt}})=-0.09069871$. This is a fixed-basis sensitivity scale, not a statistical uncertainty and not a model for the physical renormalized kernel.

Route reciprocity in the projected response must be tested with the canonical route-label exchange fixed before the kernel is inspected, rather than with an involution constructed from the kernel one wishes to symmetrize. Let $U={\sigma }_x$ after canonicalization of the source metric. This is a diagnostic exchange on route labels, not a claim of the strict microscopic chronological symmetry excluded in Appendix B.1.2. A relative phase redefinition of the two route vectors then makes the off-diagonal element real at one fixed energy whenever that phase is not already fixed by the microscopic source convention. In this canonical basis,

$$K=\left(\begin{array}{cc}a& c\\ c& b\end{array}\right),c\in \mathbb{R},$$

and direct multiplication gives

$${\sigma }_{x}K{\sigma }_x=\left(\begin{array}{cc}b& c\\ c& a\end{array}\right).$$

Hence the physical invariance condition $UK{U}^{†}=K$ reduces to the single real equation

$$a=b.$$

The off-diagonal coefficient c is unconstrained by the exchange and need not vanish. In another basis the matrix entries themselves change; the invariant statement is still $UK{U}^{†}=K$ with U transformed together with the basis. This is precisely why the anti-tautology test of Appendix B.3.1 is required: an exchange chosen only after inspecting K or the desired plane has no microscopic explanatory content.

Proposition: one-scalar condition for route locking.For a real symmetric two-route kernel,

$$[K,{\sigma }_x]=0⟺K_{11}=K_{22}.$$

After exact locking the most general kernel is

$$K_{\mathrm{sym}}=d{I}_2+f{\sigma }_x$$

with arbitrary real f. The additional condition $f=0$ defines only a special degeneracy surface and is not part of route symmetry itself.

Appendix B.7.3. General Hidden-Sector Susceptibility from Schur–Feshbach Reduction

This subsection is an exact theorem about a self-adjoint Hamiltonian/transfer-matrix completion. It must not be confused with the separately defined renormalized 1PI projection $K_0^{\mathrm{ren}}(E;\mu )$.

Let $\mathcal{H}$ be the full self-adjoint operator acting on the two visible route states together with all composite states having the same conserved quantum numbers. Let P be the orthogonal projector onto the two-dimensional visible route subspace, $Q=I_{\mathrm{full}}-P$ the projector onto its orthogonal complement, and $I_{\mathrm{full}}$ the identity on the full Hilbert space. Define

$$K_0^{\mathcal{H}}=P\mathcal{H}P,{\mathcal{H}}_H=Q\mathcal{H}Q,\mathcal{V}=P\mathcal{H}Q.$$

Exact elimination of the hidden sector does not require a one-pole approximation. For an energy E below the hidden spectral threshold,

$$K_{\mathrm{eff}}^{\mathcal{H}}\left(E\right)=K_0^{\mathcal{H}}-\mathcal{V}{({\mathcal{H}}_H-E)}^{-1}{\mathcal{V}}^{†}.$$

Dividing by the positive normalization h, define

$$M\left(E\right)\equiv {\displaystyle \frac{1}{h}}\mathcal{V}{({\mathcal{H}}_H-E)}^{-1}{\mathcal{V}}^{†}⪰0,{\displaystyle \frac{K_{\mathrm{eff}}^{\mathcal{H}}\left(E\right)}{h}}={\displaystyle \frac{K_0^{\mathcal{H}}}{h}}-M\left(E\right).$$

Positivity follows because ${({\mathcal{H}}_H-E)}^{-1}$ is positive below the hidden threshold. Explicitly, for every $x\in {\mathbb{C}}^2$,

$$x^{†}M\left(E\right)x={\displaystyle \frac{1}{h}}{∥{({\mathcal{H}}_H-E)}^{-1/2}{\mathcal{V}}^{†}x∥}^2\ge 0.$$

Thus positive semidefiniteness is not an ansatz inside this Hamiltonian/resolvent theorem. What is not proved by this identity is that an arbitrary energy-dependent renormalized 1PI inverse kernel can be decomposed with the same positive matrix $M\left(E\right)$. Establishing that bridge requires a common spectral/transfer representation and canonical source metric. Until then, the theorem proves existence and spectral compatibility of positive completions, while the physical route-resolved correlator remains the decisive dynamical object. If the hidden spectrum is discrete, denote the hidden energies by $E_a$ and their two-component visible-overlap vectors by $v_a$. Then

$$M\left(E\right)=\sum _a{\displaystyle \frac{v_a{v}_a^{†}}{h(E_a-E)}}.$$

For a continuous component introduce a positive matrix-valued spectral measure $d{\Sigma }_H\left(s\right)⪰0$ with lower support edge $s_{\mathrm{th}}$:

$$M\left(E\right)={\int }_{s_{\mathrm{th}}}^{\infty }{\displaystyle \frac{d{\Sigma }_H\left(s\right)}{h(s-E)}}.$$

Thus the hidden contribution is constrained by a positive spectral measure rather than being an arbitrary fitted matrix.

Figure A2. Schur–Feshbach reduction. Positivity of the hidden resolvent below threshold forces a positive-semidefinite susceptibility. The route-locking problem is therefore a constrained spectral problem, not an arbitrary adjustment of a $2\times 2$ matrix.

Figure A2. Schur–Feshbach reduction. Positivity of the hidden resolvent below threshold forces a positive-semidefinite susceptibility. The route-locking problem is therefore a constrained spectral problem, not an arbitrary adjustment of a $2\times 2$ matrix.

The difference between the effective diagonals is

$${(K_{\mathrm{eff}}^{\mathcal{H}}/h)}_{11}-{(K_{\mathrm{eff}}^{\mathcal{H}}/h)}_{22}=D_{\mathrm{rt}}-\left(M_{11}-M_{22}\right).$$

Exact locking in a real basis is therefore equivalent to the directly testable sum rule

$${\chi }_z\left(E\right)\equiv M_{11}\left(E\right)-M_{22}\left(E\right)=D_{\mathrm{rt}}.$$

If a physical microscopic convention fixes the route-exchange phase and a real basis cannot be chosen freely, one should additionally test

$$\Im M_{12}\left(E\right)=\Im {(K_0/h)}_{12};$$

for the real reference matrix above, the right-hand side is zero.

Appendix B.7.4. Global Minimum Theorem for the Hidden-Sector Weight

The locking sum rule permits an exact minimization with no model for the hidden spectrum. Let

$$M=\left(\begin{array}{cc}x& y\\ y^{*}& z\end{array}\right)⪰0.$$

Positive semidefiniteness implies

$$x\ge 0,z\ge {0,\left|y\right|}^2\le xz.$$

Exact locking requires

$$x-z=D_{\mathrm{rt}},$$

where no sign assumption on the future physical mismatch is necessary. Since $x,z\ge 0$,

$$TrM=x+z\ge |x-z|=|D_{\mathrm{rt}}|.$$

Introduce

$$D_{\mathrm{rt}}^{(+)}=max(D_{\mathrm{rt}},0),D_{\mathrm{rt}}^{(-)}=max(-D_{\mathrm{rt}},0).$$

Equality requires $x+z=|x-z|$, so one of the two nonnegative diagonal entries must vanish. The constraint then fixes $(x,z)=(D_{\mathrm{rt}}^{(+)},D_{\mathrm{rt}}^{(-)})$. Finally ${\left|y\right|}^2\le xz=0$, hence $y=0$.

Global minimum theorem for hidden susceptibility.Among all finite- or infinite-dimensional hidden sectors that give $M\left(E\right)⪰0$ below threshold and exactly equalize the two real route diagonals,

$$TrM\ge |D_{\mathrm{rt}}|.$$

The unique minimizing susceptibility on the visible $2\times 2$ space is

$$M_{min}=diag(D_{\mathrm{rt}}^{(+)},D_{\mathrm{rt}}^{(-)}),\parallel M_{min}{\parallel }_F=\left|D_{\mathrm{rt}}\right|.$$

For $D_{\mathrm{rt}}>0$ it is saturated by a single hidden state coupled only to route 1; for $D_{\mathrm{rt}}<0$ the roles of the two routes are interchanged; for $D_{\mathrm{rt}}=0$ no compensating susceptibility is required. The one-state solution is therefore not merely sufficient; it is globally optimal even relative to an arbitrary multi-state hidden sector.

For the declared benchmark $D_{\mathrm{rt}}>0$, so the sign-independent theorem reduces numerically to

$$M_{min}=\left(\begin{array}{cc}0.049406060606061& 0\\ 0& 0\end{array}\right),$$

and

$${\displaystyle \frac{K_{\mathrm{eff},\min }}{h}}=\left(\begin{array}{cc}{B}_{\mathrm{rt}}& C_{\mathrm{rt}}\\ C_{\mathrm{rt}}& B_{\mathrm{rt}}\end{array}\right)=B_{\mathrm{rt}}{I}_2+C_{\mathrm{rt}}{\sigma }_x.$$

If the correction is written with an explicit $1/15$ factor,

$${\displaystyle \frac{\delta K}{h}}={\displaystyle \frac{1}{15}}(r_0{I}_2+r_x{\sigma }_x+r_y{\sigma }_y+r_z{\sigma }_z),$$

the global minimum corresponds to

$$(r_0,r_x,r_y,r_z)=(-0.3705454545,0,0,-0.3705454545).$$

The stronger, overconstrained construction additionally demanded final $f=0$ and therefore required

$$T_{\mathrm{old}}=\sqrt{D_{\mathrm{rt}}^2+4C_{\mathrm{rt}}^2}=0.054597550878986.$$

The true minimum required only by route symmetry,

$$T_{min}=D_{\mathrm{rt}}=0.049406060606061,$$

is smaller by $9.50865\%$. This is not an improvement of a fit; it is a correction of the algebraic condition that the symmetry actually imposes.

Appendix B.7.5. Route-Parity Splitting and the Separate Sign-Inheritance Criterion

After exact locking,

$${\displaystyle \frac{K_{\mathrm{eff}}}{h}}=d{I}_2+f{\sigma }_x,f=C_{\mathrm{rt}}-{\chi }_x\left(E\right),{\chi }_x\left(E\right)=\Re M_{12}\left(E\right).$$

The route-even and route-odd eigenvalues of the kernel itself are

$${\lambda }_{+}=d+f,{\lambda }_{-}=d-f.$$

Hence the matrix algebra directly implies

$$f>0⟺{\lambda }_{-}<{\lambda }_{+}.$$

At the global minimum, ${\chi }_x=0$, so $f=C_{\mathrm{rt}}>0$ and

$${\lambda }_{-}-{\lambda }_{+}=-2C_{\mathrm{rt}}=-0.023236474288792.$$

This is not yet a statement about the physical energies of bound-state poles. Depending on the formulation, a “kernel” may be a Hamiltonian, an inverse propagator, an attractive interaction operator, or a component of a Bethe–Salpeter equation. Therefore the sign map

$$sign({\lambda }_{-}-{\lambda }_{+})⟶sign(E_{-}-E_{+})$$

must be derived or checked within the same physical realization.

Physical sign-inheritance criterion.For a generic effective kernel, the lower eigenvalue cannot automatically be identified with the lower physical pole mass. The exact completion theorem below nevertheless shows that there is a broad family of Hermitian completions in which the route-odd state is the exact nondegenerate ground state. Sign inheritance is therefore not obstructed structurally; the open question is whether the actual renormalized spectral measure of the confining $\mathrm{SU}{\left(15\right)}_p$ theory lies in that physical region.

Appendix B.7.6. Exact One-Hidden-State Embedding without a Feshbach Approximation

To verify that the global minimum is not an artifact of eliminating the hidden sector perturbatively, construct the full three-state Hamiltonian. Let ${\Delta }_H>0$ be the energy gap between the hidden state and the target eigenvalue. Define

$${\displaystyle \frac{\mathbb{H}}{h}}=\left(\begin{array}{ccc}{A}_{\mathrm{rt}}& C_{\mathrm{rt}}& u\\ C_{\mathrm{rt}}& B_{\mathrm{rt}}& 0\\ u& 0& E_H\end{array}\right),u^2=D_{\mathrm{rt}}{\Delta }_H,E_H=(B_{\mathrm{rt}}-C_{\mathrm{rt}})+{\Delta }_H.$$

Then direct multiplication shows that

$${\Psi }_{-}\propto \left(\begin{array}{c}1\\ -1\\ -u/{\Delta }_H\end{array}\right),{\lambda }_{*}=B_{\mathrm{rt}}-C_{\mathrm{rt}}$$

is an exact eigenstate for every ${\Delta }_H>0$. In particular, the third row gives

$$u-u{\displaystyle \frac{E_H}{{\Delta }_H}}=-{\displaystyle \frac{u}{{\Delta }_H}}(B_{\mathrm{rt}}-C_{\mathrm{rt}}),$$

while the first row closes because $u^2=D_{\mathrm{rt}}{\Delta }_H$.

The normalized probability in the hidden component is

$$P_H={\displaystyle \frac{D_{\mathrm{rt}}}{2{\Delta }_H+D_{\mathrm{rt}}}}.$$

Thus for ${\Delta }_H\gg D_{\mathrm{rt}}$ the exact odd ratio of the two visible components coexists with an arbitrarily small hidden admixture.

If every hidden spectral point lies at least a gap $g>0$ away from a reference energy $E_0$, the resolvent identity gives the stability bound

$$\parallel M(E_0+\delta E)-M\left(E_0\right)\parallel \le {\displaystyle \frac{\left|\delta E\right|}{g-\left|\delta E\right|}}\parallel M\left(E_0\right)\parallel ,\left|\delta E\right|<g.$$

For a single pole the relative drift is $\delta E/({\Delta }_H-\delta E)$. Thus an exact equality at one energy is not automatically a fine-tuning: a wide low-energy window is possible when the hidden gap is sufficiently larger than the window width.

Appendix B.7.7. Four-State Completion and Quadratically Stable Locking over an Energy Window

The preceding three-state construction proves an exact eigenvector at one selected energy, but by itself it does not eliminate every ambiguity between an eigenvalue of a reduced energy-dependent kernel and the pole ordering of the full system. A larger exact family resolves this issue without introducing a new fundamental symmetry.

For compact notation in this subsection write

$$A\equiv A_{\mathrm{rt}},B\equiv B_{\mathrm{rt}},C\equiv C_{\mathrm{rt}},D\equiv A-B=D_{\mathrm{rt}}>0,$$

and measure energies in units of the positive common scale h. Choose two nonnegative hidden susceptibilities at the target energy,

$$m_1\ge 0,m_2\ge 0,m_1-m_2=D,$$

two positive hidden gaps,

$${\Delta }_1>0,{\Delta }_2>0,$$

and couplings

$$u_1^2=m_1{\Delta }_1,u_2^2=m_2{\Delta }_2.$$

Define the target pole eigenvalue

$${\lambda }_{*}=B-C-m_2$$

and hidden energies

$$E_{H1}={\lambda }_{*}+{\Delta }_1,E_{H2}={\lambda }_{*}+{\Delta }_2.$$

The full Hermitian four-state matrix is

$$\widehat{H}=\left(\begin{array}{cccc}A& C& u_1& 0\\ C& B& 0& u_2\\ u_1& 0& E_{H1}& 0\\ 0& u_2& 0& E_{H2}\end{array}\right).$$

This is a constructive basis in which the hidden couplings are diagonal. The theorem does not assert that the microscopic hidden spectrum must be diagonal in this particular basis.

Consider

$${\Psi }_{*}=\left(\begin{array}{c}1\\ -1\\ -u_1/{\Delta }_1\\ +u_2/{\Delta }_2\end{array}\right).$$

Direct multiplication gives $(\widehat{H}-{\lambda }_{*}I){\Psi }_{*}=0$. For example, the first visible row is

$$(A-{\lambda }_{*})-C-{\displaystyle \frac{u_1^2}{{\Delta }_1}}=(D+m_2+C)-C-m_1=0,$$

because $m_1-m_2=D$; the remaining rows vanish similarly.

To prove the ordering, decompose $\widehat{H}-{\lambda }_{*}I$ into visible and hidden blocks. The hidden block

$${\Delta }_H=diag({\Delta }_1,{\Delta }_2)$$

is strictly positive. The visible block and mixing matrix are

$$V_{*}=\left(\begin{array}{cc}{m}_1+C& C\\ C& m_2+C\end{array}\right),U=diag(u_1,u_2).$$

The Schur complement is

$$S_{*}=V_{*}-U{\Delta }_H^{-1}{U}^T=C\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right).$$

For the inherited $C>0$, $S_{*}$ is positive semidefinite with a one-dimensional kernel. Since ${\Delta }_H>0$, the Schur-complement criterion gives

$$\widehat{H}-{\lambda }_{*}I⪰0,$$

and the kernel dimension of the full matrix equals that of $S_{*}$, namely one. Therefore ${\lambda }_{*}$ is the exact nondegenerate ground-state eigenvalue and the visible part of that ground state has route-odd ratio $(1,-1)$.

Proposition: exact level ordering in a completion family.For $D>0$, $C>0$, $m_{1,2}\ge 0$ with $m_1-m_2=D$, and arbitrary ${\Delta }_{1,2}>0$, the Hermitian completion above has a unique ground state whose visible route ratio is $(1,-1)$. Hence there is no structural sign ambiguity that forbids a lower physical odd pole. Such ordering is realized exactly on a positive-dimensional continuous family of hidden-composite completions, open within the constrained locking manifold. This is an existence theorem for a completion family, not a first-principles statement about the actual confining $\mathrm{SU}{\left(15\right)}_p$ spectral measure.

The one-hidden-state construction is recovered as the limiting case $m_2=0$. A new issue now becomes explicit: exact locking at one energy $E_{*}$ does not imply energy-independent symmetry over an open interval. For $E=E_{*}+x$ the two-pole susceptibility difference is

$${\chi }_z(E_{*}+x)={\displaystyle \frac{m_1{\Delta }_1}{{\Delta }_1-x}}-{\displaystyle \frac{m_2{\Delta }_2}{{\Delta }_2-x}}.$$

At the target point, ${\chi }_z\left(E_{*}\right)=m_1-m_2=D$. A generic one-pole realization has a nonzero first derivative and hence leaks linearly away from the target energy.

The minimal two-pole flattening imposes additionally

$${\chi }_z^{\prime }\left(E_{*}\right)=0.$$

For ${\Delta }_1>{\Delta }_2$ the two conditions have the unique solution

$$m_1={\displaystyle \frac{D{\Delta }_1}{{\Delta }_1-{\Delta }_2}},m_2={\displaystyle \frac{D{\Delta }_2}{{\Delta }_1-{\Delta }_2}}.$$

Substitution gives the exact leakage formula

$${\displaystyle \frac{D-{\chi }_z(E_{*}+x)}{D}}={\displaystyle \frac{x^2}{({\Delta }_1-x)({\Delta }_2-x)}}.$$

Thus the diagonal mismatch and the associated even–odd mixing begin at $O\left(x^2\right)$ rather than $O\left(x\right)$. More generally, cancelling the first m derivatives of the relevant spectral function yields leakage $O\left(x^{m+1}\right)$, where $m=0,1,2,\dots$ is the desired order of flatness and $x=E-E_{*}$.

A complete minimal construction can be given for arbitrary finite order. Let N simple hidden poles have distinct positive gaps

$${\Delta }_1>{\Delta }_2>\dots >{\Delta }_N>0.$$

Pole j has a positive physical strength $m_j>0$ and a sign $s_j=+1$ or $-1$ according to which of the two route channels it couples to. Then

$${\chi }_z(E_{*}+x)={\displaystyle \sum _{j=1}^N}{s}_j{m}_j{\displaystyle \frac{{\Delta }_j}{{\Delta }_j-x}}.$$

Locking to order m means

$${\chi }_z\left(E_{*}\right)=D_{\mathrm{rt}},{\chi }_z^{\left(n\right)}\left(E_{*}\right)=0,n=1,\dots ,m.$$

Define signed weights $w_j=s_j{m}_j$ and $y_j={\Delta }_j^{-1}>0$. The conditions become the moment equations

$${\displaystyle \sum _{j=1}^N}{w}_j=D_{\mathrm{rt}},{\displaystyle \sum _{j=1}^N}{w}_j{y}_j^n=0,n=1,\dots ,m.$$

If $N\le m$, the first N homogeneous equations contain a nonsingular Vandermonde matrix for distinct $y_j$ and force all $w_j=0$, contradicting the nonzero zeroth moment. Therefore

$$N\ge m+1.$$

For the minimal choice $N=m+1$ the solution is unique. With

$$Q_N\left(x\right)\equiv {\displaystyle \prod _{j=1}^N}({\Delta }_j-x),$$

it is conveniently written as

$${\chi }_z(E_{*}+x)=D_{\mathrm{rt}}+{(-1)}^{N+1}{D}_{\mathrm{rt}}{\displaystyle \frac{x^N}{Q_N\left(x\right)}}.$$

Partial-fraction decomposition gives

$$w_j={(-1)}^{N+1}{D}_{\mathrm{rt}}{\displaystyle \frac{{\Delta }_j^{N-1}}{{\displaystyle \prod _{k\ne j}({\Delta }_k-{\Delta }_j)}}}.$$

For ${\Delta }_1>\dots >{\Delta }_N$, the signs alternate,

$$sgn{w}_j={(-1)}^{j+1},$$

so all physical strengths can remain positive, $m_j=\left|w_j\right|>0$, with the poles coupled alternately to route 1 and route 2.

Minimal arbitrary-order locking hierarchy.Within the class of distinct simple route-diagonal hidden poles and without independent energy-dependent local counterterms, locking to order m requires at least $N=m+1$ poles. For any distinct positive gaps there is a unique minimal $N=m+1$ construction with positive physical residues; the route spectral asymmetry must alternate in sign. Spectral positivity therefore does not obstruct arbitrarily high finite-order locking, but it imposes a falsifiable interlacing/sign requirement on the route-resolved spectral weights.

For the two-pole solution with vanishing first derivative, let $P_H$ be the normalized probability in the hidden components of the exact ground state and $Z_{-}^{\mathrm{vis}}$ the norm fraction in the visible route-odd sector. Then

$$P_H={\displaystyle \frac{D}{D+{\Delta }_1-{\Delta }_2}},Z_{-}^{\mathrm{vis}}={\displaystyle \frac{{\Delta }_1-{\Delta }_2}{D+{\Delta }_1-{\Delta }_2}}.$$

Quadratic stability therefore does not require a hidden-dominated state. The condition $P_H\le p$ is equivalent to

$${\Delta }_1-{\Delta }_2\ge D{\displaystyle \frac{1-p}{p}}.$$

For the purely illustrative gaps ${\Delta }_1=1$, ${\Delta }_2=0.5$, and the inherited $D=0.049406060606061$,

$$m_1=0.098812121212122,m_2=0.049406060606061,$$

$$u_1=0.314343953675145,u_2=0.157171976837573,$$

$${\lambda }_{*}=0.186636308310149.$$

Direct diagonalization gives

$$spec\left(\widehat{H}\right)=\{0.1866363083,0.2077210756,0.7371267438,1.2865157616\}.$$

The gap above the ground-state pole is $0.02108476726$, the visible fraction is $Z_{-}^{\mathrm{vis}}=0.9100736884$, and the hidden fraction is $P_H=0.0899263116$. Substitution of the constructed eigenvector into the eigenvalue equation gives residual $2.78\times {10}^{-17}$ in the quoted numerical representation. These numbers are only a consistency benchmark in the inherited normalization; they are not a physical mass prediction.

First-principles criterion for energy stability.The minimal target-point test remains ${\chi }_z\left(E_{*}\right)=D_{\mathrm{rt}}$. Robustness over a finite energy window is tested by simultaneously measuring ${\chi }_z^{\prime }\left(E_{*}\right)$. A value consistent with zero identifies a quadratically stable branch. A large nonzero derivative does not exclude a narrow mechanism tuned to one energy, but then the linear deviation away from that point must be explicitly controlled. Exact energy-independent route symmetry over an open interval should not be attributed to a finite set of simple gapped Feshbach poles unless there is an additional local contact term, an infinite spectral support, or an appropriate hierarchy of moment cancellations.

Appendix B.7.8. Euclidean Memory Kernel: Direct Moment Rules for Route Reciprocity

The preceding formulas express hidden-sector energy dependence through poles and gaps. For a nonperturbative Euclidean calculation it is more direct to work with a time-domain kernel. Let $E_{*}$ be the energy near which route reciprocity is tested. Let ${\mathcal{H}}_H$ be the self-adjoint hidden-composite Hamiltonian, $\mathcal{V}$ the coupling to the two visible route states, and $h>0$ the same normalization used in $K_0/h$. Assume that the hidden spectrum is separated from $E_{*}$ by a positive gap $g_H>0$, and define

$${\mathcal{G}}_H\equiv {\mathcal{H}}_H-E_{*}⪰g_H{I}_H,$$

where $I_H$ is the identity on the hidden subspace.

Define the matrix Euclidean memory kernel

$$\mathcal{F}\left(t\right)\equiv {\displaystyle \frac{1}{h}}\mathcal{V}{e}^{-t{\mathcal{G}}_H}{\mathcal{V}}^{†},t\ge 0.$$

Since $e^{-t{\mathcal{G}}_H}$ is positive,

$$\mathcal{F}\left(t\right)⪰0$$

for every $t\ge 0$.

The Schur–Feshbach susceptibility is

$$M\left(E\right)={\displaystyle \frac{1}{h}}\mathcal{V}{({\mathcal{H}}_H-E)}^{-1}{\mathcal{V}}^{†}.$$

Let $M^{\left(n\right)}\left(E_{*}\right)$ denote the nth derivative with respect to E at $E=E_{*}$. Spectral calculus gives

$${\left.{\displaystyle \frac{d^n}{d{E}^n}}{({\mathcal{H}}_H-E)}^{-1}\right|}_{E=E_{*}}=n!{\mathcal{G}}_H^{-(n+1)},$$

and

$${\int }_0^{\infty }{t}^n{e}^{-t{\mathcal{G}}_H}dt=n!{\mathcal{G}}_H^{-(n+1)}.$$

Therefore

$$M^{\left(n\right)}\left(E_{*}\right)={\int }_0^{\infty }{t}^n\mathcal{F}\left(t\right)dt.$$

For

$${\chi }_z\left(E\right)=M_{11}\left(E\right)-M_{22}\left(E\right),$$

define

$$\Delta \mathcal{F}\left(t\right)={\mathcal{F}}_{11}\left(t\right)-{\mathcal{F}}_{22}\left(t\right).$$

Then

$${\chi }_z^{\left(n\right)}\left(E_{*}\right)={\int }_0^{\infty }{t}^n\Delta \mathcal{F}\left(t\right)dt.$$

Thus point locking and order-m stability,

$${\chi }_z\left(E_{*}\right)=D_{\mathrm{rt}},{\chi }_z^{\left(n\right)}\left(E_{*}\right)=0(n=1,\dots ,m),$$

are exactly equivalent to the Euclidean sum rules

$${\int }_0^{\infty }\Delta \mathcal{F}\left(t\right)dt=D_{\mathrm{rt}},{\int }_0^{\infty }{t}^n\Delta \mathcal{F}\left(t\right)dt=0,n=1,\dots ,m.$$

No pole decomposition or analytic continuation is needed for this test once the renormalized 1PI memory kernel $\mathcal{F}\left(t\right)$ is available. Here 1PI refers to the inverse-kernel/self-energy structure of the effective action. The memory kernel must not be confused with the raw two-point correlator matrix $C^{AB}\left(t\right)$.

For a worked normalization check, take one hidden state at dimensionless gap $g_H=1$ coupled only to route 1, with $|v_1{|}^2/h=D_{\mathrm{rt}}$. Then

$$\Delta \mathcal{F}\left(t\right)=D_{\mathrm{rt}}{e}^{-t},{\int }_0^{\infty }\Delta \mathcal{F}\left(t\right)dt=D_{\mathrm{rt}},$$

so point locking is exact. Numerically, using the benchmark,

$$\Delta \mathcal{F}\left(0\right)=0.0494061,\Delta \mathcal{F}\left(1\right)=0.0181755,\Delta \mathcal{F}\left(2\right)=0.0066863.$$

However,

$${\chi }_z^{\prime }\left(E_{*}\right)={\int }_0^{\infty }t\Delta \mathcal{F}\left(t\right)dt=D_{\mathrm{rt}}\ne 0.$$

Thus a single positive pole can satisfy the on-shell condition but cannot by itself satisfy the stronger first-derivative cancellation. At least two contributions with opposite route differences, or additional local/infinite-support structure, are required for quadratic stability. This small example is a check of signs and normalization, not a model of the physical hidden spectrum.

The higher moment conditions have a direct nodal consequence. Suppose the continuous, nonzero function $\Delta \mathcal{F}\left(t\right)$ has finitely many sign changes and obeys

$${\int }_0^{\infty }{t}^n\Delta \mathcal{F}\left(t\right)dt=0,n=1,\dots ,m.$$

Then it must change sign at least m times for $t>0$. If it had only $r<m$ sign changes at ${\tau }_1<\dots <{\tau }_r$, choose

$$q\left(t\right)=\epsilon t{\displaystyle \prod _{j=1}^r}(t-{\tau }_j)$$

with $\epsilon =\pm 1$ such that q and $\Delta \mathcal{F}$ have the same sign on the first interval. Then $q\left(t\right)\Delta \mathcal{F}\left(t\right)\ge 0$ and is strictly positive on a set of nonzero measure. But q is a linear combination of $t,t^2,\dots ,t^m$, so the moment conditions imply $\int q\Delta \mathcal{F}dt=0$, a contradiction. For the minimal $(m+1)$-pole exponential family, the exponentials form a Chebyshev system, so there can be at most m zeros; the minimal completion therefore has exactly m sign changes.

A finite Euclidean time window can also be controlled without assuming a specific number of poles. Define the unmeasured tail of the nth moment after $T>0$ by

$$R_n\left(T\right)={\int }_T^{\infty }{t}^n\Delta \mathcal{F}\left(t\right)dt.$$

If ${\mathcal{G}}_H⪰g_H{I}_H$, then

$$|R_n\left(T\right)|\le [{\mathcal{F}}_{11}\left(T\right)+{\mathcal{F}}_{22}\left(T\right)]{\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)T^{n-k}{\displaystyle \frac{k!}{g_H^{k+1}}}.$$

Thus each moment condition can be assigned a rigorous finite-window error provided the positive gap assumption is valid. If the hidden self-energy channel is gapless, this exponential-tail theorem cannot be applied mechanically and must be replaced by a threshold power-law bound or an explicit spectral treatment.

Physical meaning of the Euclidean representation.Route reciprocity and its energy derivatives have a direct time-domain image in the renormalized memory kernel. The hidden-sector test is thereby converted from a pole-model problem into integral conditions on the positive Hamiltonian/transfer memory kernel. They can be confronted directly with Euclidean transfer-matrix data; applying the same decomposition to a renormalized 1PI inverse kernel requires the independent spectral bridge stated above.

Appendix B.7.9. Earlier Overconstrained Rank-One Benchmark: What Remains Useful

Before the one-scalar locking proposition was identified, a stronger target was imposed: equal diagonals together with a prescribed change or cancellation of the off-diagonal entry. At the reference point

$$q=\left(\begin{array}{c}0.229727402405877\\ 0.058039803670336\end{array}\right),$$

the correction $-q{q}^T$ corresponds to

$$(r_0,r_x,r_y,r_z)=(-0.421074736697,-0.20,0,-0.370545454545)$$

and yields a negative final off-diagonal entry. This point remains useful as input to a specific contact Bethe–Salpeter benchmark where the map to $g_{\pm }$ is defined explicitly. It is no longer the minimal route-locking solution and is not used as a universal sign theorem.

Appendix B.7.10. Closure of the Full Primitive Sector

Projected $2\times 2$ route locking does not automatically guarantee that the full primitive $4\times 4$ channel has no leakage into the even complement. For the reference factorized primitive contribution, the missing vector is

$$y_{\mathrm{miss}}=\left(\begin{array}{c}-0.029\\ -0.0065\\ +0.0515\\ +0.155\end{array}\right)h,$$

defined by

$$P_{B,\mathrm{cov}}^{+}{K}_{\mathrm{miss}}{\ell }_{-}=-P_{B,\mathrm{cov}}^{+}{K}_{\mathrm{calc}}{\ell }_{-}.$$

There exists a real symmetric negative-semidefinite rank-one correction that reproduces this vector. Hence full primitive closure has no matrix-space or coefficient-size obstruction either.

The existence statement can be checked without an optimizer. In units of h, let

$$u={(-0.029,0.2215,-0.1765,-0.301)}^T,K_{\mathrm{corr}}=-u{u}^T⪯0.$$

Then, with $P_{B,\mathrm{cov}}^{+}=(I_4+S_B^T)/2$ and ${\ell }_{-}={(0,-1,1,2)}^T$,

$$P_{B,\mathrm{cov}}^{+}{K}_{\mathrm{corr}}{\ell }_{-}={(-0.029,-0.0065,+0.0515,+0.155)}^T=y_{\mathrm{miss}}/h.$$

The largest absolute entry of $K_{\mathrm{corr}}$ is $0.090601$ in this normalization. This explicit witness proves matrix-space existence; it is not asserted to be the unique or dynamically selected microscopic correction.

Proved.Failure of the old factorized subspace is not a representation-theory no-go for the full four-dimensional dynamics. A rank-one negative-semidefinite structure can supply the missing primitive direction, so neither matrix dimension nor the natural scale of the correction obstructs the required closure.

Compatibility condition.The route-space theorem $M_{11}-M_{22}=D_{\mathrm{rt}}$ and the rank-one primitive-space closure are exact existence results in different projections. The present analysis does not prove that one and the same physical hidden eigenstate or spectral measure simultaneously realizes both conditions. This compatibility must be tested in the full renormalized operator basis rather than assumed.

Appendix B.7.11. Reduced Bethe–Salpeter Window with an Odd Pole Only

The earlier overconstrained rank-one point can be inserted into a specific contact Bethe–Salpeter control model in the standard bound-state formalism [16], where the relation between matrix entries and dimensionless binding couplings $g_{\pm }$ is explicitly defined. Only within this model, not from the generic sign of f, can an odd-only pole be certified. For the benchmark point there is an open interval of a common binding parameter,

$$4.064923603089<{\eta }_h\le 4.122404222528.$$

At its midpoint,

$${\eta }_h=4.093663912808,$$

one finds

$$g_{+}=0.993028265020<1,g_{-}=1.007070319771>1.$$

In this normalization the critical value is $g_c=1$: the even channel is unbound while the odd channel supports a bound pole. For that odd pole,

$$x_{-}\equiv {\displaystyle \frac{{\kappa }_{-}}{\Lambda }}=0.004482294757819,$$

and

$${\displaystyle \frac{B_{-}}{{\Lambda }^2/\left(2{\mu }_{\mathrm{red}}\right)}}=2.009096629597169\times {10}^{-5}.$$

Constructive numerical result.The reduced Bethe–Salpeter model proves that the odd-only pole region is nonempty and compatible with a natural-sized correction. The quoted $B_{-}$ is not a physical fermion or prebaryon mass prediction because the actual $\Lambda$, reduced mass ${\mu }_{\mathrm{red}}$, contact coupling, and continuum threshold of the full $\mathrm{SU}{\left(15\right)}_p$ theory have not been computed.

Appendix B.8. Spectral Positivity, Pole Residue, and Requirements on the Hidden Sector

Appendix B.8.1. Nonzero Spectral Weight

Let $J_{-}$ be a renormalized, spatially smeared, non-null realization of the explicit 24-preon source. Reflection positivity and the transfer-operator representation give

$$C_{--}\left(t\right)={\int }_0^{\infty }{e}^{-Et}d{\mu }_{-}\left(E\right),d{\mu }_{-}\left(E\right)\ge 0.$$

In finite volume,

$$C_{--}\left(t\right)=\sum _n{\left|z_n\right|}^2{e}^{-E_{n}t}.$$

If

$$J_{-}^{†}|0\rangle \ne 0,$$

then

$$\sum _n|z_n{|}^2=\parallel J_{-}^{†}{|0\rangle \parallel }^2>0.$$

Hence at least one physical state has nonzero overlap with the source.

Proved.A non-null source $J_{-}$ cannot have identically zero total spectral weight. If the lowest point of its spectral support is an isolated pole, the pole residue of that state is strictly positive. Once an isolated pole has been established, nonvanishing of its two-point residue is therefore no longer an independent condition.

Appendix B.8.2. Why the Absolute Pole Residue Is Not a Universal Number

Under a source normalization change

$$J_{-}\to \lambda J_{-},$$

one has

$$|z_{-}{|}^2\to {\left|\lambda \right|}^2{\left|z_{-}\right|}^2.$$

Thus an absolute numerical value of $|z_{-}{|}^2$ becomes meaningful only after fixing the operator-renormalization scheme and the spatial smearing prescription. The statement that the residue is nonzero is invariant under this rescaling; its absolute magnitude is not.

Appendix B.8.3. Explicit Residue in the Reduced Model

For a canonical constant-form-factor source in the reduced model, take

$$\psi \left(\mathbf{p}\right)={\displaystyle \frac{N\theta (\Lambda -p)}{p^2+{\kappa }^2}}.$$

The normalization integral is

$$I_2={\int }_{p<\Lambda }{\displaystyle \frac{d^{3}p}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{{(p^2+{\kappa }^2)}^2}}={\displaystyle \frac{1}{4{\pi }^2\kappa }}\left[arctan{\displaystyle \frac{\Lambda }{\kappa }}-{\displaystyle \frac{\Lambda \kappa }{{\Lambda }^2+{\kappa }^2}}\right],$$

while the source-overlap integral is

$$I_1={\int }_{p<\Lambda }{\displaystyle \frac{d^{3}p}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{p^2+{\kappa }^2}}={\displaystyle \frac{1}{2{\pi }^2}}\left[\Lambda -\kappa arctan{\displaystyle \frac{\Lambda }{\kappa }}\right].$$

With $N=I_2^{-1/2}$,

$$z_{\mathrm{red}}^2={\displaystyle \frac{I_1^2}{I_2}}>0.$$

At the constructive benchmark point,

$${\displaystyle \frac{z_{\mathrm{red}}^2}{{\Lambda }^3}}=2.867126116431022\times {10}^{-4}>0.$$

This is an explicit existence result within the reduced model, not a first-principles normalized 24-preon residue.

Appendix B.9. Matching an Isolated Route-Odd Pole: When Exact Route Symmetry Implies C 12 =C 21

An isolated route-odd pole by itself does not impose $C_{12}=C_{21}$. The coefficient equality follows only when the independently defined route reflection is a symmetry of the renormalized source and matching problem and the allowed local fixed-point vertex is route even. The pole analysis below supplies the spectral part of that conditional matching theorem; it does not turn pole parity into a Wilson-coefficient identity by itself.

Appendix B.9.1. Pole-Overlap Vector

With exact route symmetry, an isolated odd pole has a two-source overlap vector proportional to the odd eigenvector,

$${\mathbf{z}}_{-}={\displaystyle \frac{z_{-}}{\sqrt{2}}}\left(\begin{array}{c}1\\ -1\end{array}\right).$$

Its pole contribution to the route correlator is

$$C^{\mathrm{pole}}\left(t\right)={\left|z_{-}\right|}^2{e}^{-E_{-}t}{P}_{-},P_{\mathrm{rt}}^{-}={\displaystyle \frac{I_2-{\sigma }_x}{2}}={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right).$$

Consequently,

$${\left(C^{\mathrm{pole}}\right)}_{12}={\left(C^{\mathrm{pole}}\right)}_{21}.$$

This is an equality of matrix entries; it does not mean that a route-odd source is transformed into a route-even state.

Appendix B.9.2. Matching Through the Normal Derivative

For the reflected relative coordinate y,

$$J_{-}(-y)=-J_{-}\left(y\right).$$

Here y is a coordinate on the routed source/configuration family; it is not by definition the physical spacetime separation between the 24 preon insertions. Hence $J_{-}\left(0\right)=0$, but

$${\partial }_y^r{J}_{-}(-y)={(-1)}^{r+1}{\partial }_y^r{J}_{-}\left(y\right)$$

shows that odd-order normal derivatives ${\partial }_n^{2m+1}{J}_{-}\left(0\right)$ are route-even and may be nonzero. As proved in Appendix B.1.14, a smooth route-fixed family can maintain a strictly positive physical point-splitting gap, so this derivative construction does not require a collapsed 24-field contact operator.

Introduce

$${\mathcal{O}}_{+}={\displaystyle \frac{{\mathcal{O}}_{12}+{\mathcal{O}}_{21}}{\sqrt{2}}},{\mathcal{O}}_{-}={\displaystyle \frac{{\mathcal{O}}_{21}-{\mathcal{O}}_{12}}{\sqrt{2}}}.$$

In a completion with exact route symmetry, the local fixed-point term

$${\mathcal{O}}_{+}^{†}{\partial }_n^{2m+1}{J}_{-}$$

is even and allowed, while

$${\mathcal{O}}_{-}^{†}{\partial }_n^{2m+1}{J}_{-}$$

is odd and forbidden. Even-order derivatives of the odd source vanish at the fixed point. Therefore any nonzero local linear matching lies in $span\left({\mathcal{O}}_{+}\right)$ and gives

$$C_{12}=C_{21}$$

in the exact-symmetry limit.

Proved selection rule.Symmetry fixes the ratio of the two crossed matching coefficients, not the absolute magnitude of the allowed matching vertex. The mass-plane mechanism therefore requires a nonzero even form factor, not merely the selection rule $F_{-}=0$.

Appendix B.9.3. Explicit Nonzero Boundary Propagator in the Composite Completion

For a free massive odd collective field on an interval with Dirichlet boundary conditions, the Green function is

$$G_D(p;y,y^{\prime })={\displaystyle \frac{sinh\left(k{y}_{<}\right)sinh\left[k(L-y_{>})\right]}{ksinh\left(kL\right)}},k=\sqrt{p_E^2+M_X^2}.$$

Between opposite boundaries,

$${\partial }_{n_0}{\partial }_{n_L}{G}_D(p;0,L)={\displaystyle \frac{k}{sinh\left(kL\right)}}.$$

Derivative matching is therefore not forced to vanish by the node. In normalized form,

$${\widehat{\Gamma }}_{+}\left(x\right)=\sqrt{2}{\displaystyle \frac{x}{sinhx}},x=kL,$$

while

$${\widehat{\Gamma }}_{-}\left(x\right)=0.$$

For every finite x, ${\widehat{\Gamma }}_{+}\left(x\right)>0$.

Appendix B.9.4. Pole-Projector Reciprocity and the Regular-Remainder Theorem

The previous selection rule can be stated directly at the order-one amplitude level. Let an isolated physical state $|X_{-}\rangle$ have the canonically normalized route-source overlap

$$\langle 0|J_A|X_{-}\rangle =z_{-}{\left(e_{-}\right)}_A,e_{-}\equiv {\displaystyle \frac{1}{\sqrt{2}}}\left({\displaystyle \genfrac{}{}{0pt}{}{1}{-1}}\right).$$

Its rank-one residue matrix is

$${\mathcal{R}}_{-}={\left|z_{-}\right|}^2{P}_{-},P_{-}=e_{-}{e}_{-}^{†}={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right).$$

Direct multiplication gives $P_{-}^2=P_{-}$ and

$${\left(P_{-}\right)}_{12}={\left(P_{-}\right)}_{21}=-{\displaystyle \frac{1}{2}}.$$

Hence every pole exchange term proportional to this residue projector has equal crossed entries, independently of the absolute pole residue and independently of the common large-N factor. When the amputated local matching vertex is route even, this gives

$${\widehat{C}}_{12}^{\mathrm{pole}}={\widehat{C}}_{21}^{\mathrm{pole}}.$$

(A207)

Exact pole-projector reciprocity theorem.An isolated physical state whose overlap vector with the unprojected route-source doublet is exactly proportional to $e_{-}$ generates an exchange tensor proportional to $P_{-}$. If the associated amputated local matching vertex is route even, its pole-mediated order-one crossed Wilson coefficients are exactly equal. The theorem is algebraic once the physical pole alignment and matching parity are established; it does not depend on the benchmark completion used to demonstrate existence elsewhere in the paper.

Now decompose the full reduced crossed coefficients into this isolated-pole contribution and the regular/background part,

$${\widehat{C}}_{ij}={\widehat{C}}_{ij}^{\mathrm{pole}}+{\widehat{C}}_{ij}^{\mathrm{reg}}.$$

Equation (A207) gives the exact identity

$${\widehat{C}}_{21}-{\widehat{C}}_{12}={\widehat{C}}_{21}^{\mathrm{reg}}-{\widehat{C}}_{12}^{\mathrm{reg}},{\widehat{C}}_{-}={\widehat{C}}_{-}^{\mathrm{reg}}.$$

(A208)

Thus the isolated odd pole cannot itself generate the reciprocity defect; every failure of equality is localized in the non-pole regular remainder.

For a quantitative future test define

$$r_{+}={\displaystyle \frac{{\widehat{C}}_{12}^{\mathrm{reg}}+{\widehat{C}}_{21}^{\mathrm{reg}}}{\sqrt{2}}},r_{-}={\displaystyle \frac{{\widehat{C}}_{21}^{\mathrm{reg}}-{\widehat{C}}_{12}^{\mathrm{reg}}}{\sqrt{2}}},$$

and let $p_{+}$ denote the nonzero even pole-mediated matching amplitude. The normalized coefficient leakage is then exactly

$${ϵ}_C={\displaystyle \frac{|{\widehat{C}}_{12}-{\widehat{C}}_{21}|}{\sqrt{|{\widehat{C}}_{12}{|}^2+{\left|{\widehat{C}}_{21}\right|}^2}}}={\displaystyle \frac{\sqrt{2}\left|r_{-}\right|}{\sqrt{|p_{+}+r_{+}{|}^2+{\left|r_{-}\right|}^2}}}.$$

If an independent calculation establishes $|r_{+}|,|r_{-}|\le \eta |p_{+}|$ with $0\le \eta <1$, then

$${ϵ}_C\le {\displaystyle \frac{\sqrt{2}\eta }{1-\eta }}.$$

(A209)

Conversely, $\eta \le {ϵ}_{*}/(\sqrt{2}+{ϵ}_{*})$ is sufficient to guarantee a target ${ϵ}_C\le {ϵ}_{*}$. No identification of this coefficient-space leakage with the phenomenological angular quantity ${ϵ}_{\Pi }$ is made without the physical matching metric.

New physical gate.The order-one strong-dynamics question has been reduced from comparing two arbitrary coefficients to one quantity: the route-odd regular remainder ${\widehat{C}}_{-}^{\mathrm{reg}}$. If this remainder is absent by a genuine confinement projection, or is parametrically suppressed in the infrared, the pole-mediated low-energy amplitude is reciprocal. If it remains unsuppressed at $O\left(1\right)$, the isolated-pole mechanism alone does not explain the plane.

Appendix B.10. Numerical Control of the Correlator Formulas and the First-Principles Identifiability Boundary

Appendix B.10.1. What Can Be Computed from the Present Inputs

The reduced-model benchmark with a single odd bound state defines a threshold-normalized control correlator,

$${\widehat{C}}^{AB}\left(t\right)=e^{E_{\mathrm{th}}t}{C}^{AB}\left(t\right),u\equiv t{\displaystyle \frac{{\Lambda }^2}{2{\mu }_{\mathrm{red}}}}.$$

If

$$b\equiv {\displaystyle \frac{B_{-}^{\mathrm{bind}}}{{\Lambda }^2/\left(2{\mu }_{\mathrm{red}}\right)}}$$

is the dimensionless binding energy of the control state, then

$${\displaystyle \frac{\widehat{C}\left(u\right)}{{\Lambda }^3}}={\displaystyle \frac{z_{\mathrm{red}}^2}{2{\Lambda }^3}}{e}^{bu}\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right).$$

For the benchmark, $b=2.009096629597169\times {10}^{-5}$. The route-Ward residual

$${ϵ}_R\left(u\right)={\displaystyle \frac{\parallel \widehat{C}-{\sigma }_x\widehat{C}{\sigma }_x{\parallel }_F}{\parallel \widehat{C}{\parallel }_F}}$$

vanishes identically in the analytic control model, while

$${\Pi }_B\left(u\right)={\displaystyle \frac{2{\widehat{C}}_{12}}{{\widehat{C}}_{11}+{\widehat{C}}_{22}}}$$

gives

$${\Pi }_B=-1.$$

The effective threshold-normalized slope

$$b_{\mathrm{eff}}(u;\Delta u)={\displaystyle \frac{1}{\Delta u}}ln{\displaystyle \frac{{\widehat{C}}_{--}(u+\Delta u)}{{\widehat{C}}_{--}\left(u\right)}}$$

returns the input value b exactly for this one-exponential analytic benchmark.

Constructive control.The control example validates the extraction formulas and demonstrates that the node of the odd source introduces no hidden algebraic obstruction. It is not a nonperturbative calculation of the unchanged chiral confining $\mathrm{SU}{\left(15\right)}_p$ theory.

Appendix B.10.2. Parity-Pure Low-Energy Projection and Normalized Matching Calibration

The route-odd numerator can be constrained further before a full four-dimensional ensemble is available. Let $H_0$ be a self-adjoint reference Hamiltonian with a diagnostic route exchange U,

$$[H_0,U]=0,U^2=I,$$

and let $W_{-}$ be route odd,

$$U{W}_{-}U=-W_{-}.$$

If the spectral projector $P_{\Lambda }={\mathbf{1}}_{[E_0,E_0+\Lambda ]}\left(H_0\right)$ contains only route-even states, then

$$P_{\Lambda }{W}_{-}{P}_{\Lambda }=P_{\Lambda }U{W}_{-}U{P}_{\Lambda }=-P_{\Lambda }{W}_{-}{P}_{\Lambda },$$

so

$$P_{\Lambda }{W}_{-}{P}_{\Lambda }=0.$$

This is an exact statement. It does not imply that the route-source kernel $K_{\perp }$ vanishes, because virtual propagation through odd states outside the low-energy projector can still generate a nonzero resolvent-weighted contribution.

In the exact gauge-reduced open-chain $SU\left(N\right)$ control introduced above,

$${\Delta }_F={\kappa }_E{\displaystyle \frac{N^2-1}{2N}},h_{\infty }=2\left(2\right|t|+\left|s\right|),$$

and the all-length gap bound gives

$${\Lambda }_{\mathrm{low}}<{\Delta }_F-h_{\infty }⟹P_{\mathrm{low}}{W}_{-}{P}_{\mathrm{low}}=0$$

uniformly in chain length. At $N=15$, ${\kappa }_E=1$, $t=s=0.1$, this guaranteed window is

$${\Lambda }_{\mathrm{low}}<6.8666667.$$

Thus an unsuppressed direct low–low route-odd term is absent throughout a broad low-energy window in the exact control, while higher odd states remain capable of contributing through the Schur–Feshbach resolvent.

A source-normalization-independent matching calibration can also be defined. Let $z_1,z_2$ be the two canonical route amplitudes of the normalized ground state and set

$$z_{\pm }={\displaystyle \frac{z_1\pm z_2}{\sqrt{2}}},r_{\Gamma }={\displaystyle \frac{z_{-}}{z_{+}}},{\eta }_F={\displaystyle \frac{\parallel K_{\perp }\parallel }{{\Delta }_{-}}}.$$

The dimensionless susceptibility

$${\widehat{\mathcal{C}}}_{\mathrm{rt}\to \mathrm{match}}^{\mathrm{ctrl}}={\left.{\displaystyle \frac{d{r}_{\Gamma }/d\lambda }{d{\eta }_F/d\lambda }}\right|}_{\lambda =0}$$

is obtained by independent first-order odd-subspace perturbation theory in the numerator and the exact linear Schur–Feshbach response in the denominator. For $SU\left(15\right)$ it has magnitude near $0.986$ for both a connector deformation and an independent hidden-pair deformation. Extending the same calculation to $L=8$ and $L=10$ gives

$$|{\widehat{\mathcal{C}}}_{\mathrm{rt}\to \mathrm{match}}^{\mathrm{ctrl}}|=0.98627733(L=8),0.98627364(L=10)$$

for the connector deformation. The relative change from $L=6$ to $L=10$ is only $1.29\times {10}^{-5}$. This is a lower-dimensional matching control, not the physical four-dimensional susceptibility.

Appendix B.10.3. A Concrete Combined Pati–Salam × Flavour Matching Control

The remaining matching ambiguity can be reduced by combining two structures that were previously established separately: the basis-covariant family selector and the unique simple Pati–Salam charge product that lies in the curvature plane.

Let

$$H_S={\lambda }^{\prime }{\lambda }^{\prime †},P_2\left(H_S\right)={\displaystyle \frac{(H_S-h_1{I}_3)(H_S-h_3{I}_3)}{(h_2-h_1)(h_2-h_3)}},$$

with $h_1<h_2<h_3$, and define

$$D_{+}=I_3-3P_2\left(H_S\right).$$

Then $D_{+}$ has eigenvalue vector

$$d_{+}=(1,-2,1),$$

and obeys

$$D_{+}^2+D_{+}-2I_3=0,Tr{D}_{+}=0,Tr{D}_{+}^2=6.$$

The construction is exact and basis covariant as long as the median eigenvalue is nondegenerate. If

$$g_S=min(h_2-h_1,h_3-h_2)>0,$$

then for $\parallel \delta H_S{\parallel }_2<g_S/2$ a standard spectral-projector estimate gives

$$\parallel D_{+}^{\prime }-D_{+}{\parallel }_2\le {\displaystyle \frac{6\parallel \delta H_S{\parallel }_2}{g_S}}.$$

This controls the selector under perturbations; it does not replace the still-open dynamical vacuum-alignment problem.

Now take the effective Pati–Salam factor

$$Q_{RT}=(B-L)T_3^R.$$

For the charged-lepton, down-quark and up-quark sectors its physical charged-sector weights are

$$q=\left(q_{\ell },q_d,q_u\right)=\left({\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{6}},{\displaystyle \frac{1}{6}}\right).$$

With the curvature-plane normal

$$n_{\Pi }=(1,2,-1),$$

one has the exact identity

$$n_{\Pi }·q=0.$$

Consider the aligned matching response

$$\delta ln{y}_{s,g}=g_M{\widehat{F}}_{+}(0;x)q_s{\left(d_{+}\right)}_g,$$

where $g_M$ is an overall reduced matching coefficient and ${\widehat{F}}_{+}$ is the normalized even form factor. Because $d_{+}·d_{+}=6$,

$$\delta A_s={\displaystyle \frac{3}{2}}{g}_M{\widehat{F}}_{+}(0;x)q_s,$$

so the combined Pati–Salam × flavour channel produces

$$(\delta A_{\ell },\delta A_d,\delta A_u)={\displaystyle \frac{g_M{\widehat{F}}_{+}(0;x)}{4}}(3,-1,1)$$

and therefore

$$\delta {\Delta }_{\Pi }=0.$$

Thus a nontrivial combined matching vertex can preserve the mass plane exactly; a family-universal threshold is not needed to obtain this cancellation, and an independent normal component is required to move the theory away from the plane.

The odd-collective boundary completion already used in the matching analysis makes the even/odd form factors explicit. For $x=kL$,

$${\widehat{F}}_{+}(0;x)=\sqrt{2}{\displaystyle \frac{x}{sinhx}}>0,{\widehat{F}}_{-}(0;x)=0.$$

Writing the matching term as $g_M{\widehat{F}}_{+}{\mathcal{O}}_{+}$ gives the canonical route coefficients

$$C_{+}^{\mathrm{ctrl}}=g_M{\widehat{F}}_{+}(0;x),C_{-}^{\mathrm{ctrl}}=0.$$

At the illustrative unit-normalization point $x=1$, $g_M=1$,

$$C_{+}^{\mathrm{ctrl}}=1.20337995742518,C_{-}^{\mathrm{ctrl}}=0,$$

and

$$(\delta A_{\ell },\delta A_d,\delta A_u)=(0.9025349681,-0.3008449894,0.3008449894),$$

whose ${\Delta }_{\Pi }$ projection vanishes exactly.

Appendix B.10.4. Scheme-stable Odd Zero and the First Signed Normal Susceptibility

The matching zero $F_{-}\left(0\right)=0$ is stronger than a single normalization convention. Let Z be any finite operator renormalization that preserves the canonical route parity,

$$[Z,U_{\mathrm{lab}}]=0.$$

In the parity basis Z is block diagonal, $Z=Z_{+}\oplus Z_{-}$. Therefore

$$F_{-}^{\mathrm{ren}}\left(0\right)=Z_{-}{F}_{-}^{\mathrm{bare}}\left(0\right)=0.$$

Hence

$$F_{-}\left(0\right)=0\mathrm{is}\mathrm{invariant}\mathrm{under}\mathrm{route}-\mathrm{parity}-\mathrm{preserving}\mathrm{finite}\mathrm{renormalizations}.$$

A prescription that mixes the parity sectors changes the route axis and must first be canonicalized before its form factors can be compared with this criterion.

The exact plane-preserving channel above has no normal displacement, so the sign of a nonzero ${\Delta }_{\Pi }$ cannot be inferred from it. To calibrate the unique physical normal direction, define

$${\widehat{n}}_{\Pi }={\displaystyle \frac{1}{\sqrt{6}}}(1,2,-1).$$

A diagnostic normal matching deformation

$$\delta ln{y}_{s,g}^{(-)}=g_M{\widehat{F}}_{+}(0;x)ϵ{\left({\widehat{n}}_{\Pi }\right)}_s{\left(d_{+}\right)}_g$$

gives

$${\displaystyle \frac{\partial {\Delta }_{\Pi }}{\partial ϵ}}={\displaystyle \frac{3\sqrt{6}}{2}}{g}_M{\widehat{F}}_{+}(0;x).$$

Because ${\widehat{F}}_{+}(0;x)>0$ for finite $x>0$, the sign of the displacement is the sign of the canonically oriented coefficient $g_Mϵ$. This is a sign calibration, not a derivation that the physical $SU{\left(15\right)}_p$ three-point vertex contains the normal deformation.

If the normal leakage is related to the canonical route-kernel coordinate by

$$ϵ={\widehat{\mathcal{C}}}_{\mathrm{rt}\to \mathrm{match}}^{\mathrm{ctrl}}\widehat{\eta }+O\left({\widehat{\eta }}^3\right),$$

then the normalized combined susceptibility is

$${\widehat{\mathcal{C}}}_{\Pi }^{\mathrm{ctrl}}(x,L)={\displaystyle \frac{3\sqrt{6}}{2}}{g}_M{\widehat{F}}_{+}(0;x){\widehat{\mathcal{C}}}_{\mathrm{rt}\to \mathrm{match}}^{\mathrm{ctrl}}\left(L\right).$$

For $N=15$, $L=10$, $x=1$ and $g_M=1$ this gives

$${\widehat{\mathcal{C}}}_{\Pi }^{\mathrm{ctrl}}=4.36080919$$

for the connector deformation, while the independent hidden-pair deformation gives $4.36078442$ in magnitude. The agreement provides a nontrivial normalization check.

Combined matching control: what is and is not closed.A concrete Pati–Salam × flavour completion now exists at the controlled EFT/reduced-model level with a basis-covariant selector, a nonzero positive even form factor, an exactly vanishing odd form factor, and an exact zero projection onto ${\Delta }_{\Pi }$ for the plane-preserving $Q_{RT}\otimes D_{+}$ channel. A diagnostic normal deformation has a fixed sign and a finite, volume-stable normalized susceptibility. The physical four-dimensional values of $g_M$, the threshold parameter replacing x, the actual normal component of the renormalized combined vertex, and the absolute $SU{\left(15\right)}_p$ three-point form factor remain open.

Appendix B.10.5. Why the Physical R μ , E - , and F + Cannot Be Invented

The actual connected correction must be extracted from a renormalized momentum-space matrix $G_{AB}\left(p_E\right)$ through

$${\Gamma }^{\left(2\right)}\left(p_E\right)=G{\left(p_E\right)}^{-1}$$

after operator mixing, external-leg amputation, and source normalization have been treated consistently. The inverse $C{\left(t_0\right)}^{-1}$ at one Euclidean time is not, in general, the 1PI kernel.

Once the physical matrix correction $\delta K_{AB}$ is known, the auxiliary coefficients are defined by

$$\begin{array}{cc}\hfill r_0& ={\displaystyle \frac{15}{2h}}(\delta K_{11}+\delta K_{22}),\hfill \end{array}$$

(267)

$$\begin{array}{cc}\hfill r_z& ={\displaystyle \frac{15}{2h}}(\delta K_{11}-\delta K_{22}),\hfill \end{array}$$

(268)

$$\begin{array}{cc}\hfill r_x& ={\displaystyle \frac{15}{h}}\Re \delta K_{12},\hfill \end{array}$$

(269)

$$\begin{array}{cc}\hfill r_y& =-{\displaystyle \frac{15}{h}}\Im \delta K_{12}.\hfill \end{array}$$

(270)

The current theoretical inputs do not determine a numerical renormalized $\delta K_{AB}$. Moreover, the structural conditions allow a continuous family of $r_0$ values, so the existence of odd-bound-state parameter regions cannot determine the actual $r_0$.

Identifiability boundary, not a no-go theorem for the theory.Assigning “physical” values to $r_0,r_x,r_y,r_z$, $E_{-}$, or the absolute $F_{+}\left(0\right)$ without renormalized two-, four-, and three-point information would amount to adding numerical assumptions, not performing a first-principles calculation. This is not an obstruction to the mechanism; it is the exact boundary of what can be inferred from the present inputs.

Appendix B.11. Final Nonperturbative Criteria for the Four-Dimensional Mechanism

It is useful to separate a kernel-level dynamical criterion from the direct pole-and-matching criterion. The first asks whether the hidden composite response restores route reciprocity. The second asks whether that reciprocity is realized by the required physical bound state and couples nontrivially to the mass-plane operator. Both levels can, in principle, be measured in a single consistent nonperturbative calculation.

Appendix B.11.1. Falsifiability, Present Computability, and the Meaning of the Test

Three levels must be kept distinct.

1.

Well-posed physical criterion. The theory supplies explicitly defined observables—the renormalized route matrix, its source metric, the pole position relative to threshold, and two matching form factors—together with explicit acceptance and exclusion conditions. A failure of any mandatory condition excludes the proposed unchanged route mechanism. This is the sense in which the proposal is falsifiable.

2.

Presently computable necessary information. Reduced Hamiltonians, positive moment problems, block-Krylov spaces, finite-time inequalities, and controlled analogue models can already test algebraic consistency and necessary spectral features. They can reveal an obstruction, but agreement at this level is not a substitute for the physical $\mathrm{SU}{\left(15\right)}_p$ correlator.

3.

Decisive first-principles calculation. This requires a regulator specialized to the chiral field content, a renormalized mixing basis containing the point-split sources, a controlled nonperturbative ensemble, continuum and finite-volume analysis, and the amputated three-point vertex. These ingredients are not constructed in the present work. No claim is made that the final numerical test is presently available as a complete first-principles calculation.

Thus practical difficulty does not make the criterion logically unfalsifiable, but it does limit the present empirical force of the program. The results established here should be read as a reduction of the physical question to explicit observables and stopping rules, not as the reported outcome of the final calculation.

Appendix B.11.2. Predefined Physical Decision Criteria for a Future SU (15) p Calculation

To prevent a posteriori adjustment of the criterion, the logical decision structure can be fixed before any physical route-correlator data are generated. Numerical tolerances must subsequently be derived from regulator, finite-volume, statistical, and continuum uncertainties; an arbitrary percentage tolerance is not part of the theory. The required criteria are:

Stage	Quantity fixed or measured	Predefined decision
Operator definition	Renormalized two-route source doublet and its complete mixing basis	If the proposed route-odd source becomes null or cannot be separated from lower-dimensional mixing in the continuum definition, the stated source realization fails.
Canonical route normalization	Positive source metric G for independently defined route sources and the whitened route basis	Require a non-null two-source sector and controlled canonical normalization. After whitening define only the diagnostic label exchange $U_{\mathrm{lab}}={\sigma }_x$ and measure $K_{\perp }=\frac{1}{2}(K-U_{\mathrm{lab}}K{U}_{\mathrm{lab}})$; $U_{\mathrm{lab}}$ is not interpreted as a microscopic Ward symmetry.
Dynamical reciprocity and gap protection	$K^{\mathrm{ren}}(E;\mu )$, ${\Delta }_{-}^{\left(15\right)}$, $\parallel K_{\perp }\parallel$, and ${\eta }_{15}=\parallel K_{\perp }\parallel /{\Delta }_{-}^{\left(15\right)}$	Exact on-shell reciprocity requires the relevant route-odd kernel component to vanish after controlled limits; approximate emergent reciprocity requires a declared small ${\eta }_{15}$ that decreases toward the infrared. If a Schur–Feshbach completion is used, its equivalent locking condition is ${\chi }_z\left(E_{*}\right)=D_{\mathrm{rt}}^{\mathrm{ren}}(E_{*};\mu )$.
Nonlocal source-level recoupling	$q_4=\sqrt{8190}/193$ and $\langle X|T_2·T_3|D\rangle =(N^2-1)/\left(2N^2\right)$	These are exact $O\left(1\right)$ source/channel group-theory factors in their stated metrics, not measurements of $\parallel K_{\perp }\parallel$. A physical use requires the route-distinguishing Wilson/Lorentz/renormalization coefficient in the same four-dimensional scheme.
Isolated odd-pole branch	Route-resolved spectrum and independently determined threshold	If this stronger pole-mediated branch is invoked, require an isolated source-accessible odd state with $E_{-}<E_{\mathrm{th}}$ and measure the ordering relative to the lowest even state. A completion-level existence proposition cannot replace this measurement.
Finite-data certificate	Matrix moments or whitened route correlators at several Euclidean times	A certificate such as $n_{+}({\mathsf{T}}_t-I)>0$ or a block-Hankel root above one is sufficient evidence for subthreshold support. Failure of a low-order sufficient certificate is recorded as inconclusive unless flat closure has been established.
Three-point matching	Amputated, mixing-subtracted $F_{+}\left(0\right)$ and $F_{-}\left(0\right)$	Require $F_{+}\left(0\right)\ne 0$ and $F_{-}\left(0\right)=0$ in the low-energy reciprocity/matching limit. If $F_{+}=0$, the pole does not generate the plane; if $F_{-}\ne 0$, the required low-energy route selection is not realized.
Controlled limits	Regulator, volume, time-window, operator-basis, and continuum studies	Every preceding condition must persist under the declared limits. A result that exists only for one smearing, one basis, or one cutoff is not classified as a physical result satisfying the criterion.

These criteria intentionally separate absence of evidence from evidence of failure. In particular, ${\theta }_N\le 1$ at finite N does not exclude a weak bound state, whereas $E_{-}\ge E_{\mathrm{th}}$ after controlled extrapolation does exclude the isolated-pole branch. Likewise, a scalar spectral reconstruction may remain valid while the route-parity criterion is not satisfied; that outcome would preserve the five-dimensional scalar representation but reject its identification with the required microscopic route symmetry.

Appendix B.11.3. Kernel-level Susceptibility Criterion

In the canonically normalized unprojected route basis one should reconstruct the energy-dependent hidden susceptibility $M\left(E\right)$, or equivalently the corresponding shift of the inverse kernel. Exact route reciprocity in a real basis requires only

$${\chi }_z\left(E\right)=M_{11}\left(E\right)-M_{22}\left(E\right)=D_{\mathrm{rt}}^{\mathrm{ren}}(E;\mu ),$$

where the mismatch must be reconstructed from the renormalized visible kernel in the same source basis and scheme. The number

$$D_{\mathrm{rt}}^{\mathrm{bench}}=0.049406060606061$$

belongs only to the declared reproducible benchmark and is not inserted as the physical renormalized mismatch. A stronger robustness diagnostic, not mandatory for a narrow on-shell mechanism, is ${\chi }_z^{\prime }\left(E_{*}\right)$. If ${\chi }_z^{\prime }\left(E_{*}\right)\simeq 0$, a two-pole completion suppresses the leading off-shell leakage to quadratic order. If the route basis is not manifestly real and its phase convention is microscopically fixed, one must also test

$$\Im M_{12}\left(E\right)=0$$

or derive an independent symmetry argument enforcing it. Then

$${\chi }_x\left(E\right)=\Re M_{12}\left(E\right),f\left(E\right)=C_{\mathrm{rt}}-{\chi }_x\left(E\right)$$

determine the parity splitting of the reduced kernel.

Remark.This test does not require resolving each hidden resonance separately. What is needed is the integrated matrix spectral response in the two visible route channels. If ${\chi }_z=D_{\mathrm{rt}}$, the route diagonals are exactly equal. The four-state completion proposition proves that $E_{-}<E_{+}$ is structurally realizable as an exact nondegenerate ground-state ordering in a positive-gap completion family. In the actual confining theory, however, the inequality $E_{-}<E_{+}$ must still be checked directly: the problem is no longer existence but phase identification.

Appendix B.11.4. Canonical Route-Label Symmetry Test in the Unprojected Basis

For the route sources $(J_1,J_2)$ compute

$$C^{AB}\left(t\right)=\int d^{3}x{\langle J_A(t,\mathbf{x})J_B^{†}\left(0\right)\rangle }_c.$$

Exact exchange symmetry of the renormalized two-source response in the canonical label basis requires

$$C\left(t\right)={\sigma }_{x}C\left(t\right){\sigma }_x,$$

i.e.,

$$C^{11}=C^{22},C^{12}=C^{21}.$$

This is a diagnostic condition on the response, not a microscopic Ward identity. Moreover, for a real symmetric transfer kernel and real sources the off-diagonal equality $C^{12}=C^{21}$ is kinematic, so the nontrivial equal-cost exchange diagnostic is the diagonal equality $C^{11}=C^{22}$. The unprojected doublet is essential. For the already projected $J_{-}$, the identity $J_{-}(-y)=-J_{-}\left(y\right)$ builds part of the parity structure in kinematically and therefore cannot by itself test whether the unprojected dynamics approaches the exchange-symmetric subspace.

Appendix B.11.5. Bound-Pole Criterion

For the projected odd correlator,

$$C_{--}\left(t\right)=\langle J_{-}\left(t\right)J_{-}^{†}\left(0\right)\rangle ,$$

define the effective energy

$$E_{-}^{\mathrm{eff}}\left(t\right)={\displaystyle \frac{1}{a_E}}ln{\displaystyle \frac{C_{--}\left(t\right)}{C_{--}(t+a_E)}}⟶E_{-}.$$

The continuum threshold $E_{\mathrm{th}}$ must be determined independently in the same finite-volume and renormalization setup. The decisive inequality is

$$E_{-}<E_{\mathrm{th}}.$$

If it fails after controlled volume and continuum extrapolations, the minimal isolated-pole four-dimensional mechanism fails under its stated assumptions.

Appendix B.11.6. Optimal Finite-Time Certificate for Subthreshold Support

The asymptotic inequality $E_{-}<E_{\mathrm{th}}$ is physically transparent, but an actual Euclidean calculation never reaches $t=\infty$. The sharper question is a sharper question: what is the strongest logically valid statement that can be extracted from finitely many positive Euclidean correlator values without a one-pole assumption?

Spectral positivity gives

$$C_{--}\left(t\right)={\int }_0^{\infty }{e}^{-Et}d{\mu }_{-}\left(E\right),d{\mu }_{-}\left(E\right)\ge 0.$$

Fix a Euclidean time step $\Delta t_E>0$, a reference time t, and an independently known physical threshold $E_{\mathrm{th}}$. Define

$${\xi }_E=e^{-\Delta t_E(E-E_{\mathrm{th}})}.$$

Then

$$E<E_{\mathrm{th}}⟺{\xi }_E>1.$$

Absorbing the positive factor $e^{-Et}$ into a new positive measure $d{\nu }_t\left({\xi }_E\right)$, define

$$m_n\left(t\right)=e^{n\Delta t_E{E}_{\mathrm{th}}}{C}_{--}(t+n\Delta t_E)={\int }_0^{\infty }{\xi }_E^{n}d{\nu }_t\left({\xi }_E\right),n=0,1,\dots .$$

A finite set of correlator values is thus converted into a truncated positive moment problem.

From $m_0,\dots ,m_{2N-1}$ construct

$${\left(H_0^{\left(N\right)}\right)}_{ij}=m_{i+j},{\left(H_1^{\left(N\right)}\right)}_{ij}=m_{i+j+1},i,j=0,\dots ,N-1.$$

For exact positive data, $H_0^{\left(N\right)}⪰0$. First assume it is positive definite and define

$${\theta }_N={\lambda }_{max}(H_1^{\left(N\right)},H_0^{\left(N\right)}).$$

Appendix B.11.11.1. Step 1: every compatible positive spectrum has upper support at least θ N .

For

$$p\left({\xi }_E\right)={\displaystyle \sum _{j=0}^{N-1}}{v}_j{\xi }_E^j,$$

one has

$$v^{†}{H}_0^{\left(N\right)}v=\int |p\left({\xi }_E\right){|}^{2}d\nu \left({\xi }_E\right),v^{†}{H}_1^{\left(N\right)}v=\int {\xi }_E{\left|p\left({\xi }_E\right)\right|}^{2}d\nu \left({\xi }_E\right).$$

If ${\xi }_{max}\left(\nu \right)=supsupp\nu$, then

$${\displaystyle \frac{v^{†}{H}_1^{\left(N\right)}v}{v^{†}{H}_0^{\left(N\right)}v}}\le {\xi }_{max}\left(\nu \right).$$

Maximizing over $v\ne 0$ yields

$${\theta }_N\le {\xi }_{max}\left(\nu \right)$$

for every positive measure reproducing the same moments.

Appendix B.11.11.2. Step 2: the bound is attainable.

The positive truncated moment functional admits an N-node Gaussian quadrature measure

$$d{\nu }_G\left({\xi }_E\right)={\displaystyle \sum _{j=1}^N}{w}_j\delta ({\xi }_E-{\theta }_j)d{\xi }_E,w_j>0,$$

where the nodes ${\theta }_j$ are the generalized eigenvalues of the same Hankel pair. The quadrature reproduces all moments through degree $2N-1$, so it belongs to the same compatibility class and has

$${\xi }_{max}\left({\nu }_G\right)={\theta }_N.$$

Therefore

$${\theta }_N=\underset{\nu \in {\mathfrak{M}}_N}{inf}supsupp\nu ,$$

where ${\mathfrak{M}}_N$ is the set of all positive Borel measures reproducing the known $2N$ moments.

Define the binding energy relative to threshold by

$$B\left(\nu \right)={\displaystyle \frac{1}{\Delta t_E}}max\{0,ln{\xi }_{max}\left(\nu \right)\}.$$

The sharp finite-data bound is

$$B_{\mathrm{sharp}}^{\left(N\right)}={\displaystyle \frac{1}{\Delta t_E}}max\{0,ln{\theta }_N\}.$$

Thus

$${\theta }_N>1$$

is a necessary and sufficient condition for the given finite data plus positivity themselves to force subthreshold spectral support. If ${\theta }_N\le 1$, there exists an explicit positive Gaussian-quadrature spectrum with no subthreshold state that reproduces exactly the same $2N$ moments. The true theory may still contain a pole, but those finite data alone do not prove it.

Appendix B.11.11.3. Optimal Euclidean filter.

Let $v_N$ be a generalized eigenvector for ${\theta }_N$, normalized by $v_N^{†}{H}_0^{\left(N\right)}{v}_N=1$, and define

$$p_N\left({\xi }_E\right)={\displaystyle \sum _{j=0}^{N-1}}{\left(v_N\right)}_j{\xi }_E^j.$$

Then

$$\int ({\xi }_E-1){\left|p_N\left({\xi }_E\right)\right|}^{2}d\nu \left({\xi }_E\right)={\theta }_N-1.$$

Thus $p_N$ is not merely diagnostic. It defines the degree-$(N-1)$ linear combination of Euclidean time shifts that maximizes the subthreshold excess relative to the chosen threshold.

If $H_0^{\left(N\right)}$ is rank deficient, the same construction is carried out after quotienting by its null space, i.e., on its positive numerical rank. This is important because a finite discrete spectrum can be recovered exactly once the Hankel rank saturates.

Optimal theorem for finite Euclidean data.Given only $2N$ exact threshold-normalized Euclidean moments, positivity, and a known threshold $E_{\mathrm{th}}$, ${\theta }_N$ is the optimal lower bound on the upper edge of the spectral support in the variable ${\xi }_E$. No method using only the same information can guarantee a larger minimal binding. Failure of ${\theta }_N>1$ means that the finite information is insufficient; it does not prove the absence of a physical pole.

Appendix B.11.11.4. Matrix generalization and source-basis invariance.

For several physical sources, replace scalar moments by Hermitian matrices

$${\mathsf{M}}_n={\int }_0^{\infty }{\xi }_E^{n}d\Sigma \left({\xi }_E\right),d\Sigma \left({\xi }_E\right)⪰0,$$

where $d\Sigma$ is a positive-semidefinite matrix-valued spectral measure and each ${\mathsf{M}}_n$ is $N_s\times N_s$ for $N_s$ independent sources. Construct the block-Hankel matrices

$${\mathbb{H}}_0={\left[{\mathsf{M}}_{i+j}\right]}_{i,j=0}^{N-1},{\mathbb{H}}_1={\left[{\mathsf{M}}_{i+j+1}\right]}_{i,j=0}^{N-1}.$$

On the positive-rank subspace of ${\mathbb{H}}_0$, define

$${\Theta }_{N,N_s}={\lambda }_{max}({\mathbb{H}}_1,{\mathbb{H}}_0).$$

For every vector polynomial $p\left({\xi }_E\right)$ of degree less than N,

$${\displaystyle \frac{p^{†}{\mathbb{H}}_{1}p}{p^{†}{\mathbb{H}}_{0}p}}={\displaystyle \frac{\int {\xi }_{E}p{\left({\xi }_E\right)}^{†}d\Sigma \left({\xi }_E\right)p\left({\xi }_E\right)}{\int p{\left({\xi }_E\right)}^{†}d\Sigma \left({\xi }_E\right)p\left({\xi }_E\right)}}\le supsupp\Sigma .$$

To see attainability without importing a scalar argument by analogy, quotient the vector-polynomial space of degree $<N$ by the null space of ${\mathbb{H}}_0$ and use ${\mathbb{H}}_0$ as its positive inner product. Multiplication by ${\xi }_E$, compressed to this finite block-Krylov space, is represented by the self-adjoint matrix

$${\mathsf{T}}_N={\mathbb{H}}_0^{-1/2}{\mathbb{H}}_1{\mathbb{H}}_0^{-1/2}$$

on the positive-rank support. Its spectral theorem, together with the source block at polynomial degree zero, defines a positive finite atomic matrix-valued measure. The block-Krylov moment identities reproduce the supplied moments through degree $2N-1$, while the largest atom is

$${\lambda }_{max}\left({\mathsf{T}}_N\right)={\Theta }_{N,N_s}.$$

Hence the Rayleigh upper-support bound is attained within the same truncated matrix-moment class, and

$${\Theta }_{N,N_s}=\underset{\Sigma \mathrm{compatible}}{inf}supsupp\Sigma .$$

This is the exact matrix analogue of the scalar extremal theorem for exact positive block moments after quotienting any Hankel null space.

The criterion is invariant under arbitrary nonsingular mixing of sources. If

$$J_{\alpha }^{\prime }=S_{\alpha \beta }{J}_{\beta },S\in GL(N_s,\mathbb{C}),$$

then ${\mathsf{M}}_n^{\prime }=S{\mathsf{M}}_n{S}^{†}$ and the block pencils transform by simultaneous congruence. Their generalized eigenvalues, including ${\Theta }_{N,N_s}$, are unchanged. Thus the sharp certificate is invariant under nonsingular matrix renormalization, changes of interpolating-operator basis, and numerical preconditioning.

Exact matrix finite-data certificate.For exact block moments and a positive matrix-valued spectral measure, ${\Theta }_{N,N_s}>1$ is necessary and sufficient for those finite data plus positivity to force source-accessible spectral support below $E_{\mathrm{th}}$. If ${\Theta }_{N,N_s}\le 1$, a compatible positive atomic matrix measure exists with no subthreshold support. Once block rank has saturated, the same moments reconstruct the source-accessible energies and pole-residue matrices exactly.

Appendix B.11.7. Total Subthreshold Weight and Number of Bound States from Finite Data

The condition ${\theta }_N>1$ answers whether finite positive Euclidean data already force some support below threshold. Two further statements provide information about total weight and multiplicity without assuming a unique pole.

Let $|\Phi \rangle$ be a normalized optimized state in a block-Krylov subspace generated by one or more renormalized sources and Euclidean time shifts. Define the threshold-shifted transfer operator

$$Y=e^{-a_E(H-E_{\mathrm{th}})},$$

where $a_E>0$ is a Euclidean time step. States below threshold have Y eigenvalues greater than one. Define

$${\mu }_Y=\langle \Phi \left|Y\right|\Phi \rangle ,{\sigma }_Y^2=\langle \Phi |Y^2|\Phi \rangle -{\mu }_Y^2,$$

and assume ${\mu }_Y>1$. Let

$$P_{>}={\mathbf{1}}_{(1,\infty )}\left(Y\right)$$

be the projector onto the entire subthreshold sector, and

$$w_{>}=\langle \Phi |P_{>}|\Phi \rangle$$

its total weight in the optimized state.

Set $m_Y={\mu }_Y-1>0$ and $X={\mu }_Y-Y$. Then

$$\langle X\rangle =0,\langle X^2\rangle ={\sigma }_Y^2,$$

while

$$Y\le 1⟺X\ge m_Y.$$

Cantelli’s one-sided inequality gives

$$Pr(X\ge m_Y)\le {\displaystyle \frac{{\sigma }_Y^2}{{\sigma }_Y^2+m_Y^2}}.$$

Therefore

$$w_{>}\ge {\displaystyle \frac{m_Y^2}{m_Y^2+{\sigma }_Y^2}}={\displaystyle \frac{{({\mu }_Y-1)}^2}{{({\mu }_Y-1)}^2+{\sigma }_Y^2}}.$$

This is a lower bound on the total weight of all states below threshold. It requires no unique-pole assumption. Only if uniqueness is established independently can $w_{>}$ be interpreted as the residue fraction of that single state.

The bound is optimal at fixed ${\mu }_Y$ and ${\sigma }_Y^2$. A positive two-atom measure with atoms at $Y=1$ and

$$Y_{★}={\mu }_Y+{\displaystyle \frac{{\sigma }_Y^2}{{\mu }_Y-1}}$$

saturates the inequality. No universally stronger lower bound follows from only these two moments.

For multiplicity, let the threshold-shifted block-Hankel Ritz values be

$${\Theta }_1\ge {\Theta }_2\ge \dots$$

and define

$$n_B=dimRan{P}_{>}$$

in the source-accessible subspace. The Courant–Fischer min–max principle gives

$${\Theta }_k>1⟹n_B\ge k.$$

If $n_B<k$, every k-dimensional trial subspace contains a nonzero vector orthogonal to the entire $Y>1$ spectral subspace, and on that vector the expectation value of Y is at most one, contradicting ${\Theta }_k>1$.

Finite-time certificates for existence, weight, and multiplicity.A single positive block-Hankel data set addresses three different questions. First, ${\Theta }_1>1$ certifies at least one source-supported subthreshold direction. Second, ${\Theta }_k>1$ certifies at least k independent subthreshold directions. Third, the first two moments of Y in an optimized normalized state give

$$w_{>}\ge {\displaystyle \frac{{({\mu }_Y-1)}^2}{{({\mu }_Y-1)}^2+{\sigma }_Y^2}}.$$

If ${\Theta }_2\le 1$ before Krylov rank has saturated, this does not prove uniqueness of the bound state. Exact counting requires closure of the source-accessible cyclic space or independent spectral information.

Appendix B.11.8. Flat Extension: When the Finite Moment Problem Becomes the Exact Physical Spectrum

The preceding moment tests are logically asymmetric. A Ritz value above one rigorously proves subthreshold support, but the absence of a second such Ritz value does not prove the absence of a second very weakly bound physical state. To obtain an exact upper multiplicity statement, one must test whether the full cyclic subspace visible to the chosen sources has already closed.

Let ${\Psi }_{\mathrm{src}}$ denote the block of source vectors and

$${\mathcal{K}}_N=span\{Y^j{\Psi }_{\mathrm{src}}:j=0,\dots ,N-1\}.$$

Define the rectangular block operator

$$V_N=({\Psi }_{\mathrm{src}},Y{\Psi }_{\mathrm{src}},\dots ,Y^{N-1}{\Psi }_{\mathrm{src}}),Ran{V}_N={\mathcal{K}}_N,$$

and the next block

$$W_N=Y^N{\Psi }_{\mathrm{src}}.$$

Then

$$V_N^{†}{V}_N={\mathbb{H}}_0^{\left(N\right)},V_N^{†}{W}_N={\mathsf{B}}_N,W_N^{†}{W}_N={\mathsf{M}}_{2N}.$$

The orthogonal projection of $W_N$ onto ${\mathcal{K}}_N$ is

$$P_N{W}_N=V_N{\left({\mathbb{H}}_0^{\left(N\right)}\right)}^{+}{\mathsf{B}}_N.$$

For the residual ${\mathsf{R}}_N=(I-P_N)W_N$,

$$\begin{array}{cc}\hfill {\mathsf{R}}_N^{†}{\mathsf{R}}_N& =W_N^{†}{W}_N-{\mathsf{B}}_N^{†}{\left({\mathbb{H}}_0^{\left(N\right)}\right)}^{+}{\mathsf{B}}_N\hfill \\ & ={\mathsf{M}}_{2N}-{\mathsf{B}}_N^{†}{\left({\mathbb{H}}_0^{\left(N\right)}\right)}^{+}{\mathsf{B}}_N\equiv {\mathsf{S}}_N⪰0.\hfill \end{array}$$

Thus

$${\mathsf{S}}_N=0⟺Y^N{\Psi }_{\mathrm{src}}\subset {\mathcal{K}}_N.$$

For all earlier generators, $Y\left(Y^j{\Psi }_{\mathrm{src}}\right)=Y^{j+1}{\Psi }_{\mathrm{src}}\in {\mathcal{K}}_N$ for $j=0,\dots ,N-2$. Therefore ${\mathsf{S}}_N=0$ implies

$$Y{\mathcal{K}}_N\subseteq {\mathcal{K}}_N.$$

Because ${\mathcal{K}}_N$ contains the initial sources and is Y-invariant, it contains every $Y^j{\Psi }_{\mathrm{src}}c$. If $\mathcal{C}$ denotes the full cyclic/source-supported subspace, then both $\mathcal{C}\subseteq {\mathcal{K}}_N$ and ${\mathcal{K}}_N\subseteq \mathcal{C}$, hence

$${\mathcal{K}}_N=\mathcal{C}.$$

Exact finite-moment closure theorem.The condition ${\mathsf{S}}_N=0$ is equivalent to Y-invariance of ${\mathcal{K}}_N$. In that case the finite block-Hankel problem is not merely variational: it is the exact representation of Y on the entire source-supported cyclic subspace. Its energies, residue matrices, route-resolved overlap data, and subthreshold multiplicities are therefore exact physical quantities within that accessible sector. Definite + or − route parity may be assigned only if an independently defined route involution is also shown to commute with the reconstructed transfer operator, equivalently with the closed correlator family. Here “exact physical means exact for the specified finite-volume/regulator transfer problem and the chosen source-supported cyclic subspace; continuum, infinite-volume, and renormalization limits remain separate controlled limits.

For the positive-semidefinite extended Gram matrix, the same condition is equivalent to flat rank extension,

$$rank{\mathbb{H}}_0^{(N+1)}=rank{\mathbb{H}}_0^{\left(N\right)}.$$

Thus, if ${\mathsf{S}}_N=0$ and exactly one exact generalized eigenvalue lies above one, the source-supported physical sector contains exactly one bound state below $E_{\mathrm{th}}$.

If ${\mathsf{S}}_N\ne 0$, the conclusion changes fundamentally. Finite moments do not fix the deep tail of the corresponding Jacobi operator: distinct positive completions can reproduce exactly the same known moments while containing different numbers of arbitrarily weak bound states. The number of Ritz nodes above threshold is then only a lower bound on physical multiplicity.

Approximate flatness nevertheless remains useful. In the scalar case let $p_N^{\mathrm{res}}\left(y\right)$ be the monic orthogonal residual polynomial. Then

$${\mathsf{S}}_N=\int {\left|p_N^{\mathrm{res}}\left(y\right)\right|}^{2}d\nu \left(y\right).$$

For any spectral region $\Omega$ with

$$d_{\Omega }=\underset{y\in \Omega }{inf}\left|p_N^{\mathrm{res}}\left(y\right)\right|>0,$$

one has

$${\mathsf{S}}_N\ge d_{\Omega }^2\nu \left(\Omega \right),\nu \left(\Omega \right)\le {\displaystyle \frac{{\mathsf{S}}_N}{d_{\Omega }^2}}.$$

Thus unsaturated finite data cannot exclude arbitrarily weak extra states, but they can rigorously bound their total source spectral weight in regions separated from the known Ritz roots. This is the genuine information boundary of finite-moment spectroscopy.

Appendix B.11.9. Three-site Control as a Finite-Moment Illustration

The 27-state three-site control above also gives a non-synthetic illustration of the finite-moment machinery, while remaining only a control model. Use the equal-cost Hamiltonian with ${\Delta }_F=1$, $t=0.30$, source $J_1=(1,0,0)$, threshold $E_{\mathrm{th}}=1$, and Euclidean step $\Delta t_E=0.10$. Define threshold-normalized scalar moments

$$m_n=e^{n\Delta t_E{E}_{\mathrm{th}}}\langle J_1|e^{-n\Delta t_{E}H}|J_1\rangle .$$

Direct evaluation gives

$$(m_0,m_1,m_2,m_3,m_4)=(1,1.001802029849,1.007232479627,1.016364662156,1.029321859462).$$

At order $N=1$,

$${\theta }_1=1.001802029849>1,$$

and at order $N=2$ the generalized eigenvalues are

$$0.948546411978,1.069873207366,$$

so

$${\theta }_2=1.069873207366>1.$$

Both orders therefore certify source-supported spectral weight below the chosen threshold in this finite control, consistently with its explicit spectrum. However the next-moment Schur residual is

$$S_2=3.55738185\times {10}^{-5}>0,$$

so the $N=2$ Krylov space has not closed and these moments do not certify uniqueness or reconstruct the full 27-state spectrum. This example is included only to demonstrate the distinction between a positive subthreshold certificate and flat-extension closure; it is not evidence for a physical $\mathrm{SU}{\left(15\right)}_p$ bound state.

Appendix B.11.10. Worked Two-Pole Example: Detection, Delayed Certification, and Exact Flat Closure

The following exact example shows how the preceding criteria work without invoking a fitted physical spectrum. In the threshold-normalized transfer variable Y, take the positive two-atom measure

$$d\nu \left(y\right)={\displaystyle \frac{1}{4}}\delta \left(y-{\displaystyle \frac{6}{5}}\right)dy+{\displaystyle \frac{3}{4}}\delta \left(y-{\displaystyle \frac{4}{5}}\right)dy.$$

The first atom represents one source-supported level below threshold because $6/5>1$; the second lies above threshold because $4/5<1$. The moments are

$$m_0=1,qquad{m}_1={\displaystyle \frac{9}{10}},qquad{m}_2={\displaystyle \frac{21}{25}},qquad{m}_3={\displaystyle \frac{102}{125}},qquad{m}_4={\displaystyle \frac{516}{625}}.$$

At Krylov order $N=1$, the only generalized Ritz value is

$${\theta }_1={\displaystyle \frac{m_1}{m_0}}={\displaystyle \frac{9}{10}}<1.$$

Thus the first two moments do not force a subthreshold state even though the generating measure contains one. This illustrates why failure of a low-order certificate is absence of information, not evidence of absence.

At order $N=2$ the Hankel pair is

$$H_0^{\left(2\right)}=\left(\begin{array}{cc}1& 9/10\\ 9/10& 21/25\end{array}\right),H_1^{\left(2\right)}=\left(\begin{array}{cc}9/10& 21/25\\ 21/25& 102/125\end{array}\right).$$

Direct evaluation gives

$$det\left(H_1^{\left(2\right)}-\theta H_0^{\left(2\right)}\right)={\displaystyle \frac{3}{100}}\left(\theta -{\displaystyle \frac{4}{5}}\right)\left(\theta -{\displaystyle \frac{6}{5}}\right).$$

Hence

$${\theta }_2={\displaystyle \frac{6}{5}}>1,B_{\mathrm{sharp}}^{\left(2\right)}={\displaystyle \frac{1}{\Delta t_E}}ln{\displaystyle \frac{6}{5}},$$

and the four moments through $m_3$ now certify subthreshold support.

The next-moment Schur residual also vanishes exactly. With

$$B_2=\left(\begin{array}{c}{m}_2\\ m_3\end{array}\right),$$

one finds

$$S_2=m_4-B_2^T{\left(H_0^{\left(2\right)}\right)}^{-1}{B}_2=0.$$

Therefore the order-two Krylov space is the full cyclic space of this source. The two generalized eigenvalues are the exact support points $4/5$ and $6/5$, and the accessible spectrum contains exactly one subthreshold state. The example separates three statements that should not be conflated in numerical work: a physical state may be present before low-order moments certify it; ${\theta }_N>1$ provides a rigorous existence certificate; and exact counting becomes possible only after the independent flat-extension test closes the cyclic space.

Figure A3. Exact behavior of the scalar two-pole validation example. At $N=1$, the state is present but not certified because ${\theta }_1=0.9<1$, and the moment problem has not closed because $S_1=0.03$. At $N=2$, ${\theta }_2=1.2$ certifies subthreshold support and $S_2=0$ proves flat closure.

Figure A3. Exact behavior of the scalar two-pole validation example. At $N=1$, the state is present but not certified because ${\theta }_1=0.9<1$, and the moment problem has not closed because $S_1=0.03$. At $N=2$, ${\theta }_2=1.2$ certifies subthreshold support and $S_2=0$ proves flat closure.

Appendix B.11.11. Exact Finite-Dimensional Validation of the Route-Resolved Moment/ Krylov Construction

The scalar two-pole example can be lifted to an exact matrix-valued test that checks not only subthreshold support but also route parity. This is a validation data set with a known answer, not a surrogate for an $\mathrm{SU}{\left(15\right)}_p$ ensemble. Let

$$P_{+}=e_{+}{e}_{+}^{†}={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right),P_{-}=e_{-}{e}_{-}^{†}={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right),P_{\mathrm{rt}}={\sigma }_x,$$

and choose one route-even level above threshold and one route-odd level below threshold. In the threshold-normalized transfer variable their exact matrix moments are

$${\mathsf{M}}_n={\displaystyle \frac{3}{4}}{\left({\displaystyle \frac{4}{5}}\right)}^n{P}_{+}+{\displaystyle \frac{1}{4}}{\left({\displaystyle \frac{6}{5}}\right)}^n{P}_{-}.$$

The first three matrices in the original route basis are

$${\mathsf{M}}_0=\left(\begin{array}{cc}\frac{1}{2}& \frac{1}{4}\\ \frac{1}{4}& \frac{1}{2}\end{array}\right),{\mathsf{M}}_1=\left(\begin{array}{cc}\frac{9}{20}& \frac{3}{20}\\ \frac{3}{20}& \frac{9}{20}\end{array}\right),{\mathsf{M}}_2=\left(\begin{array}{cc}\frac{21}{50}& \frac{3}{50}\\ \frac{3}{50}& \frac{21}{50}\end{array}\right).$$

They are positive definite, and the canonically whitened one-step transfer matrix

$$\mathsf{T}={\mathsf{M}}_0^{-1/2}{\mathsf{M}}_1{\mathsf{M}}_0^{-1/2}$$

has the exact spectral decomposition

$$\mathsf{T}={\displaystyle \frac{4}{5}}{P}_{+}+{\displaystyle \frac{6}{5}}{P}_{-},spec\mathsf{T}=\left\{{\displaystyle \frac{4}{5}},{\displaystyle \frac{6}{5}}\right\}.$$

Consequently

$$n_{+}(\mathsf{T}-I)=1,P_{\mathrm{rt}}{e}_{+}=+e_{+},P_{\mathrm{rt}}{e}_{-}=-e_{-}.$$

The reconstruction must therefore yield exactly one source-accessible subthreshold direction and identify it as route odd. The reconstructed energies are

$$E_{-}=E_{\mathrm{th}}-{\displaystyle \frac{1}{\Delta t_E}}ln{\displaystyle \frac{6}{5}}<E_{\mathrm{th}},E_{+}=E_{\mathrm{th}}-{\displaystyle \frac{1}{\Delta t_E}}ln{\displaystyle \frac{4}{5}}>E_{\mathrm{th}},$$

with residue matrices $Z_{-}=(1/4)P_{-}$ and $Z_{+}=(3/4)P_{+}$. All whitened matrices generated by this measure commute and share the independently specified reflection axis. Thus the exact common-axis residual and the odd-parity residual both vanish.

Taking only the trace gives

$$Tr{\mathsf{M}}_n={\displaystyle \frac{3}{4}}{\left({\displaystyle \frac{4}{5}}\right)}^n+{\displaystyle \frac{1}{4}}{\left({\displaystyle \frac{6}{5}}\right)}^n,$$

which is precisely the preceding scalar example. It detects the two support points after sufficient moments but contains no information telling which point is route odd. The matrix lift therefore tests the full logical distinction used in the paper: scalar moments can certify subthreshold support, whereas route-resolved data together with an independently defined $P_{\mathrm{rt}}$ are required to certify route parity. This exact data set supplies deterministic consistency checks for positivity, whitening, generalized eigenvalues, parity assignment, residues, and flat closure before any physical correlator is analyzed.

Appendix B.11.12. Three-Point Matching

Define connected three-point functions for ${\mathcal{O}}_{\pm }$ and, after state isolation and external-leg amputation, the corresponding zero-momentum-transfer form factors $F_{\pm }\left(0\right)$. Exact route symmetry requires

$$F_{-}\left(0\right)=0,$$

while physically nontrivial matching to the mass-plane operator requires

$$F_{+}\left(0\right)\ne 0.$$

Normalized ratios can be chosen so that the arbitrary source rescaling $J_{-}\to \lambda J_{-}$ cancels.

Final four-dimensional verification criterion.After regulator, operator-mixing, finite-volume, and continuum checks, a consistent set of conditions must be satisfied. At the kernel level,

$${\chi }_z\left(E\right)=D_{\mathrm{rt}},\Im M_{12}\left(E\right)=0\mathrm{if}\mathrm{not}\mathrm{enforced}\mathrm{by}\mathrm{the}\mathrm{microscopic}\mathrm{phase}\mathrm{convention},$$

and the measured sign inheritance must be compatible with $E_{-}<E_{+}$. At the directly observable level,

$${ϵ}_R\to 0,E_{-}<E_{\mathrm{th}},F_{+}\left(0\right)\ne 0,F_{-}\left(0\right)=0.$$

If ${\chi }_z$ systematically fails to reach $D_{\mathrm{rt}}$, the unchanged-field Feshbach route-locking condition is not realized. If $E_{-}\ge E_{\mathrm{th}}$ or $F_{+}\left(0\right)=0$, the minimal isolated-pole/matching mechanism fails under the stated assumptions. If $|{\chi }_z^{\prime }\left(E_{*}\right)|$ is large, the quadratically stable finite-window branch fails that stronger robustness test even though exact on-shell locking at $E_{*}$ can remain possible.

Appendix C.1. Five-Dimensional Composite Branch and Quotient Space: What They Actually Add

Reader checkpoint: interpretation of the fifth coordinate.The fifth coordinate in this section is an effective Jacobi/Stieltjes spectral coordinate on a cyclic composite sector. None of the 4D identities requires it to be a literal spacetime dimension. The reconstruction theorems below are exact only in their stated Jacobi–Robin classes; route parity still requires route-resolved matrix information.

Once the four-dimensional operator space already contains an explicit rank-one sign source, a fifth dimension is no longer needed to manufacture a parity for the fundamental preons. This changes the role of the additional coordinate in an essential way. In the most economical interpretation, the fifth coordinate is attached only to a gauge-singlet composite routing channel and represents a collective, spectral, or Krylov coordinate rather than a new fundamental spacetime direction.

What the 5D construction does not add.The Jacobi coordinate does not alter the exact identity ${\Delta }_{\Pi }=C_{21}-C_{12}$, increase its statistical significance, derive the family alignment, or replace the missing $\mathrm{SU}{\left(15\right)}_p$ correlator. Its role is narrower and useful: it supplies a local representation of a positive spectral measure and converts endpoint spectral data into a falsifiable reconstruction problem. If the one-band or locality tests are not satisfied, the simple 5D representation is excluded; the 4D operator identity remains intact.

Appendix C.12.1. Limitation of an Intrinsic Parity Acting on the Fundamental Preons

In a literal five-dimensional bulk-preon construction, an intrinsic orbifold parity would have to satisfy three requirements simultaneously:

1.

preserve the canonical chiral zero-mode spectrum of the Pati–Salam theory;

2.

avoid unwanted light mirror modes;

3.

realize the route exchange ${\mathcal{O}}_{12}\leftrightarrow {\mathcal{O}}_{21}$.

For the published irreducible preon representations these requirements are incompatible. The parity assignments that retain all required zero modes act trivially on the relevant primitive labels and therefore do not generate the needed route exchange. In higher-dimensional orbifold theories localized anomalies may be present even when the integrated four-dimensional anomaly sum vanishes, so any literal boundary/orbifold completion also requires a separate local-anomaly analysis [17,18].

Local no-go theorem and its domain of validity.An intrinsic five-dimensional parity of the published fundamental preons cannot be the origin of the required route reciprocity while preserving the canonical chiral zero-mode spectrum. This statement remains valid for that literal construction.

How the obstruction is bypassed.Composite five-dimensional completion. The fundamental preons are not assigned a new five-dimensional route parity. They remain four-dimensional. The extra coordinate is introduced only for the already gauge-singlet 24-preon route channel and is interpreted as a collective or spectral coordinate. Rank-one projection, exact Schur–Feshbach reduction, positivity of the Jacobi–Stieltjes representation, and quotienting by route reflection then produce the crossed-even relation $C_{12}=C_{21}$. The assumption that the five-dimensional parity must act directly on the fundamental preons is therefore not used.

Appendix C.12.2. Composite Quotient Space as the Minimal Interpretation

Introduce a collective relative coordinate y only for the gauge-singlet 24-preon routing channel. The physical low-energy sector is selected by the rank-one projector

$$P_{\mathrm{rt}}^{-}=e_{-}{e}_{-}^{†}={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right),e_{-}\equiv {\displaystyle \frac{1}{\sqrt{2}}}\left({\displaystyle \genfrac{}{}{0pt}{}{1}{-1}}\right).$$

Let $\zeta >0$ be a Euclidean spectral parameter and let $C\left(\zeta \right)$ denote the unprojected connected resolvent matrix of the two route sources on the covering space. Assume only positivity,

$$C\left(\zeta \right)⪰0,$$

and do not assume exact route symmetry of this unprojected matrix. Rank-one algebra then gives

$$P_{\mathrm{rt}}^{-}C\left(\zeta \right)P_{\mathrm{rt}}^{-}=\left(e_{-}^{†}C\left(\zeta \right)e_{-}\right)P_{\mathrm{rt}}^{-}\equiv c_{-}\left(\zeta \right)P_{\mathrm{rt}}^{-}.$$

If the route-odd source $J_{-}$ does not annihilate the vacuum, the scalar function $c_{-}\left(\zeta \right)$ has a positive Stieltjes representation,

$$c_{-}\left(\zeta \right)={\int }_0^{\infty }{\displaystyle \frac{d{\mu }_{-}\left(s\right)}{\zeta +s}},d{\mu }_{-}\left(s\right)\ge 0,$$

where $s\ge 0$ is the spectral variable and $d{\mu }_{-}\left(s\right)$ is the positive spectral measure seen by $J_{-}$. Consequently,

$$c_{-}\left(\zeta \right)>0(\zeta >0).$$

Exact rank-one quotient statement.If the physical low-energy completion is defined by an exact one-dimensional quotient followed by Schur–Feshbach reduction, then the projected propagator automatically has the route-odd rank-one structure for any positive dynamics on the covering space. Equalities such as $C_{11}=C_{22}$ and $C_{12}=C_{21}$ need not hold for the unprojected covering-space matrix in order for the quotient completion itself to be internally consistent. They become necessary only for the stronger claim that the quotient emerges dynamically from the unchanged four-dimensional theory without an additional low-energy datum.

Appendix C.12.3. Exact Feshbach Reduction

Simply discarding the orthogonal route channel would be incorrect if the full Hermitian route operator K mixes the odd subspace with its complement. Define

$$Q_{\mathrm{rt}}\equiv I_2-P_{\mathrm{rt}}^{-}.$$

The exact Schur–Feshbach operator acting on the one-dimensional $P_{\mathrm{rt}}^{-}$ sector is

$$K_{\mathrm{eff}}\left(E\right)=P_{\mathrm{rt}}^{-}K{P}_{\mathrm{rt}}^{-}+P_{\mathrm{rt}}^{-}K{Q}_{\mathrm{rt}}{\left(E-Q_{\mathrm{rt}}K{Q}_{\mathrm{rt}}\right)}^{-1}{Q}_{\mathrm{rt}}K{P}_{\mathrm{rt}}^{-},$$

where E is the spectral parameter. Because $P_{\mathrm{rt}}^{-}$ has rank one, $K_{\mathrm{eff}}\left(E\right)$ is necessarily a scalar operator on the physical route-odd channel. The quotient closure is therefore exact even when the raw dynamics on the covering space is not block diagonal in the even/odd basis.

Appendix C.12.4. Stieltjes–Jacobi Reconstruction and an Emergent Fifth Coordinate

Every positive scalar spectral measure $d\mu \left(s\right)$ defines a Stieltjes function

$$c\left(\zeta \right)=\int {\displaystyle \frac{d\mu \left(s\right)}{\zeta +s}},\zeta >0.$$

Starting from the normalized cyclic vector $|1\rangle$, the Lanczos algorithm constructs a tridiagonal Jacobi operator

$$H_J=\left(\begin{array}{cccc}{\alpha }_1& -{\beta }_1& 0& \dots \\ -{\beta }_1& {\alpha }_2& -{\beta }_2& \dots \\ 0& -{\beta }_2& {\alpha }_3& \ddots \\ ⋮& ⋮& \ddots & \ddots \end{array}\right),{\beta }_n>0,$$

where ${\alpha }_n\in \mathbb{R}$ are diagonal Jacobi coefficients and ${\beta }_n>0$ are nearest-neighbor links. The common minus sign of the off-diagonal entries is a phase convention. The endpoint resolvent exactly reproduces the original Stieltjes function,

$$c\left(\zeta \right)=\langle 1\left|{(\zeta I+H_J)}^{-1}\right|1\rangle .$$

Thus a one-dimensional local chain is an exact spectral representation of positive cyclic dynamics, not an arbitrarily imposed geometric analogy.

For an N-site truncation denote the Jacobi matrix by $H_J^{\left(N\right)}$ and define

$${\mathsf{A}}_J^{\left(N\right)}\left(\zeta \right)\equiv \zeta I_N+H_J^{\left(N\right)}.$$

It is an irreducible symmetric Stieltjes matrix, so its inverse is entrywise positive. In particular,

$${\left[{\mathsf{A}}_J^{\left(N\right)}{\left(\zeta \right)}^{-1}\right]}_{1N}={\displaystyle \frac{{\beta }_1{\beta }_2\dots {\beta }_{N-1}}{det{\mathsf{A}}_J^{\left(N\right)}\left(\zeta \right)}}>0.$$

Positive spectral dynamics therefore supports nonzero transfer from the endpoint source to every site reached by the Jacobi chain. In a smooth many-site limit the chain may be described by a semi-infinite continuous coordinate.

Interpretation.This is the most economical meaning of the “fifth dimension” in the present construction. It need not be a fundamental spacetime coordinate. It may instead be a local coordinate on the spectral complexity of the composite state, with the Jacobi-site number playing the role of an effective radial or orbital coordinate.

Appendix C.12.5. From the Full Endpoint Response to a One-Spectrum Robin Theorem

After constructing the quotient, one still has to determine whether the Jacobi description can be converted into a quantitative local boundary test without arbitrarily splitting the answer into “bulk” and “boundary” pieces. Two distinctions are essential. First, the first Jacobi site is exactly the normalized physical cyclic source. Second, the full inverse endpoint response cannot by itself be called a continuum Robin coefficient because it contains both the bulk Dirichlet-to-Neumann contribution and a genuine endpoint defect. In the minimal asymptotically homogeneous one-band class, however, both pieces can be reconstructed from the same physical spectrum.

Appendix C.12.12.1. Minimal one-band class with one Robin endpoint defect

Consider the semi-infinite Jacobi operator above and assume

$${\alpha }_n={\alpha }_{\infty }(n\ge 2),{\beta }_n={\beta }_{\infty }(n\ge 1),$$

while only the first diagonal entry differs from the matched Neumann endpoint value,

$${\alpha }_1={\alpha }_{\infty }-{\beta }_{\infty }+c_J.$$

The discrete endpoint defect is therefore defined by

$$c_J={\alpha }_1-({\alpha }_{\infty }-{\beta }_{\infty }).$$

This is a definition, not a fit relation.

Appendix C.12.12.2. Bulk coefficients from the essential spectral band

For the homogeneous tail, a plane-wave ansatz ${\psi }_n\propto e^{ikn}$ gives

$$s\left(k\right)={\alpha }_{\infty }-2{\beta }_{\infty }cosk,0\le k\le \pi .$$

Hence the essential spectrum is

$$[s_{-},s_{+}]=[{\alpha }_{\infty }-2{\beta }_{\infty },{\alpha }_{\infty }+2{\beta }_{\infty }],$$

which implies

$${\alpha }_{\infty }={\displaystyle \frac{s_{+}+s_{-}}{2}},{\beta }_{\infty }={\displaystyle \frac{s_{+}-s_{-}}{4}}.$$

A compact endpoint perturbation does not change this essential band, so the band edges determine the asymptotic bulk operator without an independent fit.

For the normalized spectral measure $d\nu \left(s\right)$ of the physical endpoint source, the first Jacobi diagonal coefficient is the first spectral moment,

$${\alpha }_1=\langle e_1,H_J{e}_1\rangle =\int sd\nu \left(s\right)\equiv \langle s\rangle .$$

Therefore

$$c_J=\langle s\rangle -{\displaystyle \frac{s_{+}+3s_{-}}{4}}.$$

Within this minimal class, even the sign of the endpoint defect is determined by a single physical spectrum.

Appendix C.12.12.3. Why the continuum Robin parameter is independent of an auxiliary lattice spacing

To keep dimensions explicit, begin with a local five-dimensional quadratic action discretized with spacing $a_{★}$,

$$S_2={\displaystyle \frac{1}{2}}\int d^{4}x\sum _{n\ge 1}{a}_{★}{Z}_4\left[{\left({\partial }_{\mu }{\phi }_n\right)}^2+M_5^2{\phi }_n^2\right]+{\displaystyle \frac{1}{2}}\int d^{4}x\sum _{n\ge 1}{\displaystyle \frac{Z_y}{a_{★}}}{({\phi }_{n+1}-{\phi }_n)}^2+{\displaystyle \frac{1}{2}}\int d^{4}x{c}_R{\phi }_1^2.$$

Here $M_5$ is a common bulk mass parameter, $Z_4$ multiplies the four-dimensional kinetic term, $Z_y$ multiplies the derivative along the emergent coordinate, $a_{★}$ is the auxiliary discretization spacing, and $c_R$ is the continuum endpoint coefficient. Dividing by the common factor $a_{★}{Z}_4$ gives

$${\beta }_{\infty }={\displaystyle \frac{Z_y}{a_{★}^2{Z}_4}},c_J={\displaystyle \frac{c_R}{a_{★}{Z}_4}}.$$

Therefore the canonically normalized continuum Robin parameter

$$c_{★}\equiv {\displaystyle \frac{c_R}{\sqrt{Z_y{Z}_4}}}$$

is determined directly by spectral quantities,

$$c_{★}={\displaystyle \frac{c_J}{\sqrt{{\beta }_{\infty }}}}={\displaystyle \frac{\langle s\rangle -(s_{+}+3s_{-})/4}{\sqrt{(s_{+}-s_{-})/4}}}.$$

The auxiliary spacing cancels. It is also useful to define the dimensionless defect

$${\rho }_R\equiv {\displaystyle \frac{c_J}{{\beta }_{\infty }}}={\displaystyle \frac{4\langle s\rangle -s_{+}-3s_{-}}{s_{+}-s_{-}}}.$$

The same spectrum tests the assumption that only the endpoint is modified. Writing the normalized moments as

$${\mu }_n^{\left(J\right)}\equiv \int s^{n}d\nu \left(s\right),$$

Lanczos recursion gives

$${\alpha }_1={\mu }_1^{\left(J\right)},{\beta }_1^2={\mu }_2^{\left(J\right)}-{\left({\mu }_1^{\left(J\right)}\right)}^2,$$

$${\alpha }_2={\displaystyle \frac{{\mu }_3^{\left(J\right)}-2{\alpha }_1{\mu }_2^{\left(J\right)}+{\alpha }_1^3}{{\beta }_1^2}},$$

$${\beta }_2^2={\displaystyle \frac{{\mu }_4^{\left(J\right)}-2{\alpha }_1{\mu }_3^{\left(J\right)}+{\alpha }_1^2{\mu }_2^{\left(J\right)}}{{\beta }_1^2}}-{\beta }_1^2-{\alpha }_2^2.$$

A pure one-site Robin defect therefore predicts

$${\beta }_1={\beta }_{\infty },{\alpha }_2={\alpha }_{\infty },{\beta }_2={\beta }_{\infty },\dots$$

Failure of these equalities falsifies the simple one-defect model rather than being absorbed into another arbitrary parameter.

Appendix C.12.12.4. Boundary-state phase diagram

Subtracting the lower band edge gives the exact quadratic form

$$\langle \psi ,(H_J-s_{-}I)\psi \rangle ={\beta }_{\infty }\sum _{n\ge 1}|{\psi }_{n+1}-{\psi }_n{|}^2+c_J{\left|{\psi }_1\right|}^2.$$

If $c_J\ge 0$, all terms are nonnegative and therefore

$$H_J\ge s_{-}I.$$

A positive Robin defect cannot generate an unwanted state below the band.

For the exactly homogeneous endpoint problem the exponentially localized solution yields three regimes:

$$\begin{array}{cc}{\rho }_R<0:\hfill & \mathrm{one}\mathrm{boundary}\mathrm{state}\mathrm{below}\mathrm{the}\mathrm{band},\hfill \\ 0<{\rho }_R\le 2:\hfill & \mathrm{no}\mathrm{states}\mathrm{outside}\mathrm{the}\mathrm{band},\hfill \\ {\rho }_R>2:\hfill & \mathrm{one}\mathrm{ultraviolet}-\mathrm{localized}\mathrm{state}\mathrm{above}\mathrm{the}\mathrm{band}\mathrm{only}.\hfill \end{array}$$

Thus ${\rho }_R>0$ is the relevant criterion for excluding a light below-band boundary state. An excessively large positive defect may create a high-energy state above the band, but not the unwanted subthreshold state.

In a continuum normal-direction problem, the Robin condition $({\partial }_n+c_R)\varphi =0$ gives the reflection amplitude

$$r\left(q_{\perp }\right)={\displaystyle \frac{i{q}_{\perp }-c_R}{i{q}_{\perp }+c_R}}.$$

For $c_R>0$ and $q_{\perp }\to 0$,

$$r\to -1,{\displaystyle \frac{|r+1|}{2}}\sim {\displaystyle \frac{|q_{\perp }|}{c_R}}.$$

This is an exact linear approach to the Dirichlet reflection phase at normal threshold. It is a theorem about boundary scattering in the matched local class; it is not a universal statement that an arbitrary gapped four-dimensional Euclidean correlator must show a power-law exponent one as $p_E\to 0$.

What is proved in the minimal five-dimensional spectral class.In the minimal source-anchored, asymptotically homogeneous one-band Jacobi–Robin class, a single physical two-point spectrum determines: (i) the bulk band edges, (ii) endpoint moments, (iii) the canonically normalized Robin parameter $c_{★}$, (iv) the sign of the endpoint defect, and (v) whether a below-band boundary state is forced or excluded. Neither an unknown lattice spacing nor an independent bulk fit is required for this sign test.

What this result still does not prove.The renormalized route-odd spectrum of the confining $\mathrm{SU}{\left(15\right)}_p$ theory has not yet been computed. It remains to determine whether that physical source has an infrared spectrum that is sufficiently one-band and gapped for this reconstruction, whether the recovered ${\alpha }_n$ and ${\beta }_n$ actually converge to a local homogeneous tail, whether $4\langle s\rangle -s_{+}-3s_{-}>0$, and whether relevant route-odd operators outside the crossed pair $({\mathcal{O}}_{12},{\mathcal{O}}_{21})$ remain negligible.

Appendix C.12.6. Route-Resolved Finite-Time Certificates: What a Scalar Spectrum Cannot Determine

The scalar Stieltjes–Jacobi reconstruction answers whether a positive measure can be represented by a local chain, but it cannot determine whether the lowest physical state is route-even or route-odd. The obstruction is exact: positive theories can have identical scalar energies and scalar spectral weights but opposite relative signs in the overlap vectors of the two microscopic route sources. A claim of emergent route ${\mathbb{Z}}_2$ therefore requires the matrix correlator of the two physically defined sources.

After canonicalizing the source metric, use the positive $2\times 2$ correlator $\widehat{C}\left(t\right)$ and define

$${\mathsf{T}}_t(a_E;E_{\mathrm{th}})=e^{a_E{E}_{\mathrm{th}}}\widehat{C}{\left(t\right)}^{-1/2}\widehat{C}(t+a_E)\widehat{C}{\left(t\right)}^{-1/2}.$$

The congruent numerator of ${\mathsf{T}}_t-I_2$ is

$${\Delta }_t=e^{a_E{E}_{\mathrm{th}}}\widehat{C}(t+a_E)-\widehat{C}\left(t\right).$$

Using the spectral representation, decompose it as

$${\Delta }_t=A_{<}-B_{\ge }.$$

This split is the key sign observation: every source-supported state below threshold contributes positively to $A_{<}$, whereas every state at or above threshold contributes nonpositively through $-B_{\ge }$. The inertia of ${\Delta }_t$ can therefore certify how many independent source directions must receive subthreshold support without fitting individual poles. Explicitly,

$$A_{<}=\sum _{E_n<E_{\mathrm{th}}}{e}^{-E_{n}t}\left[e^{a_E(E_{\mathrm{th}}-E_n)}-1\right]z_n{z}_n^{†}⪰0,$$

$$B_{\ge }=\sum _{E_n\ge E_{\mathrm{th}}}{e}^{-E_{n}t}\left[1-e^{-a_E(E_n-E_{\mathrm{th}})}\right]z_n{z}_n^{†}⪰0.$$

If $x\in ker{A}_{<}$, then $x^{†}{\Delta }_{t}x=-x^{†}{B}_{\ge }x\le 0$. Consequently, every subspace on which ${\Delta }_t$ is strictly positive has dimension at most $rank{A}_{<}$. By Sylvester’s law of inertia,

$$n_{+}\left({\mathsf{T}}_t-I_2\right)\le rank{A}_{<}.$$

For a finite-volume discrete spectrum, every eigenvalue of ${\mathsf{T}}_t$ larger than one therefore certifies at least one independent source-supported direction below threshold.

Equivalently, solve the generalized eigenvalue problem

$$\widehat{C}(t+a_E)v_i={\rho }_i\widehat{C}\left(t\right)v_i,E_i^{\mathrm{eff}}(t,a_E)=-{\displaystyle \frac{1}{a_E}}ln{\rho }_i.$$

Then

$$e^{a_E{E}_{\mathrm{th}}}{\rho }_i>1⟺E_i^{\mathrm{eff}}<E_{\mathrm{th}}.$$

In a discrete spectrum these principal effective energies are also variational upper bounds on the corresponding exact low energies.

The route-odd combination $e_{-}={(1,-1)}^T/\sqrt{2}$ permits an additional three-time-slice certificate. Define

$$C_{-}\left(t\right)=e_{-}^{†}\widehat{C}\left(t\right)e_{-},$$

$${\mu }_{-}=e^{a_E{E}_{\mathrm{th}}}{\displaystyle \frac{C_{-}(t+a_E)}{C_{-}\left(t\right)}},{\nu }_{-}=e^{2a_E{E}_{\mathrm{th}}}{\displaystyle \frac{C_{-}(t+2a_E)}{C_{-}\left(t\right)}},{\sigma }_{-}^2={\nu }_{-}-{\mu }_{-}^2.$$

If ${\mu }_{-}>1$, the one-sided Cantelli inequality for the spectral variable $Y=e^{-a_E(H-E_{\mathrm{th}})}$ gives

$$w_{-,<}\ge {\displaystyle \frac{{({\mu }_{-}-1)}^2}{{({\mu }_{-}-1)}^2+{\sigma }_{-}^2}}.$$

This is a rigorous lower bound on the total subthreshold spectral weight in the Euclidean-filtered route-odd trial state and does not assume a unique pole.

For a physical state $|n\rangle$ define its visible route-odd fidelity by

$$f_{n,-}={\displaystyle \frac{|e_{-}^{†}{z}_n{|}^2}{\parallel z_n{\parallel }^2}}.$$

Let $w_{-,<}^{\mathrm{LB}}$ denote the right-hand side of the Cantelli bound and define

$${\eta }_{-}^{\mathrm{LB}}\equiv w_{-,<}^{\mathrm{LB}}{\displaystyle \frac{C_{-}\left(t\right)}{Tr\widehat{C}\left(t\right)}}.$$

Then

$$\underset{E_n<E_{\mathrm{th}}}{max}{f}_{n,-}\ge {\eta }_{-}^{\mathrm{LB}}.$$

Indeed, the subthreshold odd contribution is at least $w_{-,<}^{\mathrm{LB}}{C}_{-}\left(t\right)$, whereas the total subthreshold norm of the two-component overlap vectors cannot exceed $Tr\widehat{C}\left(t\right)$. At least one subthreshold state must therefore have odd fidelity at least equal to the ratio above. In particular,

$${\eta }_{-}^{\mathrm{LB}}>{\displaystyle \frac{1}{2}}$$

rigorously certifies the existence of at least one subthreshold state whose visible route content is predominantly odd. If the chosen threshold isolates exactly one state, the same inequality becomes a direct lower bound on the odd fidelity of that state.

Information boundary of the five-dimensional interpretation.A scalar spectrum and positivity do not determine route parity. Therefore the stronger claim that the effective five-dimensional quotient emerges without an additional low-energy assumption requires the actual renormalized $2\times 2$ correlator $C^{AB}\left(t\right)$ of the two microscopically defined route sources. The scalar spectral reconstruction tests locality of the effective coordinate; the matrix correlator independently tests its route-${\mathbb{Z}}_2$ structure.

Appendix C.12.7. Common-reflection Criterion and the Two-State Anti-Tautology

Even the matrix correlator must be interpreted with care. Choose a positive reference slice $t_0$ and whiten the correlator family,

$$\mathsf{M}(t;t_0)=C{\left(t_0\right)}^{-1/2}C\left(t\right)C{\left(t_0\right)}^{-1/2}.$$

For a real symmetric $2\times 2$ family, a single nontrivial orthogonal reflection that commutes with every $\mathsf{M}(t;t_0)$ exists if and only if the family is simultaneously diagonalizable, equivalently

$$[\mathsf{M}(t_i;t_0),\mathsf{M}(t_j;t_0)]=0$$

for every measured pair of times. With uncertainties, the commutator norm must be extrapolated in time range, volume, lattice spacing, source smearing, and operator basis rather than inspected at one pair of slices.

There is a sharp anti-tautology caveat. Suppose that a full-rank $2\times 2$ correlator is exactly saturated by two nondegenerate states,

$$C\left(t\right)=Zdiag(e^{-E_{1}t},e^{-E_{2}t})Z^{†},detZ\ne 0.$$

Then $B=C{\left(t_0\right)}^{-1/2}Zdiag(e^{-E_1{t}_0/2},e^{-E_2{t}_0/2})$ is unitary and

$$\mathsf{M}(t;t_0)=Bdiag(e^{-E_1(t-t_0)},e^{-E_2(t-t_0)})B^{†}.$$

All whitened matrices therefore commute automatically and possess the spectral reflection $Bdiag(1,-1)B^{†}$. Thus “two poles plus a stable GEVP” does not distinguish a microscopic route symmetry from a generic two-state spectral decomposition.

To establish an emergent physical route symmetry one must additionally show stability when extra states and continuum contributions are resolved and determine an independent matching covector, for example an amputated vertex

$${\Gamma }_A^{\left(4\right)}={\langle {\mathcal{J}}_{\mathrm{out}}{\mathcal{O}}_A{\mathcal{J}}_{\mathrm{in}}\rangle }_{\mathrm{amp}}.$$

Only this independent vertex can identify which common spectral axis is physically even and test whether it fixes $c_{\Pi }={(1,1)}^T$ rather than a nearby normalization-dependent direction. The two-point family supplies the candidate axes; it does not name the mass-plane axis by itself.

Appendix C.12.8. Orbifold Descent and Matching

In the physical quotient space the route-odd source is an allowed section, whereas the local crossed-odd operator ${\mathcal{O}}_{-}$ does not descend as an independent local interaction. The allowed crossed-even combination

$${\mathcal{O}}_{+}={\displaystyle \frac{{\mathcal{O}}_{12}+{\mathcal{O}}_{21}}{\sqrt{2}}}$$

may couple to the odd normal derivative of the composite field. Positivity of the Jacobi transfer operator guarantees that endpoint-to-bulk transfer is generically nonzero. The equality

$$C_{12}=C_{21}$$

therefore becomes a structural consequence of this completion rather than an adjusted coefficient ratio.

For an integrated zero-momentum insertion, a Feynman–Hellmann identity can relate the even three-point function to a derivative of the two-point function with respect to a symmetric coupling. This offers a potentially useful reduction of the numerical cost of a future nonperturbative test.

Additional assumption of the five-dimensional completion.The five-dimensional/composite construction is structurally closed if the physical route-${\mathbb{Z}}_2$ quotient is accepted as one low-energy datum. The stronger claim that this quotient itself is generated by the unchanged confining $\mathrm{SU}{\left(15\right)}_p$ dynamics without a new datum remains a genuinely nonperturbative question.

Appendix D.1. Six-Dimensional Branch: Chiral Vortex, Stability, and a Nonminimal Confining No-Turn Completion

Reader checkpoint: map of the 6D branches.Four logically different six-dimensional constructions are kept separate below: the reference Einstein–Abelian–Higgs vortex; the smooth nonminimal no-turn Einstein–sigma/confining branch; the finite-radius interface fallback; and higher-curvature local-response candidates. Results proved in one branch are not silently transferred to another. The 6D analysis is prospective ultraviolet structure and is not used to prove the 4D route identity or the 5D spectral reconstruction.

Status and role of the six-dimensional analysis.The six-dimensional construction is not used to establish the principal four- and five-dimensional results. Its purpose is to test whether the four-dimensional route mechanism and the five-dimensional effective spectral representation can coexist with a smooth higher-dimensional origin of chirality and localization. Within the displayed local two-derivative nonminimal branch, the no-turn background and the complete axisymmetric scalar/radion nonnegativity theorem are established. In the regular dipole sector, the former fixed-metric value $-0.347963/R_c^2$ is removed as a physical tachyon because its eigenfunction lies in the scalar-clock gauge image and has exactly zero Schur curvature after stationary metric completion. Full six-dimensional stability is nevertheless not claimed. The constraint-reduced physical dipole operator $U_{\mathrm{phys}}$, its threshold number ${\beta }_0$, higher angular harmonics, interface-localized modes, microscopic confinement, admissible fermion/Yukawa representations, anomaly inflow, and route matching remain open. The 6D branch is therefore a prospective ultraviolet extension, not an input to the main 4D/5D conclusions.

Extra dimensions are not needed for the existence of the four-dimensional rank-one sign source, but six dimensions can play a different role: they may provide a literal smooth ultraviolet completion in which vectorlike higher-dimensional parents generate exactly chiral four-dimensional preons, while gravity and spectator gauge modes are localized without hard branes. The construction considered here is no longer based on a prescribed string-cigar metric. A self-consistent backreacted Einstein–Abelian–Higgs vortex satisfies the direct-product ${\mathrm{Spin}}^c$ Dai–Freed consistency test; a source-free top form converts homogeneous regularity selection into the choice of an integration constant; the transverse-traceless gravitational and spectator-gauge sectors have nonnegative Rayleigh quotients; the coupled vortex-vector operator factorizes and has no tachyon; and independent graviphoton zero modes and a localized massless scalar modulus are absent. The full gauge-invariant scalar analysis further separates a stable, threshold-local transverse metric–gauge channel from a background “sausage” sector that is generically rigid under regular-core and finite-volume ${\mathrm{AdS}}_6$ boundary conditions. In the natural nonminimal no-turn branch, an exact positive-square construction additionally factorizes the complete $\theta$-independent coupled scalar/radion Hamiltonian and excludes normalizable $m_4^2\le 0$. A separate finite-radius alternative closes the classical Israel–Maxwell–scalar junction but introduces an interface fluctuation problem that remains open. Thus the remaining problem is considerably narrower than the statement that an entire massive scalar Green operator is unknown, but broader than bulk stability alone.

Appendix D.13.1. Earlier Six-Dimensional Obstructions and the Smooth Alternative Branch

Several earlier no-go results remain valid within their respective assumptions:

1.

a chiral six-dimensional Weyl lift using only the published matter content has an irreducible local anomaly obstruction;

2.

an ordinary two-reflection orbifold does not simultaneously give the desired four-dimensional chirality and the required route structure;

3.

the tested full-mirror/free-${\mathbb{Z}}_2$ construction has a nontrivial global torsion obstruction;

4.

a finite hard rectangle $I_5\times I_6$ permits independent face and corner gauge-kinetic counterterms;

5.

a minimal chiral $(1,0)$ one-tensor construction does not satisfy the required irreducible quartic $\mathrm{SU}\left(15\right)$ condition;

6.

a naively prescribed separable conformal warp contains geometric defects and is not a self-consistent Einstein–Abelian–Higgs solution.

The smooth branch changes precisely these assumptions. It uses one vectorlike six-dimensional Dirac parent for each published four-dimensional preon multiplet, a smooth unit-winding codimension-two defect, a consistent ${\mathrm{Spin}}^c$ charge assignment, standard bulk Yang–Mills kinetic terms, and a self-consistent gravitating vortex. String-cigar backgrounds and gauge localization provide useful precedents for smooth codimension-two localization [19,20], but here the geometry is determined by the coupled Einstein–Abelian–Higgs equations rather than imposed analytically.

Appendix D.13.2. Consistent Normalization of the Spin c Charges

For each published four-dimensional multiplet $R_r$, introduce a vectorlike six-dimensional Dirac field whose six-dimensional chiral components are

$${\Psi }_r={\Psi }_{r,+}+{\Psi }_{r,-},{\Gamma }_7{\Psi }_{r,\pm }=\pm {\Psi }_{r,\pm }.$$

Let $V_M$ denote the auxiliary defect connection and $\Phi$ the complex vortex scalar. Both are singlets under

$$G=\mathrm{SU}{\left(15\right)}_p\times \mathrm{SU}{\left(4\right)}_{\mathrm{PS}}\times \mathrm{SU}{\left(2\right)}_L\times \mathrm{SU}{\left(2\right)}_R.$$

Use the integer normalization

$$q_V\left({\Psi }_{r,+}\right)=+1,q_V\left({\Psi }_{r,-}\right)=-1,Q_V\left(\Phi \right)=2.$$

The chirality projectors are

$$P_7^{\pm }\equiv {\displaystyle \frac{1\pm {\Gamma }_7}{2}},$$

and the vortex Yukawa coupling is

$${\mathcal{L}}_Y=-y_r\left[\Phi {\overline{\Psi }}_r{P}_7^{-}{\Psi }_r+{\Phi }^{*}{\overline{\Psi }}_r{P}_7^{+}{\Psi }_r\right].$$

Under a $U{\left(1\right)}_V$ transformation of angle $\vartheta$,

$${\Psi }_r\mapsto e^{i\vartheta {\Gamma }_7}{\Psi }_r,\Phi \mapsto e^{2i\vartheta }\Phi .$$

At $\vartheta =\pi$, every fermion acquires $-1$ while the charge-two scalar is unchanged. Identifying this central element with fermion parity leads naturally to

$${\mathrm{Spin}}^c\left(6\right)={\displaystyle \frac{\mathrm{Spin}\left(6\right)\times U{\left(1\right)}_V}{{\mathbb{Z}}_2}}.$$

Accordingly, $V_M$ is most naturally interpreted as the connection entering the ${\mathrm{Spin}}^c$ structure rather than as an unconstrained additional low-energy Abelian gauge factor.

Appendix D.13.3. Local Anomaly Polynomial and the Dai–Freed Criterion for the Direct-Product Group

For a vectorlike six-dimensional Dirac parent in representation R, the anomaly polynomial is the difference of the two six-dimensional Weyl contributions,

$$I_8^{\left(R\right)}\propto \left[e^{F_V}-e^{-F_V}\right]{\mathrm{ch}}_R\left(F_G\right)\widehat{A}\left(T\right){|}_8,$$

where $F_V$ is the ${\mathrm{Spin}}^c$ Abelian curvature, $F_G$ the non-Abelian curvature, ${\mathrm{ch}}_R$ the Chern character in representation R, and $\widehat{A}\left(T\right)$ the gravitational $\widehat{A}$ genus. Terms even in $F_V$ cancel between the two chiralities. For a product of special-unitary factors, the potentially relevant mixed term is proportional to $F_V{Tr}_R{F}_G^3$, whose coefficient is exactly the ordinary four-dimensional cubic anomaly sum. For the published preon spectrum,

$$8+8+3-19=0\left[\mathrm{SU}{\left(15\right)}_p\right],$$

$$30-30=0\left[\mathrm{SU}{\left(4\right)}_{\mathrm{PS}}\right],$$

while the perturbative cubic trace vanishes for $\mathrm{SU}{\left(2\right)}_L$ and $\mathrm{SU}{\left(2\right)}_R$. Hence

$$I_8^{\mathrm{local}}=0.$$

For the direct-product global group G, the corresponding global-anomaly check may be formulated through the Atiyah–Hirzebruch spectral sequence

$$E_{p,q}^2=H_p\left(BG;{\Omega }_q^{{\mathrm{Spin}}^c}\left(\mathrm{pt}\right)\right).$$

The relevant odd ${\mathrm{Spin}}^c$ coefficient groups vanish in the required low degrees, while the integral homology of products of $BSU\left(N\right)$ is supported in even degrees. There are therefore no nonzero $E_{p,q}^2$ entries with $p+q=7$, giving

$${\Omega }_7^{{\mathrm{Spin}}^c}\left(BG\right)=0.$$

Under the stated direct-product assumption, both the local anomaly polynomial and the torsion global-anomaly test therefore vanish. If the true global gauge group is later quotiented by common center elements or tied nontrivially to spin structure, the bordism calculation must be repeated for that actual global form.

Appendix D.13.4. Self-Consistent Einstein–Abelian–Higgs System

The central change relative to prescribed cigar geometries is to solve for the warp factors together with the defect fields. The analysis uses the standard six-dimensional gravitating Abelian-vortex system [21],

$$S=\int d^{6}x\sqrt{-g}\left[{\displaystyle \frac{M_6^4}{2}}(R-2{\Lambda }_6)-{\displaystyle \frac{1}{4}}{F}_{MN}{F}^{MN}+{\left(D_M\Phi \right)}^{*}\left(D^M\Phi \right)-{\displaystyle \frac{\lambda }{4}}{\left(\right|\Phi |}^2-v^2{)}^2\right].$$

Here $M_6$ is the six-dimensional Planck scale, R is the six-dimensional Ricci scalar, ${\Lambda }_6<0$ is the bulk cosmological constant, $F_{MN}$ is the vortex Abelian field strength, v is the symmetry-breaking scale, and $\lambda >0$ is the scalar self-coupling. The metric and vortex ansatz are

$$d{s}_6^2=M^2\left(\rho \right){\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+d{\rho }^2+L^2\left(\rho \right)d{\theta }^2,$$

$$\Phi =vf\left(\rho \right)e^{i\theta },e_{\Phi }{V}_{\theta }=1-P\left(\rho \right),$$

where $e_{\Phi }$ is the defect gauge coupling, $f\left(\rho \right)$ is the scalar profile, and $P\left(\rho \right)$ is the angular gauge profile.

Introduce the dimensionless radius and angular scale

$$x\equiv {\displaystyle \frac{m_H\rho }{\sqrt{2}}},\mathcal{L}\left(x\right)\equiv {\displaystyle \frac{m_{H}L\left(\rho \right)}{\sqrt{2}}},$$

where $m_H\equiv \sqrt{2\lambda }v$ is the Higgs mass scale, and define logarithmic warp derivatives

$$m\left(x\right)\equiv {\displaystyle \frac{dlnM}{dx}},\ell \left(x\right)\equiv {\displaystyle \frac{dln\mathcal{L}}{dx}}.$$

Introduce also the dimensionless couplings

$${\alpha }_6\equiv {\displaystyle \frac{e_{\Phi }^2}{\lambda }},{\nu }_6\equiv {\displaystyle \frac{{\kappa }_6^2{v}^2}{2}},{\mu }_6\propto {\displaystyle \frac{{\Lambda }_6}{\lambda v^4}},$$

where ${\kappa }_6$ is the six-dimensional gravitational coupling. The matter equations become

$$f^{\prime \prime }+(4m+\ell )f^{\prime }+(1-f^2)f-{\displaystyle \frac{P^2}{{\mathcal{L}}^2}}f=0,$$

$$P^{\prime \prime }+(4m-\ell )P^{\prime }-{\alpha }_6{f}^{2}P=0,$$

with primes denoting $d/dx$.

A regular unit-winding solution obeys

$$f\left(0\right)=0,P\left(0\right)=1,M\left(0\right)=1,M^{\prime }\left(0\right)=0,\mathcal{L}\left(0\right)=0,{\mathcal{L}}^{\prime }\left(0\right)=1,$$

$$f(\infty )=1,P(\infty )=0,$$

and approaches an exponentially warped throat,

$$M,\mathcal{L}\propto e^{-cx},c\equiv \sqrt{-{\mu }_6/10}>0.$$

A numerically reproduced regular control solution is

$${\alpha }_6=1.16,{\nu }_6=2.5801288700,{\mu }_{6,★}=-0.0782271150.$$

The near-core coefficients are

$$A_f\equiv f^{\prime }\left(0\right)=0.481172676974994,B_P\equiv {\displaystyle \frac{1}{2}}{P}^{\prime \prime }\left(0\right)=-0.224794977430394,$$

and the asymptotic warp exponent is

$$c_{6,★}\equiv \sqrt{-{\mu }_{6,★}/10}=0.088446093752070.$$

At $x=10$ the independent numerical integration gives

$$f=0.999979629196,P=2.6549227\times {10}^{-4},$$

$$m=-0.088445974528,\ell =-0.0884464548373,$$

while the independent Einstein-constraint residual remains below approximately $1.1\times {10}^{-13}$ over the integration interval. The asymptotic Ricci scalar has magnitude $30c_{6,★}^2$; in the sign convention used later for the $f\left(R\right)$ sector this is written as

$$R_0=-30c_{6,★}^2.$$

The Weyl-tensor square tends numerically to zero. The control therefore verifies a regular, asymptotically conformally flat warped branch without assuming an exact analytic string-cigar profile.

Appendix D.13.5. Normalization of Gauge and Gravitational Zero Modes

For an ordinary spectator Yang–Mills factor labeled by $\mathfrak{s}$,

$$S_{\mathrm{YM}}=-{\displaystyle \frac{1}{4g_{6,\mathfrak{s}}^2}}\int d^{6}x\sqrt{-g}Tr{F}_{MN}^{\left(\mathfrak{s}\right)}{F}_{\left(\mathfrak{s}\right)}^{MN}.$$

A four-dimensional gauge zero mode that is constant in the transverse coordinates has

$${\displaystyle \frac{1}{g_{4,\mathfrak{s}}^2}}={\displaystyle \frac{4\pi }{g_{6,\mathfrak{s}}^2{m}_H^2}}{I}_g,I_g\equiv {\int }_0^{\infty }\mathcal{L}\left(x\right)dx.$$

For the reproduced solution,

$$I_g=17.48736163\dots <\infty .$$

The effective four-dimensional Planck normalization analogously contains

$$I_{\mathrm{grav}}\equiv {\int }_0^{\infty }{M}^2\left(x\right)\mathcal{L}\left(x\right)dx=5.42679976\dots <\infty .$$

Thus both the spectator gauge zero mode and the graviton zero mode are normalizable on this background.

Appendix D.13.6. Turning-Point Theorem from Localization of a Standard Gauge Zero Mode

The regular vortex above has an angular radius $L\left(r\right)$ that grows away from the axis, reaches an interior maximum, and then decreases. This maximum is not merely an accident of the numerical solution. Consider the standard six-dimensional gauge action

$$S_{\mathrm{YM}}=-{\displaystyle \frac{1}{4g_6^2}}\int d^{6}x\sqrt{-g}{h}_g\left(r\right)F_{MN}{F}^{MN},$$

where $h_g\left(r\right)>0$ is a smooth gauge-kinetic weight and $h_g=1$ for ordinary Yang–Mills theory. For

$$d{s}^2=M^2\left(r\right){\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+d{r}^2+L^2\left(r\right)d{\theta }^2,$$

one has $\sqrt{-g}=M^{4}L$. A transverse-constant massless mode $A_{\mu }(x,r,\theta )=a_{\mu }\left(x\right)$ contains two inverse four-dimensional metrics, producing $M^{-4}$ and cancelling the determinant factor exactly. Therefore

$${\displaystyle \frac{1}{g_4^2}}={\displaystyle \frac{2\pi }{g_6^2}}{\int }_0^{\infty }{h}_g\left(r\right)L\left(r\right)dr.$$

The four-dimensional warp factor $M\left(r\right)$ does not enter this normalization.

Assume a regular axis,

$$L\left(0\right)=0,L^{\prime }\left(0\right)=1,L\left(r\right)>0(r>0),$$

and suppose that beyond some radius $h_g\left(r\right)\ge h_{*}>0$. If the constant bulk gauge zero mode is normalizable, then

$${\int }_0^{\infty }{h}_g\left(r\right)L\left(r\right)dr<\infty .$$

If $L^{\prime }\left(r\right)$ never vanished, continuity and the initial condition $L^{\prime }\left(0\right)=1$ would imply $L^{\prime }\left(r\right)>0$ for all $r>0$. Hence $L\left(r\right)$ would be monotonically increasing and for any fixed $r_1>0$,

$$L\left(r\right)\ge L\left(r_1\right)>0(r\ge r_1).$$

It would follow that

$${\int }_{r_1}^{\infty }{h}_{g}Ldr\ge h_{*}L\left(r_1\right){\int }_{r_1}^{\infty }dr=\infty ,$$

contradicting gauge-mode normalizability. Therefore at least one finite radius $r_c$ must satisfy

$$L^{\prime }\left(r_c\right)=0.$$

Gauge-localization turning theorem.For a smooth regular axisymmetric six-dimensional geometry with $L\left(0\right)=0$, $L^{\prime }\left(0\right)=1$, $L\left(r\right)>0$, and a standard asymptotically nondegenerate bulk gauge kinetic coefficient $h_g\left(r\right)\ge h_{*}>0$, normalizability of a transverse-constant four-dimensional gauge zero mode implies the existence of at least one interior turning point $L^{\prime }\left(r_c\right)=0$. The theorem is independent of the detailed shape of the four-dimensional warp factor because $M\left(r\right)$ cancels exactly from the gauge kinetic norm.

The distinction between gravity and gauge localization is therefore sharp. If asymptotically

$$M\left(r\right)\sim e^{-a_{6}r},L\left(r\right)\sim e^{b_{6}r},$$

then the graviton norm contains $M^{2}L\sim e^{(-2a_6+b_6)r}$ and may converge for $b_6<2a_6$, whereas the standard constant gauge mode contains only $L\sim e^{b_{6}r}$ and requires $b_6<0$. Thus in the whole interval

$$0\le b_6<2a_6$$

gravity can remain localized while the ordinary bulk gauge zero mode is not.

Formally one can evade the theorem by taking $h_g\left(r\right)\to 0$ fast enough that $\int h_{g}Ldr$ remains finite even for nondecaying L. But the locally canonically normalized non-Abelian coupling then behaves as

$$g_{6,\mathrm{loc}}\left(r\right)\sim {\displaystyle \frac{g_6}{\sqrt{h_g\left(r\right)}}}.$$

Since $\left[g_6\right]=-1$ in six dimensions, the loop expansion parameter at local energy E scales as $g_6^2{E}^2/h_g$, and the local weak-coupling cutoff scales as

$${\Lambda }_{\mathrm{loc}}\left(r\right)\propto {\displaystyle \frac{\sqrt{h_g\left(r\right)}}{g_6}}$$

up to the usual dimensionless NDA coefficient. Hence $h_g\to 0$ drives ${\Lambda }_{\mathrm{loc}}\to 0$. Such a dielectric escape is mathematically possible, but it is not a uniformly weakly coupled local bulk Yang–Mills completion.

Appendix D.13.7. Can a Four-Dimensional Vector Arise from a Tensor Field Instead?

The previous theorem concerned an ordinary six-dimensional one-form potential. In two transverse dimensions one can systematically inspect every standard two-derivative p-form potential capable of producing a four-dimensional vector. Let $C_{M_1\dots M_p}$ be a p-form potential with field strength

$$H_{p+1}=d{C}_p$$

and positive local kinetic coefficient $h_p\left(r\right)$. Its standard action is

$$S_p=-{\displaystyle \frac{1}{2(p+1)!g_p^2}}\int d^{6}x\sqrt{-g}{h}_p\left(r\right)H_{M_0\dots M_p}{H}^{M_0\dots M_p}.$$

A four-dimensional vector $a_{\mu }\left(x\right)$ arises from

$$C_{\mu a_1\dots a_{p-1}}=a_{\mu }\left(x\right){\omega }_{a_1\dots a_{p-1}}(r,\theta ),$$

where the $a_i$ are transverse indices. The two inverse four-dimensional metrics contribute $M^{-4}$, which cancels the factor $M^4$ in $\sqrt{-g}=M^{4}L$. Therefore every such vector has the universal internal norm

$${\mathcal{N}}_{p,\mathrm{vec}}\propto \int d^{2}y\sqrt{g_2}{h}_p\left(y\right){\left|{\omega }_{p-1}\left(y\right)\right|}^2.$$

Changing the four-dimensional warp factor cannot by itself localize an otherwise nonnormalizable internal form.

There are only three standard possibilities in two transverse dimensions.

Appendix D.13.13.1. One-form.

For $p=1$, the internal form has degree zero and the massless mode is constant. Thus

$${\mathcal{N}}_{1,\mathrm{vec}}\propto {\int }_0^{\infty }{h}_1\left(r\right)L\left(r\right)dr,$$

so the turning theorem applies whenever $h_1$ is asymptotically nondegenerate.

Appendix D.13.13.2. Two-form.

For $p=2$, write the axisymmetric internal one-form as

$$\omega =u\left(r\right)dr+v\left(r\right)d\theta .$$

For the standard unweighted problem, a massless zero mode is harmonic. Closure gives

$$v^{\prime }\left(r\right)=0,v\left(r\right)=C_{\theta },$$

while co-closure in $d{s}_2^2=d{r}^2+L^{2}d{\theta }^2$ gives

$${\displaystyle \frac{1}{L}}{\displaystyle \frac{d}{dr}}\left(Lu\right)=0,u\left(r\right)={\displaystyle \frac{C_r}{L\left(r\right)}}.$$

The corresponding norm is

$${\parallel \omega \parallel }^2\propto (C_r^2+C_{\theta }^2){\int }_0^{\infty }{\displaystyle \frac{dr}{L\left(r\right)}}.$$

Near a regular axis, $L\left(r\right)=r+O\left(r^3\right)$, so

$${\int }_0{\displaystyle \frac{dr}{L\left(r\right)}}\sim {\int }_0{\displaystyle \frac{dr}{r}}=\infty .$$

Thus the standard axisymmetric two-form channel has no nonzero regular normalizable harmonic internal one-form and therefore no corresponding massless four-dimensional vector zero mode.

Appendix D.13.13.3. Three-form.

For $p=3$, the natural axisymmetric internal two-form is proportional to the transverse volume form,

$${\omega }_2=C_{2}L\left(r\right)dr\wedge d\theta .$$

Its pointwise norm is constant, giving

$${\mathcal{N}}_{3,\mathrm{vec}}\propto {\int }_0^{\infty }{h}_3\left(r\right)L\left(r\right)dr.$$

This is the same radial measure as for the one-form vector and therefore inherits the same turning theorem when $h_3$ is nondegenerate.

Limit of the standard two-derivative vector/tensor escape.On a smooth regular axisymmetric geometry with two transverse dimensions, standard one-form and three-form origins of a four-dimensional vector have a norm of the form $\int h_{p}Ldr$ and therefore inherit the interior-turning theorem for asymptotically nondegenerate kinetic coefficients. The standard two-form channel has an axisymmetric harmonic internal one-form with norm $\int dr/L$, which diverges logarithmically at a regular axis. Taking $h_g\to 0$ can evade the normalization theorem only at the price of a local non-Abelian weak-coupling scale that collapses asymptotically. This does not exclude qualitatively different first-order topological sectors, composite or core-localized gauge fields, non-axisymmetric $L^2$ cohomology, or ultraviolet physics that reorganizes the gauge dynamics before the local cutoff collapses.

Appendix D.13.8. Nonminimal Escape Through Confinement in the Transverse Bulk

The turning theorem is strict only while the four-dimensional gauge boson is assumed to arise from a weakly coupled local bulk zero mode with a conventional two-derivative kinetic term. A qualitatively different localization mechanism is possible if the phase outside the defect is confining while the relevant subgroup remains Coulombic or deconfined on the defect, as in the general logic of defect gauge localization by bulk confinement [22]. In that case transverse flux penetration is controlled by the confinement gap rather than by normalizability of a constant perturbative bulk wavefunction. The assumption behind the turning theorem is changed, so there is no contradiction.

It is important first to distinguish genuine confinement from an ordinary positive local mass barrier. For an axisymmetric profile $\psi \left(r\right)$ of an exactly massless four-dimensional vector mode with $m_A^2\left(r\right)\ge 0$, consider

$$-{\displaystyle \frac{1}{L\left(r\right)}}{\displaystyle \frac{d}{dr}}\left[L\left(r\right){\displaystyle \frac{d\psi }{dr}}\right]+m_A^2\left(r\right)\psi \left(r\right)=0.$$

Multiplication by $L\psi$ and integration over r gives, for regular boundary conditions and a vanishing surface term,

$${\int }_0^{\infty }drL\left(r\right)\left[{\left({\displaystyle \frac{d\psi }{dr}}\right)}^2+m_A^2\left(r\right){\psi }^2\left(r\right)\right]=0.$$

Both terms are nonnegative. A nonzero exactly massless mode is therefore possible only in the degenerate case ${\psi }^{\prime }=0$ and $m_A^2\psi =0$ on its support. A positive local Proca/Higgs mass profile is not a substitute for confinement; it does not by itself generate a nontrivial exactly massless localized vector.

A genuine confining phase is different because its low-energy response is not represented by such a weakly coupled local Proca profile. For a structural test, assume that flux penetration far from the core is exponentially suppressed by a confinement scale ${\Lambda }_{\mathrm{conf}}>0$. If asymptotically

$$M\left(r\right)\sim e^{-a_{6}r},L\left(r\right)\sim e^{b_{6}r},$$

then gravitational localization requires

$$b_6<2a_6.$$

If the squared transverse gauge amplitude is suppressed as $e^{-2{\Lambda }_{\mathrm{conf}}r}$, its internal norm behaves as

$${\int }^{\infty }drL\left(r\right)e^{-2{\Lambda }_{\mathrm{conf}}r}\sim {\int }^{\infty }dr{e}^{(b_6-2{\Lambda }_{\mathrm{conf}})r}$$

and converges if

$$b_6<2{\Lambda }_{\mathrm{conf}}.$$

Thus there is a nonempty no-turn window

$$0\le b_6<2min(a_6,{\Lambda }_{\mathrm{conf}}),$$

in which gravity is normalizable, gauge response can be localized by a confining gap, and the angular radius can remain monotone without an interior maximum.

An explicit smooth kinematic witness demonstrates that this window is nonempty. Introduce a positive radial scale $R_c$ and a dimensionless parameter ${\kappa }_c>0$ and take

$$L\left(r\right)=R_{c}tanh\left({\displaystyle \frac{r}{R_c}}\right),M\left(r\right)={cosh}^{-{\kappa }_c}\left({\displaystyle \frac{r}{R_c}}\right).$$

Near the axis,

$$L\left(r\right)=r+O\left(r^3\right),M\left(r\right)=1-{\displaystyle \frac{{\kappa }_c}{2}}{\displaystyle \frac{r^2}{R_c^2}}+O\left(r^4\right),$$

while for every finite $r>0$,

$$L^{\prime }\left(r\right)={sech}^2\left({\displaystyle \frac{r}{R_c}}\right)>0.$$

Hence there is no interior turning point and $L\left(r\right)\to R_c$. At large radius,

$$M\left(r\right)\sim 2^{{\kappa }_c}{e}^{-({\kappa }_c/R_c)r},a_6={\displaystyle \frac{{\kappa }_c}{R_c}},$$

and the Planck normalization is exactly

$${\int }_0^{\infty }dr{M}^2\left(r\right)L\left(r\right)={\displaystyle \frac{R_c^2}{2{\kappa }_c}}<\infty .$$

Figure A4. Exact no-turn witness in the representative case ${\kappa }_c=1$. The angular radius grows monotonically to a cylinder while the four-dimensional warp factor decays. The plot illustrates the established background geometry only; it does not display the still-uncomputed physical dipole operator or determine ${\beta }_0$.

Figure A4. Exact no-turn witness in the representative case ${\kappa }_c=1$. The angular radius grows monotonically to a cylinder while the four-dimensional warp factor decays. The plot illustrates the established background geometry only; it does not display the still-uncomputed physical dipole operator or determine ${\beta }_0$.

As required by the earlier theorem, a constant perturbative bulk gauge mode is not normalizable because $\int Ldr$ diverges. But if the squared penetration amplitude is suppressed as $exp[-2\lambda r/R_c]$, with

$$\lambda \equiv {\Lambda }_{\mathrm{conf}}{R}_c>0,$$

the confinement-weighted gauge norm is finite and can be written in closed form,

$${\displaystyle \frac{I_{g,\mathrm{conf}}}{R_c^2}}={\displaystyle \frac{1}{2\lambda }}-{\displaystyle \frac{1}{2}}\left[{\psi }_0\left({\displaystyle \frac{\lambda +2}{2}}\right)-{\psi }_0\left({\displaystyle \frac{\lambda +1}{2}}\right)\right],$$

where

$${\psi }_0\left(z\right)\equiv {\displaystyle \frac{d}{dz}}ln\Gamma \left(z\right)$$

is the digamma function. Numerically,

$${\displaystyle \frac{I_{g,\mathrm{conf}}}{R_c^2}}=0.5707963268(\lambda =0.5),{\displaystyle \frac{I_{g,\mathrm{conf}}}{R_c^2}}=0.1931471806(\lambda =1).$$

The altered asymptotics are also kinematically compatible with the previously derived one-Weyl normalizability condition. If $M\sim e^{-a_{6}r}$ and the asymptotic Yukawa mass is $m_{f,\infty }$, the large-radius fermion norm reduces to

$${\int }^{\infty }dr{M}^{-1}{e}^{-2m_{f,\infty }r},$$

so convergence requires

$$m_{f,\infty }>{\displaystyle \frac{a_6}{2}}.$$

The asymptotic behavior of $L\left(r\right)$ cancels from this condition. A nondecaying asymptotic angular radius is therefore not, by itself, incompatible with the one-Weyl localization mechanism.

What the confinement escape actually proves.The minimal smooth six-dimensional program with a standard weakly coupled bulk gauge zero mode remains limited by the turning theorem and the associated rigidity of the radial breathing sector. This is not an absolute no-go for six dimensions. If four-dimensional gauge localization is instead generated by a genuine confining gap in the transverse bulk, a nonempty smooth no-turn window exists. The stronger result derived below is that a representative geometry in this class is an exact solution of a local two-field Einstein–sigma model with positive kinetic terms and regular winding. Thus the self-consistent-background problem is closed at the effective-field-theory level. The microscopic origin of the confining phase, its scale, the allowed fermion representation, non-axisymmetric perturbations, and the full anomaly/inflow analysis remain separate dynamical criteria.

Appendix D.13.9. Exact Local Matter Support of the Nonminimal No-Turn Cylinder

The confinement-based no-turn construction can be strengthened beyond the kinematic level. Consider the six-dimensional metric

$$d{s}_6^2=M^2\left(r\right){\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+d{r}^2+L^2\left(r\right)d{\theta }^2,0\le \theta <2\pi ,$$

with

$$M\left(r\right)={cosh}^{-\kappa }u,L\left(r\right)=Rtanhu,u\equiv {\displaystyle \frac{r}{R}},R>0,\kappa >0.$$

The symbol R denotes the asymptotic radius of the angular circle and $\kappa$ is the dimensionless warp exponent. Near the regular axis,

$$L\left(r\right)=r-{\displaystyle \frac{r^3}{3R^2}}+O\left(r^5\right),M\left(r\right)=1-{\displaystyle \frac{\kappa }{2}}{\displaystyle \frac{r^2}{R^2}}+O\left(r^4\right),$$

while

$$L^{\prime }\left(r\right)={sech}^2(r/R)>0$$

for every finite r. Thus the circle radius has no interior turning point. The four-dimensional Planck normalization, apart from the angular factor $2\pi$, is exactly

$$I_{\mathrm{Pl}}={\int }_0^{\infty }{M}^{2}Ldr={\displaystyle \frac{R^2}{2\kappa }}.$$

A local matter source for this geometry is provided by the two-field nonlinear sigma model

$$S_{\sigma }=\int d^{6}x\sqrt{-g}\left[{\displaystyle \frac{R_6}{2{\kappa }_6^2}}-{\displaystyle \frac{1}{2}}{(\partial \varphi )}^2-{\displaystyle \frac{1}{2}}{F}^2\left(\varphi \right){(\partial \chi )}^2-U\left(\varphi \right)\right],$$

where ${\kappa }_6$ is the six-dimensional gravitational coupling, $\varphi \left(r\right)$ is a canonically normalized radial modulus, and $\chi$ is a compact phase. For winding number $n\in \mathbb{Z}\setminus \left\{0\right\}$ take

$$\chi =n\theta .$$

The Einstein equations reconstruct positive radial and angular kinetic energies and yield the exact modulus profile

$$\varphi \left(r\right)={\displaystyle \frac{\sqrt{2(\kappa +1)}}{{\kappa }_6}}arctan\left(sinh{\displaystyle \frac{r}{R}}\right).$$

Define the dimensionless field-space angle

$$\vartheta \equiv {\displaystyle \frac{{\kappa }_6\varphi }{\sqrt{2(\kappa +1)}}}.$$

Along the solution,

$$sin\vartheta =tanh(r/R),cos\vartheta =sech(r/R).$$

The field-space angular radius and potential are then local functions of $\varphi$:

$$F^2\left(\varphi \right)={\displaystyle \frac{{sin}^2\vartheta }{{\kappa }_6^2{n}^2}}\left[4{\kappa }^2{sin}^2\vartheta +2(\kappa +1){cos}^2\vartheta \right],$$

$$U\left(\varphi \right)={\displaystyle \frac{1}{{\kappa }_6^2{R}^2}}\left[4\kappa {cos}^2\vartheta -8{\kappa }^2{sin}^2\vartheta \right].$$

Direct substitution satisfies all independent Einstein equations and both scalar equations exactly. The target-space metric is regular at the core because

$$F\left(\varphi \right)\sim {\displaystyle \frac{\varphi }{\left|n\right|}}(r\to 0),$$

so the $(\varphi ,\chi )$ field space approaches an ordinary polar plane rather than a conical singularity. In particular, no negative kinetic term is required.

The geometry also implies a useful asymptotic necessity. With $A=lnM$ and $B=lnL$, the Einstein tensor gives

$${\kappa }_6^2\left(T^r{}_r-{T^{\mu }}_{\mu }\right)={\displaystyle \frac{2(\kappa +1)}{R^2}}{sech}^2(r/R),$$

while

$${\kappa }_6^2\left(T^{\theta }{}_{\theta }-{T^{\mu }}_{\mu }\right)⟶{\displaystyle \frac{4{\kappa }^2}{R^2}}(r\to \infty ).$$

Therefore a standard localized finite-energy vortex whose anisotropic gradients vanish asymptotically, together with an isotropic cosmological term, cannot by itself support the $L\to R$, $M\sim e^{-\kappa r/R}$ cylinder. Persistent angular anisotropic stress is necessary.

The same radial modulus can be used as an effective confinement order parameter. If, as an infrared completion hypothesis,

$${\Lambda }_{\mathrm{conf}}\left(r\right)={\Lambda }_{\infty }tanh(r/R),$$

then the corresponding penetration profile is

$${\psi }_g\left(r\right)={cosh}^{-\lambda }(r/R),\lambda \equiv {\Lambda }_{\infty }R>0,$$

and its transverse norm is exactly

$$I_g={\int }_0^{\infty }L\left(r\right){\left|{\psi }_g\left(r\right)\right|}^{2}dr={\displaystyle \frac{R^2}{2\lambda }}={\displaystyle \frac{R}{2{\Lambda }_{\infty }}}<\infty .$$

The existence and finiteness of this norm are exact once the effective gap profile is assumed; the microscopic derivation of ${\Lambda }_{\mathrm{conf}}\left(\varphi \right)$ from $\mathrm{SU}{\left(15\right)}_p$ remains open.

Appendix D.13.10. Finite-backreaction Coexistence with a Unit-Winding Abelian–Higgs Vortex

The exact sigma-model source need not be interpreted as a replacement for the previously studied chirality-producing vortex. A separate numerical boundary-value calculation shows that a standard unit-winding Einstein–Abelian–Higgs vortex can coexist with the no-turn stabilizing sector at finite, order-one gravitational backreaction. In the one-parameter exact no-turn family used for this check, positivity of the residual stabilizer stress requires the asymptotic cylinder radius to exceed a gauge-tail threshold,

$$R>R_{\mathrm{tail}},R_{\mathrm{tail}}\simeq 1.093409$$

in the dimensionless normalization of that boundary-value problem. A representative radius

$$R\simeq 1.22125$$

admits a positive coexistence interval extending to approximately

$${\eta }_{max}\simeq 1.50,$$

where $\eta$ is the dimensionless gravitational weight assigned to the Abelian–Higgs vortex stress in the combined reconstruction. In particular, the order-one point $\eta =1$ has nonnegative residual radial and winding kinetic terms and a positive residual potential; the reconstructed total stress agrees with the required Einstein tensor to numerical precision.

This result is a numerical existence statement for coexistence, not a derivation of the physical value of $\eta$. The substantially larger gravitational weight associated with an earlier smooth-vortex control solution, approximately $2.58$ in the same comparison, is not supported by the scanned exact family. That failure constrains this family but is not an absolute no-go for the nonminimal six-dimensional branch, because the stabilizer changes the stress budget and other no-turn families need not have the same upper bound. The full chiral Dirac index and the absence of unwanted mirror zero modes must therefore be rechecked on the combined vortex–stabilizer background rather than inferred from kinematics alone.

Appendix D.13.11. Correction: The Compact Phase Is a Stueckelberg Coordinate, Not an Independent Scalar Zero Mode

A fixed-vector truncation appears to produce a four-dimensional shift mode. If one writes

$$\chi (x,r,\theta )=n\theta +\pi \left(x\right)$$

and sets the Kaluza–Klein vector to zero before varying the action, the coefficient of ${(\partial \pi )}^2$ is indeed finite:

$$Z_{\chi }^{\mathrm{frozen}\mathrm{KK}}={\displaystyle \frac{2\pi R^2}{{\kappa }_6^2{n}^2}}{\displaystyle \frac{5\kappa +1}{(\kappa +1)(\kappa +2)}}.$$

The integral and the displayed value are algebraically correct, but the interpretation as an additional physical scalar is not. The truncation has frozen the field that gauges angular reparametrizations.

To see the point directly, retain the circle connection in

$$d{s}_6^2=e^{2{\alpha }_{\mathrm{KK}}\phi }d{s}_5^2+R^2{e}^{2{\beta }_{\mathrm{KK}}\phi }{(d\theta +{\mathcal{A}}_{a}d{x}^a)}^2.$$

Under $\theta \mapsto \theta +\lambda (x,r)$,

$${\mathcal{A}}_a\mapsto {\mathcal{A}}_a-{\partial }_a\lambda ,\pi \mapsto \pi -n\lambda .$$

Only the combination ${\partial }_a\pi -n{\mathcal{A}}_a$ is gauge invariant. The apparent constant phase mode can be removed by unitary gauge and is the longitudinal Stueckelberg coordinate of the Kaluza–Klein vector. It must therefore be analyzed together with the vector and Einstein constraints, not counted once more as a free scalar. The regular-axis behavior $F\left(\varphi \right)\sim \varphi /\left|n\right|$ makes this gauge-invariant description smooth at the cap.

This correction does not claim that every vector or non-axisymmetric perturbation is automatically healthy. It says only that the finite frozen-vector norm does not establish an independent massless scalar. The physical $\theta$-independent spin-zero sector is the coupled radion–modulus system analyzed next.

Appendix D.13.12. Exact Positive-Square Closure of the Axisymmetric Scalar/Radion Sector

For $r>0$ reduce the natural no-turn solution on the circle using

$${\alpha }_{\mathrm{KK}}={\displaystyle \frac{1}{2\sqrt{6}}},{\beta }_{\mathrm{KK}}=-{\displaystyle \frac{3}{2\sqrt{6}}},{\lambda }_L\equiv {\displaystyle \frac{L}{R}}=e^{{\beta }_{\mathrm{KK}}\phi },$$

and define

$$q\equiv \sqrt{2}{\kappa }_6\varphi ,s_q\equiv sin\left({\displaystyle \frac{q}{2\sqrt{\kappa +1}}}\right).$$

After the Stueckelberg coordinate has been removed, the exact five-dimensional Einstein-frame action is

$$S_5={\displaystyle \frac{1}{2{\kappa }_5^2}}\int d^{5}x\sqrt{-g_5}\left[R_5-{\displaystyle \frac{1}{2}}{(\partial \phi )}^2-{\displaystyle \frac{1}{2}}{(\partial q)}^2-{\mathcal{V}}_5(\phi ,q)\right].$$

The scalar target metric is the positive matrix $G_{IJ}={\delta }_{IJ}$, and

$$\begin{array}{cc}\hfill R^2{\mathcal{V}}_5=& 8\kappa {\lambda }_L^{-2/3}\left[1-(2\kappa +1)s_q^2\right]\hfill \\ & +2{\lambda }_L^{-8/3}{s}_q^2\left[(\kappa +1)+(2{\kappa }^2-\kappa -1)s_q^2\right].\hfill \end{array}$$

On the background,

$${\lambda }_L=s_q=t,t\equiv tanh(r/R).$$

Define the two explicit functions

$$\mathcal{W}={\displaystyle \frac{1}{R}}{\lambda }_L^{-4/3}\left[4\kappa {\lambda }_L^2-2+2(\kappa +1)s_q^2\right],$$

$$\mathcal{P}={\displaystyle \frac{2\kappa }{R}}{\lambda }_L^{-4/3}({\lambda }_L^2-s_q^2).$$

Direct differentiation and coefficient matching in $s_q^2$ give the exact identity

$${\mathcal{V}}_5={\displaystyle \frac{1}{2}}{\delta }^{IJ}{\mathcal{W}}_{,I}{\mathcal{W}}_{,J}-{\displaystyle \frac{1}{3}}{\mathcal{W}}^2+{\mathcal{P}}^2.$$

This is an algebraic identity for every $\kappa >0$, not a fit to the background. Along the solution,

$$\mathcal{P}=0,{\delta }^{IJ}{\mathcal{W}}_{,I}{\mathcal{P}}_{,J}=0.$$

If $\zeta$ is the five-dimensional Einstein-frame proper coordinate,

$$d\zeta ={\lambda }_L^{1/3}dr,A=lnM+{\displaystyle \frac{1}{3}}ln{\lambda }_L,$$

the background obeys the first-order flow

$${\dot{\Phi }}^I={\mathcal{W}}_{,I},\dot{A}=-{\displaystyle \frac{1}{6}}\mathcal{W},{\Phi }^I=(\phi ,q),$$

where a dot denotes $d/d\zeta$.

In the conformal coordinate $dz=e^{-A}d\zeta =dr/M$, write the canonically rescaled physical scalar amplitudes as

$$\Psi ={(f_0,f_{\phi },f_q)}^T.$$

For the normalization above, the fake-supergravity part of the coupled metric–scalar operator is generated by

$$\mathbb{S}=e^A{\left(\begin{array}{cc}\mathcal{W}/12& {\mathcal{W}}_{,J}/\sqrt{3}\\ {\mathcal{W}}_{,I}/\sqrt{3}& {\mathcal{W}}_{,IJ}-{\delta }_{IJ}\mathcal{W}/4\end{array}\right)}_{\mathrm{bg}}.$$

The extra square ${\mathcal{P}}^2$ vanishes to first order on the background and contributes only

$$2e^{2A}{\mathcal{P}}_{,I}{\mathcal{P}}_{,J}$$

to the scalar potential block. Therefore the full $\theta$-independent coupled Hamiltonian is

$$\mathbb{H}={\mathbb{Q}}^{†}\mathbb{Q}+{\mathbb{K}}^{†}\mathbb{K},\mathbb{Q}=-{\partial }_z+\mathbb{S},\mathbb{K}\Psi =\sqrt{2}{e}^A{\mathcal{P}}_{,I}{f}_I.$$

This is the standard gravity–multi-scalar factorization specialized to the present conventions, with the additional positive rank-one block displayed explicitly [29]. Its nonzero scalar-block eigenvalue is

$$R^2{\lambda }_{\perp }\left(r\right)=M^2\left(r\right)\left[12{\kappa }^2{t}^2+{\displaystyle \frac{8{\kappa }^2}{\kappa +1}}(1-t^2)\right]>0$$

at every finite interior point.

For a normalizable eigenmode, integration by parts yields

$$\begin{array}{cc}\hfill m_4^2\int dz{\Psi }^{†}\Psi =& {\int dz\left|\mathbb{Q}\Psi \right|}^2+\int dz{\left|\mathbb{K}\Psi \right|}^2\hfill \\ & +{\left[{\Psi }^{†}\mathbb{Q}\Psi \right]}_{\mathrm{boundary}}.\hfill \end{array}$$

The five-dimensional circle chart degenerates at $r=0$, so the endpoint cannot be assigned an arbitrary self-adjoint extension. Smooth six-dimensional cap regularity requires

$$\delta L=O\left(r^3\right),\delta \phi =O\left(r^2\right),\delta q=O\left(r\right),$$

and hence

$$f_0=O\left(r^{1/2}\right),f_{\phi }=O\left(r^{5/2}\right),f_q=O\left(r^{3/2}\right).$$

These powers make the cap boundary current vanish. At infinity a normalizable mode has zero flux as well. Consequently

$$m_4^2\ge 0.$$

The threshold case requires a separate check because a positive factorization alone does not generally remove multi-scalar zero modes [30]. If $m_4^2=0$, both squares vanish. The condition $\mathbb{K}\Psi =0$ makes the matter fluctuation tangent to $\mathcal{P}=0$; by $\nabla \mathcal{W}·\nabla \mathcal{P}=0$ it can be written as

$$f_I=a_T\left(u\right)w_I\left(u\right),w_I\equiv R{\mathcal{W}}_{,I},u\equiv r/R.$$

The remaining first-order equations $\mathbb{Q}\Psi =0$ reduce exactly to

$${\displaystyle \frac{d{f}_0}{du}}=t^{1/3}\left({\displaystyle \frac{w}{12}}{f}_0+{\displaystyle \frac{w_I{w}_I}{\sqrt{3}}}{a}_T\right),{\displaystyle \frac{d{a}_T}{du}}=t^{1/3}\left({\displaystyle \frac{f_0}{\sqrt{3}}}-{\displaystyle \frac{w}{4}}{a}_T\right),$$

where $w=R\mathcal{W}$. Near the cap,

$$w=-2t^{-4/3}+O\left(t^{2/3}\right),w_I{w}_I={\displaystyle \frac{8}{3}}{t}^{-8/3}+O\left(t^{-2/3}\right).$$

With the Frobenius ansatz

$$f_0=A_0{t}^p,a_T=B_0{t}^{p+4/3},$$

the indicial equations give

$$p={\displaystyle \frac{1}{2}}\mathrm{or}p=-{\displaystyle \frac{3}{2}},A_0=\sqrt{3}\left(p+{\displaystyle \frac{5}{6}}\right)B_0.$$

Moreover,

$$w_{\phi }=-{\displaystyle \frac{2\sqrt{6}}{3}}{t}^{-4/3}+O\left(t^{2/3}\right),f_{\phi }=a_T{w}_{\phi }=O\left(t^p\right).$$

The $p=-3/2$ branch is singular, while the $p=1/2$ branch violates the smooth-cap requirement $f_{\phi }=O\left(t^{5/2}\right)$. Setting $B_0=0$ forces $A_0=0$. Thus no nontrivial smooth normalizable zero mode exists.

The asymptotic potentials fall as $1/z^2$, so the essential spectrum begins at $m_4^2=0$. Zero is a continuum threshold, not a normalizable scalar eigenstate. It is therefore established

$${spec}_{\mathrm{normalizable}}\left(m_4^2\right)\cap (-\infty ,0]=⌀$$

for the natural $\theta$-independent scalar/radion sector. The benchmark value $\kappa =0.08844609375207024$ gives $R^2{\lambda }_{\perp }\left(0\right)=0.0574963632643$; this number checks the regular core limit but is not used in the proof.

Scope of the axisymmetric bulk result.The natural no-turn branch has a positive physical kinetic metric and no normalizable tachyon or scalar zero mode in its full coupled $\theta$-independent radion–modulus sector. This establishes the absence of an axisymmetric two-derivative effective-field-theory instability for that branch. It does not prove stability of non-axisymmetric $n_{\theta }\ne 0$ harmonics, does not supply an allowed microscopic $\mathrm{SU}{\left(15\right)}_p$ fermion representation or Yukawa vertex, and does not replace the full six-dimensional anomaly, inflow, confinement, or route-matching calculations.

Appendix D.13.13. Dipole Scalar Clocks and Exact Schur-null Cancellation

The first non-axisymmetric sector, $|n_{\theta }|=1$, is special because it contains the two translations of the smooth defect. A regular real perturbation of the complex sigma field can be written as

$$\delta \Phi =u\left(r\right)+v\left(r\right)e^{2i\theta }.$$

Relative to the winding background $\overline{\Phi }=F\left(r\right)e^{i\theta }$, its modulus and phase components are

$$\delta F=(u+v)cos\theta ,F\delta \chi =(v-u)sin\theta .$$

Whenever $F^{\prime }\left(r\right)>0$ on the open radial domain, this perturbation is a scalar clock for a transverse diffeomorphism. With the convention $\delta \Phi ={\mathcal{L}}_X\overline{\Phi }$,

$$X^r={\displaystyle \frac{u+v}{F^{\prime }}}cos\theta ,X^{\theta }={\displaystyle \frac{v-u}{F}}sin\theta .$$

The apparent factors $1/F$ and polar harmonics do not create a core singularity: the regular Frobenius domain for u and v is precisely the polar representation of a smooth Cartesian vector field. For the benchmark domain, the numerically reconstructed clock also preserves the outer normalizability condition. The negative fixed-metric dipole direction is therefore movable into the metric sector by a globally admissible gauge transformation; it is not, by itself, a gauge-invariant tachyon.

This conclusion can be made exact without assuming a particular projected completion. Let the full on-shell quadratic Hessian in matter and metric variables be

$$\mathcal{H}=\left(\begin{array}{cc}A& B\\ B^{†}& C\end{array}\right),$$

and let an admissible infinitesimal diffeomorphism generated by X define

$$q_X={\mathcal{L}}_X\overline{\Phi },g_X={\mathcal{L}}_X\overline{g}.$$

The exact diffeomorphism Ward identity is

$$\mathcal{H}\left({\displaystyle \genfrac{}{}{0pt}{}{q_X}{g_X}}\right)=0.$$

Its metric component gives

$$B^{†}{q}_X+C{g}_X=0.$$

Thus $g_X$ is the stationary metric completion at fixed $q_X$. The full quadratic action on $(q_X,g_X)$ vanishes exactly. Whenever the metric constraint operator admits the inverse required to define the Schur complement,

$$A_{\mathrm{Schur}}=A-B{C}^{-1}{B}^{†},$$

one obtains the exact clock-sector identity

$$A_{\mathrm{Schur}}{q}_X=0.$$

The globally reconstructed negative fixed-metric eigenfunction is an admissible scalar-clock image of precisely this type. Consequently its numerical fixed-slice eigenvalue

$$-0.34796305/R_c^2$$

is not a candidate physical mass. Stationary metric completion removes its fixed-metric Morse index by an exact rank-one cancellation on that direction.

The same statement can be expressed as a finite-dimensional theorem after discretization. Let

$$K\psi =\lambda N\psi ,N=L{L}^{†}>0,A=L^{-1}K{L}^{-†},$$

be the fixed-metric generalized matter Hessian in kinetic-whitened variables, and let $|t\rangle$ be the normalized exact translation vector. With

$$P_T=I-|t\rangle \langle t|,$$

the unique self-adjoint operator that annihilates the translation and leaves the quadratic form on $t^{\perp }$ unchanged is

$$A_W=P_{T}A{P}_T.$$

Its correction has rank at most two,

$$A_W-A=-A|t\rangle \langle t|-|t\rangle \langle t|A+\langle t\left|A\right|t\rangle |t\rangle \langle t|,rank(A_W-A)\le 2,$$

and

$$spec\left(A_W\right)=\left\{0\right\}\cup spec\left(P_{T}A{P}_T{|}_{t^{\perp }}\right).$$

This is an exact Ward-completion theorem; it contains no fitted stabilizing term.

For the stated benchmark discretization, the six lowest fixed-metric eigenvalues in units of $R^{-2}$ are

$$(-0.34796305,0.49343951,0.62642984,1.04592596,1.13416008,1.57320081),$$

whereas the completed spectrum is

$$(0,0.48904980,0.61698396,1.04168556,1.13180235,1.57294869).$$

The normalized Ward residual is $1.44\times {10}^{-12}$, the low-spectrum reconstruction residual is $1.27\times {10}^{-11}$, and the correction has numerical rank two. The fixed negative mode is strongly localized in the finite core, while its norm beyond $r/R=4$ is about $5.97\times {10}^{-6}$. These are numerical properties of the displayed benchmark, not universal spectral constants.

The canonical projection remains useful as an independent numerical illustration, but it is not the definition of the physical metric reduction. The Ward identity proves exact Schur-null curvature on the admissible clock image; it does not imply the operator identity

$$A_{\mathrm{Schur}}=P_{T}A{P}_T$$

on the full transverse space. Constraint elimination can change the finite-core quadratic form orthogonal to the clock direction and can, in principle, generate a different metric bound state. The remaining question is therefore not whether the former fixed-metric matter mode survives: it does not. The remaining question is whether the independently derived physical metric operator contains another negative normalizable state.

Remaining smooth-branch calculation.One must retain all $n_{\theta }=\pm 1$ Kaluza–Klein vector and Einstein lapse/shift variables, solve their regular constraint equations, construct the self-adjoint physical Schur complement with its induced boundary form, and determine its spectrum without identifying it in advance with $P_{T}A{P}_T$. A nonnegative operator would establish dipole stability of the smooth no-turn branch under the stated two-derivative assumptions. A negative normalizable eigenvalue would exclude that branch under the same assumptions.

Appendix D.13.14. Exact Birman–Schwinger Reduction of the Remaining Metric Problem

After quotienting the scalar-clock directions and canonically normalizing the physical amplitudes, write the $n_{\theta }=\pm 1$ problem as

$$S_{\pm 1,\mathrm{phys}}^{\left(2\right)}={\displaystyle \frac{1}{2}}\int d^{4}xdz\left({\partial }_{\mu }{\Psi }^{†}{\partial }^{\mu }\Psi -{\Psi }^{†}{H}_{\mathrm{phys}}\Psi \right),H_{\mathrm{phys}}=-I{\partial }_z^2+U_{\mathrm{phys}}\left(z\right),$$

on its regular Friedrichs domain. The core regularity, asymptotic, and principal-part results already established above imply

$$m_4^2<0⟹\mathrm{a}\mathrm{discrete},\mathrm{normalizable},\mathrm{finite}-\mathrm{core}\mathrm{eigenstate}.$$

Thus a negative continuum, a cap instability, and a remnant of the removed clock direction are excluded.

Choose a self-adjoint nonnegative reference operator $H_0\ge 0$ with the same principal part and asymptotic Friedrichs class as $H_{\mathrm{phys}}$. Decompose the finite-core form perturbation as

$$G=H_{\mathrm{phys}}-H_0=G_{+}-G_{-},G_{\pm }\ge 0,$$

and define

$$H_{+}=H_0+G_{+}\ge 0,H_{\mathrm{phys}}=H_{+}-G_{-}.$$

The explicit constrained Einstein second variation must determine $G_{-}$; it is not fixed by the preceding symmetry argument. Under the standard condition that $G_{-}$ is relatively form-compact with respect to $H_{+}$, define for ${\zeta }_{\mathrm{BS}}>0$

$$\mathcal{B}\left({\zeta }_{\mathrm{BS}}\right)=G_{-}^{1/2}{(H_{+}+{\zeta }_{\mathrm{BS}})}^{-1}{G}_{-}^{1/2}.$$

This is a positive compact operator. If $H_{\mathrm{phys}}\psi =-\mu \psi$ and $\varphi =G_{-}^{1/2}\psi$, then

$$\mathcal{B}\left(\mu \right)\varphi =\varphi .$$

The converse follows from the same resolvent identity. Hence

$$-\mu \in spec\left(H_{\mathrm{phys}}\right)⟺1\in spec\mathcal{B}\left(\mu \right),{\zeta }_{\mathrm{BS}}>0,$$

with equality of multiplicities. Moreover,

$$N_{(-\infty ,-\mu ]}\left(H_{\mathrm{phys}}\right)=N_{[1,\infty )}\left(\mathcal{B}\left(\mu \right)\right).$$

Because ${(H_{+}+{\zeta }_{\mathrm{BS}})}^{-1}$ decreases in the operator order as $\mu$ increases,

$$\beta \left(\mu \right)={\lambda }_{max}\mathcal{B}\left(\mu \right)$$

is nonincreasing. If the threshold limit

$${\beta }_0=\underset{\mu \downarrow 0}{lim}\beta \left(\mu \right)$$

exists, possibly as $+\infty$, then

$${\beta }_0<1⟹N_{-}\left(H_{\mathrm{phys}}\right)=0,$$

whereas

$${\beta }_0>1⟹N_{-}\left(H_{\mathrm{phys}}\right)\ge 1.$$

The equality ${\beta }_0=1$ is a genuine threshold case: the endpoint analysis must distinguish a normalizable zero mode, a zero-energy resonance, and a limiting continuum state. If $\mathcal{B}(0+)$ is trace class, either $Tr\mathcal{B}(0+)<1$ or the stronger readily evaluated bound ${\parallel \mathcal{B}(0+)\parallel }_{\mathrm{HS}}<1$ is sufficient to exclude negative modes.

There is no universal replacement for this threshold susceptibility by a statement that the gravitational correction is small. The essential spectrum of the nonnegative reference problem begins at zero. For any $\epsilon >0$, choose a normalized state with $\langle \psi ,H_0\psi \rangle <\epsilon /2$ and set

$$V_{\epsilon }=-\epsilon |\psi \rangle \langle \psi |.$$

Then $\parallel V_{\epsilon }\parallel =\epsilon$, but

$$\langle \psi ,(H_0+V_{\epsilon })\psi \rangle <-\epsilon /2.$$

An arbitrarily small attractive operator-norm perturbation can therefore create a negative Rayleigh quotient when the continuum touches zero. The spatial structure of the zero-energy resolvent, encoded by ${\beta }_0$, is indispensable.

The collapse of the first nongauge finite-box level confirms the relevance of this threshold. In units of $R_c^{-2}$, its values for $r_{max}/R_c=4,6,8,10,12,16,20,25,30$ are respectively

$$0.63477,0.52616,0.49344,0.39065,0.27524,0.13565,0.06686,0.02761,0.01140,$$

consistent with $inf{spec}_{\mathrm{ess}}{H}_0=0$. A positive level in a finite radial box is therefore not a uniform stability margin.

For a controlled finite-core calculation, let ${\mathcal{B}}_R$ be a truncated operator and establish independently

$$\parallel \mathcal{B}(0+)-{\mathcal{B}}_R\parallel \le {ϵ}_R,{\beta }_R={\lambda }_{max}\left({\mathcal{B}}_R\right).$$

Weyl’s inequality gives

$$|{\beta }_0-{\beta }_R|\le {ϵ}_R.$$

Consequently,

$${\beta }_R+{ϵ}_R<1$$

rigorously excludes negative modes, while

$${\beta }_R-{ϵ}_R>1$$

rigorously establishes at least one negative mode. If neither inequality holds, the core size, resolution, or tail estimate is insufficient to decide the spectrum. The numerical eigenvalue and its truncation bound must therefore be reported together.

If the attractive part has finite rank,

$$G_{-}={\displaystyle \sum _{a=1}^r}{g}_a|w_a\rangle \langle w_a|,g_a>0,$$

the nonzero spectrum of $\mathcal{B}\left(\mu \right)$ equals the spectrum of the positive $r\times r$ matrix

$$K_{ab}\left(\mu \right)=\sqrt{g_a{g}_b}\langle w_a|{(H_{+}+{\zeta }_{\mathrm{BS}})}^{-1}|w_b\rangle .$$

For rank one, $G_{-}=g|w\rangle \langle w|$ with $\langle w|w\rangle =1$,

$$\beta \left(\mu \right)=g\langle w|{(H_{+}+{\zeta }_{\mathrm{BS}})}^{-1}|w\rangle ,g_c={\displaystyle \frac{1}{\langle w\left|H_{+}^{-1}\right|w\rangle }},$$

whenever the threshold inverse exists on the support of w. This is an exact simplification if the constrained Einstein kernel is found to be of low rank; no such rank assumption is made in the physical analysis.

The six-dimensional smooth-branch question is therefore reduced to a single well-defined spectral quantity, ${\beta }_0$. Its value is presently unresolved because $U_{\mathrm{phys}}\left(z\right)$ and $G_{-}$ require the explicit gauge-invariant Einstein second variation and exact elimination of the nondynamical variables. No fixed-metric matter eigenvalue can substitute for that calculation.

Appendix D.13.15. Finite-Radius Interface Completion and the Boundary-Microphysics Limit

The exact no-turn cylinder above is a consistent effective background when transverse gauge response is attributed directly to bulk confinement. A different question arises if its compact winding is instead gauged by an ordinary local Maxwell–Stueckelberg sector with finite positive kinetic coefficients. In that class the infinite-cylinder stress split cannot be maintained asymptotically: the radial kinetic budget vanishes while a nonzero angular budget remains. Rewriting the same system by bulk Hodge duality as a finite-coupling higher-form BF theory does not change this conclusion, because the BF term is metric independent and the propagating kinetic sectors are locally equivalent.

A conditional classical escape is to retain the same smooth bulk geometry only on a finite interval $0\le r\le r_b$ and let an effective five-dimensional interface carry the junction stress. Write

$$u_b={\displaystyle \frac{r_b}{R}},t_b={tanh}^2{u}_b,$$

and use the gauge-invariant boundary action

$$S_b=-{\int }_{r=r_b}{d}^{5}x\sqrt{-\gamma }{e}^{{\lambda }_b(\varphi -{\varphi }_b)}\left[T_b+{\displaystyle \frac{1}{2}}{f}_b^2{\left(D\chi \right)}^2\right].$$

For the background covariant winding $W_b=D_{\theta }\chi$ define

$$Q_b\equiv {\displaystyle \frac{f_b^2{W}_b^2}{L_b^2}},{\rho }_b\equiv T_b+{\displaystyle \frac{Q_b}{2}}.$$

The induced stress is $S^{\mu }{}_{\nu }=-{\rho }_b{{\delta }^{\mu }}_{\nu }$ and $S^{\theta }{}_{\theta }=-T_b+Q_b/2$. With the outward orientation selected by $f_b^2>0$, the Israel equations give exactly

$$Q_b={\displaystyle \frac{2[1+(\kappa -1)t_b]}{{\kappa }_6^{2}R\sqrt{t_b}}},$$

$$T_b={\displaystyle \frac{1-(7\kappa +1)t_b}{{\kappa }_6^{2}R\sqrt{t_b}}},{\rho }_b={\displaystyle \frac{2[1-(3\kappa +1)t_b]}{{\kappa }_6^{2}R\sqrt{t_b}}}.$$

Thus the interface data are not three freely adjustable coefficients. If

$$x_b\equiv {\displaystyle \frac{T_b}{Q_b}},$$

then

$$x_b\left(t\right)={\displaystyle \frac{1-(7\kappa +1)t}{2[1+(\kappa -1\left)t\right]}},{\displaystyle \frac{d{x}_b}{dt}}=-{\displaystyle \frac{4\kappa }{{[1+(\kappa -1)t]}^2}}<0,$$

and hence

$$t_b={\displaystyle \frac{1-2x_b}{(7\kappa +1)+2(\kappa -1)x_b}}.$$

Positive bare tension is equivalent to

$$0<t_b<{\displaystyle \frac{1}{7\kappa +1}},0<x_b<{\displaystyle \frac{1}{2}}.$$

Within this window one microscopic equation-of-state ratio fixes the interface radius uniquely.

Maxwell matching also removes the integration constant left by the bulk inverse reconstruction. For

$$W\left(r\right)=n\left[1-\delta {tanh}^2(r/R)\right],0<\delta <1,$$

one obtains the exact value

$$C_{★}=-{\displaystyle \frac{1}{{\kappa }_6^2}}\left[1+{sech}^{4\kappa }{u}_b\left(\kappa +(1-\kappa ){sech}^2{u}_b\right)\right]<0.$$

Indeed, if

$$I\left(r\right)=C_{★}+{\int }_0^r{M}^4\left(\overline{r}\right)X_0\left(\overline{r}\right)L\left(\overline{r}\right)d\overline{r},$$

where $X_0$ is the positive angular stress budget of the ungauged exact reconstruction, Maxwell matching is precisely

$$I\left(r_b\right)=-M_b^4{L}_b{Q}_b<0.$$

Since $I^{\prime }\left(r\right)=M^4{X}_{0}L>0$, it follows that $I\left(r\right)<0$ throughout $0\le r\le r_b$. The reconstructed Maxwell stress is proportional to $W^{\prime }I/\left(M^{4}LW\right)$ and is therefore positive for the decreasing $W\left(r\right)$. By continuity on the compact interval, a nonempty sufficiently small range of $\delta >0$ keeps all reconstructed kinetic functions finite and positive.

This completion is also economical in its independent data. Once $(\kappa ,R,{\kappa }_6,q,n)$ and a microscopic ratio $x_b=T_b/Q_b$ are fixed, the inversion theorem fixes $u_b$. If the microscopic boundary stiffness $f_b^2$ is known as well, then

$$f_b^2=Q_b{\displaystyle \frac{R^2{t}_b}{n^2{(1-\delta t_b)}^2}}$$

determines

$$\delta ={\displaystyle \frac{1-{\displaystyle \frac{R\sqrt{t_b}}{\left|n\right|}}\sqrt{Q_b/f_b^2}}{t_b}}.$$

The scalar junction then determines ${\lambda }_b$. Thus the classical interface is not an arbitrary functional fit: its remaining difficulty is microscopic origin, not coefficient counting.

The interface scalar equation also closes with no second independent brane function. For outward normal $n^r=+1$ it is

$${\varphi }_b^{\prime }+{\lambda }_b{\rho }_b=0,{\lambda }_b=-{\displaystyle \frac{{\varphi }_b^{\prime }}{{\rho }_b}}.$$

The fixed-background curvature ${\lambda }_b^2{\rho }_b$ is positive in the positive-tension window. This is a local background statement, not a proof of the coupled interface fluctuation spectrum.

The exact interface is necessarily flux dominated:

$$Q_b-{\rho }_b={\displaystyle \frac{8\kappa tanh{u}_b}{{\kappa }_6^{2}R}}>0(u_b>0).$$

In fact $1<Q_b/{\rho }_b<2$ throughout the positive-tension window. The boundary winding term has a five-dimensional dual description,

$$S_{\mathrm{dual},b}=-{\displaystyle \frac{1}{2}}\int f_b^{-2}{G}_4\wedge {*}_5{G}_4\pm q\int C_3\wedge F_2,G_4=d{C}_3,$$

in which the four-form carries energy density ${\rho }_G=Q_b/2$ and the BF term carries no stress. Therefore ${\rho }_G>T_b$ at every finite positive-tension radius.

This stress inequality gives a sharp microscopic no-go theorem. Consider any number of canonical wall scalars ${\sigma }^A\left(y\right)$ with positive target metric $G_{AB}$ in a controlled thin-wall decoupling limit, with

$$\mathcal{E}={\displaystyle \frac{1}{2}}{G}_{AB}{\sigma }^{A\prime }{\sigma }^{B\prime }+V\left(\sigma \right)+{\displaystyle \frac{1}{2}}K\left(\sigma \right)w^2,V\ge 0,K\ge 0,$$

and vanishing vacuum wall energy. The one-dimensional first integral gives

$${\displaystyle \frac{1}{2}}{G}_{AB}{\sigma }^{A\prime }{\sigma }^{B\prime }=V+{\displaystyle \frac{1}{2}}K{w}^2.$$

Consequently

$${\rho }_{\mathrm{can}}=2\int Vdy+w^2\int Kdy,Q_{\mathrm{can}}=w^2\int Kdy,$$

so

$${\rho }_{\mathrm{can}}-Q_{\mathrm{can}}=2\int Vdy\ge 0.$$

This contradicts the exact required inequality $Q_b>{\rho }_b$. Hence the finite interface cannot be the thin limit of an ordinary positive-potential canonical scalar wall. The theorem does not exclude an intrinsic topological flux brane, a strongly backreacted wall whose thickness is comparable with R, noncanonical or higher-derivative matter, or genuinely nonperturbative confining interface stress.

For the dimensionless benchmark $\kappa =R={\kappa }_6=q=n=1$, $u_b=0.30$, and $\delta =0.05$, the exact values are

$$(Q_b,T_b,{\rho }_b)=(6.865476860643,1.102237530709,4.534975961031),$$

$${\displaystyle \frac{T_b}{Q_b}}=0.160547847307,{\displaystyle \frac{Q_b}{{\rho }_b}}=1.513894873895,{\lambda }_b=-0.411160278028.$$

These numbers are unit checks of the exact formulas, not predictions of the microscopic $\mathrm{SU}{\left(15\right)}_p$ theory. Positive tension also truncates the cigar early: the retained Planck fraction obeys

$$f_{\mathrm{Pl}}<1-{\left({\displaystyle \frac{7\kappa }{7\kappa +1}}\right)}^{\kappa }<0.1332.$$

Boundary-microphysics implication.The finite-radius background and its Israel–Maxwell–scalar junction are classically closed with finite positive couplings, and the interface equation of state is one-dimensional rather than freely tunable. However, the required stress cannot originate from an ordinary canonical positive-potential thin wall. Moreover, the smooth infinite-background factorization proved above does not automatically control new interface-localized scalar, vector, or radion modes. The remaining viable origins are a flux-dominated topological interface, a strongly backreacted thick defect, or nonperturbative confining stress. Their flux quantization, global gauge and Dai–Freed consistency, and coupled interface spectrum remain open. Without such microscopic input, further small local scalar, vector, or BF deformations do not decide the six-dimensional theory.

Appendix D.13.16. Exactly One Four-Dimensional Weyl Mode on the Backreacted Geometry

Unit winding invokes the Jackiw–Rossi/Weinberg vortex-index mechanism [23,24]; related higher-dimensional vortex constructions localize chiral generations and fermions [25,26]. Backreaction changes the radial norm but not the angular index. After the standard spin-connection redefinition, the zero-mode profile behaves asymptotically as

$${\psi }_0\sim M^{-2}{L}^{-1/2}exp\left[-{\int }^{\rho }{m}_f\left({\rho }^{\prime }\right)d{\rho }^{\prime }\right],m_{f,\infty }=y_{r}v.$$

Combining the kinetic norm with $M\sim e^{-cx}$ and $x=m_H\rho /\sqrt{2}$ gives

$$-2{\displaystyle \frac{y_{r}v}{m_H/\sqrt{2}}}+c<0.$$

Since $m_H=\sqrt{2\lambda }v$,

$${\displaystyle \frac{y_r}{\sqrt{\lambda }}}>{\displaystyle \frac{c}{2}}.$$

At the reproduced background,

$${\displaystyle \frac{y_r}{\sqrt{\lambda }}}>0.044223046876\dots .$$

Together with the unit-winding index and regularity analysis, this leaves exactly one normalizable four-dimensional Weyl zero mode per vectorlike six-dimensional Dirac parent over a broad Yukawa-coupling range.

Appendix D.13.17. Regularity Surface and the Limit of the Ordinary BPS Mechanism

The self-consistent Einstein–Abelian–Higgs solution is not generic in the full bulk parameter space: the regular gravity-localizing branch lies on a codimension-one regularity surface. For the numerical background, the core relation

$$B_P=-{\displaystyle \frac{{\alpha }_6}{2{\nu }_6}}$$

is satisfied. Using only the rounded values ${\alpha }_6=1.16$ and ${\nu }_6=2.5801288700$ printed above gives $-0.22479497313$, which differs from the separately quoted shooting value by $4.30\times {10}^{-9}$. The agreement is therefore at the level allowed by the unreported parameter digits, not at every displayed digit of $B_P$.

Could the ordinary Abelian–Higgs Bogomolnyi limit explain this selection automatically? Direct analysis gives a negative answer. At the critical coupling where the usual first-order BPS equations emerge, the warped branch simultaneously approaches

$${\mu }_6\to 0,c=\sqrt{-{\mu }_6/10}\to 0,M\to 1.$$

The exponential gravitational localization then disappears and the four-dimensional Planck integral diverges.

Local no-go theorem and its scope.The ordinary minimal Abelian–Higgs BPS limit does not select the required warped gravity-localizing Einstein–Abelian–Higgs branch. This is a conditional no-go theorem for that specific BPS realization, not for all possible first-order or topological completions.

Appendix D.13.18. Top Form: From a Tuned Local Parameter to an Integration Constant

Introduce a five-form potential $C_5$ with six-form field strength

$$F_6=d{C}_5=q{\mathrm{vol}}_6.$$

A top form in six dimensions has no local propagating degree of freedom; its equation of motion makes q an integration constant. In physical normalization it shifts the effective cosmological constant according to

$${\Lambda }_{\mathrm{eff}}={\Lambda }_0+{\displaystyle \frac{q^2}{2}}.$$

Equivalently, introduce dimensionless flux parameters ${\widehat{q}}_{\mathfrak{a}}$ through

$${\mu }_{6,\mathrm{eff}}={\mu }_{6,0}+{\displaystyle \sum _{\mathfrak{a}=1}^{N_F}}{\widehat{q}}_{\mathfrak{a}}^2,$$

where ${\mu }_{6,0}$ is the bare dimensionless cosmological parameter and $N_F$ is the number of independent top-form sectors. The regular branch reproduced above requires

$${\mu }_{6,★}=-0.0782271150.$$

For the illustrative bare value ${\mu }_{6,0}=-0.10$,

$$\Delta {\mu }_{6,★}\equiv {\mu }_{6,★}-{\mu }_{6,0}=0.021772885000000006,$$

so one continuous top form requires

$${\widehat{q}}_{★}=\sqrt{\Delta {\mu }_{6,★}}=0.14755637905560032.$$

The matter and metric profiles are then exactly those already solved at ${\mu }_{6,\mathrm{eff}}={\mu }_{6,★}$. A codimension-one choice of a local Lagrangian parameter has therefore been converted into the selection of an integration-constant sector. This is a structural improvement, but not an automatic mechanism of radiative self-tuning.

Appendix D.13.19. Quantum Bifurcation: When Must the Flux Be Quantized?

The answer depends on the global higher-form structure.

Appendix D.13.13.1. Source-free noncompact effective-theory branch.

The smooth background has topology

$${\mathcal{M}}_6\simeq {\mathbb{R}}^{1,3}\times {\mathbb{R}}^2.$$

There is no nontrivial compact six-cycle ${\Sigma }_6$ on which to impose a period condition

$${\displaystyle \frac{1}{2\pi }}{\int }_{{\Sigma }_6}{F}_6\in \mathbb{Z}.$$

Thus, if (Q1) $C_5$ is treated as a source-free top-form sector of the low-energy theory, (Q2) no electrically charged four-branes are introduced, and (Q3) no additional compact six-cycle is added, the constant q remains continuous. Under Q1–Q3 the top form can select the regular background at the level of the effective theory. This does not assert that an unknown quantum-gravity completion must realize precisely this noncompact branch.

Appendix D.13.13.2. Compact higher-form gauge branch.

If $C_5$ is a compact gauge field and electrically charged four-branes exist, the flux takes discrete values

$$\widehat{q}={\widehat{q}}_0+n\widehat{e},n\in \mathbb{Z},$$

where ${\widehat{q}}_0$ is an offset and $\widehat{e}$ the charge spacing. Exact regularity for one top form requires

$${({\widehat{q}}_0+n\widehat{e})}^2=\Delta {\mu }_{6,★}.$$

For fixed ${\widehat{q}}_0$ and $\widehat{e}$, a generic real value of $\Delta {\mu }_{6,★}$ does not belong to this discrete set. A single compact quantized top form therefore does not generically hit the exact regularity surface.

Appendix D.13.20. Four Equal Fluxes Fill Arithmetic Shells but Do Not Remove the Radial Spacing

Take $N_F$ equal-charge top forms with zero offsets and common spacing $\widehat{e}$,

$$\Delta {\mu }_{\mathrm{flux}}={\widehat{e}}^2{\displaystyle \sum _{\mathfrak{a}=1}^{N_F}}{n}_{\mathfrak{a}}^2.$$

Define

$$m_{\mathrm{shell}}\equiv {\displaystyle \sum _{\mathfrak{a}=1}^{N_F}}{n}_{\mathfrak{a}}^2\in {\mathbb{Z}}_{\ge 0}.$$

For $N_F=1$ only perfect squares occur. For $N_F=2$ not every integer is a sum of two squares. For $N_F=3$, Legendre’s theorem excludes integers of the form $4^a(8b+7)$. Lagrange’s four-square theorem guarantees that every nonnegative integer is a sum of four squares. Therefore four equal-charge top forms are sufficient for universal shell coverage,

$$N_F^{\mathrm{universal}}=4.$$

This is an exact arithmetic statement about angular degeneracy in flux space.

However, the allowed radial values are still

$$\Delta {\mu }_{\mathrm{flux}}={\widehat{e}}^2{m}_{\mathrm{shell}}.$$

The nearest shell obeys

$$\left|\Delta {\mu }_{\mathrm{flux}}-\Delta {\mu }_{6,★}\right|\le {\displaystyle \frac{{\widehat{e}}^2}{2}}.$$

Adding more equal-charge forms increases shell degeneracy but does not reduce the radial spacing ${\widehat{e}}^2$. Thus many equal-charge fluxes do not generically solve exact regularity unless the charge scale itself is appropriately aligned. Incommensurate charges can generate a much denser discretuum, but density alone is not a theorem of exact intersection with the required regularity surface.

Appendix D.13.21. Linear Stability Theorem: Transverse-Traceless Gravitational Mode

Consider a four-dimensional transverse-traceless perturbation

$$h_{\mu \nu }(x,\rho ,\theta )={ϵ}_{\mu \nu }{e}^{ip·x}{e}^{in\theta }{h}_n\left(\rho \right),p^2=-m_4^2.$$

The radial equation is

$$-{\displaystyle \frac{d}{d\rho }}\left(M^{4}L{\displaystyle \frac{d{h}_n}{d\rho }}\right)+n^2{\displaystyle \frac{M^4}{L}}{h}_n=m_4^2{M}^{2}L{h}_n.$$

Multiplying by $h_n^{*}$, integrating, and using regular and normalizable boundary conditions gives

$$m_4^2={\displaystyle \frac{{\displaystyle {\int }_0^{\infty }d\rho \left[M^{4}L|h_n^{\prime }{|}^2+n^2\frac{M^4}{L}{\left|h_n\right|}^2\right]}}{{\displaystyle {\int }_0^{\infty }d\rho M^{2}L{\left|h_n\right|}^2}}}\ge 0.$$

Thus the transverse-traceless tensor channel has no tachyons. For $n=0$, the constant profile is an exact zero mode and is normalizable because $I_{\mathrm{grav}}<\infty$.

For completeness, define the asymptotic conformal coordinate $z_{\mathrm{as}}$ by

$$d{z}_{\mathrm{as}}={\displaystyle \frac{d\rho }{M}},S\left(z_{\mathrm{as}}\right)\equiv {\displaystyle \frac{3lnM+lnL}{2}},$$

and write $h=e^{-S}\psi$. Then

$$\left[-{\displaystyle \frac{d^2}{d{z}_{\mathrm{as}}^2}}+V_T\left(z_{\mathrm{as}}\right)\right]\psi =m_4^2\psi ,$$

$$V_T=S^{\prime 2}+S^{\prime \prime }=\left({\displaystyle \frac{d}{d{z}_{\mathrm{as}}}}+S^{\prime }\right)\left(-{\displaystyle \frac{d}{d{z}_{\mathrm{as}}}}+S^{\prime }\right),$$

which gives a second proof of nonnegativity. Because $M,L\propto e^{-cx}$ asymptotically,

$$V_T\left(z_{\mathrm{as}}\right)\sim {\displaystyle \frac{6}{z_{\mathrm{as}}^2}},$$

and the numerical reconstruction gives $z_{\mathrm{as}}^2{V}_T\to 5.99972\dots$. Since $V_T\to 0$, the tensor continuum begins at $m_4^2=0$: the localized graviton is a normalizable threshold zero mode rather than a state separated from the continuum by a positive tensor gap.

Appendix D.13.22. Linear Stability Theorem: Spectator non-Abelian Gauge Fields

For a radial mode $a_n\left(\rho \right)$ of a spectator gauge factor,

$$-{\displaystyle \frac{d}{d\rho }}\left(M^{2}L{\displaystyle \frac{d{a}_n}{d\rho }}\right)+n^2{\displaystyle \frac{M^2}{L}}{a}_n=m_4^{2}L{a}_n.$$

The same integration-by-parts argument yields

$$m_4^2={\displaystyle \frac{{\displaystyle {\int }_0^{\infty }d\rho \left[M^{2}L|a_n^{\prime }{|}^2+n^2\frac{M^2}{L}{\left|a_n\right|}^2\right]}}{{\displaystyle {\int }_0^{\infty }d\rho L{\left|a_n\right|}^2}}}\ge 0.$$

Hence the spectator gauge sector contains no tachyonic modes. For $n=0$ the constant gauge mode is normalizable because $I_g<\infty$.

Appendix D.13.23. Exact Positivity of the Coupled vortex-U(1) V Vector Channel

The gauge perturbation of the vortex-forming Abelian field is more subtle because it mixes with spin-one metric perturbations. The physical variables must therefore be gauge invariant. The analysis uses the standard gauge-invariant reduction of vector perturbations of a six-dimensional Abelian vortex [27].

Introduce a conformal radial coordinate w by

$$dx=\mathcal{L}\left(w\right)dw,$$

where x is the dimensionless radial coordinate introduced above. To avoid confusion with the logarithmic warp derivative $\ell \left(x\right)=dln\mathcal{L}/dx$, denote the angular Fourier number by $n_{\theta }\in \mathbb{Z}$. A gauge-invariant vector master mode $N_{n_{\theta }}\left(w\right)$ satisfies

$$\left[-{\displaystyle \frac{d^2}{d{w}^2}}+{\displaystyle \frac{{\left[M\left(w\right)P\left(w\right)\right]}^{\prime \prime }}{M\left(w\right)P\left(w\right)}}\right]N_{n_{\theta }}=\left(m_4^2-n_{\theta }^2\right)N_{n_{\theta }},$$

where primes in this subsection denote $d/dw$. Define

$$W_V\left(w\right)\equiv {\displaystyle \frac{{\left[M\left(w\right)P\left(w\right)\right]}^{\prime }}{M\left(w\right)P\left(w\right)}}={\displaystyle \frac{d}{dw}}ln\left[M\left(w\right)P\left(w\right)\right].$$

Then

$${\displaystyle \frac{{\left(MP\right)}^{\prime \prime }}{MP}}=W_V^{\prime }+W_V^2,$$

and the one-dimensional operator factorizes exactly,

$$-{\displaystyle \frac{d^2}{d{w}^2}}+{\displaystyle \frac{{\left(MP\right)}^{\prime \prime }}{MP}}=\left({\displaystyle \frac{d}{dw}}+W_V\right)\left(-{\displaystyle \frac{d}{dw}}+W_V\right).$$

For every regular mode in the operator domain,

$$\langle N,H_{V}N\rangle ={∥\left(-{\displaystyle \frac{d}{dw}}+W_V\right)N∥}^2\ge 0,$$

where $H_V$ denotes the factorized vector operator. Therefore

$$m_4^2\ge n_{\theta }^2\ge 0.$$

This is an exact no-tachyon theorem for the coupled vortex-vector channel.

For $n_{\theta }=0$, the regular zero-mode solution is proportional to the background gauge profile,

$$A_{\mu }^{\left(0\right)}\left(w\right)\propto P\left(w\right).$$

Its norm on the regular self-consistent background is

$$I_V\equiv {\int }_0^{\infty }\mathcal{L}\left(x\right)P^2\left(x\right)dx=0.9525446502\dots <\infty .$$

Thus the perturbation problem contains a localized massless vector associated with the vortex $U{\left(1\right)}_V$. Higgsing the defect field does not, by itself, prove the absence of a four-dimensional massless vector mode. Whether this mode is phenomenologically acceptable, can be identified with an allowed low-energy field, or is lifted by additional ultraviolet structure is a separate model-building question. The theorem establishes only its existence and normalizability within the stated system.

Appendix D.13.24. Independent Graviphoton Zero Modes Are Not Localized

Spin-one metric perturbations contain two independent gauge-invariant combinations, which denote by $V_{\mu }$ and $Z_{\mu }$. Their homogeneous zero-mode solutions behave as

$$V_{\mu }^{\left(0\right)}\propto {\displaystyle \frac{1}{M^{3}L}},Z_{\mu ,\mathrm{hom}}^{\left(0\right)}\propto {\displaystyle \frac{M}{L}}.$$

The Einstein–Hilbert norm of $V_{\mu }$ contains

$$I_{V_{\mathrm{grav}}}\propto \int dx{\displaystyle \frac{1}{L{M}^4}}.$$

Since $M,L\propto e^{-cx}$ at large x,

$${\displaystyle \frac{1}{L{M}^4}}\propto e^{5cx},$$

so $I_{V_{\mathrm{grav}}}$ diverges. For the homogeneous $Z_{\mu }$ solution the norm contains

$$I_{Z_{\mathrm{grav}}}\propto \int dx{\displaystyle \frac{M^4}{L}}.$$

Near the regular core, $M\to 1$ and $L\sim x$, hence $M^4/L\sim 1/x$ and the integral diverges logarithmically. A separate $Z_{\mu }$ contribution is algebraically tied to the localized vortex gauge mode, but it is not an additional independent graviphoton degree of freedom. Therefore there are no independent localized graviphoton vector zero modes.

Appendix D.13.25. Absence of a Localized Massless Scalar Modulus

The scalar sector mixes scalar metric perturbations, the real and imaginary Higgs perturbations, and scalar components of the vortex gauge field. A zero-mode analysis alone does not establish positivity of the entire massive scalar spectrum, but the gauge-invariant asymptotics exclude a localized massless modulus [27]. The full scalar problem must also distinguish this statement from the separate stable transverse metric–gauge channel and from the background radial-breathing, or “sausage”, sector whose boundary-value problem is generically rigid [28].

One gauge-invariant scalar combination contains an off-diagonal metric perturbation $\Pi$ with

$$\Pi \sim M^{-4}.$$

The corresponding canonically weighted field grows and is not normalizable. A diagonal scalar variable $\Xi$ behaves as

$$\Xi \sim M^{-2},$$

so its canonical combination tends to a nonzero constant and its norm diverges linearly in the conformal coordinate. A second metric scalar $\Psi$ has an acceptable constant asymptotic tail but a nonnormalizable core behavior. The remaining scalar variables are algebraically related to these combinations and do not produce an independent normalizable zero state. Hence no localized massless radion, dilaton, or breathing scalar zero mode exists under the stated regularity and asymptotic assumptions.

This is a theorem about zero modes. It does not imply that every massive scalar eigenvalue is nonnegative; that requires the gauge-invariant channel-by-channel analysis discussed below.

Appendix D.13.26. Numerical Nondegeneracy of the Static Boundary-Value Problem

An independent diagnostic asks whether the regular background lies on an obvious continuous family of static solutions. Fix ${\alpha }_6$ and ${\nu }_6$ and vary the regular-core parameters

$$p_j=(A_f,B_P,{\mu }_6).$$

At a finite matching radius $x_{\mathrm{end}}=8$, define the residual vector

$${\mathcal{R}}_4(A_f,B_P,{\mu }_6)=\left(\begin{array}{c}f\left(x_{\mathrm{end}}\right)-1\\ P\left(x_{\mathrm{end}}\right)\\ m\left(x_{\mathrm{end}}\right)+c\left({\mu }_6\right)\\ \ell \left(x_{\mathrm{end}}\right)+c\left({\mu }_6\right)\end{array}\right),c\left({\mu }_6\right)=\sqrt{-{\mu }_6/10}.$$

The finite-dimensional shooting Jacobian is

$${\left(J_{\mathrm{shoot}}\right)}_{ij}={\displaystyle \frac{\partial {\mathcal{R}}_i}{\partial p_j}}.$$

After dimensionless row and column scaling, the numerical singular values are

$$s\left(J_{\mathrm{shoot}}\right)=(1.6225154,1.14600875,0.23261066),$$

and at fixed ${\mu }_6$ the $(A_f,B_P)$ block gives

$$s\left(J_{AB}\right)=(1.6344161,1.15268557).$$

All displayed singular values are nonzero. Within this finite-dimensional shooting subspace there is therefore no infinitesimal static direction that leaves all asymptotic residuals unchanged. This is a strong numerical nondegeneracy check, but it is not a spectral positivity theorem for the full infinite-dimensional scalar Hessian.

Appendix D.13.27. Local Continuation of the Regular Vortex and Top-Form Tracking of the Regular Branch

The absence of a normalizable massless scalar modulus does not imply that the regular vortex is isolated under changes of microscopic parameters. A different question is whether a neighboring regular solution exists when the homogeneous parameters of the Einstein–Abelian–Higgs system are varied slightly.

Near the core write

$$f\left(x\right)=A_{f}x+O\left(x^3\right),P\left(x\right)=1+B_P{x}^2+O\left(x^4\right).$$

Together with the effective cosmological parameter ${\mu }_6$, the coefficients $A_f$ and $B_P$ define three shooting variables,

$$p=(A_f,B_P,{\mu }_6).$$

The two dimensionless microscopic parameters controlling the matter and gravitational sectors are

$${\lambda }_{\mathrm{micro}}=({\alpha }_6,{\nu }_6).$$

At a large but finite matching radius $x_{\mathrm{end}}$, define the independent residual map

$${\mathcal{R}}_3(A_f,B_P,{\mu }_6;{\alpha }_6,{\nu }_6)=\left(\begin{array}{c}f\left(x_{\mathrm{end}}\right)-1\\ P\left(x_{\mathrm{end}}\right)\\ m\left(x_{\mathrm{end}}\right)-\ell \left(x_{\mathrm{end}}\right)\end{array}\right).$$

The third residual is sufficient because once $f\to 1$, $P\to 0$, and $m=\ell$, the Einstein constraint fixes the common AdS slope,

$$m=\ell =-\sqrt{-{\mu }_6/10}.$$

Thus the regular finite-radius boundary-value problem is locally represented by ${\mathcal{R}}_3=0$.

At the refined numerical solution

$$(A_f,B_P,{\mu }_6)\simeq (0.48117259,-0.22479497,-0.07822627),$$

the finite-difference Jacobian

$$J_{\mathcal{R}}\equiv {\displaystyle \frac{\partial {\mathcal{R}}_3}{\partial (A_f,B_P,{\mu }_6)}}$$

has

$$det{J}_{\mathcal{R}}=-6.07899134497\times {10}^9,\kappa \left(J_{\mathcal{R}}\right)\simeq 541.77,$$

where $\kappa$ is the matrix condition number in the stated dimensionless scaling. The determinant is emphatically nonzero in the reproduced finite-radius problem, so no infinitesimal direction in $(A_f,B_P,{\mu }_6)$ leaves all three residuals unchanged.

The implicit-function theorem then has a direct physical interpretation. If the exact infinite-radius boundary map is continuously differentiable and its Jacobian remains nonsingular in the continuum limit, a unique smooth local regularity surface exists,

$$A_f=A_f({\alpha }_6,{\nu }_6),B_P=B_P({\alpha }_6,{\nu }_6),{\mu }_{6,★}={\mu }_{6,★}({\alpha }_6,{\nu }_6).$$

The theorem is exact; the persistence of the nonsingularity hypothesis in the physical infinite-radius problem is supported numerically rather than proved analytically.

This branch was tested by reintegrating the full nonlinear equations under independent variations of ${\alpha }_6$ and ${\nu }_6$ by up to approximately $\pm 2\%$. The regular solution persists throughout the tested window, with local slopes

$${\displaystyle \frac{d{\mu }_{6,★}}{d{\nu }_6}}=-0.14985878\dots ,{\displaystyle \frac{d{\mu }_{6,★}}{d{\alpha }_6}}=+0.06115664\dots .$$

The exact core identity

$$B_P=-{\displaystyle \frac{{\alpha }_6}{2{\nu }_6}}$$

provides an independent derivative check,

$${\displaystyle \frac{\partial B_P}{\partial {\nu }_6}}={\displaystyle \frac{{\alpha }_6}{2{\nu }_6^2}},{\displaystyle \frac{\partial B_P}{\partial {\alpha }_6}}=-{\displaystyle \frac{1}{2{\nu }_6}},$$

and the numerical continuation reproduces these derivatives within the stated integration accuracy.

Now add a source-free six-form with

$$F_6=\widehat{q}{\mathrm{vol}}_6,$$

where $\widehat{q}$ is a spacetime-independent integration constant in each source-free sector. It shifts

$${\mu }_{6,\mathrm{eff}}={\mu }_{6,\mathrm{bare}}+{\widehat{q}}^2.$$

For the illustrative value ${\mu }_{6,\mathrm{bare}}=-0.10$, the reference regular solution requires

$${\widehat{q}}_{★}=\sqrt{{\mu }_{6,★}-{\mu }_{6,\mathrm{bare}}}=0.14755637905560032.$$

Along the regularity surface, with ${\mu }_{6,\mathrm{bare}}$ fixed,

$$2{\widehat{q}}_{★}d{\widehat{q}}_{★}=d{\mu }_{6,★},$$

so

$${\displaystyle \frac{d{\widehat{q}}_{★}}{d{\nu }_6}}=-0.50780178\dots ,{\displaystyle \frac{d{\widehat{q}}_{★}}{d{\alpha }_6}}=+0.20723143\dots .$$

A single continuous nonpropagating integration constant therefore has exactly the local codimension required to follow the regularity surface under sufficiently small homogeneous changes of the vortex parameters.

What the regularity servo solves.Provided the boundary map remains nondegenerate in the continuum and infinite-radius limit, the regular vortex is not an isolated tuned point but lies on a locally smooth regularity surface. A continuous source-free top-form integration constant can track that surface and compensate homogeneous shifts of ${\alpha }_6$ and ${\nu }_6$ without introducing a new local propagating degree of freedom.

Exact limitation.The quantity $\widehat{q}$ is a global integration constant, not a field $\widehat{q}\left(x^{\mu }\right)$. It therefore cannot provide a Green function for an arbitrary spacetime-dependent perturbation $\delta T_{\mu \nu }\left(x\right)$. Homogeneous regularity tracking and local dynamical scalar response are different problems. The gauge-invariant scalar analysis below makes this distinction explicit.

Appendix D.13.28. Full Gauge-Invariant Scalar Decomposition: Transverse Stability, Threshold Locality, and Radial-Breathing Rigidity

Restoring the Einstein scalar constraints changes the interpretation of fixed-geometry matter fluctuations. Two gauge-invariant sectors must be distinguished because they have different physical behavior.

Appendix D.13.13.1. Transverse metric–gauge scalar channel.

For one transverse gauge-invariant combination, all constraint variables can be eliminated exactly. Its master equation reduces to a second-order operator with the factorized positive form

$$\left(A_T^{†}{A}_T+Z_T^2\left(\rho \right)\right)u_T=0,Z_T^2\left(\rho \right)>0$$

on the physical bounded domain. This excludes a tachyonic branch, and the stated regularity conditions also exclude a normalizable zero mode.

In an asymptotic conformal coordinate $z_T$, the operator behaves as

$$H_T=-{\partial }_{z_T}^2+{\displaystyle \frac{{\nu }_T^2-\frac{1}{4}}{z_T^2}}+o\left(z_T^{-2}\right),{\nu }_T={\displaystyle \frac{5}{2}}.$$

For a compact-core probe, a regular delta-normalized continuum mode therefore behaves at $k\to 0$ as

$${\psi }_k\left(z_T\right)=O\left(k^{{\nu }_T+1/2}\right)=O\left(k^3\right),$$

so the threshold spectral measure is

$$d{\mu }_T\left(k\right)=O\left(k^6\right)dk.$$

The static susceptibility

$${\chi }_T\left(0\right)={\int }_0^{\infty }{\displaystyle \frac{d{\mu }_T\left(k\right)}{k^2}}$$

has an infrared integrand $O\left(k^4\right)dk$ and converges. The inverse moments

$$M_n^{\left(T\right)}\equiv \langle s_T|H_T^{-n}|s_T\rangle$$

have threshold integrand $k^{6-2n}dk$, so $n=1,2,3$ are guaranteed finite. Consequently, the low-momentum core response has the local expansion

$${\chi }_T\left(Q\right)=a_0+a_1{Q}^2+a_2{Q}^4+O\left(Q^5\right),$$

where Q is the magnitude of the four-dimensional Euclidean momentum and the first generic continuum nonanalyticity enters at order $Q^5$.

Strict result for the transverse scalar channel.Under the stated regular-core assumptions, the full gauge-invariant transverse metric–gauge scalar channel is non-tachyonic, has no normalizable zero mode, has exact asymptotic index ${\nu }_T=5/2$, possesses a finite static compact-core response, and admits a local derivative expansion through $Q^4$. This is a result for the full metric-coupled channel, not a Cowling approximation.

Appendix D.13.13.2. Why the fixed-geometry scalar continua are not the full sausage spectrum.

If one artificially imposes $\delta g_{MN}=0$, the Higgs-amplitude and gauge-profile fluctuations form a useful matter/Cowling Hessian. Their large inverse-square indices and strong threshold suppression are mathematically meaningful diagnostics. Once the Einstein constraints are restored, however, these matter variables are not an independent physical pair in the background-field sector. The fixed-geometry spectra must therefore not be identified with the complete gauge-invariant scalar spectrum.

Appendix D.13.13.3. Background “sausage” sector.

Gauge-invariant perturbations of fields already nonzero in the background – the Higgs amplitude, the angular vortex gauge profile, and diagonal warp factors – form the system conventionally called the sausage sector in the six-dimensional hyperstring literature [28]. The local first-order solution space has dimension

$$dim\mathcal{S}=6.$$

Core regularity and finiteness leave

$$dim{\mathcal{C}}_{\mathrm{core}}=3.$$

Acceptable asymptotics at infinity remove three growing or divergent branches, so before imposing the internal turning-point regularity condition,

$$dim{\mathcal{A}}_{\infty }=3.$$

For a regular finite-volume vortex there is an interior point ${\rho }_c$ at which the relevant logarithmic angular derivative satisfies

$$\ell \left({\rho }_c\right)=0.$$

Regularity of derivative perturbations imposes one further independent linear relation, leaving

$$dim{\mathcal{A}}_{\infty ,{\rho }_c}=2.$$

The generic intersection in the six-dimensional local solution space is therefore

$${dim}_{\mathrm{gen}}\left({\mathcal{C}}_{\mathrm{core}}\cap {\mathcal{A}}_{\infty ,{\rho }_c}\right)=max(0,3+2-6)=0.$$

Generically, only the zero homogeneous sausage perturbation satisfies all core, asymptotic, and turning-point conditions. Published gauge-invariant hyperstring matching likewise found no exceptional acceptable nonzero mode over a broad scanned mass range [28]. This is not a theorem that a positive massive sausage spectrum exists; it is a stronger statement of generic rigidity or nonpropagation of the homogeneous sausage boundary-value problem under the stated assumptions.

Appendix D.13.13.4. Exact limitation of the top-form repair.

For a source-free top form

$$F_6=q{\mathrm{vol}}_6,d(*F_6)=0.$$

Since $*F_6$ is a zero-form in six dimensions,

$$dq=0.$$

After Fourier decomposition along the four-dimensional coordinates,

$$\delta q\left(k_{\mu }\right)=0\mathrm{for}\mathrm{every}{k}_{\mu }\ne 0.$$

The top-form integration constant can move the $Q=0$ background/flux sector and follow the homogeneous regularity surface, but it does not add a local propagating scalar degree of freedom at nonzero four-dimensional momentum. This is an exact limitation: the top form is a background selector, not a $Q\ne 0$ sausage Green function.

Correction of the scalar-sector interpretation.The fixed-geometry Higgs–gauge continuum remains mathematically useful, but only as a matter-subsector diagnostic. The full transverse scalar channel has an independent structural stability result. The minimal Einstein–Abelian–Higgs background sausage sector, by contrast, is generically rigid, and the top form cannot repair this rigidity for $Q\ne 0$. The remaining question is therefore specific: does phenomenology require a nonzero local radial-breathing response, and if so, what minimal local structural modification supplies it without reintroducing arbitrary brane coefficients, anomalies, unwanted zero modes, or route-odd spurions?

Appendix D.13.29. Minimal Local Scalar Response from Quadratic Curvature

The rigidity of the background sausage sector motivates a sharply defined question: can one add a single local scalar response channel without introducing an arbitrary brane field or spoiling the fermion and gauge properties already obtained?

The first purely geometric possibility is the Gauss–Bonnet combination

$${\mathcal{R}}_{\mathrm{GB}}^2=R_{MNPQ}{R}^{MNPQ}-4R_{MN}{R}^{MN}+R^2.$$

Here $R_{MNPQ}$ is the six-dimensional Riemann tensor, $R_{MN}$ the Ricci tensor, and R the Ricci scalar. In six dimensions this term is dynamical but belongs to the Lovelock class [31]; its field equations remain second order and, on a nondegenerate background, it does not introduce a new local scalar degree of freedom. It can deform the background, but it does not by itself supply the missing local radial-breathing response.

The minimal curvature modification that does introduce exactly one additional local scalar degree of freedom without an independent massive spin-two ghost is metric $f\left(R\right)$ gravity [32,33],

$$S={\displaystyle \frac{1}{2{\kappa }_6^2}}\int d^{6}x\sqrt{-g}f\left(R\right)+S_{\mathrm{matter}},f\left(R\right)=R-2{\Lambda }_{\mathrm{eff}}+\beta R^2.$$

Here $\beta$ has dimensions of inverse curvature. Define

$$\mathfrak{F}\left(R\right)\equiv {\displaystyle \frac{df}{dR}}=1+2\beta R.$$

The metric field equations are

$$\mathfrak{F}{R}_{MN}-{\displaystyle \frac{1}{2}}f{g}_{MN}+\left(g_{MN}{\square }_6-{\nabla }_M{\nabla }_N\right)\mathfrak{F}={\kappa }_6^2{T}_{MN},$$

where $T_{MN}$ is the six-dimensional energy–momentum tensor and ${\square }_6\equiv g^{MN}{\nabla }_M{\nabla }_N$. With

$$T\equiv g^{MN}{T}_{MN},$$

the trace equation in D dimensions is

$$\mathfrak{F}R-{\displaystyle \frac{D}{2}}f+(D-1){\square }_D\mathfrak{F}={\kappa }_D^{2}T,$$

so in six dimensions

$$\mathfrak{F}R-3f+5{\square }_6\mathfrak{F}={\kappa }_6^{2}T.$$

The ${\square }_6\mathfrak{F}$ term is precisely the new local scalar dynamics absent in Einstein gravity and in the nondegenerate Gauss–Bonnet modification.

Let $R_0$ and $T_0$ be constant background values and write

$$R=R_0+\delta R,T=T_0+\delta T.$$

In the asymptotic vacuum region $T_0=0$, and

$$\delta \mathfrak{F}=2\beta \delta R.$$

Linearizing the trace equation gives

$$10\beta {\square }_6\delta R+\left(2\beta R_0-2{\mathfrak{F}}_0\right)\delta R={\kappa }_6^2\delta T,$$

where

$${\mathfrak{F}}_0\equiv 1+2\beta R_0.$$

Dividing by $10\beta$ yields the scalaron Klein–Gordon equation

$$\left({\square }_6-m_s^2\right)\delta R={\displaystyle \frac{{\kappa }_6^2}{10\beta }}\delta T,m_s^2\equiv {\displaystyle \frac{1+\beta R_0}{5\beta }}.$$

Unlike the top form, this scalar responds locally to $\delta T\left(x\right)$ and therefore remains dynamical at nonzero four-dimensional momentum.

For the asymptotic ${\mathrm{AdS}}_6$ throat, use

$$R_0=-30c_{6,★}^2$$

and define

$$b_{R^2}\equiv \beta c_{6,★}^2.$$

Then

$${\mathfrak{F}}_0=1-60b_{R^2},{\displaystyle \frac{m_s^2}{c_{6,★}^2}}={\displaystyle \frac{1-30b_{R^2}}{5b_{R^2}}}={\displaystyle \frac{1}{5b_{R^2}}}-6.$$

The correct sign of the massless graviton kinetic term requires ${\mathfrak{F}}_0>0$, and stability of the $R^2$ sector requires $\beta >0$. Together,

$$0<b_{R^2}<{\displaystyle \frac{1}{60}}.$$

Throughout this interval,

$${\displaystyle \frac{m_s^2}{c_{6,★}^2}}>6,$$

so the scalaron is neither a massless modulus nor a tachyon. As $\beta \to 0^{+}$, $m_s^2\sim 1/\left(5\beta \right)\to \infty$, continuously recovering Einstein gravity at low energy.

As an interior illustrative point, not a fit, take

$$b_{R^2}={\displaystyle \frac{1}{120}}.$$

Then

$${\mathfrak{F}}_0={\displaystyle \frac{1}{2}},{\displaystyle \frac{m_s^2}{c_{6,★}^2}}=18,{\displaystyle \frac{m_s}{c_{6,★}}}=\sqrt{18}\simeq 4.24264.$$

Define the asymptotic scalaron index

$${\nu }_s^2\equiv {\displaystyle \frac{25}{4}}+{\displaystyle \frac{m_s^2}{c_{6,★}^2}}={\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{5b_{R^2}}}.$$

At $b_{R^2}=1/120$,

$${\nu }_s\simeq 4.92443.$$

The first generic continuum nonanalyticity therefore scales as $Q^{2{\nu }_s}\simeq Q^{9.84886}$; the low-momentum expansion is analytic through the ordinary even powers below that threshold. More generally, $0<b_{R^2}<1/60$ implies ${\nu }_s>7/2$, so the compact-core scalaron response is strongly threshold suppressed and infrared finite through several derivative orders.

What is proved for $R+\beta R^2$.Within linearization about the asymptotic background, metric $f\left(R\right)$ gravity adds exactly one local scalar channel and no independent massive spin-two ghost. For $0<\beta c_{6,★}^2<1/60$, the massless graviton has the correct kinetic sign, the scalaron satisfies $m_s^2>6c_{6,★}^2>0$, and it couples locally to the trace perturbation $\delta T\left(x\right)$. This is the first minimal purely gravitational mechanism considered here that can, in principle, provide a nonzero local scalar response at $Q\ne 0$.

What is not yet proved.This is a linear statement about the asymptotic background. A complete six-dimensional $f\left(R\right)$ completion would require a self-consistent nonlinear vortex solution in $R+\beta R^2$ gravity, a reanalysis of its fermion zero modes and gauge localization, and the full gauge-invariant scalar boundary-value problem. The healthy scalaron removes the previous absence of any local $Q\ne 0$ scalar candidate; it does not by itself solve the full sausage problem.

Current structural results in the minimal smooth six-dimensional branch.Under the stated assumptions – the ${\mathrm{Spin}}^c$ charge assignment, direct-product non-Abelian global group, a self-consistent regular Einstein–Abelian–Higgs vortex, and, where invoked, a source-free noncompact top form – local anomalies cancel and the corresponding direct-product ${\mathrm{Spin}}^c$ bordism obstruction is absent. A backreacted regular vortex is reproduced numerically; exactly one normalizable Weyl zero mode is obtained per vectorlike Dirac parent under the stated Yukawa bound; the graviton and spectator gauge zero modes are normalizable; the tensor, spectator-gauge, and coupled vortex-vector channels are non-tachyonic; independent graviphoton zero modes and a localized massless scalar modulus are absent; and the full transverse scalar channel is stable and threshold-local. The homogeneous background radial-breathing sector remains generically rigid, and a source-free top form changes only the $Q=0$ sector.

What remains open in six dimensions.If the ultraviolet completion requires a compact higher-form gauge field and charged four-branes, its charge lattice and exact intersection with the regularity surface must be derived; a dense discretuum is not by itself sufficient. The minimal Einstein–Abelian–Higgs system plus a source-free top form does not provide a general nonzero-Q local response of the background breathing sector. The $\beta R^2$ scalaron is a healthy local candidate, but its sufficiency must be tested in a complete nonlinear vortex. If the non-Abelian global group is quotiented by centers, the ${\mathrm{Spin}}^c$ bordism analysis must be repeated. The microscopic origin of the defect $U{\left(1\right)}_V$, the top-form ultraviolet choice, and the orbital involution $X_{\mathrm{orb}}$ are not derived from unchanged confining $\mathrm{SU}{\left(15\right)}_p$. Finally, the physical route correlator, GEVP poles, and matching residues remain part of the same low-energy nonperturbative test as in four dimensions.

Appendix D.13.30. Why a Healthy Scalaron Does Not Remove the Geometric Turning Condition

The term $\beta R^2$ does create a healthy local scalar degree of freedom in the interval $0<b_{R^2}<1/60$. This does not imply that it removes the extra regularity condition in the background radial-breathing sector.

On the healthy branch of metric $f\left(R\right)$ gravity,

$$\mathfrak{F}\left(R\right)={\displaystyle \frac{df}{dR}}>0,$$

so one may perform a regular conformal transformation to the Einstein frame. In six dimensions,

$$g_{MN}^{\left(E\right)}={\mathfrak{F}}^{1/2}{g}_{MN}.$$

The proper radial element and angular radius become

$$d{r}_E={\mathfrak{F}}^{1/4}dr,L_E={\mathfrak{F}}^{1/4}L.$$

Because $d{r}_E/dr>0$, a stationary point of the Einstein-frame circle radius obeys

$${\displaystyle \frac{d{L}_E}{d{r}_E}}=0⟺{\displaystyle \frac{L^{\prime }}{L}}+{\displaystyle \frac{{\mathfrak{F}}^{\prime }}{4\mathfrak{F}}}=0.$$

The old Jordan-frame condition $L^{\prime }=0$ is therefore not conformally invariant: the scalaron can move the turning point. But if $L_E\left(0\right)=0$, $L_E>0$ away from the regular axis, and $L_E\to 0$ at infinity in a finite-volume localizing geometry, continuity forces at least one interior maximum. The turning point is displaced, not eliminated.

In the Einstein frame, healthy $f\left(R\right)$ gravity is Einstein gravity plus one ordinary scalaron obeying second-order equations. Adding an ordinary second-order scalar increases both the number of local integration constants and the number of regular/asymptotic boundary data. If the independent compatibility condition at the interior turning point remains, the boundary-value deficiency does not change. A healthy scalaron is therefore a genuine local response channel but not automatically a generic completion of the sausage sector.

Appendix D.13.31. Exact Limit of a Finite Local Hamilton–Jacobi Completion

A stronger possibility is to ask whether a finite local Hamilton–Jacobi function can encode the anisotropic angular radius and reduce the radial system to first order. Define

$$A_6\left(r\right)\equiv lnM\left(r\right),B_6\left(r\right)\equiv lnL\left(r\right),\mathcal{E}\left(r\right)\equiv e^{4A_6+B_6},t_L\equiv e^{-B_6}={\displaystyle \frac{1}{L}}.$$

Let f and P be the scalar and angular gauge profiles. Consider the most general finite local ansatz involving nonnegative powers of the inverse circle radius,

$$S_{\mathrm{HJ}}=\mathcal{E}{\displaystyle \sum _{n=0}^{N_{\mathrm{HJ}}}}{t}_L^n{W}_n(f,P),$$

where $N_{\mathrm{HJ}}$ is the highest power and the $W_n$ are real local functions.

For $N_{\mathrm{HJ}}\ge 2$, the highest power $t_L^{2N_{\mathrm{HJ}}}$ in the Hamilton–Jacobi equation has coefficient

$${\displaystyle \frac{(N_{\mathrm{HJ}}-1)(3N_{\mathrm{HJ}}+5)}{16}}{W}_{N_{\mathrm{HJ}}}^2+{\displaystyle \frac{1}{4{\nu }_6}}{\left({\displaystyle \frac{\partial W_{N_{\mathrm{HJ}}}}{\partial f}}\right)}^2+\mathrm{additional}\mathrm{nonnegative}\mathrm{squares}=0.$$

For ${\nu }_6>0$ and $N_{\mathrm{HJ}}\ge 2$, every term is nonnegative and the coefficient multiplying $W_{N_{\mathrm{HJ}}}^2$ is strictly positive. Hence $W_{N_{\mathrm{HJ}}}=0$. Iterating the argument reduces every finite ansatz to $N_{\mathrm{HJ}}\le 1$.

The case $N_{\mathrm{HJ}}=0$ is a scalar-factorized pseudo-superpotential and forces $A_6^{\prime }=B_6^{\prime }$, which is incompatible with a regular codimension-two axis. The only remaining nontrivial case is $N_{\mathrm{HJ}}=1$. Core regularity then fixes the functional form almost completely, while the ${\mathrm{AdS}}_6$ vacuum condition reduces to

$${\displaystyle \frac{C_{\mathrm{HJ}}^2{e}^{{\nu }_6/2}({\nu }_6^2-{\nu }_6+4)}{16{\nu }_6}}+{\displaystyle \frac{{\alpha }_6{({\nu }_6+2)}^2}{4{\nu }_6}}-2{\mu }_6=0,$$

where $C_{\mathrm{HJ}}$ is a real integration constant of the one-power ansatz. On the physical branch ${\alpha }_6>0$, ${\nu }_6>0$, and ${\mu }_6<0$, the first term is nonnegative, the second strictly positive, and $-2{\mu }_6>0$. The left-hand side cannot vanish.

Local no-go theorem for the tested finite Hamilton–Jacobi class.For the displayed Einstein–Abelian–Higgs potential, including finitely many positive-metric scalar extensions whose highest-power Hamilton–Jacobi contributions enter only through the nonnegative squares shown above, no finite real local ansatz polynomial in nonnegative integer powers of $1/L$ is simultaneously compatible with a regular codimension-two axis and a localizing ${\mathrm{AdS}}_6$ vacuum. The statement is not a no-go theorem for arbitrary scalar potentials or for noninteger, nonpolynomial, or infinite Hamilton–Jacobi functionals.

This theorem does not exclude nonpolynomial or genuinely infinite Hamilton–Jacobi functionals, new tensor or topological sectors, a new symmetry that changes the constraint algebra, or qualitatively different transverse geometry. It shows only that adding another ordinary scalar or another finite local term has no established mathematical reason to cure the minimal sausage obstruction.

Current physical meaning of the six-dimensional branch.The minimal smooth six-dimensional construction remains strong as a completion of chirality and several zero-mode localization and stability requirements, but its generic local radial-breathing response to an arbitrary four-dimensional source reaches a structural limit. The nonminimal bulk-confinement mechanism changes the assumption responsible for the gauge-localization turning theorem and goes substantially further: the no-turn geometry has an exact positive-kinetic local Einstein–sigma realization, finite-backreaction coexistence with a unit-winding Abelian–Higgs vortex occurs in a nonempty numerical window, and the complete coupled $\theta$-independent radion–modulus operator is nonnegative with no normalizable zero mode. The earlier frozen-KK-vector phase mode is not an independent scalar. In the regular dipole sector the former fixed-metric negative eigenfunction is an admissible scalar-clock image, and the exact Ward identity makes its stationary metric completion Schur-null; hence $-0.347963/R_c^2$ is not a physical mass. The remaining gauge-invariant metric problem has been reduced exactly to the threshold Birman–Schwinger number ${\beta }_0$, but the explicit constraint-reduced potential $U_{\mathrm{phys}}$ and the value of ${\beta }_0$ have not yet been computed. The six-dimensional branch is therefore prospective rather than complete: the physical dipole and higher harmonics, interface-localized modes, microscopic confining sector, admissible fermion representation and Yukawa coupling, full anomaly/inflow analysis, and route matching remain open.

Appendix E.1. Spinor Decomposition and Seven-Dimensional Zero-Mode Multiplicity

The tested all-preon construction lifts the preons to seven dimensions and introduces mirror fermion copies on a $T^3/{\mathbb{Z}}_2$ geometry together with an order-two holonomy line and a route involution. Its gauge multiplicity is 405, but the physical four-dimensional zero-mode count also depends on the internal spinor multiplicity of each seven-dimensional parent. A direct Clifford-algebra check gives that multiplicity explicitly.

Appendix E.14.1. Seven-dimensional Spinor Decomposition

A complex Dirac spinor in seven dimensions has

$$2^{\lfloor 7/2\rfloor }=8$$

complex components. One may choose

$${\Gamma }^{\mu }={\gamma }^{\mu }\otimes I_2,$$

while the spin lift of simultaneous inversion of the three extra coordinates is, up to an irrelevant phase,

$$-i{\Gamma }^4{\Gamma }^5{\Gamma }^6=\pm {\gamma }_5\otimes I_2.$$

Projection onto one four-dimensional chirality therefore leaves complex rank four. A single four-dimensional Weyl spinor has complex dimension two, so under the stated orbifold assumptions

$$1\mathrm{seven}-\mathrm{dimensional}\mathrm{Dirac}\mathrm{field}⟶2\mathrm{four}-\mathrm{dimensional}\mathrm{Weyl}\mathrm{zero}\mathrm{modes}\mathrm{of}\mathrm{the}\mathrm{selected}\mathrm{chirality}.$$

The gauge multiplicity is

$$120+120+45+120=405$$

therefore becomes

$$N_{0,\mathrm{original}}=2\times 405=810.$$

Antiperiodic mirror fields do not contain constant zero modes and therefore do not remove the extra copy already present in the original sector.

Appendix E.14.2. Why a Majorana Condition Is Not an Immediate Repair

The published preons belong to complex gauge representations, including the fundamental $\mathbf{15}$ of $\mathrm{SU}{\left(15\right)}_p$ and the $\mathbf{4}$ of $\mathrm{SU}{\left(4\right)}_{\mathrm{PS}}$. A simple Majorana reality condition would require a gauge-compatible antilinear structure relating each representation to its conjugate. Such a structure is not available for the individual complex fields. The multiplicity therefore cannot be halved simply by declaring a Majorana condition.

Consequence for the all-preon seven-dimensional branch.Under the assumptions of the tested seven-dimensional bulk construction, the selected-chirality zero-mode count is 810. The number 405 is only the gauge-component multiplicity and cannot be used as the four-dimensional zero-mode count. Repair of the resulting factor-of-two excess requires an additional projector, flux, conjugation/reality structure, or another ultraviolet mechanism that has not yet been constructed and checked.

Appendix E.14.3. Kaluza–Klein Tower Burden

For the shifted mirror tower with equal radii $R_7$,

$$M_{n_y,n_z,n_w}^2={\displaystyle \frac{1}{R_7^2}}\left[n_y^2+{\left(n_z+{\displaystyle \frac{1}{2}}\right)}^2+n_w^2\right].$$

The cumulative number of modes scales asymptotically as

$$N_{\mathrm{KK}}^{7D}\propto {\left(\Lambda R_7\right)}^3,$$

where $\Lambda$ is a four-dimensional energy cutoff. Even if the zero-mode doubling were repaired, the ultraviolet tower would remain much denser than in a single compact dimension. This is not an independent no-go theorem, but it makes the seven-dimensional branch substantially less economical than the active four-, five-, and six-dimensional alternatives.

Appendix F.1. Local No-Go Theorems, Their Domains of Validity, and the Mechanisms That Bypass Them

Throughout this work, a no-go theorem means the impossibility of a mechanism under an explicitly stated set of assumptions. A later construction does not “invalidate” such a theorem; it succeeds only by changing at least one assumption. The main results can therefore be organized as follows.

Obstruction or failed criterion	What it excludes	Status and possible bypass
Local onsite automorphism ${\mathcal{O}}_{12}\leftrightarrow {\mathcal{O}}_{21}$	multiplicity-one irreducible representations cannot be locally interchanged by an internal symmetry, by Schur’s lemma	Bypassed in 4D: nonlocal Wilson-network route reflection plus the explicit rank-one sign source
Full primitive holonomy $S_B$ from ordinary Wilson transport	ordinary transport gives $U_{\tau }=I_4$, not a matrix basis-equivalent to $S_B$	Bypassed in 4D: full $S_B$ is stronger than required; the rank-one $B_{-}$ line already carries ${\rho }_{-}\left(\tau \right)=-1$
Factorized closure in the span $G_4+G_R+I$	this restricted matrix span lacks the required primitive locking direction	Bypassed in 4D: a hidden-composite Schur–Feshbach susceptibility lies outside that span; exact route reciprocity requires only $M_{11}-M_{22}=D_{\mathrm{rt}}$
Simple equal-weight mediator	the specific projector/matching response is wrong	Mediator rejected, not the 4D theory: use a hidden composite sector and Schur–Feshbach reduction
Intrinsic 5D parity of the fundamental preons	cannot preserve the published chiral zero-mode spectrum and simultaneously generate route exchange	Bypassed in the composite 5D branch: the preons remain 4D and the quotient acts on the composite route sector
Exact quotient reached after finite generic RG time	a generic nonzero odd coupling does not become exactly zero under a smooth finite RG flow	Not required for the explicit quotient completion: an interacting IR fixed point or a boundary-universality limit may realize approximate/emergent suppression; actual finite-scale reciprocity remains a correlator test
Positive total endpoint inverse response $\Rightarrow c_R>0$ without bulk subtraction	false in general because the full endpoint response contains the bulk Dirichlet-to-Neumann part	Corrected: in the explicit one-band/one-defect class the same spectrum fixes the bulk band and the excess Robin defect; outside that class the one-parameter Robin claim must be generalized or rejected
Over-strong 4D condition $f=0$	vanishing off-diagonal splitting was incorrectly imposed as part of route symmetry	Corrected: in a real basis exact reflection requires equality of the two diagonals; for either sign of the mismatch the global minimum is $M_{min}=diag(D_{\mathrm{rt}}^{(+)},D_{\mathrm{rt}}^{(-)})$ with $Tr{M}_{min}=\left|D_{\mathrm{rt}}\right|$
Minimal chiral 6D Weyl lift	irreducible local six-dimensional anomaly	Bypassed in 6D: vectorlike six-dimensional Dirac parents plus vortex-generated four-dimensional chirality
Hard-boundary 6D matching	independent face and corner counterterms can reintroduce arbitrary coefficients	Bypassed in the smooth branch; the separate finite-interface branch is conditional and must satisfy the exact Israel–Maxwell equation of state rather than use free matching coefficients
Minimal one-tensor or naive conformal-warp branch	quartic anomaly or singular/inconsistent geometry	Bypassed: ordinary Yang–Mills on a self-consistent regular gravitating vortex
Healthy $R+\beta R^2$ scalaron as a complete cure of the sausage sector	the scalaron is healthy but the Einstein-frame circle still has an interior turning point; the boundary-value deficiency persists if the turning compatibility condition remains independent	Not solved in the minimal scalar branch: a new structural identity or a different constraint sector would be required
Finite local 6D Hamilton–Jacobi ansatz	a finite ansatz in nonnegative powers of $1/L$ cannot satisfy both regular-axis and localizing ${\mathrm{AdS}}_6$ conditions on the physical branch	Only qualitatively different routes remain: nonpolynomial/infinite functionals, new topological or tensor sectors, or a symmetry changing the constraint algebra
Removing the 6D circle turning point while retaining standard perturbative bulk gauge localization	regular axis plus finite $\int h_{g}Ldr$ with $h_g\ge h_{*}>0$ forces $L^{\prime }\left(r_c\right)=0$	No escape within the standard bulk zero-mode assumption
Standard 6D p-form or dielectric vector escape	one- and three-form channels inherit $\int h_{p}Ldr$; the two-form harmonic one-form has $\int dr/L$ core divergence; $h_g\to 0$ collapses the uniformly weak-coupling scale	No controlled escape in the standard local two-derivative weak-coupling class
Turning obstruction as an absolute 6D no-go	assumes the gauge boson is a perturbative bulk zero mode	Bypassed structurally in a nonminimal branch: genuine bulk confinement can localize gauge response without requiring a decaying circle; the self-consistent microscopic confining completion remains open
Frozen-KK-vector phase norm interpreted as a physical scalar zero mode	omits the circle connection that gauges angular reparametrizations	Corrected: the phase is a Stueckelberg coordinate; the physical zero-harmonic scalars are the radion and radial modulus
Possible hidden tachyon in the natural axisymmetric no-turn scalar/radion core	asymptotic masses alone do not exclude a core bound state	Closed at two-derivative bulk EFT level: $\mathbb{H}={\mathbb{Q}}^{†}\mathbb{Q}+{\mathbb{K}}^{†}\mathbb{K}$ and smooth-cap conditions exclude normalizable $m_4^2\le 0$
Ordinary finite-positive-coupling gauging of the winding on the infinite no-turn cylinder	the asymptotic radial kinetic budget vanishes while the required angular budget remains nonzero; a regular bulk BF rewrite is Hodge-dual to the same system	No escape in this infinite-cylinder local class: the metric-independent BF term carries no compensating stress
Finite-radius interface treated as freely adjustable boundary data	arbitrary interface coefficients would merely relocate the tuning	Reduced to a conditional theorem: Israel and Maxwell matching fix $T_b,Q_b,C_{★}$; $T_b/Q_b$ fixes the radius and one common exponential slope closes the scalar background junction
Canonical positive-potential thin wall as the microscopic origin of the finite interface	every such wall obeys $Q_{\mathrm{can}}\le {\rho }_{\mathrm{can}}$, whereas the exact junction requires $Q_b>{\rho }_b$	Excluded under the stated thin-wall assumptions: only a flux/topological interface, strong backreaction, noncanonical matter, or nonperturbative confining stress can evade the theorem
Bulk factorization applied automatically to finite-interface modes	a finite interface changes the endpoint action and boundary conditions	Not established: coupled interface-localized scalar, radion, and vector fluctuations require a separate analysis
Old 405 zero-mode count in the all-preon 7D construction	genuine 7D Dirac spinors give $405\to 810$	Not yet bypassed: no explicit repair has passed the required checks
Missing physical 4D values of $E_{-}$ and $F_{+}$	analytic reduced-model inputs do not determine nonperturbative observables	Cannot be bypassed: actual confining-theory correlator data are required

Interpretation of the no-go map.The large number of negative statements is not evidence that the entire program has failed. It is a map of simple mechanisms that have been eliminated. In four dimensions, the main algebraic obstructions are bypassed within the same Pati–Salam chiral-fermion content by rank-one composite dynamics; the scalar flavour assignment is updated separately below to the representation-theory-consistent two-spurion convention. In five dimensions, the surviving construction acts on a composite spectral coordinate rather than on the fundamental preons. In six dimensions, the minimal smooth perturbative bulk-zero-mode program reaches a structural limit, but a nonminimal bulk-confinement localization mechanism reopens a qualitatively different route. If an ordinary finite-coupling gauging is demanded, a finite interface gives a conditional classical completion, but its flux-dominated microscopic origin and fluctuation spectrum remain unresolved. In seven dimensions, no tested repair of the zero-mode doubling is yet available. The remaining four-dimensional nonperturbative question is not a no-go theorem but the decisive physical calculation.

Appendix F.2. Additional Phenomenological Matching to Low-Energy Parameters

The central route/sign theorem does not depend on the detailed structure of flavour thresholds. Nevertheless, connecting the microscopic construction to measured Yukawa parameters requires several additional and logically independent checks.

Appendix F.16.1. Universal Threshold Corrections Cannot Generate the Normal to the Curvature Plane

If all three Yukawa eigenvalues in a sector s are rescaled by the same factor,

$$y_{s,g}\to c_s{y}_{s,g},$$

then

$$A_s\to A_s.$$

Likewise, any correction that is linear in the generation label at the level of logarithms is annihilated by the second finite difference. Therefore ultraviolet matching can shift ${\Delta }_{\Pi }$ only if it contains genuinely generation-curved information. Pure family-singlet gauge thresholds are insufficient by themselves.

Appendix F.16.2. What Would Be Required for an Independent CKM Test

The curvature plane uses only the singular values of the charged-fermion Yukawa matrices. It therefore contains no information by itself about their left singular vectors. Writing

$$Y_u=U_{uL}{\widehat{Y}}_u{U}_{uR}^{†},Y_d=U_{dL}{\widehat{Y}}_d{U}_{dR}^{†},$$

the observable mixing matrix is

$$V_{\mathrm{CKM}}=U_{uL}^{†}{U}_{dL}.$$

A common alignment of the up- and down-sector family tensors would give $U_{uL}=U_{dL}$, and hence $V_{\mathrm{CKM}}=I$, even if the nine singular values obeyed the curvature relation exactly. Conversely, realistic CKM angles require a relative orientation that is not fixed by the identity ${\Delta }_{\Pi }=C_{21}-C_{12}$.

This limitation agrees with the published $\mathrm{SU}{\left(15\right)}_p$ flavour analysis: right-handed-isospin symmetry tightly correlates the up and down Yukawa matrices and gives a trivial CKM matrix in the absence of additional spontaneous symmetry breaking; a realistic benchmark uses nontrivial flavour-spurion textures [7]. Such textures make a future mass–mixing relation structurally conceivable, but they do not make it a consequence of route reciprocity.

A genuine CKM test of the same microscopic mechanism would require, before comparison with mixing data:

1.

separate dynamical definitions of the vacuum-aligned tensors $D_{+}^{\left(u\right)}$ and $D_{+}^{\left(d\right)}$ and their relative orientation;

2.

a derivation showing whether the route-odd operator rotates the left singular vectors or only changes the singular values;

3.

a model-fixed CKM combination whose dependence on $C_{21}-C_{12}$ follows from the renormalized matching vertex; and

4.

a joint likelihood or covariance treatment that does not count the mass eigenvalues twice.

Until these steps are supplied, Gatto–Sartori–Tonin-type mass–mixing relations are useful historical analogies to flavour textures, not evidence for the present plane. CKM is therefore retained as a separate flavour-alignment problem in the outlook and is not combined with $Z_{\Pi }$.

Appendix F.16.3. The Published Two-Spurion Flavour Sector as an Independent Experimental Gate

The corrected scalar-spurion completion is useful for more than mass textures. In Ref. [7] the same two flavour-breaking spurions that enter the Yukawa matrices also generate flavour-changing neutral currents and dipole operators. In their numerical benchmark, present electron-EDM and neutral-kaon constraints probe compositeness scales of order ${10}^4\mathrm{TeV}$, while projected electron-EDM sensitivity can extend toward ${10}^6\mathrm{TeV}$. These numbers belong to that specific four-singlet $\mathrm{SU}{\left(4\right)}_F$ realization and its fitted operator coefficients; they are not transferred as numerical bounds on the Pati–Salam route branch.

The structural lesson is nevertheless direct. Once a concrete Pati–Salam matching identifies the scalar spurions that realize the family tensor and the route operators, the same parameter point must satisfy two logically independent tests:

1.

the route-sector conditions leading to the charged-fermion curvature relation, including the renormalized $2\times 2$ correlator and the even/odd three-point matching test; and

2.

flavour-safety constraints obtained after rotating the same spurions to the physical quark and charged-lepton mass bases.

A parameter choice that reproduces the plane but violates the resulting EDM, neutral-meson, or charged-lepton-flavour bounds is not a viable microscopic completion.

There is also a notation safeguard. Ref. [7] parametrizes several diagram classes by nonperturbative coefficients such as F, $F^{\prime }$, G, I, J, and K. The present route coefficients $C_{12}$ and $C_{21}$ are defined by an independent operator basis. They must not be identified with any pair of those coefficients merely because two diagrams look “crossed”. Such an identification is legitimate only after the external states, spurion contractions, preon-line interchanges, renormalization scheme, and operator normalization have been matched explicitly.

New independent stopping rule.A future microscopic embedding of route reciprocity into the two-spurion flavour sector must satisfy a joint route-and-flavour test. Failure of the flavour constraints excludes that embedding without invalidating the abstract route identity ${\Delta }_{\Pi }=C_{21}-C_{12}$.

Appendix F.16.4. The Crossed Matching Coefficient as the Final Low-Energy Bridge

Even if a microscopic route-odd pole exists, the physical low-energy coefficients must still be determined by an independent matching vertex. Appendix B.1.14 proves that this vertex can be extracted with a finite-separation interpolating source after pole isolation and external-leg amputation; a collapsed 24-preon source is not required. In a renormalized scheme let

$$\omega =({\omega }_{12},{\omega }_{21})$$

be the Wilson-coefficient covector in the two crossed-route directions. In the exact route-symmetry limit it must satisfy

$${\sigma }_x\omega =\omega ,$$

where ${\sigma }_x$ exchanges the two route entries. A convenient normalized leakage diagnostic is

$${ϵ}_{\Gamma }={\displaystyle \frac{|{\omega }_{12}-{\omega }_{21}|}{\sqrt{|{\omega }_{12}{|}^2+{\left|{\omega }_{21}\right|}^2}}}.$$

The exact-symmetry extrapolation target is therefore

$${ϵ}_{\Gamma }\to 0.$$

This condition is independent of the absolute normalization of the matching vertex.

Appendix F.16.5. Small Departures from the Exact Plane

Suppose that the physical route symmetry is weakly broken by a spurion ${\epsilon }_{\mathrm{orb}}$ that is odd under route reflection,

$${\epsilon }_{\mathrm{orb}}\to -{\epsilon }_{\mathrm{orb}}.$$

Then the lowest crossed-odd coefficient has an expansion containing only odd powers,

$$C_{-}\equiv {\displaystyle \frac{C_{21}-C_{12}}{\sqrt{2}}}={\epsilon }_{\mathrm{orb}}{\kappa }_1+O\left({\epsilon }_{\mathrm{orb}}^3\right),$$

where ${\kappa }_1$ is the linear symmetry-breaking matching coefficient in the chosen renormalization scheme. Consequently,

$${\Delta }_{\Pi }=\sqrt{2}{\epsilon }_{\mathrm{orb}}{\kappa }_1+O\left({\epsilon }_{\mathrm{orb}}^3\right).$$

At the matching-EFT level, if an exact parent route symmetry exists, the renormalized matching functional is analytic in a single route-odd spurion near ${\epsilon }_{\mathrm{orb}}=0$, and ${\kappa }_1$ has no singular enhancement, the exact limit is restored at ${\epsilon }_{\mathrm{orb}}=0$ and a small leakage is technically natural in the usual symmetry sense. These assumptions are explicit: the existence of the exact parent route symmetry in the unchanged confining theory remains open.

Appendix F.16.6. An Emergent Alternative: the Regular Route-Odd Remainder as an IR-irrelevant Deformation

Equation (A208) suggests a weaker dynamical mechanism that does not require an exact microscopic route exchange: the regular odd remainder can flow toward zero near an infrared-symmetric fixed manifold. Let $u_{-}\left(\mu \right)$ be a canonically renormalized coordinate for the leading physically coupled route-odd regular deformation, and assume a locally Lipschitz Wilsonian flow with invariant symmetric manifold,

$$\mu {\displaystyle \frac{d{u}_{-}}{d\mu }}={\beta }_{-}(u_{-},u_{+}),{\beta }_{-}(0,u_{+})=0.$$

Finite-RG-time no-hit theorem.If $u_{-}\left({\mu }_0\right)\ne 0$ on a regular branch, a locally unique smooth RG trajectory cannot first reach $u_{-}=0$ at a finite scale and then remain on the invariant manifold. Exact finite-scale reciprocity therefore requires either that the trajectory already starts on the symmetric manifold or that a nonanalytic phase transition, confinement projection, constraint, or genuine quotient invalidates the smooth-flow assumptions.

Approximate reciprocity can nevertheless emerge. Linearization around an attractive symmetric fixed point gives

$$u_{-}\left(\mu \right)=u_{-}\left(\Lambda \right){\left({\displaystyle \frac{\mu }{\Lambda }}\right)}^{\delta }+O\left(u_{-}^2\right),\delta ={\Delta }_{-}-4.$$

If $\delta >0$, the route-odd deformation is infrared-irrelevant. A required suppression from ${ϵ}_{\mathrm{UV}}$ to ${ϵ}_{\mathrm{IR}}$ across $R=\Lambda /\mu$ corresponds to the explicit target

$${\delta }_{\mathrm{req}}={\displaystyle \frac{ln({ϵ}_{\mathrm{UV}}/{ϵ}_{\mathrm{IR}})}{lnR}}.$$

This is not a predicted anomalous dimension. It converts the zero-new-datum branch into a first-principles step-scaling test of the renormalized route-odd remainder. A measured $\delta \le 0$ for the leading physically coupled odd mode closes the smooth IR-attractor route; sufficiently positive $\delta$ keeps approximate emergent reciprocity viable.

Appendix F.3. Sharp Determinant-Tolerant Global Family Bound

This appendix records the joint determinant maximization used in the main family section. After permuting the median index to the last position,

$$\Sigma =\left(\begin{array}{cc}M& v\\ v^T& x\end{array}\right),M=M^T\in {\mathbb{C}}^{2\times 2},$$

and the transverse norm satisfies

$$y^2={\parallel M\parallel }_F^2+2{\parallel v\parallel }^2.$$

Let ${\sigma }_1\ge {\sigma }_2\ge 0$ be the Takagi singular values of M. A unitary congruence preserves $|det\Sigma |$ and the Frobenius norm, while

$$|det\Sigma |\le \left|x\right|{\sigma }_1{\sigma }_2+{\sigma }_1{\parallel v\parallel }^2.$$

The direction and phase of v can saturate the last inequality. Since

$${\parallel v\parallel }^2={\displaystyle \frac{y^2-{\sigma }_1^2-{\sigma }_2^2}{2}},$$

the exact constrained maximum is

$$D_{max}(x,y)=\underset{{\sigma }_1\ge {\sigma }_2\ge 0}{max}\left[\left|x\right|{\sigma }_1{\sigma }_2+{\displaystyle \frac{{\sigma }_1}{2}}(y^2-{\sigma }_1^2-{\sigma }_2^2)\right],$$

with ${\sigma }_1^2+{\sigma }_2^2\le y^2$. Elementary optimization gives

$$D_{max}(x,y)=\left\{\begin{array}{cc}\left|x\right|y^2/2,\hfill & \left|x\right|/y\ge 1/\sqrt{2},\hfill \\ {\left(\right|x|}^2+y^2{)}^{3/2}/\left(3\sqrt{3}\right),\hfill & \left|x\right|/y\le 1/\sqrt{2}.\hfill \end{array}\right.$$

For $\left|x\right|=r\sqrt{1-q^2}$ and $y=rq$,

$${\displaystyle \frac{2D_{max}}{r^3}}=\left\{\begin{array}{cc}{q}^2\sqrt{1-q^2},\hfill & q^2\le 2/3,\hfill \\ 2/\left(3\sqrt{3}\right),\hfill & q^2\ge 2/3.\hfill \end{array}\right.$$

Combining this exact determinant envelope with the radial quartic minimization yields

$$k_{\mathrm{crit}}\left(s\right)=\underset{0<q\le 1}{inf}{\displaystyle \frac{\varphi (s{q}^2-1)}{g_{det}\left(q\right)}},$$

where

$$s={\displaystyle \frac{\delta }{a}},k={\displaystyle \frac{|{\kappa }_{det}|}{\sqrt{a({\lambda }_1+{\lambda }_2)}}},$$

$$\varphi \left(A\right)=z_A+{\displaystyle \frac{A}{z_A}}+{\displaystyle \frac{1}{4z_A^3}},z_A^2={\displaystyle \frac{A+\sqrt{A^2+3}}{2}}.$$

For the benchmark mass window

$$s\ge {\displaystyle \frac{0.156998596111}{0.156599862399}}=1.002546194523,$$

and numerical minimization of this explicit one-dimensional function gives the uniform sufficient coefficient $1.4178$ used in the main text. This supersedes the looser triangle estimate that treated the quadratic-minor and residual cubic determinant bounds as independently saturable.

Appendix F.4. The Confining Phase and the Scalar Vacuum: An Independent Limitation

The route-operator program assumes that the published chiral $\mathrm{SU}{\left(15\right)}_p$ theory realizes the required confining phase. Anomaly cancellation is necessary for consistency, but it is not sufficient to determine the dynamically selected phase. Statements about gauge-noninvariant condensates or gauge-fixed effective potentials must also be interpreted with the restrictions implied by Elitzur’s theorem and the Nielsen identities [34,35].

The corrected two-scalar completion contains an antisymmetric precolor channel $A\in \overline{\mathbf{105}}$ and a symmetric precolor channel $A^{\prime }\in \overline{\mathbf{120}}$, together with antisymmetric and symmetric flavour spurions, respectively. Ref. [7] identifies the second representation as a correction to the assignment used in earlier $\mathrm{SU}{\left(15\right)}_p$ studies. This corrected representation data do not specify the complete scalar potential, a renormalization prescription for all scalar masses, or ultraviolet boundary values for the quadratic terms. Therefore the signs of

$$m_{A,R}^2\left(\mu \right),m_{A^{\prime },R}^2\left(\mu \right)$$

are not determined by the available published data; here $m_{A,R}^2\left(\mu \right)$ and $m_{A^{\prime },R}^2\left(\mu \right)$ denote the renormalized quadratic coefficients of the antisymmetric and symmetric scalar channels at renormalization scale $\mu$.

At quadratic order the mixed invariant vanishes,

$$A_{\alpha \beta }^{†}{A}^{\prime \alpha \beta }=0,$$

because one tensor is antisymmetric and the other is symmetric. Hence the Hessian at the origin is block diagonal in these two channels until additional backgrounds are turned on. Single-field quartic alignment can be classified algebraically, whereas the relative orientation of a mixed vacuum depends on cross-quartic couplings that are not fixed by the present input.

Open question.The actual strong-coupling phase selection remains nonperturbative. Scalar mass signs, gauge-invariant phase correlators, and downstream response ratios must therefore be treated as independent observables rather than replaced by assumed “natural” numbers. None of them is inserted implicitly into the proof of the route-sign theorem.

Appendix F.5. Domain of Validity and Discarded Alternative Mechanisms

This section records the domains of validity of the main statements and prevents local no-go theorems from being extended beyond their assumptions.

1.

The article does not claim that natural exact locking is impossible in the unchanged four-dimensional theory. What is excluded is the factorized, multiplicity-one scalar-irrep ansatz without hidden connected spectral directions. In the general analysis, exact real route locking is the single scalar condition $M_{11}-M_{22}=D_{\mathrm{rt}}$, and the globally minimal positive-semidefinite susceptibility is $diag(D_{\mathrm{rt}}^{(+)},D_{\mathrm{rt}}^{(-)})$ with $D_{\mathrm{rt}}^{(+)}=max(D_{\mathrm{rt}},0)$ and $D_{\mathrm{rt}}^{(-)}=max(-D_{\mathrm{rt}},0)$.

2.

The corrected two-scalar flavour completion is not identified with the full four-singlet $\mathrm{SU}{\left(4\right)}_F$ model of Ref. [7]. The analysis imports the corrected $\overline{\mathbf{105}}/\overline{\mathbf{120}}$ precolor tensor assignment and derives its consequences in the three-dimensional Pati–Salam singlet-flavour space. The benchmark spurion entries, CKM fit, and flavour bounds of the four-singlet realization are external comparison data, not parameters of the present route proof.

3.

The full holonomy $S_B$ is not required. Ordinary published Wilson transport indeed does not produce the full matrix $S_B$, but the physical rank-one sign character already exists explicitly in the operator space.

4.

Two route-visible low-energy poles are not required as a minimal criterion. The minimal four-dimensional sign mechanism needs only an isolated route-odd pole below threshold with nonzero route-even matching. A two-parity spectral rank would be a stronger model-dependent statement.

5.

Benchmark correlators and residues are not presented as first-principles data. They are deterministic controls of the extraction formulas and demonstrate that no hidden algebraic obstruction is built into the proposed estimators.

6.

The five-dimensional construction is not interpreted as an intrinsic parity of the fundamental preons. The branch used here is a composite quotient or emergent coordinate. An earlier overly strong Robin interpretation is also rejected: positivity of the full endpoint response does not by itself determine the sign of the Robin parameter $c_R$, and a linear approach to Dirichlet reflection is a statement about the normal threshold rather than an automatic four-dimensional renormalization-group law with exponent one.

7.

A healthy $R^2$ scalaron is not presented as a complete solution of the background radial-breathing problem. It is a genuine local degree of freedom, but by itself it does not remove the unavoidable turning point of the angular radius in the minimal localization class and does not alter the boundary-condition counting without an additional structural identity.

8.

Changing only the asymptotic circle geometry is not presented as a rescue of the minimal six-dimensional theory. Under standard bulk gauge localization, a regular axis together with normalizability forces an interior stationary point $L^{\prime }\left(r_c\right)=0$.

9.

The early anomaly no-go for a minimal chiral six-dimensional Weyl lift is not a no-go theorem for all six-dimensional completions. A vectorlike six-dimensional Dirac parent on a smooth vortex background avoids the chiral bulk-anomaly obstruction and, under the assumptions stated in the six-dimensional section, yields exactly one four-dimensional Weyl zero mode.

10.

The positive-square bulk factorization is not a theorem about every six-dimensional perturbation. It covers the complete coupled $\theta$-independent scalar/radion sector of the displayed local two-derivative no-turn Einstein–sigma action and excludes normalizable $m_4^2\le 0$ there. In the $|n_{\theta }|=1$ sector, the exact clock-sector Schur-null theorem removes the former fixed-metric negative eigenvalue as a physical mass. It does not determine the full constraint-reduced metric operator orthogonal to the clock image. Any negative mode of that operator would have to be a discrete normalizable finite-core state and is governed by the Birman–Schwinger threshold value ${\beta }_0$. The explicit value of ${\beta }_0$, the $|n_{\theta }|\ge 2$ sectors, higher-derivative terms, microscopic representations, anomalies/inflow, and confinement dynamics remain unresolved.

11.

The finite-radius interface result is a background theorem, not a full boundary-stability theorem. It fixes the classical junction data and proves a canonical thin-wall no-go. It does not transfer the smooth infinite-background self-adjoint domain to an interface with new localized kinetic terms; those scalar, radion, and vector boundary conditions require a separate analysis.

12.

The original seven-dimensional count of 405 zero-mode components does not survive an independent spinor check. A genuine seven-dimensional Dirac spinor gives 810 four-dimensional components of the selected chirality under the inversion used in that construction.

Appendix F.6. Falsifiable Consequences and a Practical Measurement Program

The construction is scientifically meaningful only if its decisive criteria can be tested independently.

Appendix F.20.1. Four-Dimensional Correlator Program

One must construct a renormalized operator basis containing $J_1$, $J_2$, and all operators with the same quantum numbers, and evaluate the corresponding matrix correlator with a regulator suitable for a chiral gauge theory. Standard variational and generalized-eigenvalue methods [36] can then be applied after operator renormalization and positivity checks.

The minimal null tests and dynamical extractions are:

1.

determine the positive renormalized source metric G for independently defined route sources, canonically normalize that two-source space, and only then introduce the diagnostic label exchange $U_{\mathrm{lab}}={\sigma }_x$. This $U_{\mathrm{lab}}$ is not a microscopic Ward transformation. Measure the nontrivial diagonal route defect and the route-odd inverse-kernel component

$$K_{\perp }={\displaystyle \frac{1}{2}}\left(K-U_{\mathrm{lab}}K{U}_{\mathrm{lab}}\right),\parallel K_{\perp }\parallel ,$$

while remembering that $G_{12}=G_{21}$ is kinematic for a real symmetric transfer kernel;

2.

extract the physical even ground energy and lowest odd energy from parity-projected correlators and form

$${\Delta }_{-}^{\left(15\right)}=E_{-}-E_0,{\eta }_{15}={\displaystyle \frac{\parallel K_{\perp }\parallel }{{\Delta }_{-}^{\left(15\right)}}}.$$

A decreasing ${\eta }_{15}$ toward the infrared is the direct test of gap-protected emergent reciprocity;

3.

reconstruct the Schur–Feshbach inverse-kernel or self-energy response and test, where that completion is applicable,

$${\chi }_z\left(E_{*}\right)\equiv M_{11}\left(E_{*}\right)-M_{22}\left(E_{*}\right)=D_{\mathrm{rt}}$$

in the physical matching window; in addition, estimate ${\chi }_z^{\prime }\left(E_{*}\right)$ as a measure of quadratic stability in energy;

4.

if the basis is not manifestly real, verify $\Im M_{12}\left(E\right)\to 0$ and measure ${\chi }_x\left(E\right)=\Re M_{12}\left(E\right)$;

5.

compare the kernel splitting $f=C_{\mathrm{rt}}-{\chi }_x$ with the measured pole-energy difference $E_{-}-E_{+}$ in order to establish the sign inheritance between the effective kernel and the physical poles;

6.

test complete monotonicity of the positive projected correlators;

7.

identify the finite-volume multiparticle threshold and verify its stability under changes of the spatial volume;

8.

extrapolate the extracted $E_{-}$ to the continuum and test $E_{-}<E_{\mathrm{th}}$;

9.

on a finite Euclidean-time grid construct the threshold-normalized moments $m_n$, the Hankel matrices $H_0^{\left(N\right)}$, $H_1^{\left(N\right)}$, and

$${\theta }_N={\lambda }_{max}(H_1^{\left(N\right)},H_0^{\left(N\right)}).$$

The condition ${\theta }_N>1$ is a rigorous finite-data certificate of subthreshold spectral support, while ${\theta }_N\le 1$ means only that this particular finite moment set does not force such support;

10.

add the next block-moment layer ${\mathsf{M}}_{2N}$ and compute the flat-extension residual ${\mathsf{S}}_N$. If ${\mathsf{S}}_N=0$ in a controlled continuum and infinite-volume limit, the finite Krylov space has closed and the multiplicity of source-accessible states is determined exactly. If ${\mathsf{S}}_N\ne 0$, do not claim uniqueness; instead use residual-polynomial bounds on hidden spectral weight;

11.

determine the plateau-normalized three-point form factors and require $F_{+}\left(0\right)\ne 0$ and $F_{-}\left(0\right)\to 0$ in the exact route-symmetry limit.

Appendix F.20.2. Testing the Emergence of an Effective Five-Dimensional Description

If the five-dimensional picture is to be interpreted not as an additional completion datum but as an effective description derived from the unchanged four-dimensional theory, one must compute the actual normalized spectral measure of the physical cyclic source

$$D_{\partial }={\partial }_y{J}_{-}\left(0\right)$$

and test the single-spectrum reconstruction contract. One first identifies a controlled essential spectral band $[s_{-},s_{+}]$ and then reconstructs the asymptotic bulk Jacobi coefficients

$${\alpha }_{\infty }={\displaystyle \frac{s_{+}+s_{-}}{2}},{\beta }_{\infty }={\displaystyle \frac{s_{+}-s_{-}}{4}}.$$

Next one determines the first four Jacobi moments ${\mu }_1^{\left(J\right)},\dots ,{\mu }_4^{\left(J\right)}$ and reconstructs ${\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2$. A pure or nearly pure Robin branch requires convergence toward the bulk tail and the sign condition

$$4\langle s\rangle -s_{+}-3s_{-}>0.$$

If these tests are passed, the canonically normalized boundary parameters $c_{★}$ and ${\rho }_R$ are fixed without an arbitrary discretization step; a positive boundary defect does not generate a state below the essential band, and reflection near the normal threshold approaches the Dirichlet sign. Failure of the one-band, locality, or sign conditions does not invalidate the four-dimensional existence of the source; it invalidates only the simple one-parameter Robin interpretation.

Even if all scalar spectral conditions are satisfied, route parity must still be tested with matrix data. After canonically normalizing the $2\times 2$ correlator $C^{AB}\left(t\right)$, one evaluates the inertia $n_{+}({\mathsf{T}}_t-I)$, the generalized effective energies $E_i^{\mathrm{eff}}$, and the three-slice lower bound on the total route-odd subthreshold weight $w_{-,<}$. Only the combination of local Jacobi reconstruction with a route-resolved matrix correlator can support the stronger statement that the effective five-dimensional quotient emerges from the unchanged four-dimensional theory.

Appendix F.20.3. Testing the Six-Dimensional Ultraviolet Completion

For the smooth six-dimensional branch the decisive chain is longer and should be regarded as a sequence of independent tests rather than a single compound condition.

First, vectorlike six-dimensional Dirac parents with the corrected ${\mathrm{Spin}}^c$ assignment and local anomaly cancellation give, for the direct-product global form,

$${\Omega }_7^{{\mathrm{Spin}}^c}\left(BG\right)=0.$$

Second, the coupled Einstein–Abelian–Higgs equations possess an independently reproduced regular warped solution. Third,

$$wind\left(\Phi \right)=1,{\displaystyle \frac{y_a}{\sqrt{\lambda }}}>{\displaystyle \frac{c}{2}}$$

leaves exactly one normalizable four-dimensional Weyl mode per parent Dirac multiplet. Finite normalization integrals $I_{\mathrm{grav}}$ and $I_g$ localize the graviton and the spectator-gauge zero mode. The transverse-traceless and spectator spectra are non-tachyonic. The coupled bosonic analysis additionally proves positivity of the vortex-vector master operator, excludes independent graviphoton zero modes and a localized massless scalar modulus, and finds no static flat direction in the finite-dimensional shooting space. The branch of the top-form integration constant determines whether regularity can be selected continuously at the effective-field-theory level or becomes an arithmetic compact-flux problem.

The scalar sector is not described by a single universal operator. The transverse gauge-invariant scalar channel and the background radial “breathing” or “sausage” sector have different structures. In the transverse channel the master operator factorizes into a nonnegative form; no normalizable zero mode is present under the stated conditions; the asymptotic index is ${\nu }_T=5/2$; and the response to a compact-core source is infrared finite and analytic through terms of order $Q^4$, with the first generic continuum contribution of order $Q^5$. By contrast, the background radial-breathing sector is generically rigid when regular-core, asymptotic, and interior-turning-point conditions are imposed simultaneously. Spectra computed on a fixed background are therefore only Cowling-type diagnostics for that sector. A source-free top form cannot repair the nonzero-momentum response, because $d(*F_6)=0$ implies $\delta q\left(k\right)=0$ for $k\ne 0$ [28].

The latest structural theorems narrow the possible repairs further. For standard bulk gauge localization,

$$g_4^{-2}\propto \int h_g\left(r\right)L\left(r\right)dr,$$

so a regular axis and asymptotically nondegenerate $h_g$ force an interior stationary point $L^{\prime }\left(r_c\right)=0$. A healthy $R+\beta R^2$ scalaron supplies a local scalar degree of freedom, but in the Einstein frame it shifts rather than removes this stationary point. A finite local Hamilton–Jacobi completion in nonnegative powers of $1/L$ is likewise excluded in the stated regular ${\mathrm{AdS}}_6$ class.

This does not close six dimensions absolutely. A qualitatively different nonminimal possibility is to localize the four-dimensional gauge response through a genuinely confining transverse bulk rather than through a weakly propagating constant bulk zero mode. In that case the gauge-flux penetration amplitude is controlled by a confinement gap, and a no-turn geometry can coexist with finite gravitational and gauge-response norms. The representative no-turn metric is already supported by an exact local positive-kinetic Einstein–sigma action. For its physical $\theta$-independent radion–modulus sector, the positive-square identity and smooth-cap boundary theorem give

$$\mathbb{H}={\mathbb{Q}}^{†}\mathbb{Q}+{\mathbb{K}}^{†}\mathbb{K},{spec}_{\mathrm{normalizable}}\left(m_4^2\right)\cap (-\infty ,0]=⌀.$$

For the regular dipole, the global clock domain and the exact diffeomorphism Ward identity now give a stronger result. The actual fixed-metric negative eigenfunction lies in the admissible scalar-clock image. Its stationary metric completion has exactly zero full quadratic action, and the physical Schur complement annihilates that clock direction whenever it is defined. The value $-0.347963/R_c^2$ is therefore not a physical mass. After quotienting the clock image, any remaining $m_4^2<0$ mode must be a discrete normalizable finite-core metric state.

The independently derived constraint-reduced operator must then be decomposed as $H_{\mathrm{phys}}=H_{+}-G_{-}$ and tested through

$$\mathcal{B}\left({\zeta }_{\mathrm{BS}}\right)=G_{-}^{1/2}{(H_{+}+{\zeta }_{\mathrm{BS}})}^{-1}{G}_{-}^{1/2},{\beta }_0=\underset{\mu \downarrow 0}{lim}{\lambda }_{max}\mathcal{B}\left(\mu \right).$$

The cases ${\beta }_0<1$ and ${\beta }_0>1$ respectively exclude and establish a negative physical mode; ${\beta }_0=1$ requires a separate threshold analysis. Because the continuum begins at zero, a small operator norm of the Einstein correction is not sufficient. A finite-core calculation is conclusive only when a bound $\parallel \mathcal{B}(0+)-{\mathcal{B}}_R\parallel \le {ϵ}_R$ is supplied together with the computed eigenvalue. This explicit Einstein calculation, rather than another projected matter Hessian, is the remaining smooth-branch spectral problem.

If the winding sector is instead gauged by an ordinary finite-positive-coupling Maxwell–Stueckelberg field, the infinite cylinder encounters a separate asymptotic stress obstruction. The finite-radius alternative obeys the exact conditions

$$x_b={\displaystyle \frac{T_b}{Q_b}}\in (0,1/2),t_b={\displaystyle \frac{1-2x_b}{(7\kappa +1)+2(\kappa -1)x_b}},$$

$$Q_b-{\rho }_b={\displaystyle \frac{8\kappa tanh{u}_b}{{\kappa }_6^{2}R}}>0,{\lambda }_b=-{\displaystyle \frac{{\varphi }_b^{\prime }}{{\rho }_b}}.$$

These formulas close the classical radius and scalar junction once the microscopic interface equation of state is supplied. They also rule out an ordinary canonical positive-potential thin wall, which has the opposite inequality $Q_{\mathrm{can}}\le {\rho }_{\mathrm{can}}$. The bulk factorization above is not a substitute for the missing interface fluctuation analysis.

The remaining task is therefore not to invent another metric ansatz. It is to compute the physical Einstein-reduced dipole Schur complement and its Birman–Schwinger threshold value, derive either the confining phase or the flux-dominated interface from the preon theory, establish an allowed microscopic fermion/Yukawa representation, analyze higher-harmonic and interface-localized modes, complete the anomaly and inflow calculation for the actual global group, and compute route matching. A failure of this optional six-dimensional ultraviolet completion would not invalidate the four-dimensional mass mechanism.

Appendix G.1. External Symmetric-Spurion Benchmark

For the printed corrected-spurion benchmark projected to three light directions, the Hermitian matrix $H_S={\lambda }^{\prime }{\lambda }^{\prime †}$ has

$$h_1=2.0688090\times {10}^{-6},h_2=1.99703865\times {10}^{-2},h_3=3.941178595.$$

The adjacent gaps are

$$g_{12}=1.99683177\times {10}^{-2},g_{23}=3.92120821.$$

The median eigenvector has squared overlap $0.99703349$ with the second displayed benchmark basis direction. The polynomial selector $J_{+}\left(H_S\right)={\chi }_S^{\prime }\left(H_S\right)$ has eigenvalues

$$j_1=+0.07869866,j_2=-0.07829993,j_3=+15.45417374.$$

The tensor-lifted median coefficient is

$$2j_2=-0.156599862399,$$

which yields the illustrative determinant-free window quoted in the main text. This benchmark is a robustness check on the corrected symmetric-spurion construction; its matrix entries are not parameters fitted in the Pati–Salam route calculation.

A ${10}^4$-sample rounding test, varying every independent real and imaginary printed matrix entry by $\pm 5\times {10}^{-5}$ while preserving symmetry, keeps the minimum adjacent eigenvalue gap above $1.99439\times {10}^{-2}$ and the squared overlap with the nominal median direction above $0.99999958$. Thus the selector is not an artifact of four-decimal printing.

Appendix G.2. Channel-resolved Matching Sensitivities

In the ordered channel basis $(E_{00},E_{03},E_{150},E_{153})$, the explicitly defined linear benchmark uses

$${\mathbf{j}}_{\Pi }=(-5.2174599612,-5.2592962711,4.7375121485,5.7392440838),s_{\Pi B}=40.947129303807.$$

The unit-channel bridge coefficients are

channel	$j_{\Pi ,i}$	$j_{\Pi ,i}/s_{\Pi B}$
$E_{00}$	$-5.2174599612$	$-0.1274194321$
$E_{03}$	$-5.2592962711$	$-0.1284411474$
$E_{150}$	$+4.7375121485$	$+0.1156982731$
$E_{153}$	$+5.7392440838$	$+0.1401623064$

For a one-percent microscopic deformation, the current one-standard-deviation mass-plane tolerance requires $|{\chi }_X|<0.02977010$. Cauchy–Schwarz then gives the sufficient bounds

$$\parallel a_X^{\prime }{\parallel }_2<0.17221325\mathrm{in}(E_{03},E_{150}),$$

$$\parallel a_X^{\prime }{\parallel }_2<0.13376787\mathrm{in}(E_{03},E_{150},E_{153}),$$

$$\parallel a_X^{\prime }{\parallel }_2<0.11608733\mathrm{in}\mathrm{the}\mathrm{full}\mathrm{four}-\mathrm{channel}\mathrm{space}.$$

These are sufficient norm bounds, not necessary ones; vectors nearly orthogonal to ${\mathbf{j}}_{\Pi }$ can be larger.

Appendix G.3. Infinitesimal Open-Chain Propagation Through Thirty Links

For the representation-matched $\mathrm{SU}\left(15\right)$ control with ${\kappa }_E=1$ and $t=s=0.1$, exact sparse Schur–Feshbach differentiation at the symmetric pole gives

L	${\Delta }_{-}$	${\mathcal{T}}_{\delta t}^{\mathrm{lin}}$	${\mathcal{T}}_y^{\mathrm{lin}}$
10	7.368810652920	0.013787376619	0.045539868565
16	7.368769910590	0.013183440645	0.045567603795
25	7.368755060818	0.012952821072	0.045578188839
30	7.368751974595	0.012903760207	0.045580440205

Quadratic $1/L$ fits over all four points and over the last three points give

$$0.012787612517\lesssim {\mathcal{T}}_{\delta t,\infty }^{\mathrm{lin},\mathrm{ctrl}}\lesssim 0.012790598924,$$

$$0.045585628863\lesssim {\mathcal{T}}_{y,\infty }^{\mathrm{lin},\mathrm{ctrl}}\lesssim 0.045585756787.$$

At $L=25$, the finite one-percent connector ratio differs from the exact infinitesimal susceptibility by only about $0.69\%$, confirming that the one-percent benchmark lies close to the linear regime.

Appendix G.4. Mode-resolved Cancellation Conditioning

The connector response can be written $K_{\perp }^{\left(1\right)}={\sum }_i{c}_i$. The cancellation factor

$${\mathcal{C}}_{\mathrm{canc}}={\displaystyle \frac{|{\sum }_i{c}_i|}{{\sum }_i\left|c_i\right|}}$$

was evaluated from the complete single- and double-resolvent spectral decomposition in the finite control. The extended sequence is

L	${\mathcal{C}}_{\mathrm{canc}}$	$1/{\mathcal{C}}_{\mathrm{canc}}$	top-two absolute-weight fraction
6	0.017723850	56.42	0.87810
8	0.016378855	61.05	0.87618
10	0.015745251	63.51	0.87501
12	0.015398287	64.94	0.87416
14	0.015188424	65.84	0.87343
16	0.015051848	66.44	0.87274

Several quadratic/cubic $1/L$ fits over $L\ge 6$ place the finite-control asymptotic factor in the range

$$0.0145582\lesssim {\mathcal{C}}_{\mathrm{canc}}^{(\infty ),ctrl}\lesssim 0.0146225.$$

The corresponding conditioning factor is approximately 68–69. This finite saturation is encouraging but remains a numerical property of the lower-dimensional control.

Appendix G.5. Transverse-Dimensional Local Strong-Coupling Control

For an additive local route detuning $\left|x\right|/{\kappa }_E=0.01$, the exact shared-link retention bounds give

$$0.004999945356\le {\eta }_{\mathrm{loc}}^{2+1D}\le 0.005000072465,$$

$$0.004999885060\le {\eta }_{\mathrm{loc}}^{3+1D}\le 0.005000155044.$$

The local parity denominator is therefore extremely stable under transverse plaquette dressing, while the additive numerator is not filtered. This numerical control is deliberately kept separate from the physical four-dimensional ${\eta }_{15}$.

Appendix H.1. Phenomenological Observable

For a charged sector $s\in \{\ell ,d,u\}$ and generation $g=1,2,3$, let $y_{s,g}\left(\mu \right)$ be the renormalized Yukawa eigenvalue at common scale $\mu$. The logarithmic curvature is

$$A_s\left(\mu \right)={\displaystyle \frac{1}{4}}ln{\displaystyle \frac{y_{s,1}\left(\mu \right)y_{s,3}\left(\mu \right)}{y_{s,2}^2\left(\mu \right)}}.$$

The fixed plane residual is

$${\Delta }_{\Pi }=A_{\ell }+2A_d-A_u,n_{\Pi }=(1,2,-1).$$

The covariance-dependent diagnostic is $Z_{\Pi }=\left|{\Delta }_{\Pi }\right|/{\sigma }_{{\Delta }_{\Pi }}$. It measures standardized distance from the point null; it is not a probability that the exact plane is true.

Appendix H.2. Route-Source Metric and Canonicalization

For renormalized route sources $J_{Ai}$, with route label $A\in \{12,21\}$ and profile label i, define

$$C_{ij}^{AB}\left(t\right)={\langle J_{Ai}\left(t\right)J_{Bj}^{†}\left(0\right)\rangle }_c.$$

At a fixed positive reference time $t_0$ after contact subtraction,

$$G=C\left(t_0\right)\succ 0,\widehat{C}\left(t\right)=G^{-1/2}C\left(t\right)G^{-1/2}.$$

The route exchange is fixed independently as $U={\sigma }_x$ on the canonical two-route subspace. Source recombinations act on the whitened family by unitary similarity and therefore do not change its generalized eigenvalues.

A useful exact source-algebra control is provided by the six-channel parent vectors of the explicit crossed construction. Their coefficient-space Gram matrix is

$${\widehat{G}}_{\mathrm{src}}=\left(\begin{array}{cc}56& 24\\ 24& 56\end{array}\right),det{\widehat{G}}_{\mathrm{src}}=2560,$$

with eigenvalues 80 and 32 and hence condition number $2.5$. This proves that the two canonical source directions are non-null and not nearly linearly dependent in the declared coefficient metric. It is not yet the Euclidean metric $G=C\left(t_0\right)$ of the confining theory, but it is a useful preconditioner and sanity check for the variational basis.

The same lesson was tested dynamically in the long-chain control. A local odd source had a small target residue, about $5.46\times {10}^{-5}$, because it was dominated by a nearby contaminating mode. Enlarging the profile basis and choosing the exact contaminant-null linear combination increased the target odd residue to about $0.498$ while driving the designated contaminant to numerical zero. This demonstrates that a small residue of one local interpolator need not imply a physically absent pole; the multi-profile canonical GEVP is an essential part of the extraction protocol.

Appendix H.3. Kernel-Level Defect

For a Hermitian effective route kernel at the tracked physical pole,

$$K=\left(\begin{array}{cc}{K}_{11}& K_{12}\\ K_{12}^{*}& K_{22}\end{array}\right),$$

define

$$K_{\perp }={\displaystyle \frac{1}{2}}(K-UK{U}^{†}),$$

$$\parallel K_{\perp }{\parallel }_2=\sqrt{{\displaystyle \frac{{(K_{11}-K_{22})}^2}{4}}+{(\Im K_{12})}^2}.$$

The denominator used for normalized route breaking is the independently resolved route-odd spectral gap ${\Delta }_{-}^{\left(15\right)}$ of the same pole branch. The physical normalized defect is

$${\eta }_{15}={\displaystyle \frac{\parallel K_{\perp }{\parallel }_2}{{\Delta }_{-}^{\left(15\right)}}}.$$

No benchmark visible matrix, static source overlap, or local color recoupling coefficient is substituted for these two observables.

Appendix H.4. Pole and Threshold Decision

If an isolated route-odd pole is used, its energy $E_{-}$ must satisfy

$$E_{-}<E_{\mathrm{th}},$$

where $E_{\mathrm{th}}$ is obtained independently from the lowest continuum or multiparticle channel with the same quantum numbers. The pole must have nonzero source-accessible residue in the canonically normalized correlator. A source formula by itself does not establish this spectral statement.

Appendix H.5. Matching Decision

The normalized crossed operators are

$${\mathcal{O}}_{+}={\displaystyle \frac{{\mathcal{O}}_{12}+{\mathcal{O}}_{21}}{\sqrt{2}}},{\mathcal{O}}_{-}={\displaystyle \frac{{\mathcal{O}}_{21}-{\mathcal{O}}_{12}}{\sqrt{2}}}.$$

After pole amputation and subtraction of allowed operator mixing, the zero-momentum matching form factors must obey

$$F_{+}\left(0\right)\ne 0,F_{-}\left(0\right)=0.$$

The first condition excludes a trivial decoupled solution; the second enforces vanishing normal matching. Both are zero/nonzero statements invariant under finite invertible parity-preserving scheme changes.

Appendix H.6. Cancellation Conditioning

For a spectral decomposition of the linear route-breaking response,

$$K_{\perp }^{\left(1\right)}=\sum _i{c}_i,{\mathcal{C}}_{\mathrm{canc}}={\displaystyle \frac{|{\sum }_i{c}_i|}{{\sum }_i\left|c_i\right|}}.$$

The inverse ${\kappa }_{\mathrm{canc}}=1/{\mathcal{C}}_{\mathrm{canc}}$ is the linear error-amplification condition number. A cancellation-protected limit is accepted only if either ${\mathcal{C}}_{\mathrm{canc}}$ approaches a nonzero limit or the absolute control of the individual contributions improves sufficiently that the amplified uncertainty tends to zero.

Appendix H.7. Finite-Volume and Continuum PASS

A complete positive result requires one common state to survive the following sequence:

$$\begin{array}{c}\mathrm{source}\mathrm{canonicalization}\to \mathrm{parity}-\mathrm{resolved}\mathrm{GEVP}\to \mathrm{pole}\mathrm{tracking},\\ \mathrm{pole}\mathrm{tracking}\to \mathrm{controlled}L\to \infty \to \mathrm{regulator}/\mathrm{continuum}\mathrm{limit}\to \mathrm{three}-\mathrm{point}\mathrm{matching}.\end{array}$$

The quantities ${\eta }_{15}$, $E_{-}-E_{\mathrm{th}}$, the pole residue, and $F_{\pm }\left(0\right)$ must refer to that same branch. Mixing values taken from different source normalizations, different matching energies, or different spectral branches does not constitute a physical test of route reciprocity.

Appendix J.1. Intermediate Derivations: Route-To-Primitive Matrix Algebra

This appendix repeats the key matrix steps without omissions, so that signs and basis conventions can be checked directly.

Appendix J.33.1. Crossed-route Permutation

In the route ordering $(11,12,21,22)$,

$$P_{\times }=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right),P_{\times }^2=I_4.$$

The route-to-primitive matrix is

$$R=\left(\begin{array}{cccc}-1& 2& 4& -8\\ 1& -1& -4& 4\\ 1& -2& -3& 6\\ -1& 1& 3& -3\end{array}\right).$$

A direct determinant expansion gives $detR=1$, so $R^{-1}$ has integer entries. The induced primitive action

$$S_B=R^{-1}{P}_{\times }R$$

automatically satisfies

$$S_B^2=R^{-1}{P}_{\times }^{2}R=I_4.$$

Explicit multiplication gives the matrix quoted in the main text.

Appendix J.33.2. Odd Vector and Dual Covector

For the right eigenvector,

$$S_B\left(\begin{array}{c}2\\ -1\\ 1\\ 0\end{array}\right)=\left(\begin{array}{c}-2\\ 1\\ -1\\ 0\end{array}\right)=-\left(\begin{array}{c}2\\ -1\\ 1\\ 0\end{array}\right).$$

The dual covector obeys

$$S_B^T\left(\begin{array}{c}0\\ -1\\ 1\\ 2\end{array}\right)=\left(\begin{array}{c}0\\ 1\\ -1\\ -2\end{array}\right)=-\left(\begin{array}{c}0\\ -1\\ 1\\ 2\end{array}\right).$$

Hence the operator combination selected by ${\ell }_{-}$ is

$${\ell }_{-}^{T}B=-B_R+B_L+2B_X.$$

The difference of the crossed-route rows is

$$R_{21}-R_{12}=(1,-2,-3,6)-(1,-1,-4,4)=(0,-1,1,2)={\ell }_{-}^T.$$

Therefore

$${\mathcal{O}}_{21}-{\mathcal{O}}_{12}={\ell }_{-}^T{\mathcal{O}}_{\mathrm{prim}}$$

with no additional assumption.

Appendix J.33.3. Projectors

Since $S_B^2=I$, the right-vector spectral projectors and dual coefficient-covector projectors are, respectively,

$$P_{B,\mathrm{vec}}^{\pm }={\displaystyle \frac{I\pm S_B}{2}},P_{B,\mathrm{cov}}^{\pm }={\displaystyle \frac{I\pm S_B^T}{2}}.$$

They satisfy

$${\left(P_{B,\mathrm{vec}}^{\pm }\right)}^2=P_{B,\mathrm{vec}}^{\pm },P_{B,\mathrm{vec}}^{+}{P}_{B,\mathrm{vec}}^{-}=0,P_{B,\mathrm{vec}}^{+}+P_{B,\mathrm{vec}}^{-}=I,$$

and the analogous identities hold in the dual space. Because $Tr{S}_B=Tr{S}_B^T=2$,

$$rank{P}_{B,\mathrm{vec}}^{-}=rank{P}_{B,\mathrm{cov}}^{-}=1,rank{P}_{B,\mathrm{vec}}^{+}=rank{P}_{B,\mathrm{cov}}^{+}=3.$$

The source coefficient ${\ell }_{-}$ belongs to $im{P}_{B,\mathrm{cov}}^{-}$, whereas the right eigenvector $v_{-}$ belongs to $im{P}_{B,\mathrm{vec}}^{-}$.

Appendix J.2. Intermediate Derivations: Route Symmetry, Feshbach Susceptibility, and the Global Minimum

Appendix J.34.1. General Hermitian Kernel and the One-Scalar Symmetry Condition

Let

$$K=\left(\begin{array}{cc}a& c-id\\ c+id& b\end{array}\right)=k_0{I}_2+k_x{\sigma }_x+k_y{\sigma }_y+k_z{\sigma }_z,$$

where

$$k_0={\displaystyle \frac{a+b}{2}},k_x=c,k_y=d,k_z={\displaystyle \frac{a-b}{2}}.$$

Conjugation by $P_{\mathrm{rt}}={\sigma }_x$ gives

$${\sigma }_x{\sigma }_x{\sigma }_x={\sigma }_x,{\sigma }_x{\sigma }_y{\sigma }_x=-{\sigma }_y,{\sigma }_x{\sigma }_z{\sigma }_x=-{\sigma }_z.$$

Hence

$${\sigma }_{x}K{\sigma }_x=K\iff k_y=k_z=0\iff a=b,d=0.$$

For a canonically real symmetric kernel $d=0$ automatically, so only $a=b$ remains. The symmetric family is

$$K_{\mathrm{sym}}=d_0{I}_2+f{\sigma }_x=\left(\begin{array}{cc}{d}_0& f\\ f& d_0\end{array}\right),$$

with

$${\lambda }_{+}=d_0+f,{\lambda }_{-}=d_0-f.$$

Thus at the level of kernel eigenvalues,

$$f>0\iff {\lambda }_{-}<{\lambda }_{+}.$$

No statement about physical pole or mass ordering follows until the sign inheritance of the actual dynamical equation has been established.

Appendix J.34.2. Positive-Semidefinite Susceptibility Theorem in Full Detail

For hidden states above the energy at which the visible sector is reduced,

$$M\left(E\right)={\displaystyle \frac{1}{h}}\mathcal{V}{({\mathcal{H}}_H-E)}^{-1}{\mathcal{V}}^{†}⪰0.$$

Write

$$M=\left(\begin{array}{cc}x& y\\ y^{*}& z\end{array}\right).$$

Positive semidefiniteness is equivalent to

$$x\ge 0,z\ge 0,xz-{\left|y\right|}^2\ge 0.$$

The known route mismatch is

$$D_{\mathrm{rt}}=A_{\mathrm{rt}}-B_{\mathrm{rt}}=0.049406060606061.$$

For a general future visible block, exact locking requires $x-z=D_{\mathrm{rt}}$ without an assumption on the sign of $D_{\mathrm{rt}}$. Hence

$$TrM=x+z\ge |x-z|=|D_{\mathrm{rt}}|.$$

Equality holds only if one of the two nonnegative diagonal entries vanishes. With

$$D_{\mathrm{rt}}^{(+)}=max(D_{\mathrm{rt}},0),D_{\mathrm{rt}}^{(-)}=max(-D_{\mathrm{rt}},0),$$

the constraint fixes $(x,z)=(D_{\mathrm{rt}}^{(+)},D_{\mathrm{rt}}^{(-)})$. Determinant positivity then forces $y=0$, so

$$M_{min}=\left(\begin{array}{cc}{D}_{\mathrm{rt}}^{(+)}& 0\\ 0& D_{\mathrm{rt}}^{(-)}\end{array}\right),Tr{M}_{min}=\parallel M_{min}{\parallel }_F=\left|D_{\mathrm{rt}}\right|.$$

For the numerical benchmark $D_{\mathrm{rt}}>0$, this reduces to $M_{min}=diag(D_{\mathrm{rt}},0)$. The proof does not assume rank one; rank one emerges as a property of the global minimum.

Appendix J.34.3. Relation to a Single-Hidden-State Parametrization

For one real hidden state, define

$$M=q{q}^T,q={(q_1,q_2)}^T.$$

Then

$$q{q}^T={\displaystyle \frac{q_1^2+q_2^2}{2}}{I}_2+q_1{q}_2{\sigma }_x+{\displaystyle \frac{q_1^2-q_2^2}{2}}{\sigma }_z.$$

Let

$$p=q_1{q}_2.$$

Locking requires

$$q_1^2-q_2^2=D_{\mathrm{rt}}.$$

Therefore

$$q_1^2+q_2^2=\sqrt{D_{\mathrm{rt}}^2+4p^2},$$

$$q_1^2={\displaystyle \frac{\sqrt{D_{\mathrm{rt}}^2+4p^2}+D_{\mathrm{rt}}}{2}},q_2^2={\displaystyle \frac{\sqrt{D_{\mathrm{rt}}^2+4p^2}-D_{\mathrm{rt}}}{2}}.$$

The final off-diagonal coefficient is

$$f=C_{\mathrm{rt}}-p.$$

If route reciprocity is the only target, f remains unconstrained and the global minimum occurs at $p=0$:

$$q_1^2=D_{\mathrm{rt}},q_2=0,{\parallel q\parallel }^2=D_{\mathrm{rt}}.$$

If one additionally imposes the special degeneracy $f=0$, then $p=C_{\mathrm{rt}}$ and the required strength becomes

$${\parallel q\parallel }^2=\sqrt{D_{\mathrm{rt}}^2+4C_{\mathrm{rt}}^2}=0.054597550878986.$$

This explains exactly why the older target was $9.50865\%$ stronger than necessary.

Using the convention in which the correction is written in units of $1/15$,

$${\displaystyle \frac{\delta K}{h}}=-q{q}^T,$$

the coefficients are

$$r_0=-{\displaystyle \frac{15}{2}}(q_1^2+q_2^2),r_x=-15p,r_z=-{\displaystyle \frac{15}{2}}{D}_{\mathrm{rt}},r_y=0.$$

At the global minimum $p=0$,

$$(r_0,r_x,r_y,r_z)=(-0.3705454545,0,0,-0.3705454545).$$

The older contact Bethe–Salpeter control point with $r_x=-0.20$ is nonminimal and is used only within that particular mapping to the reduced couplings $g_{\pm }$.

Appendix J.34.4. Exact Three-State Construction and Stability over an Energy Window

Choose ${\Delta }_H>0$ and

$${\displaystyle \frac{\mathbb{H}}{h}}=\left(\begin{array}{ccc}{A}_{\mathrm{rt}}& C_{\mathrm{rt}}& u\\ C_{\mathrm{rt}}& B_{\mathrm{rt}}& 0\\ u& 0& (B_{\mathrm{rt}}-C_{\mathrm{rt}})+{\Delta }_H\end{array}\right),u^2=D_{\mathrm{rt}}{\Delta }_H.$$

The vector

$${\Psi }_{-}={(1,-1,-u/{\Delta }_H)}^T$$

satisfies

$$\mathbb{H}{\Psi }_{-}=h(B_{\mathrm{rt}}-C_{\mathrm{rt}}){\Psi }_{-}.$$

The equality can be checked row by row. Its normalized hidden-state fraction is

$$P_H={\displaystyle \frac{D_{\mathrm{rt}}}{2{\Delta }_H+D_{\mathrm{rt}}}}.$$

For a general positive hidden spectral measure whose support is separated from $E_0$ by a gap g, the scalar resolvent identity gives

$$\left|{\displaystyle \frac{1}{s-E_0-\delta E}}-{\displaystyle \frac{1}{s-E_0}}\right|\le {\displaystyle \frac{\left|\delta E\right|}{g-\left|\delta E\right|}}{\displaystyle \frac{1}{s-E_0}},$$

and integration of the positive-semidefinite measure yields

$$\parallel M(E_0+\delta E)-M\left(E_0\right)\parallel \le {\displaystyle \frac{\left|\delta E\right|}{g-\left|\delta E\right|}}\parallel M\left(E_0\right)\parallel .$$

This provides a quantitative stability bound over a finite physical matching window.

Appendix J.34.5. Full Four-State Schur Proof and Quadratic Spectral Flattening

Let $A=B+D$ with $D>0$, $C>0$, $m_1-m_2=D$, $m_i\ge 0$, ${\Delta }_i>0$, $u_i^2=m_i{\Delta }_i$, and ${\lambda }_{*}=B-C-m_2$. For

$$\widehat{H}-{\lambda }_{*}I=\left(\begin{array}{cc}{V}_{*}& U\\ U^T& {\Delta }_H\end{array}\right),$$

with

$${\Delta }_H=diag({\Delta }_1,{\Delta }_2),U=diag(u_1,u_2),V_{*}=\left(\begin{array}{cc}{m}_1+C& C\\ C& m_2+C\end{array}\right),$$

the Schur complement is

$$V_{*}-U{\Delta }_H^{-1}{U}^T=C\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)⪰0.$$

Since ${\Delta }_H>0$ and the Schur complement has rank one, $\widehat{H}-{\lambda }_{*}I⪰0$ and its kernel is one dimensional. A kernel vector is

$${(1,-1,-u_1/{\Delta }_1,+u_2/{\Delta }_2)}^T.$$

Thus the visible route-odd combination is the unique full ground state of this completion family.

For $E=E_{*}+x$, the diagonal hidden susceptibility is

$${\chi }_z\left(E\right)={\displaystyle \frac{m_1{\Delta }_1}{{\Delta }_1-x}}-{\displaystyle \frac{m_2{\Delta }_2}{{\Delta }_2-x}}.$$

The conditions ${\chi }_z\left(E_{*}\right)=D$ and ${\chi }_z^{\prime }\left(E_{*}\right)=0$ imply

$$m_1={\displaystyle \frac{D{\Delta }_1}{{\Delta }_1-{\Delta }_2}},m_2={\displaystyle \frac{D{\Delta }_2}{{\Delta }_1-{\Delta }_2}},$$

for ${\Delta }_1>{\Delta }_2$, and direct simplification yields

$$D-{\chi }_z(E_{*}+x)=D{\displaystyle \frac{x^2}{({\Delta }_1-x)({\Delta }_2-x)}}.$$

This is the exact quadratic-flattening identity used in the main text.

Appendix J.3. Intermediate Derivations: Spectral Positivity and the Pole Residue

For a Hilbert space with a positive transfer matrix,

$$C\left(t\right)=\langle 0\left|J{e}^{-Ht}{J}^{†}\right|0\rangle .$$

Inserting a complete set of energy eigenstates gives

$$C\left(t\right)=\sum _n\langle 0\left|J\right|n\rangle \langle n|J^{†}|0\rangle e^{-E_{n}t}=\sum _n{\left|z_n\right|}^2{e}^{-E_{n}t}.$$

At $t=0^{+}$, after the chosen regularized smearing and contact subtraction,

$$C\left(0^{+}\right)=\sum _n|z_n{|}^2=\parallel J^{†}{|0\rangle \parallel }^2.$$

Therefore $J^{†}|0\rangle \ne 0$ implies strictly positive total spectral weight.

Suppose the lowest point of the support is isolated, with energy $E_0$ below the next state or the continuum. Then

$$e^{E_{0}t}C\left(t\right)={\left|z_0\right|}^2+O\left(e^{-(E_1-E_0)t}\right),$$

so

$$|z_0{|}^2=\underset{t\to \infty }{lim}{e}^{E_{0}t}C\left(t\right)>0.$$

The theorem does not establish isolation of the lowest support; it establishes positivity of the residue conditional on isolation.

Appendix J.4. Intermediate Derivations: the Five-Dimensional Jacobi/Stieltjes Transfer Theorem

Let $d\mu \left(s\right)$ be a positive measure with enough finite moments to run the Lanczos recursion. The associated orthogonal polynomials generate a Jacobi matrix with positive off-diagonal coefficients ${\beta }_n>0$. After the alternating rephasing $|n\rangle \to {(-1)}^n|n\rangle$, the off-diagonal elements are $-{\beta }_n$.

For $\zeta >0$ and a positive-semidefinite finite Jacobi truncation $H_J^{\left(N\right)}$, the matrix

$${\mathsf{A}}_J^{\left(N\right)}\left(\zeta \right)=\zeta I_N+H_J^{\left(N\right)}$$

has positive diagonal entries, nonpositive off-diagonal entries, and is an irreducible positive-definite Stieltjes matrix. Its inverse is therefore strictly positive entrywise.

For finite N, the endpoint cofactor can be evaluated explicitly. Deleting row N and column 1 leaves a lower-bidiagonal chain contribution. Apart from the cofactor sign, the product of off-diagonal elements is ${(-1)}^{N-1}{\beta }_1\dots {\beta }_{N-1}$; the cofactor sign cancels this factor. Hence

$${\left[A^{-1}\right]}_{1N}={\displaystyle \frac{{\beta }_1{\beta }_2\dots {\beta }_{N-1}}{detA}}>0.$$

No special free-interval form is required. This is an exact nonzero-transfer theorem for every finite cyclic positive spectral measure. The semi-infinite limit follows whenever the corresponding local resolvent matrix elements converge.

Appendix J.5. Intermediate Derivations: Five-Dimensional Robin Reconstruction

This appendix derives the one-spectrum formulas consecutively so that the main text does not require jumps between definitions.

Appendix J.37.1. From Band Edges to Bulk Coefficients

The homogeneous Jacobi recurrence

$$-{\beta }_{\infty }{\psi }_{n-1}+{\alpha }_{\infty }{\psi }_n-{\beta }_{\infty }{\psi }_{n+1}=s{\psi }_n$$

with ${\psi }_n=e^{ikn}$ gives

$$s={\alpha }_{\infty }-2{\beta }_{\infty }cosk.$$

At $k=0$ and $k=\pi$,

$$s_{-}={\alpha }_{\infty }-2{\beta }_{\infty },s_{+}={\alpha }_{\infty }+2{\beta }_{\infty }.$$

Solving the two equations gives

$${\alpha }_{\infty }={\displaystyle \frac{s_{+}+s_{-}}{2}},{\beta }_{\infty }={\displaystyle \frac{s_{+}-s_{-}}{4}}.$$

Appendix J.37.2. From the First Moment to the Boundary Defect

Because $e_1$ is the normalized physical cyclic endpoint,

$${\alpha }_1=\langle e_1,J{e}_1\rangle =\int sd\nu \left(s\right)=\langle s\rangle .$$

For a Neumann-matched endpoint,

$${\alpha }_1^{\left(N\right)}={\alpha }_{\infty }-{\beta }_{\infty }.$$

Therefore

$$c_J={\alpha }_1-{\alpha }_1^{\left(N\right)}=\langle s\rangle -{\displaystyle \frac{s_{+}+3s_{-}}{4}}.$$

Division by ${\beta }_{\infty }$ gives the dimensionless Robin defect ${\rho }_R$.

Appendix J.37.3. Canonical Normalization of the Continuum Limit

The discretization identities

$${\beta }_{\infty }={\displaystyle \frac{Z_y}{a_{★}^2{Z}_4}},c_J={\displaystyle \frac{c_R}{a_{★}{Z}_4}}$$

imply

$${\displaystyle \frac{c_J}{\sqrt{{\beta }_{\infty }}}}={\displaystyle \frac{c_R}{\sqrt{Z_y{Z}_4}}}.$$

The auxiliary lattice spacing $a_{★}$ therefore cancels exactly in the canonically normalized Robin parameter.

Appendix J.37.4. Moment Reconstruction

The first four moments give

$${\alpha }_1={\mu }_1^{\left(J\right)},{\beta }_1^2={\mu }_2^{\left(J\right)}-{\left({\mu }_1^{\left(J\right)}\right)}^2,$$

$${\alpha }_2={\displaystyle \frac{{\mu }_3^{\left(J\right)}-2{\alpha }_1{\mu }_2^{\left(J\right)}+{\alpha }_1^3}{{\beta }_1^2}},$$

$${\beta }_2^2={\displaystyle \frac{{\mu }_4^{\left(J\right)}-2{\alpha }_1{\mu }_3^{\left(J\right)}+{\alpha }_1^2{\mu }_2^{\left(J\right)}}{{\beta }_1^2}}-{\beta }_1^2-{\alpha }_2^2.$$

These are direct Lanczos identities. No physical $\mathrm{SU}{\left(15\right)}_p$ numbers enter until a real source spectral measure has been computed.

Appendix J.6. Intermediate Derivations: Anomalies, Regularity, and Stability in Six Dimensions

Appendix J.38.1. Spin C Anomaly Logic

The corrected charges $+1$ and $-1$ imply that the anomaly difference is proportional to $e^{F_V}-e^{-F_V}$, so even powers of the auxiliary field strength $F_V$ cancel. The remaining mixed cubic non-Abelian term is proportional to the same published four-dimensional cubic anomaly sum, which vanishes. For the direct-product gauge group G, the Atiyah–Hirzebruch spectral sequence contains no total-degree-seven contribution because the odd ${\mathrm{Spin}}^c$ coefficient groups vanish and $H_{*}(BG;\mathbb{Z})$ is concentrated in even degree. Consequently,

$${\Omega }_7^{{\mathrm{Spin}}^c}\left(BG\right)=0.$$

Appendix J.38.2. Reference Einstein–Abelian–Higgs Solution and Top-Form Selection

The regular branch has

$${\mu }_{6,★}=-0.0782271150,c_{6,★}=\sqrt{-{\mu }_{6,★}/10}.$$

A source-free top form changes only the effective cosmological parameter,

$${\mu }_{6,\mathrm{eff}}={\mu }_{6,0}+{\widehat{q}}^2.$$

Thus choosing

$${\widehat{q}}^2={\mu }_{6,★}-{\mu }_{6,0}$$

reproduces the same Einstein–Abelian–Higgs background equations. For ${\mu }_{6,0}=-0.10$,

$${\widehat{q}}_{★}=0.14755637905560032.$$

Appendix J.38.3. The Four-Square Theorem and Why It Does Not Imply Continuous Tuning

With four equal compact fluxes,

$$\Delta {\mu }_6={\widehat{e}}^2(n_1^2+n_2^2+n_3^2+n_4^2).$$

Lagrange’s four-square theorem guarantees that every integer $m_{\mathrm{shell}}\ge 0$ can occur, but the allowed radii remain ${\widehat{e}}^2{m}_{\mathrm{shell}}$. Hence holes in the integer shell label disappear while the radial spacing remains. Exact regularity is not guaranteed unless the required $\Delta {\mu }_{6,★}$ lies on that flux lattice.

Appendix J.38.4. Rayleigh Positivity

For the transverse-traceless tensor and spectator-gauge radial Sturm–Liouville problems, multiply the equations by the complex-conjugate mode, integrate, and drop the boundary terms using regularity and normalizability. The resulting numerators are sums of nonnegative terms, so $m_4^2\ge 0$. The coupled vortex-vector analysis extends this argument by exact factorization of its master operator. Independent graviphoton and scalar zero-mode loopholes are then excluded by non-normalizability. The full scalar decomposition sharpens the statement further: the transverse metric–gauge scalar channel is itself non-tachyonic and threshold-local, whereas the background radial-breathing sector is generically rigid under the stated regularity conditions.

Appendix J.38.5. Local Continuation of the Regular Six-Dimensional Branch by the Implicit-Function Theorem

Let

$$\mathcal{R}(p;\lambda )=0,p=(A_f,B_P,{\mu }_6),\lambda =({\alpha }_6,{\nu }_6).$$

At a regular point $(p_{★},{\lambda }_{★})$ define

$$J_{\mathcal{R}}=D_p\mathcal{R}(p_{★};{\lambda }_{★}).$$

If $det{J}_{\mathcal{R}}\ne 0$, the implicit-function theorem gives a unique local function $p=p\left(\lambda \right)$. Differentiating the identity $\mathcal{R}\left(p\right(\lambda );\lambda )=0$ with respect to a microscopic parameter ${\lambda }_i$ gives

$$J_{\mathcal{R}}{\displaystyle \frac{dp}{d{\lambda }_i}}+{\displaystyle \frac{\partial \mathcal{R}}{\partial {\lambda }_i}}=0,$$

so

$${\displaystyle \frac{dp}{d{\lambda }_i}}=-J_{\mathcal{R}}^{-1}{\displaystyle \frac{\partial \mathcal{R}}{\partial {\lambda }_i}}.$$

The third component is $d{\mu }_{6,★}/d{\lambda }_i$. If a continuous source-free top form shifts

$${\mu }_{6,\mathrm{eff}}={\mu }_{6,\mathrm{bare}}+{\widehat{q}}^2,$$

then along the regularity surface, at fixed ${\mu }_{6,\mathrm{bare}}$,

$$2{\widehat{q}}_{★}{\displaystyle \frac{d{\widehat{q}}_{★}}{d{\lambda }_i}}={\displaystyle \frac{d{\mu }_{6,★}}{d{\lambda }_i}},$$

therefore

$${\displaystyle \frac{d{\widehat{q}}_{★}}{d{\lambda }_i}}={\displaystyle \frac{1}{2{\widehat{q}}_{★}}}{\displaystyle \frac{d{\mu }_{6,★}}{d{\lambda }_i}}.$$

This explains why one continuous integration constant can satisfy one homogeneous regularity condition, but also why it cannot replace a local scalar Green function: one global constant cannot generate a functional derivative with respect to an arbitrary spacetime-dependent source $\delta T_{\mu \nu }\left(x\right)$.

Appendix J.38.6. Factorization of the Vector Operator and Zero-Mode Norms

Write

$$H_V=-{\partial }_w^2+{\displaystyle \frac{{\left(MP\right)}^{\prime \prime }}{MP}},W_V={\partial }_{w}ln\left(MP\right).$$

Since

$$W_V^{\prime }+W_V^2={\displaystyle \frac{{\left(MP\right)}^{\prime \prime }}{MP}},$$

one has

$$H_V=({\partial }_w+W_V)(-{\partial }_w+W_V).$$

Thus for a regular normalizable master mode,

$$\langle N,H_{V}N\rangle ={\parallel (-{\partial }_w+W_V)N\parallel }^2\ge 0.$$

The eigenvalue equation

$$H_{V}N=(m_4^2-n_{\theta }^2)N$$

immediately gives

$$m_4^2\ge n_{\theta }^2.$$

This is stronger than a numerical eigenvalue scan because it follows directly from operator factorization.

For the independent homogeneous graviphoton candidates, non-normalizability follows from the asymptotic behavior at the two ends of the geometry. At infinity $M,L\sim e^{-cx}$, hence

$${\int }^{\infty }dx{\displaystyle \frac{1}{L{M}^4}}\sim {\int }^{\infty }dx{e}^{5cx}=\infty .$$

Near the core $L\sim x$ and $M\to 1$, so

$${\int }_{0}dx{\displaystyle \frac{M^4}{L}}\sim {\int }_0{\displaystyle \frac{dx}{x}}=\infty .$$

Neither homogeneous graviphoton candidate is therefore a normalizable independent four-dimensional vector zero mode.

Appendix J.38.7. Why the Shooting Jacobian Is Only a Finite-Dimensional Nondegeneracy Diagnostic

The residual map $\mathcal{R}:{\mathbb{R}}^3\to {\mathbb{R}}^4$ depends only on the finite-dimensional regular-core parameters $(A_f,B_P,{\mu }_6)$ at fixed $({\alpha }_6,{\nu }_6)$. Full column rank of the shooting Jacobian implies that there is no infinitesimal displacement inside this three-dimensional shooting family that leaves all four residuals unchanged. It does not imply positivity of the functional second variation with respect to arbitrary spatially dependent scalar perturbations. This is why the shooting calculation is a static finite-dimensional nondegeneracy diagnostic rather than a proof of the full scalar spectrum. The gauge-invariant scalar reduction must be used for the latter; it yields a stable transverse channel and a separate background-breathing rigidity obstruction.

Appendix J.38.8. Threshold Locality of the Full Transverse Scalar Channel

For the physical gauge-invariant transverse scalar master operator,

$$H_T=-{\partial }_{z_T}^2+{\displaystyle \frac{{\nu }_T^2-\frac{1}{4}}{z_T^2}}+o\left(z_T^{-2}\right),{\nu }_T={\displaystyle \frac{5}{2}}.$$

The regular continuum Bessel branch on a fixed compact core interval scales as

$${\psi }_k=O\left(k^{{\nu }_T+1/2}\right)=O\left(k^3\right).$$

A compact core source therefore has spectral measure

$$d{\mu }_T\left(k\right)=O\left(k^6\right)dk.$$

For

$$M_n=\langle s,H_T^{-n}s\rangle ,$$

the infrared integral is proportional to

$${\int }_0{k}^{6-2n}dk,$$

which converges for $n=1,2,3$. Expanding ${(H_T+Q^2)}^{-1}$ in inverse powers of $H_T$ therefore gives regular coefficients through $Q^4$. The threshold continuum can first generate the generic fractional or nonanalytic contribution at order

$$Q^{2{\nu }_T}=Q^5.$$

Appendix J.38.9. Dimension Counting in the Background Breathing Sector and the Limitation of the Top Form

Let the complete gauge-invariant first-order system of the background radial-breathing sector have a local solution space $\mathcal{S}$ of dimension six. Core regularity leaves a three-dimensional subspace ${\mathcal{C}}_{\mathrm{core}}$. The acceptable asymptotic conditions leave a three-dimensional subspace, and regularity at the interior turning point removes one further dimension, so

$$dim{\mathcal{A}}_{\infty ,r_c}=2.$$

Generic linear algebra then gives

$${dim}_{\mathrm{gen}}({\mathcal{C}}_{\mathrm{core}}\cap {\mathcal{A}}_{\infty ,r_c})=max(0,3+2-6)=0.$$

Thus nonzero homogeneous breathing modes require exceptional rather than generic matching.

For

$$F_6=q{\mathrm{vol}}_6,$$

the source-free field equation $d(*F_6)=0$ implies $dq=0$, because $*F_6$ is a zero-form in six dimensions. After a four-dimensional Fourier transform,

$$\delta q\left(k\right)=0(k\ne 0).$$

The top form can shift the homogeneous $Q=0$ background regularity condition, but it cannot provide a local propagating response of the radial-breathing sector at nonzero four-dimensional momentum.

Appendix J.7. Intermediate Derivations: Uniqueness of the Six-Dimensional Vortex Zero Mode

Appendix J.39.1. Index

For a complex mass profile

$$m(r,\theta )=\left|m\left(r\right)\right|e^{in\theta },$$

the transverse two-dimensional Dirac operator belongs to the Jackiw–Rossi class. Under the standard regularity and gap assumptions, its index equals the winding number [23,24]:

$$ind{\mathcal{D}}_{\perp }=n.$$

For $n=1$,

$$N_L-N_R=1.$$

This alone would still allow $(N_L,N_R)=(1,0),(2,1),\dots$.

Appendix J.39.2. Angular Regularity

Near a regular cigar tip the transverse metric is polar,

$$d{s}_2^2=d{r}^2+r^{2}d{\theta }^2+O\left(r^4\right).$$

Spinor components admit angular harmonics $e^{im\theta }$. Unit winding couples harmonics whose angular numbers differ by one. Regularity eliminates harmonics with singular radial powers. For winding one, exactly one angular pairing admits regular normalizable zero-mode behavior at the core.

Appendix J.39.3. First-order Radial System

Within that angular channel, the zero-energy equations reduce to a first-order system

$${\displaystyle \frac{d}{dr}}\mathbf{f}\left(r\right)=M\left(r\right)\mathbf{f}\left(r\right).$$

A regular initial datum at the tip fixes the solution up to one overall normalization, so the regular solution space has dimension at most one. The index theorem guarantees at least one chiral solution. Therefore exactly one zero mode exists and no opposite-chirality zero mode remains.

Appendix J.39.4. Asymptotic Norm

At large radius the vortex mass approaches $m_{\infty }$, while the string-cigar measure contributes an exponential warp factor. Schematically,

$$f\left(r\right)\sim e^{-m_{\infty }r}\times e^{\mathrm{spin}-\mathrm{connection}\mathrm{contribution}}.$$

Combining the asymptotic spin connection with the norm measure gives the convergence condition

$$m_{\infty }>{\displaystyle \frac{c}{4}}.$$

This is the normalizability condition quoted in the main text for the corresponding backreacted branch.

Appendix J.8. Intermediate Derivations: Spinor Multiplicity in Seven Dimensions

The minimal complex Dirac representation of the seven-dimensional Clifford algebra has dimension eight. Under

$$\mathrm{Spin}(1,6)\supset \mathrm{Spin}(1,3)\times \mathrm{Spin}\left(3\right),$$

one may write schematically

$${\mathbf{8}}_{\mathbb{C}}\simeq ({\mathbf{2}}_L\otimes {\mathbf{2}}_{\mathrm{int}})\oplus ({\mathbf{2}}_R\otimes {\mathbf{2}}_{\mathrm{int}}).$$

The internal factor ${\mathbf{2}}_{\mathrm{int}}$ is precisely the twofold multiplicity missed by a naive chirality count. The simultaneous inversion operator is proportional to

$${\gamma }_5\otimes I_2.$$

It selects one four-dimensional chirality but acts trivially on the internal doublet. The projected subspace therefore has complex dimension

$$2\times 2=4,$$

while one four-dimensional Weyl spinor has complex dimension two. Hence there are two four-dimensional Weyl zero modes per gauge component.

For the published multiplets the gauge multiplicities are

$$15·4·2=120,15·4·2=120,3·15=45,{\displaystyle \frac{15·16}{2}}=120.$$

Their sum, 405, counts gauge components but not the internal spinor multiplicity. The corrected number of selected four-dimensional Weyl components is therefore

$$2\times 405=810.$$

Appendix J.9. Independent Checks of the Central Analytic and Numerical Results

The following checks were rerun independently from the formulas printed in this manuscript. The purpose of the table is to separate exact algebraic reproduction from numerical consistency and from results that still require the original boundary-value or nonperturbative data.

Quantity or claim	Method	Outcome
Route-to-primitive map R, $S_B$, $v_{-}$, and ${\ell }_{-}$	exact integer matrix algebra	$detR=1$, $S_B^2=I_4$, and both odd eigenvector equations vanish exactly
Pati–Salam unitary commutant	Schur decomposition of the nineteen fundamental copies	$U{\left(1\right)}_W\times U{\left(1\right)}_{W^{\prime }}\times U{\left(3\right)}_{\psi }$, with determinant-one subgroup $S(U{\left(1\right)}_W\times U{\left(1\right)}_{W^{\prime }}\times U{\left(3\right)}_{\psi })$; no element exchanges the inequivalent W and $W^{\prime }$ blocks
Corrected scalar-spurion symmetry	direct Weyl-statistics and precolor-index exchange	$\mathcal{A}\in \overline{\mathbf{105}}$ selects ${\lambda }^T=-\lambda$, while ${\mathcal{A}}^{\prime }\in \overline{\mathbf{120}}$ selects ${\left({\lambda }^{\prime }\right)}^T=+{\lambda }^{\prime }$
Three-flavour antisymmetric-spurion no-go	Youla normal form / singular-value check	any nonzero $3\times 3$ complex antisymmetric $\lambda$ gives $spec\left(\lambda {\lambda }^{†}\right)=\{{\sigma }^2,{\sigma }^2,0\}$ and cannot define a unique median eigendirection
Symmetric-spurion family reconstruction	Takagi plus Lagrange–Sylvester projector	for nondegenerate $H_S={\lambda }^{\prime }{\lambda }^{\prime †}$, $D_{+}=I_3-3P_2\left(H_S\right)$ has eigenvalues $(1,-2,1)$ and obeys $D_{+}^2+D_{+}-2I_3=0$ exactly
Two-spurion misalignment invariant	positive Hilbert–Schmidt norm of the commutator	${\mathcal{I}}_{AS}=Tr\left({[H_A,H_S]}^{†}[H_A,H_S]\right)\ge 0$, with equality iff $H_A$ and $H_S$ commute
Texture-curvature bridge	direct second-difference algebra	${\Delta }_{\Pi }={\Delta }_{\Pi }^{\left(c\right)}+{\displaystyle \frac{1}{4}}ln\kappa (q_{\ell }+2q_d-q_u)$; the stated effective exponent assignments give $(q_{\ell },q_d,q_u)=(1,0,1)$ and cancel the exponent contribution
Right-handed-isospin no-go	sector-wide rescaling invariance of $A_s$	exact $Y_u\propto Y_d$ gives $A_u=A_d$ and would require $A_{\ell }+A_d=0$; the displayed central value is $-0.3829327$
Metric-reflection anti-tautology	exact symbolic multiplication	$U_c=2Zc{c}^T/\left(c^{T}Zc\right)-I$ satisfies $U_{c}Z{U}_c^T=Z$, $U_c^2=I$, and $U_c^{T}c=c$
Component-counting exchange coefficient	exact $2\times 2$ multiplication	$U\left(a\right)Z_{\mathrm{PS}}U{\left(a\right)}^T=Z_{\mathrm{PS}}$ gives $a=\sqrt{6/7}$ and the fixed-covector ratio $C_{21}/C_{12}=\sqrt{6/7}$
Factorized-kernel no-go	exact symbolic multiplication	$P_{B,\mathrm{cov}}^{+}(a{G}_4+b{G}_R){\ell }_{-}=0$ has only $a=b=0$
Natural Fierz graph counts	exhaustive enumeration of all perfect matchings of twelve labels	$N_0=10395$, degree 30, $N_1=155925$, $N_{▵}=51975$, and $N_{\square }=1715175$
Rank-one primitive correction	direct exact multiplication	the displayed $K_{\mathrm{corr}}=-u{u}^T$ reproduces $y_{\mathrm{miss}}$ exactly
Four-state completion benchmark	direct Hermitian diagonalization	${\lambda }_1=0.1866363083$, ${\lambda }_2=0.2077210756$, ${\lambda }_3=0.7371267438$, and ${\lambda }_4=1.2865157616$; eigenvector residual below ${10}^{-16}$
Reduced-model source residue	direct evaluation of $I_1^2/I_2$	$z_{\mathrm{red}}^2/{\Lambda }^3=2.86712611643102\times {10}^{-4}$
Two-sided significance conversion	Gaussian tail at $Z_{\Pi }=1.556$	$p=0.1197081$, consistent with the quoted $0.120$
Bazavov correlation subblock	direct construction ${\Sigma }_B=diag\left({\sigma }_B\right){\rho }_{B}diag\left({\sigma }_B\right)$ and Jacobian propagation	$J_B{\Sigma }_B{J}_B^T=6.55160\times {10}^{-5}$ and ${\sigma }_{{\Delta }_{\Pi },B}=0.00809420$; this is one source block, not the full uncertainty
Sign-independent susceptibility minimum	positivity plus $x-z=D_{\mathrm{rt}}$	$TrM\ge |D_{\mathrm{rt}}|$ with unique $M_{min}=diag(D_{\mathrm{rt}}^{(+)},D_{\mathrm{rt}}^{(-)})$
Two-pole finite-moment example	exact rational Hankel algebra	generalized eigenvalues $4/5$ and $6/5$; ${\theta }_1=9/10$ does not certify, whereas ${\theta }_2=6/5$ certifies one subthreshold state and $S_2=0$ proves exact closure
Route-resolved finite-dimensional construction	exact $2\times 2$ matrix moments and canonical whitening	$spec\mathsf{T}=\{4/5,6/5\}$, $n_{+}(\mathsf{T}-I)=1$, and the unique subthreshold eigendirection is $e_{-}$; taking the trace removes the parity information
Route step-scaling map	exact linear-response composition	for the same microscopic perturbation, $\mathbf{u}\left(sL\right)=\mathcal{R}\left(sL\right)\mathcal{R}{\left(L\right)}^{-1}\mathbf{u}\left(L\right)$; at a one-channel fixed point ${\theta }_{-}=-ln\left|{\Sigma }_{--}\right|/lns$
Sharp finite-difference moment counterexample	binomial finite-difference identity with positive $2\times 2$ residues	route-breaking ${\sigma }_z$ weight can vanish in moments $0,\dots ,2N-1$ and first appear in $M_{2N}$, proving sharpness of the $2N+1$ count
Earliest regular flat closure	direct $2\times 2$ block-Schur algebra	$S_1=M_2-M_1{M}_0^{-1}{M}_1=0$ plus $[M_0,U]=[M_1,U]=0$ fixes all $M_n=M_0^{1/2}{T}_1^n{M}_0^{1/2}$ and gives exact regular reciprocity
Jacobi–Robin formulas	exact moment and band-edge algebra	$c_J=\langle s\rangle -(s_{+}+3s_{-})/4$ and $c_{★}=c_J/\sqrt{{\beta }_{\infty }}$
No-turn Planck and confinement norms	exact integration	$I_{\mathrm{Pl}}/R^2=1/\left(2\kappa \right)$ and the quoted penetration integrals at $\lambda =0.5,1$ are reproduced
Frozen-KK phase integral	exact integration before gauge reduction	the coefficient proportional to $(5\kappa +1)/\left[\right(\kappa +1\left)\right(\kappa +2\left)\right]$ is correct; its interpretation as an independent scalar is not
$R+\beta R^2$ scalaron	exact trace linearization	$m_s^2/c_6^2=1/\left(5b_{R^2}\right)-6$ and ${\nu }_s^2=1/4+1/\left(5b_{R^2}\right)$
Axisymmetric scalar/radion potential	exact symbolic differentiation	the residual of ${\mathcal{V}}_5-{\left[\right|\nabla \mathcal{W}|}^2/2-{\mathcal{W}}^2/3+{\mathcal{P}}^2]$ is identically zero
Axisymmetric flow and positive block	exact symbolic differentiation	first-order flow and $\nabla \mathcal{W}·\nabla \mathcal{P}=0$ hold; the added scalar block has rank one and positive eigenvalue
Axisymmetric zero-mode core test	explicit Frobenius reduction	indices $p=-3/2,1/2$; neither nontrivial branch satisfies smooth-cap scalar regularity
Dipole scalar-clock map	exact harmonic decomposition and core-domain check	$\delta F=(u+v)cos\theta$ and $F\delta \chi =(v-u)sin\theta$ are generated by the displayed smooth transverse vector field
Dipole canonical Ward completion	independent numerical reconstruction of $A_W=P_{T}A{P}_T$	Ward residual $1.44\times {10}^{-12}$, rank-two correction, reconstruction residual $1.27\times {10}^{-11}$, and nonnegative projected benchmark spectrum with one translation zero mode
Finite-interface Israel–Maxwell matching	direct evaluation of the exact boundary formulas	$C_{★}=-1.83747565890127$ for the stated unit benchmark; Maxwell, Israel, and $I\left(r_b\right)$ residuals vanish to floating-point precision
Interface equation-of-state inversion	exact algebra and numerical substitution	$d{x}_b/d{t}_b<0$ and the reconstructed $t_b=0.08486303817337079$ equals ${tanh}^2\left(0.3\right)$
Interface flux dominance	exact symbolic subtraction	$Q_b-{\rho }_b=8\kappa tanh{u}_b/\left({\kappa }_6^{2}R\right)>0$; the benchmark gives $Q_b/{\rho }_b=1.51389487389543$
Scalar interface closure	direct boundary variation and numerical substitution	${\lambda }_b=-{\varphi }_b^{\prime }/{\rho }_b=-0.411160278027790$ and ${\varphi }_b^{\prime }+{\lambda }_b{\rho }_b=0$
Canonical thin-wall ceiling	exact first-integral reduction	${\rho }_{\mathrm{can}}-Q_{\mathrm{can}}=2\int Vdy\ge 0$; the quartic example gives $Q_{\mathrm{can}}/{\rho }_{\mathrm{can}}=2/3$
Seven-dimensional spinor count	Clifford-dimension decomposition	$8_{\mathbb{C}}\to 2$ selected four-dimensional Weyl spinors per gauge component, hence $405\to 810$

The nonlinear Einstein–Abelian–Higgs shooting solution, its finite-radius Jacobians, the $\pm 2\%$ continuation scan, and the vortex–stabilizer coexistence window are numerical boundary-value results rather than consequences of the closed formulas in the table. Their quoted values are internally consistent with the displayed constraints, with one precision qualification: the rounded values ${\alpha }_6=1.16$ and ${\nu }_6=2.5801288700$ reproduce the listed $B_P$ only up to $4.30\times {10}^{-9}$. An independent numerical reproduction requires the complete unrounded parameter set, boundary conditions, and numerical tolerances. The quoted values are therefore retained as numerical results and are not promoted to analytic theorems.

Appendix J.10. Practical Numerical Procedure for a First-Principles Calculation

This appendix gives a minimal sequence of steps that would close the four-dimensional criterion without mixing bare, renormalized, projected, and physical quantities.

Appendix J.42.1. Step 1: Regularization and Definition of the Chiral Gauge Theory

A concrete regulator for the chiral $\mathrm{SU}{\left(15\right)}_p$ theory is required, and it must reproduce the published anomaly-free fermion spectrum in the continuum limit. Modern constructions of chiral gauge theories based on boundaries between topological phases demonstrate that such formulations are possible in principle [39,40], but a regulator specialized to the present model still has to be defined.

Appendix J.42.2. Step 2: Renormalized Operator Basis

One must determine the renormalization matrix ${\mathcal{Z}}_{ij}$ for all local and point-split operators with the same quantum numbers,

$$J_i^{\mathrm{ren}}={\mathcal{Z}}_{ij}{J}_j^{\mathrm{bare}}.$$

For the surviving emergent-reciprocity branch no microscopic intertwiner is assumed. The route doublet must first be renormalized and canonically normalized, after which the diagnostic route-label exchange is used only to decompose the response into $K_{\Vert }$ and $K_{\perp }$. If an additional exact symmetry completion were proposed, its bare and renormalized representations would have to intertwine with the mixing matrix, $\mathcal{Z}{U}_{\mathrm{bare}}=U_{\mathrm{ren}}\mathcal{Z}$; the often quoted commutator $[\mathcal{Z},{\sigma }_x]=0$ is only a special case in which the same literal swap has been independently justified in both bases. For emergent reciprocity, the amount of route-odd response must instead be measured and extrapolated toward the appropriate infrared limit.

Appendix J.42.42.1. Route-resolved step scaling before the spectroscopy fit.

Introduce the crossed deformation doublet ${\mathcal{Q}}_{\pm }$ and two finite-volume response probes with nonsingular matrix $\mathcal{R}\left(L\right)$. On matched ensembles at L and $sL$, construct

$${\Sigma }_{\mathrm{rt}}(s,L)=\mathcal{R}\left(sL\right)\mathcal{R}{\left(L\right)}^{-1}.$$

The first smooth-emergence test is ${\left({\Sigma }_{\mathrm{rt}}\right)}_{-+}=0$ in manifold-adapted coordinates. Only after that test passes should a transverse exponent be quoted; in the one-real-channel fixed-point limit, ${\theta }_{-}=-ln\left|{\Sigma }_{--}\right|/lns$. This calculation is logically independent of the later pole fit.

Appendix J.42.3. Step 3: Unprojected Two-Point Matrix and the Canonical Route Metric

A nonperturbative ensemble must be generated and the connected matrix correlator evaluated,

$$C^{AB}\left(t\right)=\sum _{\mathbf{x}}{\langle J_A(t,\mathbf{x})J_B^{†}\left(0\right)\rangle }_c,A,B=1,2.$$

Before interpreting an inverse kernel, the positive source-overlap metric G must be determined and canonically normalized. The two route sources are defined independently of the mass-plane fit; after whitening, introduce the diagnostic label exchange

$$U_{\mathrm{lab}}={\sigma }_x,K_{\perp }={\displaystyle \frac{1}{2}}\left(K-U_{\mathrm{lab}}K{U}_{\mathrm{lab}}\right).$$

This $U_{\mathrm{lab}}$ is a bookkeeping operation on the canonically normalized two-route response, not a microscopic Ward transformation; the strict closure-preserving microscopic exchange has already been excluded. The physical quantities to measure are the route-odd response $K_{\perp }$, the parity-resolved spectrum, and their controlled limits. If some genuinely new exact symmetry completion is proposed, it must independently satisfy the metric and fixed-covector conditions $UG{U}^{†}=G$, $U^2=I$, and $U^T{c}_{\Pi }=c_{\Pi }$; constructing such a U after choosing $c_{\Pi }$ is the tautological reflection discussed in the main text.

Appendix J.42.4. Step 4: The Matrix Susceptibility M(E) as the Shortest Reciprocity Test

From the connected momentum-space matrix function, its resolvent, or an equivalent variational self-energy construction, one must extract the renormalized shift of the inverse kernel in the same canonical basis. The inverse of $C\left(t_0\right)$ at a single Euclidean time must not be identified with a one-particle-irreducible kernel. The required object is the energy-dependent matrix susceptibility

$$M\left(E\right)={\displaystyle \frac{1}{h}}\mathcal{V}{({\mathcal{H}}_H-E)}^{-1}{\mathcal{V}}^{†},$$

or a directly reconstructed equivalent.

The principal scalar test is

$${\chi }_z\left(E\right)=M_{11}\left(E\right)-M_{22}\left(E\right)\stackrel{?}{=}{D}_{\mathrm{rt}}=0.049406060606061.$$

If the canonical basis is not manifestly real, one also tests

$$\Im M_{12}\left(E\right)\stackrel{?}{=}0.$$

Then define

$${\chi }_x\left(E\right)=\Re M_{12}\left(E\right),f\left(E\right)=C_{\mathrm{rt}}-{\chi }_x\left(E\right).$$

The test must be repeated across the physical matching window rather than at one isolated energy. The derivative ${\chi }_z^{\prime }\left(E_{*}\right)$ should also be estimated. On-shell locking requires only ${\chi }_z\left(E_{*}\right)=D_{\mathrm{rt}}$; the stronger condition ${\chi }_z^{\prime }\left(E_{*}\right)\simeq 0$ measures quadratic stability in energy. The exact four-state completion proposition proves that a nondegenerate route-odd ground state is structurally realizable for positive gaps and $C_{\mathrm{rt}}>0$, but only the physical correlator can determine whether the actual $\mathrm{SU}{\left(15\right)}_p$ dynamics realizes that branch.

Appendix J.42.5. Step 5: Physical Sign Inheritance and the Threshold

Kernel splitting alone is insufficient. Whenever possible, the spectral energies of both route-parity channels should be extracted directly and the measured sign of f compared with the physical inequality

$$E_{-}<E_{+}.$$

This is the sign-inheritance test.

The threshold $E_{\mathrm{th}}$ must not be replaced by a sum of constituent masses without an analysis of interactions. It should be determined from the lowest multi-particle or multi-composite state with the same quantum numbers in the same finite volume. The bound-state criterion is

$$E_{-}<E_{\mathrm{th}}.$$

Appendix J.42.6. Step 6: Three-Point Function

One must compute

$$G_{\pm }^{\left(3\right)}(T,t)=\sum _{\mathbf{x},\mathbf{y}}\langle 0|J_{-}(T,\mathbf{y}){\mathcal{O}}_{\pm }(t,\mathbf{x})J_{-}^{†}\left(0\right){|0\rangle }_c.$$

After plateau isolation, subtraction of operator mixing, and amputation of external legs, the form factors $F_{\pm }\left(0\right)$ are obtained. Normalized ratios should be used where possible to cancel source-overlap factors. The required result in the low-energy reciprocity/matching limit is

$$F_{+}\left(0\right)\ne 0,F_{-}\left(0\right)=0.$$

Appendix J.42.7. Step 7: Combined Criterion

The final classification should be made only after regulator control, finite-volume analysis, and continuum extrapolation. In the canonically normalized route basis the primary dynamical test is

$$K_{\perp }\left(E_{*}\right)\to 0\mathrm{for}\mathrm{exact}\mathrm{on}-\mathrm{shell}\mathrm{reciprocity},$$

whereas gap-protected approximate reciprocity is tested by

$${\eta }_{15}={\displaystyle \frac{\parallel K_{\perp }\parallel }{{\Delta }_{-}^{\left(15\right)}}}\ll 1.$$

If the Schur–Feshbach representation is used, ${\chi }_z\left(E_{*}\right)=D_{\mathrm{rt}}^{\mathrm{ren}}(E_{*};\mu )$ is the corresponding completion-level locking condition, with $\Im M_{12}=0$ in a real canonical convention. The benchmark number $D_{\mathrm{rt}}^{\mathrm{bench}}=0.049406060606061$ is used only in declared illustrative completions. The stronger isolated-pole matching branch additionally requires

$$E_{-}<E_{\mathrm{th}},F_{+}\left(0\right)\ne 0,F_{-}\left(0\right)=0.$$

No microscopic Ward involution is part of this surviving criterion. The exclusion conditions are checked separately:

• if the renormalized route-odd kernel component remains nonzero beyond the declared physical tolerance, or ${\eta }_{15}=O\left(1\right)$ with no independent spectral-purification mechanism, the gap-protected emergent-reciprocity branch fails;
• if a Schur–Feshbach completion is used and ${\chi }_z-D_{\mathrm{rt}}^{\mathrm{ren}}$ remains parametrically nonzero, that particular route-locking realization fails;
• if $E_{-}\ge E_{\mathrm{th}}$, the isolated-pole branch is excluded by the bound-state criterion;
• if $F_{+}=0$, the mass-plane matching branch is not realized;
• if $F_{-}\ne 0$ in the low-energy reciprocity/matching limit, the required route selection rule is not realized.

If locking holds but the measured pole ordering is opposite to the route-odd ground-state branch admitted by the exact completion theorem, then the realization selected by the unchanged theory is excluded by that physical phase criterion even though the abstract completion-level existence theorem remains mathematically valid.