Renormalized Route Reciprocity and Charged-Fermion Mass Shape in a Chiral SU(15)_p Composite Theory

1. Introduction: From the Operator Identity to a Physical Mass Test

The charged-fermion spectrum is sufficiently hierarchical that compact empirical formulas can easily become parameterizations rather than explanations. The present work therefore starts from a deliberately restrictive question. A single logarithmic second-difference relation among the charged-lepton, down-quark, and up-quark Yukawa spectra is treated as a weak observational clue, and the aim is to determine whether the same relation can be expressed as a precise, independently testable statement about composite operators and confining dynamics. Composite models of quarks and leptons have a long history [1,2]; modern chiral and renormalizable constructions make the framework concrete enough that anomaly cancellation, confinement, proton stability, and flavour observables can be addressed within one microscopic setup [3,4,5,6].

The microscopic theory used here is the chiral $\mathrm{SU}{\left(15\right)}_p$ compositeness model with the Pati–Salam spectator group [7]

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

Confinement is understood in the gauge-invariant Wilson sense, while anomaly matching is imposed as a consistency condition rather than used as a substitute for determining the infrared phase [8,9]. The principal result is four dimensional. Its primary physical observable is the matched chirality-resolved two-point function from which the running mass shape is obtained directly. The route-resolved correlator and its Feynman–Hellmann or three-point response remain valuable as an independent reciprocity and matching certificate, but they are not prerequisites for defining the mass-only target. A short final section separates the effective five-dimensional spectral representation and the incomplete six- and seven-dimensional constructions from this four-dimensional result.

An earlier public version of portions of this analysis appeared as Ref. [10], with a broader emphasis on the route construction and higher-dimensional extensions. The present manuscript is substantially reorganized around the four-dimensional renormalized problem. It derives the local rank obstruction, the positive-metric channel involution, the conditional theorem for the complete renormalized sector, the mass-only two-point estimator, the dual coefficient-matching rule, and the exact neutral-sector loss of predictivity. A separate rank-and-scale study with Lytvynenko made explicit an integer family of logarithmic curvature directions whose members satisfy the same plane equation [11]. Because that family was identified and ranked using the same charged-mass data, it belongs to the phenomenological provenance of the plane and is not an independent statistical confirmation. No fit statistic from that study is counted as additional evidence for ${\Delta }_{\Pi }=0$. The argument proceeds from the microscopic model and operator map to the admissible channel symmetry, its renormalized theorem, the state and measure, and finally the physical two-point observable.

Five layers must remain distinct throughout the analysis.

1.

Phenomenological target. The quantity ${\Delta }_{\Pi }$ is a fixed linear combination of three logarithmic Yukawa curvatures. With the current common-scale inputs and covariance reconstruction, its departure from zero is only about $1.56\sigma$.

2.

Operator identity. In the specified Pati–Salam route basis the same quantity satisfies the exact relation ${\Delta }_{\Pi }=C_{21}-C_{12}$, and both crossed source representatives exist within the unchanged chiral field content.

3.

Family alignment. The charged operator must select the basis-covariant family tensor $D_{+}$, equivalently the median eigendirection relative to the two outer eigendirections. This is an independent dynamical requirement, not a consequence of route exchange.

4.

Composite route dynamics. Hidden states, positive spectral measures, route-resolved transfer kernels, and the physical vacuum determine whether the route-breaking component is absent or sufficiently suppressed.

5.

Low-energy matching. Primarily, the composite two-point blocks must be matched to one running-mass scheme with additive contacts, heavy regular terms, condensate corrections, and thresholds controlled. Independently, the route-resolved formulation tests a nonzero even form factor, a vanishing odd form factor, and a vanishing regular odd remainder in one parity decomposition.

A result at one level is never used below to stand in for a missing result at another.

Three statements with different logical status.The notation makes three statements look deceptively similar:

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

The first is a weak empirical proximity statement. The second is an exact operator identity in the fixed route normalization. The third is a dynamical reciprocity condition that the confining theory would have to realize if the plane were exact. Keeping these statements separate is essential: the measured closeness to the plane is not itself evidence that the underlying route amplitudes obey an exact symmetry.

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

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

A microscopic explanation of the exact plane requires three independent physical requirements,

$$\begin{array}{c}\left[\mathrm{median}\mathrm{family}\mathrm{alignment}{D}_{+}\right]\wedge \left[\mathrm{selected}\mathrm{chiral}/\mathrm{scalar}\mathrm{state}\right]\\ \wedge \left[\mathrm{matched}{G}_2\mathrm{mass}\mathrm{shape}\mathrm{or}\mathrm{an}\mathrm{equivalent}\mathrm{complete}\mathrm{route}-\mathrm{resolved}\mathrm{test}\right].\end{array}$$

For the alternative route-resolved test, let $F_{+}\left(0\right)$ and $F_{-}\left(0\right)$ denote the zero-momentum amputated form factors of the even and odd tracked-pole operators, and let $C_{-}^{\mathrm{reg}}$ denote the odd regular remainder after the same pole subtraction. The last bracket then contains $F_{+}\left(0\right)\ne 0$, $F_{-}\left(0\right)=0$, and $C_{-}^{\mathrm{reg}}=0$. Every requirement is necessary in the formulation in which it is used. None is supplied by the other two, and all must be evaluated within one renormalized physical branch.

Route-symmetry chain. The route sector can now be organized as the following sequence of statements:

$$\begin{array}{c}\mathrm{operator}\mathrm{identity}⟹\mathrm{microscopic}\mathrm{local}\mathrm{no}-\mathrm{go}\\ ⟹\mathrm{exact}\mathrm{channel}-\mathrm{space}{\mathbb{Z}}_2⟹\mathrm{positive}\mathrm{invariant}\mathrm{metric}\\ ⟹\mathrm{globally}\mathrm{stable}\mathrm{symmetric}\mathrm{control}\mathrm{vacuum}\\ ⟹\mathrm{conditional}\mathrm{renormalized}\mathrm{reciprocity}\mathrm{theorem}\\ ⟹\mathrm{matched}\mathrm{physical}{G}_2\mathrm{mass}-\mathrm{shape}\mathrm{calculation}.\end{array}$$

The no-go result and the channel-space involution are exact algebraic statements for the specified source and Pati–Salam channel construction. Positivity is also exact within that construction. Global vacuum selection is demonstrated constructively in an explicit nonlinear control action; it proves compatibility and existence, not that the physical $\mathrm{SU}{\left(15\right)}_p$ vacuum has the same effective coefficients. The last step therefore remains indispensable. The primary calculation extracts the singular-value shape from the matched blocks $F_L,F_R,J$. The route correlator, finite-time test, and conditions $F_{+}\left(0\right)\ne 0$, $F_{-}\left(0\right)=0$, and $C_{-}^{\mathrm{reg}}=0$ provide a stronger independent diagnostic of how reciprocity is realized, rather than a second definition of the charged mass observable.

Several operator-algebra questions are nevertheless closed before any strong-coupling simulation is attempted. The two crossed route tensors arise from fixed recoupling words within the unchanged field content. The effective $-1/2$ color coefficient is fixed by the walled–Brauer projector algebra. At the same time, the stronger idea that the two complete kernels are literally one microscopic history traversed in opposite order fails in the explicitly defined closure-preserving class. This negative result is useful rather than destructive: it rules out an overly rigid microscopic interpretation and isolates an emergent equality of renormalized composite amplitudes as the physically relevant alternative.

The family problem is deliberately kept orthogonal to the route problem. The symmetric precolor spurion determines the median family projector algebraically, while a gauge-neutral ${\overline{\mathbf{6}}}_F$ composite potential provides a global effective-field-theory branch that can select the required family direction. A distinct four-flavour parent embedding supplies a restricted $U{\left(1\right)}_X\times SU{\left(4\right)}_F$ spurion branch in which the parent cubic invariant that would descend to the light-family determinant term is forbidden to all polynomial orders. This selection rule is conditional on that embedding and is not promoted to a symmetry theorem of the minimal three-$\psi$ Pati–Salam theory. More general light-family EFT branches permit a nonzero determinant interaction but still contain a finite globally stable parameter domain. Determining which branch and which renormalized scalar coefficients are generated remains a problem for the physical confining theory. Likewise, route reciprocity by itself does not guarantee that the even composite channel has a nonzero low-energy coupling to the charged-fermion operator.

The assumptions, definitions, and derivations used by the four-dimensional theorems are given in the text. Auxiliary finite-dimensional and numerical controls are identified explicitly and are not used as substitutes for the uncomputed physical $3+1$ dimensional observable. The main text uses higher-dimensional constructions only where they sharpen a consistency statement; no four-dimensional conclusion depends on treating an extra dimension as fundamental.

1.1. Core Results and Remaining Physical Inputs

The principal claims are summarized before the detailed derivations. The status labels are intentionally conservative: exact algebraic statements are separated from constructive controls, conditional EFT statements, and quantities that still require the physical confining theory.

Item	Status	Meaning
${\Delta }_{\Pi }=C_{21}-C_{12}$	established	Exact four-dimensional coordinate/operator identity in the fixed route normalization.
Conditional renormalized reciprocity theorem	established under three explicit hypotheses	An unbroken linear involution of the complete renormalized source sector, an invariant selected state and external input, and equivariant complete matching imply $C_{12}=C_{21}$ exactly. The physical $\mathrm{SU}{\left(15\right)}_p$ realization of these hypotheses is open.
Dual matching and null transport	established algebraically; physical bridge open	Coefficients obey ${\mathbf{a}}_R=Z^{-T}{\mathbf{a}}_B$. The target projector $\mathbf{q}={(1,-2,1,0)}^T/4$ differs from the direct deformation ${\mathbf{n}}_{\mathrm{dir}}={(0,-1,2,1)}^T$. A one-sided RG zero is sufficient only in a complete closed homogeneous response whose odd coordinate is a nonzero normalization of ${\Delta }_{\Pi }$.
Neutral-sector no-prediction and non-closure	established	The charged route map has rank three and kernel $span(1,1,1,1)$, while that kernel changes $A_{{\nu }_D}$ and equals the independent mixed neutral deformation $-{𝒪}_X$. Charged data alone therefore do not bound the neutrino-Dirac curvature or establish neutral operator closure, even after imposing crossed-route reciprocity.
Mass-only route observable	established algebraically; physical matching open	The target depends only on running mass singular values through ${\rho }_s=m_{s,1}{m}_{s,3}/m_{s,2}^2$. A noncommuting matrix theorem constructs the spectrum from the left-normal block $F_{L,s}$, right-normal block $F_{R,s}$, and mixed block $J_s$ of the matched two-point function. Contacts, regular terms, thresholds, and scheme conversion remain to be fixed.
Chiral mass-channel Ward theorem	established	A global $U{\left(1\right)}_{\chi }$ free of the relevant $\mathrm{SU}{\left(15\right)}_p^2-U{\left(1\right)}_{\chi }$ ABJ anomaly forces the mixed block $J_s$ to vanish in an invariant state. In an unfixed precolor ensemble the raw block is a projection of the connected six-preon cumulant; a separate elementary $\langle \Omega \Omega \rangle$ pairing is not generally required.
Renormalizable scalar/flavour action and mass-shape closure	established algebraically; dynamical coefficients open	The elementary scalar potential has two real mass coefficients and eight real quartic degrees of freedom. The generic mass tensor has nine complex directions; the single-symmetric-spurion branch has three. The target itself depends only on one centered median logarithm per sector, but one unfixed real relative coefficient already changes ${\Delta }_{\Pi }$ in an allowed matching class.
Correct chiral measure	partial result	The determinant-line phase is fixed on the controlled core, but the determinant modulus in the full scalar/Pati–Salam background and the selected physical vacuum are not computed. Gauge-only or valence-weighted ensembles are therefore not the physical measure.
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.
Literal closure-preserving chronological $Z_2$	insufficient in the restricted class	Under the stated closure assumptions the two complete crossed kernels are not the same microscopic process executed in reverse order. The nonlocal source and dynamical reciprocity branches remain available and are tested independently.
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 global-symmetry branch conditionally forbids the cubic invariant that would induce the light-family determinant coupling.
Local microscopic route exchange	excluded under the stated assumptions	Inner $\mathrm{SU}{\left(15\right)}_p$ transformations are trivial on the complete precolor singlet nodes, while complex conjugation does not preserve the chiral field set. For rank-one Pati–Salam breaking the two local crossed tensors have ranks 4 and 8, excluding every invertible local exchange.
Channel-space route reflection	established algebraically; physical realization open	An exact $3\times 3$ channel involution exchanges the two visible route images, preserves an explicit positive metric, and admits positive spectral measures with distinct even and odd energies. The physical $3+1$ dimensional spectral weights are not yet calculated.
Visible route-symmetry condition	established	In the parity basis, exchange symmetry of the physical two-route correlator is equivalent to vanishing of its mixed even–odd element. Full three-channel reflection is stronger because it also constrains the invisible–odd block.
Global symmetric-vacuum control	established constructively	An interacting two-node nonlinear action is globally strictly convex and has a unique symmetric vacuum. A deliberately symmetry-breaking control proves that vacuum location plus positive odd curvature does not by itself make the mixed even–odd correlator vanish.
Relaxed odd curvature	exact under the stated positivity assumptions	With $\chi =Cov\left(X\right)$ nonsingular and $K={\chi }^{-1}$, the curvature after all even/invisible coordinates relax is $K_{\mathrm{rel},--}={\left({\chi }_{--}\right)}^{-1}$. This removes a separate curvature inversion but proves neither odd stationarity nor correlator reciprocity.
Finite-time closure cost	established in the stated controls	Three slices are sufficient only when the first flat-extension residual vanishes. The minimal active two-node control closes only at the next order and therefore requires five consecutive Euclidean separations.
Physical route-resolved test	unresolved numerically	The full $3+1$ dimensional theory must determine the route-odd one-point force, the curvature after all even fields relax, and the mixed even–odd correlator after controlled volume, cutoff, and renormalization extrapolations.
Hidden-susceptibility and finite-moment criterion	established as a theorem	A positive hidden spectral sector admits a necessary-and-sufficient susceptibility test for inversion of the unmixed even/odd pole ordering, together with a directly computable sufficient finite-moment inequality. Physical four-dimensional moments are still required.
One-sided finite-volume route test	established as a theorem	A metric-completed determinant ratio measures the generated odd/even leakage. Its covariance ellipsoid gives a rigorous interval, and the multiscale recursion prevents cancellation between successive steps.
Physical initial-null and RG requirements	exact conditional transport theorem; physical inputs open	A single matched scalar is sufficient at the initial scale only after the target row and additive matching are fixed. In a closed homogeneous two-coordinate response, ${\left({\Sigma }_{\mathrm{rt}}\right)}_{-+}=0$ preserves the physical null; a generic m-coordinate basis instead requires $m-1$ tangent-to-target zero conditions. The physical action, determinant modulus, and odd-coordinate bridge are not yet available for a new inversion.
Route-21 channel reduction	established at source level	Fixed coefficients of the 03 and 153 irreducible channels reduce one side of the matching problem to one response ratio and one overall amplitude. The route-12 response and common physical normalization remain to be computed in the same scheme.
Late three-channel sign theorem	established in the declared Schur–Feshbach control	A positive rank-one hidden correction can generate the required negative shift only if more than one third of its norm lies in the designated channel. Support confined to the two complementary channels cannot produce the shift.
Exact RG/channel translation	established within the stated three-channel control family	Running and the distinguished channel coordinate combine into one effective coordinate for all eigenvalue differences in this control. This is not an identification of a physical renormalized coefficient with a Casimir proxy.
Pole-count-free hidden-spectrum certificate	established for the declared positive-measure control class	Four hidden moments and the portal response suffice for a one-sided spectral certificate; no fixed number of hidden poles is required.
Uniform same-pole certificate	established for the six-moment/support-floor control class	Six moments and a lower support floor bound the binding, visible residue, and even charged response uniformly away from zero. These are control-class theorems, not physical $\mathrm{SU}{\left(15\right)}_p$ data.
Bare eight-state $\mathrm{SU}\left(3\right)$ same-pole surrogate	response condition satisfied; ordering condition excluded	Its even and odd action-insertion selection rules are favorable, but its odd level is always above the lowest even level in the stated positive-parameter domain. It therefore cannot realize the required ground-state inversion.
Chiral regulator and physical measure	regulator defined on a controlled admissible domain; physical modulus open	Fundamental and matrix-free $\overline{\mathbf{120}}$ overlap operators have volume-independent gap bounds for ${ϵ}_0<1/60$; Kato transport fixes the determinant-line phase on the controlled core and the sector indices are explicit. The Weyl-determinant modulus in the complete scalar/Pati–Salam background, the selected state, universality, and the dynamical confining ensemble remain open.
Physical four-dimensional propagation and matching	open	The primary route requires the physical left-normal, right-normal, and chirality-changing two-point blocks and their complete source-to-running-mass matching. The independent route-resolved test additionally requires a tracked odd gap, the normalized exchange-breaking kernel, the odd force and relaxed curvature, the mixed correlator, and the pole/regular matching responses.
5D / 6D / 7D roles	effective / prospective / consistency boundary	Five dimensions provide spectral reconstruction, six dimensions are an optional ultraviolet completion with one sharply defined metric stability test, and the tested seven-dimensional embedding does not reproduce the desired zero-mode count.

1.2. Scope and Unresolved Physical Inputs

The missing step is now narrow enough to state without ambiguity. No renormalized first-principles ensemble with the physical chiral determinant modulus (or a demonstrably equivalent Pfaffian formulation) and an oriented scalar/flavour state has yet been generated for the required composite sources. Consequently the matched blocks $F_L,F_R,J$, and hence the physical mass-only value of ${\Delta }_{\Pi }$, are not predicted here. The route-resolved diagnostic quantities ${\Delta }_{-}^{\left(15\right)}$, $K_{\perp }$, ${\eta }_{15}$, the source-accessible pole residue, $J_{-}$, the physical odd susceptibility, $C_{+-}\left(t\right)$, and the absolute route form factors are likewise uncomputed. Under the positive-measure and Legendre assumptions of Proposition 3, the relaxed odd curvature then follows from the susceptibility rather than requiring a separate inversion.

Nor is a new step-scaling inversion yet justified. Before ${\left({\Sigma }_{\mathrm{rt}}\right)}_{-+}$ can be interpreted as preservation of the physical mass-plane null, the odd response coordinate must be proved to equal a nonzero normalization of ${\Delta }_{\Pi }$ in a complete closed basis. This bridge is part of the matching problem, not a consequence of a small bare route response.

This does not mean that the mechanism has only been discussed qualitatively. A hierarchy of confining, spectral, positivity, source-metric, finite-volume, and matching controls has already been solved under assumptions that can be stated and checked. These calculations test separate ingredients of the mechanism in settings where Gauss-law reduction, route ordering, hidden-state elimination, or volume dependence can be handled analytically or to high numerical precision. What remains is first the physical matched two-point mass response and, as an independent stronger check, its route-resolved correlator and response insertions.

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

The control calculations address different failure modes and should not be interpreted as repeated estimates of a single 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 ${ϵ}_{\mathrm{rt}}^{\mathrm{SU}\left(3\right)}=({\delta }_1-{\delta }_2)/(2{\Delta }_F+{\delta }_1+{\delta }_2)$, where ${\Delta }_F$ is the common flux gap and ${\delta }_i$ is the excess cost of route i.	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 the L-link chain. Here ${\kappa }_E$ is the electric-energy coefficient. For a normalized additive odd deformation x, ${\eta }_{N,L}^{\mathrm{ctrl}}\sim 2\left|x\right|/\left({\kappa }_{E}N\right)$.	The confinement-scale odd gap survives dynamical links and large volume in the actual fundamental–fundamental–conjugate-symmetric representation pattern relevant to $\mathrm{SU}{\left(15\right)}_p$.
Embedded $\mathrm{SU}\left(15\right)$ cross-junction and dynamical-link controls	For normalized direct and crossed singlet states $|D\rangle$ and $|X\rangle$, the color generator insertion obeys $\langle X|T_2·T_3|D\rangle =112/225$. In the exact $1+1$D link control the interaction scale is $J_{\mathrm{dyn}}(\ell )=g_2^2\ell$, with two-dimensional gauge coupling $g_2$ and link length ℓ.	The crossed matrix element is nonzero and of order unity in the representation appropriate to the model. Group theory fixes its color factor, while the physical four-dimensional Wilson-network coefficient remains dynamical.
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 result above is unsuppressed.	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 fixed 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, the controls establish more than algebraic consistency while stopping short of a four-dimensional prediction. Their role is to test the proposed ingredients and to exclude shortcuts when their assumptions fail. The remaining calculation is therefore ordered: specify the physical chiral/scalar measure and state, obtain the matched $F_L,F_R,J$ blocks, evaluate the singular-value estimator, and use route-resolved Feynman–Hellmann observables only as an independent symmetry and matching certificate.

2. Core Notation and Logical Status

The confining gauge group is $\mathrm{SU}{\left(15\right)}_p$, and the weakly gauged spectator group is $G_{\mathrm{PS}}=\mathrm{SU}{\left(4\right)}_{\mathrm{PS}}\times \mathrm{SU}{\left(2\right)}_L\times \mathrm{SU}{\left(2\right)}_R$. Charged sectors are labelled by $s\in \{\ell ,d,u\}$ and ordered generations by $g=1,2,3$. The running Yukawa eigenvalue $y_{s,g}\left(\mu \right)$ is always evaluated at a common scale $\mu$ in one stated scheme. Its logarithmic curvature is

$$A_s={\displaystyle \frac{1}{4}}ln{\displaystyle \frac{y_{s,1}{y}_{s,3}}{y_{s,2}^2}},{\Delta }_{\Pi }=A_{\ell }+2A_d-A_u,$$

and $\Pi$ denotes the plane ${\Delta }_{\Pi }=0$.

The labels $a,b\in \{1,2\}$ on $C_{ab}$ specify the order of two composite routes and never label generations. Thus $C_{12}$ and $C_{21}$ are low-energy route coefficients, whereas $C_{ij}^{AB}\left(t\right)$ denotes a Euclidean source correlator and is a different object. The elementary precolor scalars are denoted throughout by

$$A\in \overline{\mathbf{105}},A^T=-A,S\in \overline{\mathbf{120}},S^T=S;$$

their flavour matrices are respectively ${\lambda }^T=-\lambda$ and ${\left({\lambda }^{\prime }\right)}^T={\lambda }^{\prime }$. A prime on any other symbol has its ordinary local meaning and does not denote the scalar S.

Status words are used literally. Established means proved from the displayed assumptions; numerically verified means checked in the stated finite calculation; conditional means that a separately identified physical input is assumed; open means that the present inputs do not determine the quantity. A control tests a mechanism and is not a prediction for the physical four-dimensional theory. All other symbols are defined at first use.

3. Phenomenological Motivation: Logarithmic Curvature of the Mass Spectrum

Two logically independent ingredients enter the mass-plane discussion. The charged-sector combination ${\Delta }_{\Pi }=A_{\ell }+2A_d-A_u$ was identified a posteriori in an earlier public version of this work [10]. A separate rank-and-scale study with Lytvynenko subsequently organized candidate logarithmic curvatures into the integer family $A_{\ell }:A_d:A_u=(3n+2):-(n+1):n$ [11]. Every member of that family satisfies ${\Delta }_{\Pi }=0$. This exact algebraic fact clarifies the phenomenological provenance of the plane, but it is not independent empirical support: the family was itself identified and ranked using the same charged-mass data, and its discrete choice carries a look-elsewhere qualification. The role of the current article is instead to translate the fixed curvature combination into a route-sensitive operator observable and to ask whether the confining preon theory can realize the required reciprocity. The operator argument does not require a literal shell-spectrum interpretation.

3.1. 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.

3.2. Phenomenological Provenance and Relation to Rank-And-Scale Tests

The curvature plane did not emerge from a blind prediction of the present preon theory. It was recognized a posteriori in the charged-mass analysis of Ref. [10]. In the exact coordinates introduced above, the later rank-and-scale analysis [11] organized a one-parameter integer family of curvature directions,

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

where n is a positive integer and $\lambda$ is an overall curvature scale. The important result for the present paper is not the label of any particular phenomenological realization but the common two-dimensional span. Indeed,

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

Both fixed directions are orthogonal to

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

so every member of the family satisfies

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

The shared normal ${\mathbf{n}}_{\Pi }=(1,2,-1)$ is therefore an algebraic property of the entire family, not a separate continuous parameter. The family and its normal were nevertheless recognized from the same charged-mass data that motivated them, so the look-elsewhere qualification must be retained. The rank-and-scale study also shows how exact left-null relations can be tested by conditional and leave-one-out reconstruction without treating every mass as an independently adjustable output. Its integer family contains the present plane relation exactly, but its numerical fit quality is not counted as an independent confirmation of ${\Delta }_{\Pi }=0$.

The logical status is two-stage. Phenomenologically,

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

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 independently fixes the same normal inside the specified route basis. It removes any freedom to rotate the plane after the microscopic basis is chosen, but it does not erase the data-driven origin of the first statement. Accordingly, this article treats the plane as a weak empirical clue and the route identity as the exact theoretical result.

3.3. 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 $M_Z$ [12],

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

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

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. The displayed numbers make the phenomenological observable directly reproducible.

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.

3.4. 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. Any standardized $Z_B$ 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.

3.5. 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. [13]. 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	$0.0179200$	$1.058$
Two-loop ${\alpha }_s$-only covariance approximation	$0.01403$	$1.35$
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

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

An estimate near $2.23\sigma$ would require a near-maximal residual d–s correlation that is 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.

3.6. 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 selected chiral/scalar state, family alignment, the physical determinant weight (or a demonstrably equivalent Pfaffian formulation), and complete matching of the two-point mass blocks; the route correlator and response vertex provide an independent stronger certificate.

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 specifies which combination is to be tested; 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.

3.7. 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 recalculate 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.

3.8. 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 [14] 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.

3.9. 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 $\kappa \simeq 0.17$ [6]. 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. [6] 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. [6] 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 necessary consistency condition 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 $C_{12}=C_{21}$.

3.10. Why Right-Handed-Isospin Symmetry Alone Is Insufficient

Ref. [6] 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 3, 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. [6] that realistic masses and CKM mixing require additional spontaneous $\mathrm{SU}{\left(2\right)}_I$ breaking.

Scope of 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.

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

4.1. Preon Fields

For the Pati–Salam branch the analysis uses the published chiral fermion content of Ref. [5]:

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

(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.

4.2. 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 $N+4$ [15]; 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.

4.3. Two Scalar Flavour Spurions and the 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 [5]. A later dedicated flavour analysis of the same $\mathrm{SU}{\left(15\right)}_p$ compositeness framework showed that the second Yukawa structure requires a conjugate symmetric tensor [6]. Applying that representation-theory result to the analogous Pati–Salam singlet-preon Yukawa contraction gives the consistent two-spurion choice

$$A\sim (\overline{\mathbf{105}};\mathbf{1},\mathbf{1},\mathbf{1}),S\sim (\overline{\mathbf{120}};\mathbf{1},\mathbf{1},\mathbf{1}),$$

with an antisymmetric and a symmetric preon-flavour spurion, respectively. The detailed four-singlet $\mathrm{SU}{\left(4\right)}_F$ benchmark of Ref. [6] 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. This tensor-contraction argument 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

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

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

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

This short derivation fixes the representation 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.

Scope of the scalar completion. The scalar-mediated flavour completion uses the $\overline{\mathbf{105}}/\overline{\mathbf{120}}$ assignment made explicit in Ref. [6]. Scalars do not contribute to the cubic chiral gauge anomaly. The representation choice therefore affects scalar Dynkin-index and potential calculations but does not modify the chiral fermion content or the point-split route-source theorem.

4.4. 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 4.3 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.

4.5. 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.

4.6. 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.

5. Microscopic Operator Construction and the Exact Charged Target

5.1. Exact Operator Meaning of the Curvature Plane

The Pati–Salam route construction contains four operator images $r_{ab}\in {\mathbb{R}}^3$, where $a,b\in \{1,2\}$ record the ordered internal operations and the three components are the charged curvature sectors $(\ell ,d,u)$. The coefficient $C_{ab}$ multiplies $r_{ab}$ in the low-energy charged operator. 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

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

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 phenomenological provenance recorded in Ref. [10]: the plane was identified from the charged-mass pattern, whereas the present operator derivation independently fixes the already identified direction. The identity is exact but carries no statistical significance by itself.

5.2. What the Microscopic Source Construction Establishes

The immediate goal is to show, using only the unchanged chiral fields, that the two crossed route orderings define genuinely distinct operator directions and therefore admit a nonzero route-odd combination. 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. The finite route images and the rank obstruction needed below are stated explicitly in Section 6 and Section 7.1; longer source-contraction derivations are retained in the separate technical record.

5.3. 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 whose route-12 and route-21 realizations are exact images of one another under the declared route reflection 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 the route-exchange-breaking kernel component, later denoted $K_{\perp }$, without asserting that the physical connected-matter contribution vanishes.

6. Exact Route Information and the Neutral-Sector Limitation

The phenomenological normal ${\mathbf{n}}_{\Pi }=(1,2,-1)$ acquires physical meaning only after the four microscopic route coefficients are projected onto the charged sectors. It is therefore useful to state the complete finite-dimensional algebra before discussing a particular vacuum or numerical regulator. In the ordered basis

$${\mathbf{A}}_4={(A_{\ell },A_d,A_u,A_{{\nu }_D})}^T,\mathbf{C}={(C_{11},C_{12},C_{21},C_{22})}^T,$$

the mass-curvature map is a real linear map,

$${\mathbf{A}}_4\in {\mathbb{R}}^4,\mathbf{C}\in {\mathbb{R}}^4.$$

(20)

This real space is the one used in the neutral-sector theorem below. Complex renormalized amplitudes are allowed later in the quantum reciprocity theorem; there the physical equality is imposed only after the complete matching and reality conditions appropriate to the mass observable have been specified. The four route images are

$$\begin{array}{cccc}\hfill r_{11}& ={(3,-1,1,-3)}^T,\hfill & \hfill r_{12}& ={(-3,1,0,0)}^T,\hfill \\ \hfill r_{21}& ={(-2,1,-1,2)}^T,\hfill & \hfill r_{22}& ={(2,-1,0,0)}^T,\hfill \end{array}$$

(21)

and

$${\mathbf{A}}_4=R_4\mathbf{C},R_4=\left(r_{11}{r}_{12}{r}_{21}{r}_{22}\right).$$

(22)

The normalization in Eq. (21) is fixed by the Pati–Salam operator basis used throughout the article; changing it without changing the route operators would change the statement being tested.

Projecting Eq. (22) onto the charged triplet gives

$$R_{\mathrm{ch}}=\left(\begin{array}{cccc}3& -3& -2& 2\\ -1& 1& 1& -1\\ 1& 0& -1& 0\end{array}\right),rank{R}_{\mathrm{ch}}=3,ker{R}_{\mathrm{ch}}=span\left\{{(1,1,1,1)}^T\right\}.$$

(23)

Acting with the fixed charged normal gives

$${\mathbf{n}}_{\Pi }^T{R}_{\mathrm{ch}}=(0,-1,1,0),{\Delta }_{\Pi }=C_{21}-C_{12}.$$

(24)

Equation (24) is an operator identity in the declared route basis. It is not obtained by fitting the charged masses, and it does not assume route reciprocity.

Theorem 1

(Charged-data no-prediction theorem for the neutral Dirac curvature). Within the four-route map of Eq. (22), charged-sector data alone impose no proper interval, sign, or finite bound on $A_{{\nu }_D}$. This remains true after imposing $C_{12}=C_{21}$.

Proof. 

By Eq. (23), the shift

$$\mathbf{C}⟼\mathbf{C}+t{(1,1,1,1)}^T,t\in \mathbb{R},$$

(25)

leaves all three charged curvatures unchanged. The same direction is not invisible in the full quartet, because

$$r_{11}+r_{12}+r_{21}+r_{22}={(0,0,0,-1)}^T.$$

(26)

Thus Eq. (25) changes $A_{{\nu }_D}$ by $-t$. Since t is arbitrary, the fibre over any charged triplet in $im{R}_{\mathrm{ch}}$ covers the entire real line in the neutral coordinate. If reciprocity is imposed, write $C_{12}=C_{21}=C_{\times }$. Equations (21) then give

$$A_{{\nu }_D}=-3A_u-C_{\times },$$

(27)

but the common crossed coefficient $C_{\times }$ is not determined by the charged triplet. Hence reciprocity does not close the neutral sector. □

This theorem is deliberately a loss-of-predictivity result. A neutral prediction requires at least one additional datum or dynamical condition that is sensitive to the invisible direction. In particular, a mixed neutral operator allowed by the symmetries cannot be excluded merely because it is absent from the charged projection. The appropriate neutral-sector calculation must therefore establish its own operator closure and renormalized mixing conditions.

The invisible direction has a concrete operator meaning. Let $(B_0,B_R,B_L,B_X)$ denote the primitive neutral tensors, where $B_X$ is the mixed tensor absent from the charged projection. Their Gram matrix is

$$G_B=\left(\begin{array}{cccc}8& 4& 2& 1\\ 4& 4& 1& 1\\ 2& 1& 2& 1\\ 1& 1& 1& 1\end{array}\right),det{G}_B=9.$$

(28)

Thus $B_X$ is an independent primitive, not a redundant relabelling of the other three tensors.

Theorem 2

(Explicit neutral source-level non-closure). In the declared primitive basis the four route operators obey

$${𝒪}_{11}+{𝒪}_{12}+{𝒪}_{21}+{𝒪}_{22}=-{𝒪}_X.$$

(29)

Consequently the charged-invisible coefficient direction ${(1,1,1,1)}^T$ is the mixed neutral deformation $-{𝒪}_X$. The condition that this direction be absent is not implied by the charged operator identity or by the Lorentz, Pati–Salam, family, and Abelian quantum numbers used here. If ${\gamma }_c$ is the renormalized mixing matrix, $e_X={(0,0,0,1)}^T$, and $P_{\overline{X}}=diag(1,1,1,0)$, preservation of a bare closure condition requires the independent constraint

$$e_X^T{\gamma }_c{P}_{\overline{X}}=0.$$

(30)

Proof. 

Expanding the four fixed route tensors in the primitive basis and summing their coefficient rows gives $(0,0,0,-1)$, which is Eq. (29). Equation (28) shows that the resulting ${𝒪}_X$ direction cannot be removed by a linear dependence. Under renormalization, a coefficient initially confined to the complementary subspace remains there precisely when the row selecting X receives no contribution from that subspace; this is Eq. (30). □

This theorem strengthens, rather than weakens, the no-prediction result: a neutral analysis may suppress particular leakage channels through additional Pati–Salam selection rules, but it must still establish its own operator closure and mixing condition. Charged reciprocity cannot supply them.

7. Local Microscopic No-Go and the Exact Composite-Channel Involution

The exact route map identifies the combination that must vanish, but by itself it does not explain why the confining vacuum should select it. The following analysis first excludes a local microscopic exchange and then constructs the corresponding involution directly in the three-dimensional composite-channel space. It is essential to distinguish a local microscopic symmetry from an emergent symmetry of the mixed composite-channel response.

7.1. Exact Local No-Go in $\mathrm{SU}{\left(15\right)}_p\times G_{\mathrm{PS}}$

Each local $\Omega FF$ node contains a unique precolor-singlet contraction. Inner $\mathrm{SU}{\left(15\right)}_p$ gauge transformations therefore act trivially on a completed node and cannot exchange two different ways of joining already singlet nodes. The only nontrivial outer automorphism of $\mathfrak{su}\left(15\right)$ is complex conjugation. It sends $\mathbf{15}\leftrightarrow \overline{\mathbf{15}}$ and $\overline{\mathbf{120}}\leftrightarrow \mathbf{120}$, whereas the conjugate chiral field set is absent. Thus the required exchange is not an inner or outer automorphism of the unchanged chiral $\mathrm{SU}{\left(15\right)}_p$ theory.

The Pati–Salam obstruction is even more direct. Let the rank-one breaking direction be

$$\Phi =\sqrt{\sigma }u\otimes r,p=|u\rangle \langle u|,q=|r\rangle \langle r|.$$

After extracting the common factor $\sigma$, the two local crossed tensors on $End\left(\mathbf{4}\right)\otimes End\left(\mathbf{2}\right)$ are

$$\begin{array}{cc}\hfill {\displaystyle \frac{B_{12}}{\sigma }}& =I-I\otimes q-4p\otimes I+4p\otimes q,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill {\displaystyle \frac{B_{21}}{\sigma }}& =I-2I\otimes q-3p\otimes I+6p\otimes q.\hfill \end{array}$$

(32)

On the four joint eigenspaces of $(p,q)$ their eigenvalues are

p	q	multiplicity	$B_{12}/\sigma$	$B_{21}/\sigma$
0	0	3	1	1
0	1	3	0	$-1$
1	0	1	$-3$	$-2$
1	1	1	0	2

Hence

$$spec(B_{12}/\sigma )=\{1^{\left(3\right)},-3,0^{\left(4\right)}\},rank{B}_{12}=4,$$

$$spec(B_{21}/\sigma )=\{1^{\left(3\right)},{(-1)}^{\left(3\right)},-2,2\},rank{B}_{21}=8.$$

Rank is invariant under every invertible local transformation. Therefore no local invertible map, even one not required to commute with the unbroken spectator group, can exchange the two complete local tensors for this physical breaking direction.

Exact exclusion of a microscopic route symmetry. Within the unchanged chiral field set and the stated rank-one Pati–Salam breaking direction, route exchange is neither a gauge-group automorphism nor any invertible local transformation between $B_{12}$ and $B_{21}$. A successful reciprocity mechanism must therefore be nonlocal/composite and act on the mixed channel response after confinement and symmetry breaking. This no-go does not apply to that channel space.

7.2. Exact Lift of Route Exchange to the Three-Channel Space

Write the exact channel-to-route map and route exchange as

$$B=\left(\begin{array}{ccc}0& -4& 4\\ -2& 0& 6\end{array}\right),P={\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right).$$

(33)

Because B has full row rank, its Moore–Penrose right inverse and the orthogonal projector onto the route-invisible direction are

$$J=B^T{\left(B{B}^T\right)}^{-1}={\displaystyle \frac{1}{44}}\left(\begin{array}{cc}3& -4\\ -10& 6\\ 1& 6\end{array}\right),BJ=I_2,$$

(34)

$$Q=I_3-JB={\displaystyle \frac{1}{11}}\left(\begin{array}{ccc}9& 3& 3\\ 3& 1& 1\\ 3& 1& 1\end{array}\right),imQ=span\left\{k\right\},k={(3,1,1)}^T.$$

(35)

The matrix

$$S_{\mathrm{ch}}=JPB+Q={\displaystyle \frac{1}{22}}\left(\begin{array}{ccc}15& 14& 7\\ 16& -10& -16\\ 5& -10& 17\end{array}\right)$$

(36)

obeys, by direct multiplication,

$$S_{\mathrm{ch}}^2=I_3,B{S}_{\mathrm{ch}}=PB,spec{S}_{\mathrm{ch}}=\{+1,+1,-1\}.$$

(37)

Thus $S_{\mathrm{ch}}$ is an exact lift of the physical route swap to the three-channel space. It is fixed by the independently derived channel map B, the predetermined route swap P, and the convention that the invisible kernel direction is even; it is not reconstructed from the observed mass-plane normal.

The lift has an explicit positive invariant metric,

$$H_{\mathrm{ch}}=B^{T}B+Q={\displaystyle \frac{1}{11}}\left(\begin{array}{ccc}53& 3& -129\\ 3& 177& -175\\ -129& -175& 573\end{array}\right),$$

(38)

for which

$$x^T{H}_{\mathrm{ch}}x={\parallel Bx\parallel }^2+{\parallel Qx\parallel }^2>0(x\ne 0),S_{\mathrm{ch}}^T{H}_{\mathrm{ch}}{S}_{\mathrm{ch}}=H_{\mathrm{ch}}.$$

(39)

A convenient parity basis is

$$\begin{array}{cccccc}\hfill w_{+}& ={\displaystyle \frac{1}{44}}{(-1,-4,7)}^T,\hfill & \hfill B{w}_{+}& ={(1,1)}^T,\hfill & \hfill S_{\mathrm{ch}}{w}_{+}& =+w_{+},\hfill \end{array}$$

(40)

$$\begin{array}{cccccc}\hfill w_{-}& ={\displaystyle \frac{1}{44}}{(7,-16,-5)}^T,\hfill & \hfill B{w}_{-}& ={(1,-1)}^T,\hfill & \hfill S_{\mathrm{ch}}{w}_{-}& =-w_{-},\hfill \end{array}$$

(41)

$$\begin{array}{cccccc}\hfill k& ={(3,1,1)}^T,\hfill & \hfill Bk& =0,\hfill & \hfill S_{\mathrm{ch}}k& =k.\hfill \end{array}$$

(42)

Their metric norms are $2,2,11$ and their mutual $H_{\mathrm{ch}}$ products vanish.

A constructive positive spectral class follows immediately. For any nonnegative Laplace transforms $a_{+}\left(t\right),a_{-}\left(t\right),a_0\left(t\right)$ of positive spectral measures,

$${\Pi }_{\mathrm{ch}}\left(t\right)=a_{+}\left(t\right)w_{+}{w}_{+}^T+a_{-}\left(t\right)w_{-}{w}_{-}^T+a_0\left(t\right)k{k}^T⪰0$$

(43)

is $S_{\mathrm{ch}}$ invariant, and the route correlator is

$$C_{\mathrm{rt}}\left(t\right)=B{\Pi }_{\mathrm{ch}}\left(t\right)B^T=\left(\begin{array}{cc}{a}_{+}+a_{-}& a_{+}-a_{-}\\ a_{+}-a_{-}& a_{+}+a_{-}\end{array}\right).$$

(44)

The even and odd eigenvalues are $2a_{+}$ and $2a_{-}$. They may contain different pole energies, for example $a_{+}=e^{-E_{+}t}$ and $a_{-}=e^{-E_{-}t}$ with $E_{+}\ne E_{-}$. Exact route symmetry therefore does not require an even–odd degeneracy, a zero state, or a singular source metric.

Constructive possibility theorem in the actual channel space. The exact Pati–Salam channel map admits a nonempty positive spectral class with an exact route ${\mathbb{Z}}_2$, a nondegenerate positive metric, and distinct even and odd spectra. Hence the known $\mathrm{SU}{\left(15\right)}_p\times G_{\mathrm{PS}}$ representation structure contains no algebraic or positivity no-go against emergent route reciprocity. The theorem establishes possibility only; the physical $3+1$ dimensional spectral weights remain to be calculated.

7.3. The Minimal Visible Condition Is One Even–Odd Matrix Element

Let $W=(w_{+},w_{-},k)$ and $L=W^{-1}$. Explicitly,

$$L=\left(\begin{array}{ccc}-1& -2& 5\\ 1& -2& -1\\ 3/11& 1/11& 1/11\end{array}\right),L{S}_{\mathrm{ch}}W=diag(1,-1,1).$$

(45)

For a positive Hermitian channel correlator $\Pi \left(t\right)$ define the parity-basis correlator

$$\Sigma \left(t\right)=L\Pi \left(t\right)L^{†}.$$

Denote $a={\Sigma }_{++}$, $b={\Sigma }_{--}$, and $z={\Sigma }_{+-}$. Direct projection through B gives

$$C_{\mathrm{rt}}\left(t\right)=\left(\begin{array}{cc}a+b+2\Re z& a-b-2i\Im z\\ a-b+2i\Im z& a+b-2\Re z\end{array}\right).$$

(46)

Therefore

$$P{C}_{\mathrm{rt}}\left(t\right)P=C_{\mathrm{rt}}\left(t\right)⟺{\Sigma }_{+-}\left(t\right)=0.$$

(47)

For a real symmetric channel response this condition is

$${\Sigma }_{+-}=-{\Pi }_{03,03}+4{\Pi }_{150,150}-5{\Pi }_{153,153}+6{\Pi }_{03,153}-8{\Pi }_{150,153}=0,$$

(48)

with an exact cancellation of the 03–150 mixing term. The diagonal result $4{\Pi }_{150}-{\Pi }_{03}-5{\Pi }_{153}=0$ is recovered as a special case, not assumed generally.

Positivity gives the normalized diagnostic

$${\rho }_{+-}\left(t\right)={\displaystyle \frac{|{\Sigma }_{+-}\left(t\right)|}{\sqrt{{\Sigma }_{++}\left(t\right){\Sigma }_{--}\left(t\right)}}}\in [0,1],$$

(49)

and the route-diagonal asymmetry obeys

$${ϵ}_{\mathrm{rt}}\left(t\right)={\displaystyle \frac{2|{\Sigma }_{+-}\left(t\right)|}{{\Sigma }_{++}\left(t\right)+{\Sigma }_{--}\left(t\right)}}\le {\rho }_{+-}\left(t\right).$$

(50)

Thus the charged plane requires only the visible even–odd condition ${\Sigma }_{+-}=0$. Exact invariance of the entire three-channel object is stronger: it additionally requires the invisible–odd block ${\Sigma }_{0-}$ to vanish. A completely diagonal parity basis is stronger still and is not needed for the charged relation.

8. Conditional Reciprocity of the Complete Renormalized Sector

The exact identity in Eq. (24) turns the empirical plane into a sharp question: under which conditions is $C_{12}=C_{21}$ a theorem rather than a numerical coincidence? A literal local exchange of the two crossed Pati–Salam tensors is unavailable in the unchanged microscopic theory, as shown in Section 7.1. The relevant structure instead acts on the confined channel space.

For the three-channel coordinate used below, define

$$\mathsf{B}=\left(\begin{array}{ccc}0& -4& 4\\ -2& 0& 6\end{array}\right),P=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),S_{\mathrm{ch}}={\displaystyle \frac{1}{22}}\left(\begin{array}{ccc}15& 14& 7\\ 16& -10& -16\\ 5& -10& 17\end{array}\right).$$

(51)

Direct multiplication gives

$$S_{\mathrm{ch}}^2=I_3,\mathsf{B}{S}_{\mathrm{ch}}=P\mathsf{B}.$$

(52)

For N collective nodes the lift is $S=I_N\otimes S_{\mathrm{ch}}$, while the visible map has the form $\mathcal{B}={\ell }^T\otimes \mathsf{B}$ for the chosen node functional ℓ. This is an algebraic route exchange, not yet a symmetry of the quantum theory.

Theorem 3

(Conditional renormalized route reciprocity). Let Φ denote the complete set of regulated fields and composite sources relevant to the crossed route doublet, and let ${\mathcal{D}}_{\mathrm{ch}}$ denote the corresponding renormalized channel data supplied to the low-energy matching map. Suppose that:

A1.

the complete renormalized functional, including the measure, insertions, finite terms, and counterterms, obeys an exact linear involution,

$${\Gamma }_R[S\Phi ]={\Gamma }_R[\Phi ];$$

(53)

A2.

the selected state, boundary conditions, and external input are invariant under the same involution;

A3.

the full matching map is equivariant,

$${\mathcal{L}}_R\left(S{\mathcal{D}}_{\mathrm{ch}}\right)=P{\mathcal{L}}_R\left({\mathcal{D}}_{\mathrm{ch}}\right),$$

(54)

where ${\mathcal{L}}_R$ includes the vertex, operator mixing, amputation, pole subtraction, and the regular contribution.

Then the renormalized crossed coefficient doublet

$$\omega =\left({\displaystyle \genfrac{}{}{0pt}{}{C_{12}}{C_{21}}}\right)$$

(55)

obeys $P\omega =\omega$, and therefore

$$C_{12}=C_{21}.$$

(56)

The statement permits complex coefficients. It neither fixes their common value nor requires it to be nonzero.

Proof. 

Assumptions A1 and A2 imply invariance of the source-generating functional and hence of the complete renormalized response under simultaneous action of S on the channel data. Applying the equivariant matching map of A3 gives

$$\omega ={\mathcal{L}}_R\left({\mathcal{D}}_{\mathrm{ch}}\right)={\mathcal{L}}_R\left(S{\mathcal{D}}_{\mathrm{ch}}\right)=P{\mathcal{L}}_R\left({\mathcal{D}}_{\mathrm{ch}}\right)=P\omega .$$

(57)

The fixed subspace of P is one dimensional and consists of vectors proportional to ${(1,1)}^T$. Equation (56) follows. If the microscopic operation is antiunitary rather than the stated linear involution, the same argument generally gives a conjugation relation and does not by itself prove equality of two arbitrary complex coefficients. □

Assumption A3 is not a restatement of the desired equality. It can be verified before the two crossed coefficients are evaluated, by checking the individual maps from regulated sources to the matched coefficient doublet.

Proposition 1

(Componentwise criterion for equivariant matching). Let the linearized matching response be the composition

$${\mathcal{L}}_R={\mathcal{R}}_{\mathrm{reg}}\circ {\mathcal{P}}_{\mathrm{pole}}\circ {\mathcal{A}}_{\mathrm{amp}}\circ Z_{𝒪}\circ Z_J.$$

(58)

Here $Z_J$ renormalizes the source multiplet, $Z_{𝒪}$ is the operator mixing matrix, ${\mathcal{A}}_{\mathrm{amp}}$ amputates the external legs, ${\mathcal{P}}_{\mathrm{pole}}$ implements the common pole subtraction, and ${\mathcal{R}}_{\mathrm{reg}}$ restores the matched regular part, including finite counterterms. Let $S_0=S$ act on the regulated data, let $S_5=P$ exchange the two output routes, and let $S_k$ be specified involutions on the four intermediate spaces. If every stage $L_k$ of Eq. (58) satisfies

$$S_{k+1}{L}_k=L_k{S}_k,k=0,\dots ,4,$$

(59)

and every additive finite term is fixed by the corresponding $S_{k+1}$, then ${\mathcal{L}}_{R}S=P{\mathcal{L}}_R$ and assumption A3 holds.

Proof. 

Move $S_0$ through the five factors successively by Eq. (59). The last involution is $S_5=P$, so ${\mathcal{L}}_{R}S=P{\mathcal{L}}_R$. An additive finite term transforms equivariantly precisely when it lies in the fixed affine subspace stated in the hypothesis. No value of $C_{12}$ or $C_{21}$ enters these checks. □

The proposition separates five independently falsifiable requirements. A failure of source renormalization, mixing, amputation, pole subtraction, or the regular finite remainder is enough to invalidate A3 even if the bare route correlator happens to be symmetric.

The renormalization requirement in this theorem is substantive. If bare and renormalized channel operators are related by

$${𝒪}^R=Z_{𝒪}{𝒪}^B,$$

(60)

then a bare cancellation need not survive a generic $Z_{𝒪}$.

Proposition 2

(Dual coefficient projector and inverse-transpose matching). Order the four Pati–Salam channel coefficients as $\mathbf{a}={(a_{00},a_{03},a_{150},a_{153})}^T$ and the route coefficients as $\mathbf{d}={(C_{11},C_{12},C_{21},C_{22})}^T$. In the normalization of the route construction their exact map is

$$\mathbf{a}={\mathsf{K}}_{\mathrm{rt}}^T\mathbf{d},{\mathsf{K}}_{\mathrm{rt}}=\left(\begin{array}{cccc}0& 0& 0& -8\\ 0& 0& -4& 4\\ 0& -2& 0& 6\\ -1& 1& 3& -3\end{array}\right).$$

(61)

Since $det{\mathsf{K}}_{\mathrm{rt}}=64$, the map is invertible, and

$$C_{21}-C_{12}={\mathbf{q}}^T\mathbf{a},\mathbf{q}={\displaystyle \frac{1}{4}}{(1,-2,1,0)}^T.$$

(62)

The direct deformation that changes the two crossed route rows oppositely is

$${\mathbf{n}}_{\mathrm{dir}}={\displaystyle \frac{1}{2}}{\left[{\left({\mathsf{K}}_{\mathrm{rt}}\right)}_{21}-{\left({\mathsf{K}}_{\mathrm{rt}}\right)}_{12}\right]}^T={(0,-1,2,1)}^T,{\mathbf{q}}^T{\mathbf{n}}_{\mathrm{dir}}=1,$$

(63)

but ${\mathbf{n}}_{\mathrm{dir}}\ne \mathbf{q}$.

For a closed operator basis, let ${\mathbf{O}}_R=Z{\mathbf{O}}_B$ and let ${\mathbf{p}}_A=\langle f|{\mathbf{O}}_A|i\rangle$, $A\in \{B,R\}$. Then

$${\mathbf{p}}_R=Z{\mathbf{p}}_B,{\mathbf{a}}_R=Z^{-T}{\mathbf{a}}_B.$$

(64)

Consequently the renormalized target is represented in bare coordinates by

$${\Delta }_C^R\equiv {\mathbf{q}}_R^T{\mathbf{a}}_R={\left(Z^{-1}{\mathbf{q}}_R\right)}^T{\mathbf{a}}_B\equiv {\mathbf{w}}_B^T{\mathbf{a}}_B.$$

(65)

If projected NPR vertices obey $Z\Lambda =I$, only the target-specific column ${\mathbf{w}}_B=\Lambda {\mathbf{q}}_R$ is required; this is a solve in the complete basis, not a single propagator inversion. Under a finite scheme change ${\mathbf{O}}_{R^{\prime }}=F{\mathbf{O}}_R$, the covector column transforms as ${\mathbf{q}}_{R^{\prime }}=F{\mathbf{q}}_R$, so ${\mathbf{q}}_{R^{\prime }}^T{\mathbf{a}}_{R^{\prime }}={\mathbf{q}}_R^T{\mathbf{a}}_R$.

Proof. 

Direct inversion of Eq. (61) gives

$$C_{12}=-{\displaystyle \frac{1}{4}}(3a_{00}+a_{150}),C_{21}=-{\displaystyle \frac{1}{2}}(a_{00}+a_{03}),$$

(66)

which proves Eq. (62); subtraction of the two route rows gives Eq. (63). Finally, invariance of the interaction, ${\mathbf{a}}_B^T{\mathbf{O}}_B={\mathbf{a}}_R^T{\mathbf{O}}_R$, implies ${\mathbf{a}}_R=Z^{-T}{\mathbf{a}}_B$, while matrix elements transform in the same representation as the operators. Equation (65) follows by substitution. □

The normalized coordinate reflection

$$S_a=I_4-2{\mathbf{n}}_{\mathrm{dir}}{\mathbf{q}}^T,S_a^2=I_4,{\mathbf{q}}^T{S}_a=-{\mathbf{q}}^T,$$

(67)

exchanges the two crossed route rows. It is an exact coordinate identity, not by itself a symmetry of the microscopic functional. In particular, a Feynman–Hellmann response along ${\mathbf{n}}_{\mathrm{dir}}$ calibrates a derivative; it does not determine the baseline scalar ${\mathbf{q}}^T{\mathbf{a}}_R$ without the physical matching map.

Here $\mathbf{a}$ is the coefficient vector after matching into the declared mass-curvature route basis. If the same letter is used for a raw Yukawa, vertex, or microscopic deformation amplitude, ${\mathbf{q}}^T\mathbf{a}$ is not automatically ${\Delta }_C$: the nonlinear Yukawa-to-curvature Jacobian and its additive terms must first be established. This distinction is precisely why the mass-only estimator is primary and an unmatched Feynman–Hellmann slope is only a calibration response.

A sufficient intertwining condition for operator renormalization is

$$S_R{Z}_{𝒪}=Z_{𝒪}{S}_B.$$

(68)

For the odd left covector ${\nu }_{\mathrm{ch}}=(-1,2,1)$ of the three-channel representation,

$${\nu }_{\mathrm{ch}}{S}_{\mathrm{ch}}=-{\nu }_{\mathrm{ch}}.$$

(69)

The odd left eigenspace is one dimensional. Consequently, if $[Z_{𝒪},S_{\mathrm{ch}}]=0$, then

$${\nu }_{\mathrm{ch}}{Z}_{𝒪}=z_{-}{\nu }_{\mathrm{ch}}$$

(70)

for some finite $z_{-}$. An exact zero in the odd coordinate is then preserved. Without Eq. (68) or an equivalent physical matching statement, geometric parity of the source or a bare correlator cancellation is not a proof about $C_{21}-C_{12}$.

There are also two logically distinct reciprocity targets. The scalar statement $C_{12}=C_{21}$ is one constraint on the matched coefficients. Equality of complete Euclidean correlators,

$$c_{12}\left(t\right)=c_{21}\left(t\right)\mathrm{for}\mathrm{all}t,$$

(71)

requires the entire antisymmetric spectral measure, including contact terms at $t=0$, to vanish. Equation (56) follows from the full symmetry hypotheses, but an isolated integrated or pole coefficient equal to zero does not conversely establish Eq. (71).

The sufficient hypotheses are nonempty. Section 9.1 gives an interacting collective action with nonzero inter-node dynamics, an invariant positive channel metric, a quartic interaction, and a globally strictly convex potential. Its unique vacuum is fixed by S, and at an invariant finite regulator the full functional obeys the involution. Section 9.2 supplies the complementary counterexample: a stationary vacuum on the symmetric plane with positive odd curvature can nevertheless have a nonzero even–odd correlator when the action is not invariant. Vacuum location is therefore necessary for an unbroken realization but is not sufficient by itself.

9. Symmetric Vacuum: Constructive Realization and Why Vacuum Location Is Insufficient

9.1. An Interacting Globally Stable Realization

The preceding results identify the correct symmetry but do not yet show that a stable interacting effective action can select it. To provide a nontrivial constructive control, introduce two channel fields ${\varphi }_0\left(x\right)$ and ${\varphi }_1\left(x\right)$, where the node label is an auxiliary spectral coordinate rather than a second physical time. With $P_{+},P_{-},P_0$ the $H_{\mathrm{ch}}$-orthogonal projectors onto $w_{+},w_{-},k$, define

$$\begin{array}{cc}\hfill A_0& ={\displaystyle \frac{3}{4}}{P}_{+}+{\displaystyle \frac{3}{5}}{P}_{-}+{\displaystyle \frac{1}{3}}{P}_0,\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill L_5& ={\displaystyle \frac{1}{8}}{P}_{+}+{\displaystyle \frac{1}{10}}{P}_{-}+{\displaystyle \frac{1}{12}}{P}_0,\hfill \end{array}$$

(73)

$$\begin{array}{cc}\hfill A_1& ={\displaystyle \frac{1}{2}}{P}_{+}+{\displaystyle \frac{2}{5}}{P}_{-}+{\displaystyle \frac{1}{4}}{P}_0,\hfill \end{array}$$

(74)

and

$$Y=\left(\begin{array}{cc}{A}_0& L_5\\ L_5& A_1\end{array}\right),K=I_6-Y,G=I_2\otimes H_{\mathrm{ch}},\mathbb{S}=I_2\otimes S_{\mathrm{ch}}.$$

(75)

For $\lambda \ge 0$ consider the collective Euclidean effective action

$${\Gamma }_{\lambda }[\Phi ;\mathcal{J}]=\int d^{4}x\left[{\displaystyle \frac{1}{2}}{\partial }_{\mu }{\Phi }^{T}G{\partial }_{\mu }\Phi +{\displaystyle \frac{1}{2}}{\Phi }^{T}GK\Phi +{\displaystyle \frac{\lambda }{4}}{({\Phi }^{T}G\Phi )}^2-{\mathcal{J}}^{T}G\Phi \right].$$

(76)

The rational coefficients are selected inside a positive domain and include a nonzero coupling $L_5$ to the second spectral node. They are not derived values of the physical $\mathrm{SU}{\left(15\right)}_p$ theory.

For an even source, $\mathbb{S}\mathcal{J}=\mathcal{J}$, the action obeys

$${\Gamma }_{\lambda }[\mathbb{S}\Phi ;\mathcal{J}]={\Gamma }_{\lambda }[\Phi ;\mathcal{J}],$$

(77)

which in the normalized parity coordinates is $X_{-}\mapsto -X_{-}$. The three two-node quadratic blocks are

$$Y_{+}=\left(\begin{array}{cc}3/4& 1/8\\ 1/8& 1/2\end{array}\right),Y_{-}=\left(\begin{array}{cc}3/5& 1/10\\ 1/10& 2/5\end{array}\right),Y_0=\left(\begin{array}{cc}1/3& 1/12\\ 1/12& 1/4\end{array}\right).$$

(78)

The corresponding eigenvalues of $K_a=I_2-Y_a$ are

$$\begin{array}{cc}\hfill spec{K}_{+}& =\left\{{\displaystyle \frac{3-\sqrt{2}}{8}},{\displaystyle \frac{3+\sqrt{2}}{8}}\right\},\hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill spec{K}_{-}& =\left\{{\displaystyle \frac{5-\sqrt{2}}{10}},{\displaystyle \frac{5+\sqrt{2}}{10}}\right\},\hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill spec{K}_0& =\left\{{\displaystyle \frac{17-\sqrt{5}}{24}},{\displaystyle \frac{17+\sqrt{5}}{24}}\right\}.\hfill \end{array}$$

(81)

Hence

$${\mu }_{\mathrm{all}}^2={\displaystyle \frac{3-\sqrt{2}}{8}}=0.1982233047\dots >0,{\mu }_{-}^2={\displaystyle \frac{5-\sqrt{2}}{10}}=0.3585786438\dots >0.$$

(82)

In canonical coordinates the quartic Hessian is

$$\lambda \left[\left(X^{T}X\right)I_6+2X{X}^T\right]⪰0.$$

The full potential is therefore globally strictly convex. For $\mathcal{J}=0$ it has the unique vacuum $X=0$; for any even source the unique vacuum satisfies $X_{-}^{\mathrm{vac}}=0$.

The odd linear susceptibility of the control is also explicit,

$$K_{-}^{-1}=\left(\begin{array}{cc}60/23& 10/23\\ 10/23& 40/23\end{array}\right),{\parallel K_{-}^{-1}\parallel }_2={\displaystyle \frac{1}{{\mu }_{-}^2}}=2.7887885054\dots .$$

(83)

Thus a small odd source obeys $\parallel X_{-}^{\mathrm{vac}}\parallel \le \parallel j_{-}\parallel /{\mu }_{-}^2$ at linear order. In a full theory in which the even and invisible fields are allowed to relax, the relevant stability object is the Schur-complement curvature

$$K_{\mathrm{rel},--}={\Gamma }_{--}^{\left(2\right)}-{\Gamma }_{-e}^{\left(2\right)}{\left({\Gamma }_{ee}^{\left(2\right)}\right)}^{-1}{\Gamma }_{e-}^{\left(2\right)},$$

(84)

where e denotes all non-odd coordinates.

Proposition 3

(Conditional susceptibility form of the relaxed curvature). Assume that the physical Euclidean measure is positive, the connected susceptibility $\chi =Cov\left(X\right)$ is nonsingular, and the Legendre map exists on the selected state. If $K={\Gamma }^{\left(2\right)}={\chi }^{-1}$ and the same odd/even block decomposition is used in both matrices, then

$$K_{\mathrm{rel},--}={\left({\chi }_{--}\right)}^{-1}.$$

(85)

Proof. 

Apply the block-inverse formula to $K={\chi }^{-1}$. The inverse of the $--$ block of $\chi$ is the Schur complement of the complementary $ee$ block of K, which is precisely Eq. (84). □

Thus, under the stated assumptions, a separate expensive inversion for the relaxed curvature is unnecessary once the physical odd susceptibility has been measured. The identity does not establish the assumptions for the full chiral/scalar measure, and it proves neither odd stationarity $J_{-}=0$ nor the mixed-correlator condition $C_{+-}=0$.

What the nonlinear control proves. There exists an interacting positive effective action, compatible with the exact channel lift and with an active second spectral node, for which route parity is exact, the odd curvature is strictly positive, and the vacuum is uniquely and globally selected on the charged plane. This removes a possible consistency obstruction. It does not determine the coefficients of the physical $3+1$ dimensional $\mathrm{SU}{\left(15\right)}_p$ effective action.

9.2. Why Vacuum Location Is Not Enough: A Symmetry-Breaking Countercontrol

A crucial logical check is obtained by adding a small forbidden even–odd mixing. There is a finite interval of the breaking parameter in which the origin remains the unique stable vacuum and the odd curvature stays positive, while the mixed correlator is nonzero. In the explicit two-node control, stability survives for

$$\left|\epsilon \right|<0.2895758\dots ,$$

whereas the first mixed moments are

$$\begin{array}{cc}\hfill {\left(M_1\right)}_{+-}& =\epsilon ,\hfill \end{array}$$

(86)

$$\begin{array}{cc}\hfill {\left(M_2\right)}_{+-}& ={\displaystyle \frac{27}{20}}\epsilon ,\hfill \end{array}$$

(87)

$$\begin{array}{cc}\hfill {\left(M_3\right)}_{+-}& ={\displaystyle \frac{2237}{1600}}\epsilon +{\epsilon }^3,\hfill \end{array}$$

(88)

$$\begin{array}{cc}\hfill {\left(M_4\right)}_{+-}& ={\displaystyle \frac{2619}{2000}}\epsilon +{\displaystyle \frac{27}{10}}{\epsilon }^3.\hfill \end{array}$$

(89)

Therefore “the vacuum lies on the plane” plus “the odd curvature is positive” is not a proof of route reciprocity. The correlator symmetry is an independent physical statement and must be measured.

9.3. Finite-Time Closure: Three Slices Are the Earliest Case, Five Are Required for the Minimal Active Two-Node Control

Let a two-node transfer operator have block form

$$Y=\left(\begin{array}{cc}{A}_0& B_5^{†}\\ B_5& A_1\end{array}\right),E=\left({\displaystyle \genfrac{}{}{0pt}{}{I}{0}}\right).$$

Its first boundary moments are

$$M_0=I,M_1=A_0,M_2=A_0^2+B_5^{†}{B}_5,$$

so the first flat-extension residual is

$$S_1=M_2-M_1^2=B_5^{†}{B}_5.$$

(90)

Thus the three-slice test closes the spectrum only if the boundary source has no active coupling to the second node. If $B_5$ is full rank and nonzero, $S_1>0$ and three time separations cannot establish flat closure. When $B_5$ is invertible, two block-Krylov steps span the complete two-node accessible space and

$$S_2=0$$

(91)

exactly. The minimal active two-node control therefore requires the five consecutive moments $M_0,\dots ,M_4$, equivalently five Euclidean time separations.

This result refines rather than contradicts the finite-moment theorem already used in this paper. The general adaptive cost is $2N+1$ time slices for closure at order N. Three slices are the exact minimum when $S_1=0$; the first genuinely active two-node case instead closes at $N=2$ and needs five.

9.4. Minimal Physical Certificate for the Unchanged $3+1$ Dimensional Theory

The channel-space construction separates three route-sector questions that should be tested independently in the physical theory. Let $X_{-}$ denote the canonically normalized route-odd collective coordinate. Define the odd one-point force on the plane,

$$J_{-}(X_{+},X_0)={\left.{\displaystyle \frac{\delta {\Gamma }_{\mathrm{phys}}}{\delta X_{-}}}\right|}_{X_{-}=0},$$

(92)

the relaxed odd curvature $K_{\mathrm{rel},--}$ of (84), and the mixed even–odd two-point function or equivalently the mixed inverse two-point vertex. The minimal exact certificate is

$$J_{-}=0,{\lambda }_{min}\left(K_{\mathrm{rel},--}\right)\ge {\mu }_{-,\mathrm{phys}}^2>0,{\Gamma }_{+-}^{\left(2\right)}\left(p\right)=0⟺C_{+-}\left(t\right)=0.$$

(93)

The first condition says that the plane is stationary; the second says that it is stable after all even/invisible fields are allowed to readjust; the third is the stronger route-symmetry condition that the countercontrol shows cannot be inferred from the first two. If the exact physical effective action has the $S_{\mathrm{ch}}$ symmetry, all three follow, but a numerical calculation should test them separately.

Operationally, the two renormalized route sources should be converted into their sum and difference after the same source normalization. The $2\times 2$ matrix correlator should be measured at least five consecutive Euclidean separations in the first active two-node analysis. One should compute ${\rho }_{+-}$ with the full covariance, test $S_1$, proceed to $S_2$ if $S_1\ne 0$, and increase the order further if closure has still not occurred. Independent even and odd Feynman–Hellmann deformations then determine the one-point force and relaxed curvature. The entire sequence must be repeated across volume, cutoff, and renormalization choices before a continuum statement is made.

Matching remains a separate gate.Equation (93) is a certificate for route-sector stationarity, stability, and accessible two-point symmetry. It is not by itself an explanation of the charged masses. The same physical branch must still satisfy the pole-resolved matching conditions

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

or an exactly equivalent statement in the fully renormalized parity-resolved correlator and vertex. This separation is essential: a symmetric route vacuum with no nonzero even matching to the charged operator would not generate the observed mass-plane relation.

10. 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. [6]. 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 two-spurion completion defines the ordered eigendirections without inserting the observed generation labels by hand.

10.1. Why a Single Antisymmetric Spurion Is Insufficient

With the two-scalar assignment of Section 4.3, 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 limitation and its completion. 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 $D_{+}=diag(1,-2,1)$. The symmetric spurion ${\lambda }^{\prime }$ introduced in the next subsection removes this degeneracy and supplies the required projector without changing the route construction.

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.

10.2. 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.

This family projector must not be identified with the median projector of a physical charged mass matrix. With ${\mathcal{H}}_s=M_s{M}_s^{†}$ as defined in Section 13, one generally has

$$P_2\left(H_S\right)\ne P_2\left({\mathcal{H}}_s\right)$$

(94)

unless an additional alignment theorem or matched dynamical calculation is provided. The left-hand side selects the family direction from the scalar spurion; the right-hand side is a spectral projector of the physical mass observable. Their notational similarity does not supply the missing physical alignment.

There is also a precise limitation. 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.

10.3. 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.

10.4. 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. [6] 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 symmetric spurion can define $D_{+}$ directly in the three-family Pati–Salam flavour space, without an auxiliary four-flavour projection. 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.

10.5. 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

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

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 over the normalized field orbit. 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 stated two-spurion benchmark. They are illustrative values, not universal predictions of the Pati–Salam branch.

10.6. 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

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

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,

$${\kappa }_{det}=0$$

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 ${\kappa }_{det}$.

11. Mass-Only Formulation and a Direct Two-Point Estimator

The route identity can be tested through a physical mass observable without reconstructing a separate Higgs vertex for each charged sector. Let $M_s^R\left(\mu \right)$ be the renormalized mass matrix in a common low-energy scheme, and suppose that the Yukawa definition used in Section 3 satisfies

$$M_s^R\left(\mu \right)={\kappa }_s\left(\mu \right)Y_s^R\left(\mu \right),{\kappa }_s\ne 0,$$

(95)

with one scalar factor for the whole sector. If $m_{s,1}\le m_{s,2}\le m_{s,3}$ are the singular values, the factor $|{\kappa }_s|$ cancels from

$${\rho }_s={\displaystyle \frac{|det{M}_s^R|}{m_{s,2}^3}}={\displaystyle \frac{m_{s,1}{m}_{s,3}}{m_{s,2}^2}}.$$

(96)

Hence, when the charged route map is complete in the ${\mathbf{n}}_{\Pi }$ direction,

$$C_{21}-C_{12}={\displaystyle \frac{1}{4}}{\left[ln{\displaystyle \frac{m_e{m}_{\tau }}{m_{\mu }^2}}+2ln{\displaystyle \frac{m_d{m}_b}{m_s^2}}-ln{\displaystyle \frac{m_u{m}_t}{m_c^2}}\right]}_{\mu ,\mathrm{scheme}}.$$

(97)

All nine masses in Eq. (97) must be running parameters at the same scale and in the same scheme. Pole, constituent, and mixed-threshold masses cannot be substituted without changing the observable. If the low-energy theory contains an additional contribution ${\mathbf{A}}_{\mathrm{extra}}$ not represented by the four-route map, the right-hand side instead measures $C_{21}-C_{12}+{\mathbf{n}}_{\Pi }^T{\mathbf{A}}_{\mathrm{extra}}$. Thus the mass-only reduction removes an unnecessary vertex reconstruction, not the operator-completeness requirement.

The two-point version of the result is matrix-valued and does not assume commuting flavour structures. Here $G_2$ denotes this matched fermionic two-point observable, whose sector-s momentum-space representative is written as $G_s\left(p\right)$. After the required source and contact renormalization, write the Euclidean inverse propagator and propagator at $p^2>0$ as

$$\begin{array}{cc}\hfill {\Gamma }_s^{\left(2\right)}\left(p\right)=G_s{\left(p\right)}^{-1}& =\left(\begin{array}{cc}{B}_s^{†}& i\sigma ·p{A}_{R,s}\\ i\overline{\sigma }·p{A}_{L,s}& B_s\end{array}\right),\hfill \end{array}$$

(98)

$$\begin{array}{cc}\hfill G_s\left(p\right)& =\left(\begin{array}{cc}{J}_s& -i\sigma ·p{F}_{L,s}\\ -i\overline{\sigma }·p{F}_{R,s}& J_s^{†}\end{array}\right).\hfill \end{array}$$

(99)

Here $A_{L,s}$ and $A_{R,s}$ are positive Hermitian flavour matrices, while $B_s$ is the chirality-changing self-energy. Hypercubic tensor structures, when present at finite cutoff, require their own projectors and cannot simply be discarded.

Theorem 4

(Direct $G_2$ mass-spectrum theorem). Assume that the blocks in Eqs. (98)–(99) obey the displayed adjoint convention, that

$$F_{L,s}\succ 0,F_{R,s}\succ 0,$$

(100)

and that the remaining inverses used below exist on the selected momentum window. For the logarithmic mass-shape estimator, assume in addition that $J_s$ has rank three. Then

$${\mathcal{T}}_s\equiv J_s{F}_{R,s}^{-1}{J}_s^{†}{F}_{L,s}^{-1}=A_{L,s}^{-1}{B}_s{A}_{R,s}^{-1}{B}_s^{†}\sim {\mathcal{M}}_s{\mathcal{M}}_s^{†},$$

(101)

where

$${\mathcal{M}}_s=A_{L,s}^{-1/2}{B}_s{A}_{R,s}^{-1/2}.$$

(102)

The eigenvalues of ${\mathcal{T}}_s$ are therefore the squared singular values of the canonical mass function. Under arbitrary consistent invertible changes of external flavour normalization, the matrix ${\mathcal{T}}_s$ changes by similarity and its spectrum is invariant.

Proof. 

The off-diagonal block equations in ${\Gamma }_s^{\left(2\right)}{G}_s=I$ are

$$A_{L,s}{J}_s=B_s{F}_{R,s},A_{R,s}{J}_s^{†}=B_s^{†}{F}_{L,s}.$$

(103)

Substitution gives the equality in Eq. (101). Conjugation by $A_{L,s}^{1/2}$ shows similarity to ${\mathcal{M}}_s{\mathcal{M}}_s^{†}$. If the interpolators transform as ${\Psi }_L\mapsto U_L{\Psi }_L$ and ${\Psi }_R\mapsto U_R{\Psi }_R$, then

$$F_L\mapsto U_L{F}_L{U}_L^{†},F_R\mapsto U_R{F}_R{U}_R^{†},J\mapsto U_{L}J{U}_R^{†},$$

(104)

and direct substitution yields $\mathcal{T}\mapsto U_L\mathcal{T}{U}_L^{-1}$. □

A numerically positive representative with the same spectrum is

$${\mathcal{T}}_s^{(+)}=F_{L,s}^{-1/2}{J}_s{F}_{R,s}^{-1}{J}_s^{†}{F}_{L,s}^{-1/2}⪰0.$$

(105)

Equivalently, define

$$W_s=F_{L,s}^{-1/2}{J}_s{F}_{R,s}^{-1/2},$$

(106)

and compute its ordered singular values ${\sigma }_{s,1}\le {\sigma }_{s,2}\le {\sigma }_{s,3}$. Once the two-point function has been matched to the same running-mass convention as Eq. (97), the stable estimator is

$$\begin{array}{cc}\hfill {\Delta }_{\Pi }& ={\displaystyle \frac{1}{4}}\sum _{s=\ell ,d,u}{\eta }_s\left(ln{\sigma }_{s,1}-2ln{\sigma }_{s,2}+ln{\sigma }_{s,3}\right),\eta =(1,2,-1),\hfill \end{array}$$

(107)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{1}{8}}\sum _{s=\ell ,d,u}{\eta }_{s}ln{\displaystyle \frac{det{\mathcal{T}}_s}{{\lambda }_{s,2}^3}},\hfill \end{array}$$

(108)

where ${\lambda }_{s,2}={\sigma }_{s,2}^2$ is the median eigenvalue of ${\mathcal{T}}_s$. Equation (107) is the primary numerical definition: it uses positive singular values and never requires a branch choice for a complex logarithm. The determinant expression Eq. (108) is its corollary. Its factor $1/8$ is required because the eigenvalues are masses squared.

The theorem is exact for the displayed renormalized blocks, but the physical interpretation is conditional on matching. A composite-current correlator has the schematic form

$$G_{\mathrm{comp}}\left(p\right)=\mathcal{Z}\left(p\right)G_{\mathrm{EFT}}\left(p\right)\overline{\mathcal{Z}}\left(p\right)+P_{\mathrm{contact}}\left(p\right)+R_{\mathrm{heavy}}\left(p\right).$$

(109)

Multiplicative flavour normalizations cancel from the spectrum as stated, whereas additive contact and heavy regular terms do not. For local three-preon currents of canonical dimension $9/2$, continuum power counting permits scalar and vector contact polynomials through ${\left(p^2\right)}^2$ before symmetry reductions. Condensate power corrections and threshold conversion also separate a raw mass function from an $\overline{\mathrm{MS}}$ running mass. If a closed set of momentum responses has the form

$$B_k=c_{k}M+\sum _a{N}_{ka}{X}_a+R_k,$$

(110)

then a linear mass projector $\widehat{M}={\sum }_k{w}_k{B}_k$ exists precisely when

$$w^{T}c=1,w^{T}N=0,$$

(111)

equivalently $c\notin spanN$. Its controlled remainder satisfies

$$\parallel \widehat{M}-M\parallel \le \sum _k|w_k|\parallel R_k\parallel .$$

(112)

Equations (111)–(112) give a concrete identifiability gate rather than an assumption that two arbitrary momenta always suffice.

For statistical inference the nonlinear operations must follow, not precede, the ensemble average and matching. The three charged sectors should be evaluated on the same configurations with common weights. One first forms the joint averaged block correlators, repeats source subtraction, matching, and Eqs. (108)–(107) inside every joint resample, and only then constructs the covariance of the three logarithmic shapes. Averaging a per-configuration logarithm is a different estimator and is not used here.

12. Chiral State, Scalar Action, and the Connected Six-Preon Mass Channel

The mass-only theorem reduces the required observable but also exposes a state-selection issue. The microscopic field content relevant to this issue contains three fundamental Weyl fields ${\psi }_i$, the fundamental Pati–Salam fields $W,W^{\prime }$, the conjugate-symmetric Weyl field $\Omega$, and two elementary flavour scalars: an antisymmetric tensor $A\in \overline{\mathbf{105}}$ and a symmetric tensor $S\in \overline{\mathbf{120}}$. Their elementary Yukawa couplings are

$${\mathcal{L}}_Y={\displaystyle \frac{1}{2}}{\lambda }_{ij}{A}_{ab}{ϵ}^{\alpha \beta }{\psi }_{i\alpha }^a{\psi }_{j\beta }^b+{\displaystyle \frac{1}{2}}{\lambda }_{ij}^{\prime }{S}_{ab}{ϵ}^{\alpha \beta }{\psi }_{i\alpha }^a{\psi }_{j\beta }^b+\mathrm{h}.\mathrm{c}.,{\lambda }^T=-\lambda ,{\left({\lambda }^{\prime }\right)}^T={\lambda }^{\prime }.$$

(113)

The complete renormalizable scalar potential for these two tensors can be written with

$$a=Tr{A}^{†}A,s=Tr{S}^{†}S$$

(114)

as

$$\begin{array}{cc}\hfill V(A,S)=& V_0+m_A^{2}a+m_S^{2}s+u_A{a}^2+v_{A}Tr\left[{\left(A^{†}A\right)}^2\right]+u_S{s}^2+v_{S}Tr\left[{\left(S^{†}S\right)}^2\right]\hfill \\ \hfill & +w_{1}as+w_{2}Tr\left(A^{†}A{S}^{†}S\right)+\left\{\eta Tr\left(A^{†}S{A}^{†}S\right)+\mathrm{h}.\mathrm{c}.\right\}.\hfill \end{array}$$

(115)

Thus the scalar sector contains two real mass coefficients and eight real quartic degrees of freedom: six real coefficients and one complex $\eta$. There are no cubic invariants and no holomorphic quadratic invariants for this field content; the $\mathrm{SU}\left(15\right)$ center charge excludes them. Equation (115) is a specification of the allowed action, not a determination of its physical parameter point or vacuum.

Proposition 4

(Completeness of the renormalizable scalar basis). For $A^T=-A$ in $\overline{\mathbf{105}}$ and $S^T=S$ in $\overline{\mathbf{120}}$, both Pati–Salam singlets, the potential in Eq. (115) exhausts all $\mathrm{SU}{\left(15\right)}_p$-invariant scalar monomials of canonical dimension at most four. The elementary Yukawa terms in Eq. (113) are likewise the complete dimension-four scalar–fermion couplings for the stated field content.

Proof. 

A lower rank-two tensor has $\mathrm{SU}\left(15\right)$ centre charge $-2$. At degree at most four, centre neutrality permits only equal numbers of holomorphic and antiholomorphic tensors; a single ${ϵ}_{15}$ cannot occur because too few indices are present. Quadratic contractions give only $Tr{A}^{†}A$ and $Tr{S}^{†}S$, since $Tr{A}^{†}S=0$. Every quartic delta contraction has either two double-trace components or one four-cycle. Tensor symmetry, cyclicity, and Hermitian conjugation reduce these graphs to the seven structures displayed in Eq. (115), the last of which comes with its complex conjugate. This gives six real and one complex quartic coefficients. Centre charge and the spectator representations then leave only the $A\psi \psi$ and $S\psi \psi$ Yukawa contractions. Their flavour symmetries follow from Fermi statistics. An independent exact-rank enumeration of the allowed contraction graphs gives the same basis and checks the reduction. □

The displayed interactions possess a global $U{\left(1\right)}_{\chi }$ symmetry

$$q_{\chi }(W,W^{\prime },\psi ,\Omega ,A,S)=(0,0,17,-3,-34,-34).$$

(116)

It is free of the gauge anomaly relevant to the Ward identity used below. With $T\left(\mathbf{15}\right)=1/2$ and $T\left(\mathbf{120}\right)=(15+2)/2=17/2$,

$${\mathcal{A}}_{\mathrm{SU}{\left(15\right)}_p^2-U{\left(1\right)}_{\chi }}=3\left(17\right)T\left(\mathbf{15}\right)-3T\left(\mathbf{120}\right)=3\left(17\right){\displaystyle \frac{1}{2}}-3{\displaystyle \frac{17}{2}}=0.$$

(117)

The Pati–Salam-charged fields W and $W^{\prime }$ have zero $\chi$ charge. No claim is made that cubic-global or gravitational ’t Hooft anomalies vanish; no background gauge or gravitational source for $U{\left(1\right)}_{\chi }$ is introduced in the flat-space Ward identity. Let $L_i\sim \Omega W{\psi }_i$ and $R_j\sim \Omega W^{\prime }{\psi }_j$ denote the left- and right-sector three-preon currents, with all precolor, spectator, and spin contractions taken as in Section 5. Each current carries $\chi$ charge 14, and their chirality-changing product $L_i{R}_j$ has charge 28.

Theorem 5

($U{\left(1\right)}_{\chi }$ Ward obstruction and the connected mass channel). In a $U{\left(1\right)}_{\chi }$-invariant state, the chirality-changing two-point block $J_s$ of Eq. (99) vanishes. In a precolor-unfixed formulation in which the center and fermion-parity selection rules are imposed before gauge fixing, the surviving contribution to a nonzero $J_s$ is the appropriate spin, spectator, and flavour projection of the connected six-preon cumulant

$$K_c^{\left(6\right)}[\Omega W\psi ;\Omega W^{\prime }\psi ].$$

(118)

A separate elementary anomalous propagator $\langle \Omega \Omega \rangle$ is not a general prerequisite.

Proof. 

The Ward identity for Eq. (116) multiplies the mixed current product by $e^{28i\alpha }$. In an invariant state its expectation value must therefore vanish. For the cumulant statement, expand the six-field moment associated with the two three-preon currents into connected partitions. The six fields contain two $\Omega$ tensors of centre charge $-2$ and four fundamentals of charge $+1$. A block of size n containing $k_{\Omega }$ copies of $\Omega$ has centre charge

$$q_{15}(n,k_{\Omega })=n-3k_{\Omega }(mod15).$$

(119)

Every proper partition is eliminated by the following exhaustive classification:

Partition type	Possible block charge	Reason for vanishing
Any partition with an odd block	not needed	every odd connected fermion cumulant vanishes by fermion parity
$2+4$	for $n=2$: $2,-1,-4$; for $n=4$: $4,1,-2$	no even block is neutral modulo 15
$2+2+2$	each block has charge $2,-1$, or $-4$	every two-field connected factor is centre charged

These are all proper partitions of six. The precolor-centre Ward identity therefore removes every proper even factor, and fermion parity removes every partition containing an odd factor. Only the connected six-field cumulant and its prescribed tensor projection remain. The statement concerns the full projected channel and does not require factorization through a two-field $\Omega \Omega$ expectation value. □

A nonzero fermion mass consequently requires a selected state that breaks $U{\left(1\right)}_{\chi }$ in the relevant channel, together with a nonzero rank-three projection of Eq. (118). Averaging symmetry-related vacua can erase $J_s$ even when each oriented vacuum is massive, while leaving the normal blocks unchanged. Vacuum orientation must therefore be fixed before the ensemble average used in the estimator.

The chiral measure is an independent part of this state specification. The microscopic fermion content contains nineteen left-handed fundamentals ${\mathbf{15}}_L$ and one left-handed ${\overline{\mathbf{120}}}_L$. On the controlled admissible gauge core, the determinant-line phase can be anchored continuously at the identity. In six checked $6^3\times 12$ backgrounds the maximal link-coordinate norm was

$$\underset{e}{max}{\parallel X_e\parallel }_2=0.0314119<0.05<r_{★},$$

(120)

and the overlap gaps in both representations remained positive. This is a finite numerical check of the controlled core, not a proof for the complete scalar and Pati–Salam background space. The modulus of the physical Weyl weight,

$$w_{\chi }\left(U\right)\propto |det{\mathcal{D}}_{\mathrm{Weyl},15}{\left(U\right)|}^{19}|det{\mathcal{D}}_{\mathrm{Weyl},\overline{120}}\left(U\right)|,$$

(121)

has not been computed. When the scalar Yukawa pairings in Eq. (113) are active, the appropriate finite-volume object is more generally a consistently oriented Pfaffian in an enlarged Nambu space. Anomaly cancellation fixes neither its modulus nor the selected vacuum. Existing gauge-only or valence ensembles are therefore controls, not the physical chiral measure required by Eqs. (108) and (107).

13. Relative Mass-Shape Closure and Identifiability

The allowed scalar action does not by itself determine the relative flavour shape. It does, however, permit a finite algebraic classification. Let

$$D={\lambda }^{\prime },B=\lambda ,H=D{D}^{†},G=B{B}^{†},$$

(122)

and suppose that H has three simple eigenvalues with spectral projectors $P_i$. On the generic patch for which the required off-diagonal elements of G in the H eigenbasis are nonzero, the nine matrices

$$\mathfrak{B}=\{D,HD,H^{2}D\}\cup \{P_{i}G{P}_{j}D:i\ne j\}$$

(123)

have complex rank nine and span ${\mathrm{Mat}}_3\left(\mathbb{C}\right)$.

Theorem 6

(Generic rank-nine mass-tensor basis). Let D be invertible with three distinct Takagi singular values, and let all off-diagonal entries of G be nonzero in the Takagi basis of D. Then the nine matrices in Eq. (123) form a basis of ${\mathrm{Mat}}_3\left(\mathbb{C}\right)$. The rank-nine locus contains a Zariski-open dense subset of the corresponding nondegenerate parameter space.

Proof. 

Choose the Takagi frame $D=diag(d_1,d_2,d_3)$ with $d_i>0$ and distinct $h_i=d_i^2$. The diagonal entries of $D,HD,H^{2}D$ are the three vectors $\left(d_i\right)$, $\left(h_i{d}_i\right)$, and $\left(h_i^2{d}_i\right)$. Their determinant is, up to its ordering sign,

$$d_1{d}_2{d}_3(h_2-h_1)(h_3-h_1)(h_3-h_2)\ne 0.$$

(124)

They therefore span all diagonal matrices. For $i\ne j$,

$$P_{i}G{P}_{j}D=G_{ij}{d}_j{E}_{ij},$$

(125)

where $E_{ij}$ is the corresponding off-diagonal matrix unit. The assumed $G_{ij}\ne 0$ supplies all six off-diagonal units, proving rank nine.

For the generic statement, express the nine matrices in any fixed coordinate basis and clear the nonzero eigenvalue-gap denominators in the projectors. The relevant $9\times 9$ minor becomes a polynomial in the real and imaginary tensor coordinates. The explicit frame above, and independently an exact rational witness used in the verification, makes that minor nonzero. It is therefore not the zero polynomial; its nonvanishing set is Zariski open and dense. Degenerate spectra or vanishing connectors belong to the proper algebraic exceptional set and require a separate rank analysis. □

A generic covariant mass matrix can consequently be expanded exactly in this local basis. The coefficients are outputs of the fully specified dynamics and matching; the basis theorem does not turn them into arbitrary new ultraviolet couplings.

On the restricted branch involving only the symmetric spurion, Cayley–Hamilton reduction is stronger:

$$M_s={\kappa }_s{a}_s[I+r_{s}H+t_s{H}^2]D.$$

(126)

After an overall scale is removed, two relative complex coefficients remain. Dropping the $H^{2}D$ term requires a dynamical bound; it is not an algebraic consequence of three generations.

The target quotient is lower dimensional than the full ordered mass spectrum. Define

$$u_s=ln{\displaystyle \frac{m_{s,2}}{{\left(m_{s,1}{m}_{s,2}{m}_{s,3}\right)}^{1/3}}},v_s={\displaystyle \frac{1}{2}}ln{\displaystyle \frac{m_{s,3}}{m_{s,1}}}.$$

(127)

For ordered positive masses, $v_s\ge \frac{3}{2}\left|u_s\right|$, and the pair $(u_s,v_s)$ is a complete scale-free coordinate. Nevertheless

$${\rho }_s=e^{-3u_s},{\Delta }_{\Pi }=-{\displaystyle \frac{3}{4}}(u_{\ell }+2u_d-u_u),$$

(128)

so the hierarchy coordinate $v_s$ cancels exactly from the route target.

The same fact has a basis-invariant matrix form. Let ${\mathcal{H}}_s=M_s{M}_s^{†}$ and let $P_2\left({\mathcal{H}}_s\right)$ be the spectral projector onto its median eigenvalue. Then

$${\Delta }_{\Pi }={\displaystyle \frac{1}{8}}\sum _s{\eta }_{s}Tr\left\{[I-3P_2\left({\mathcal{H}}_s\right)]log{\mathcal{H}}_s\right\}.$$

(129)

Alternatively set $Q_s={\mathcal{H}}_s/{(det{\mathcal{H}}_s)}^{1/3}$, with ${\tau }_{1,s}=Tr{Q}_s$ and ${\tau }_{2,s}=Tr{Q}_s^{-1}$. Its eigenvalues are the roots of

$$q^3-{\tau }_{1,s}{q}^2+{\tau }_{2,s}q-1=0,$$

(130)

and if $q_{s,2}$ denotes the median root,

$${\Delta }_{\Pi }=-{\displaystyle \frac{3}{8}}\sum _s{\eta }_{s}ln{q}_{s,2}.$$

(131)

Thus two scale-free invariants per sector suffice to reconstruct the target, although only one combination per sector survives in the final logarithm.

This reduction is not a prediction of the remaining coordinate. A one-parameter counterexample is already sufficient. Take

$$D=diag(1,2,4),M_{\ell }\left(t\right)=[I+t{\left(D{D}^{†}\right)}^2]D,$$

(132)

and keep the other sectors fixed. Then

$${\Delta }_{\Pi }\left(t\right)-{\Delta }_{\Pi }\left(0\right)={\displaystyle \frac{1}{4}}ln{\displaystyle \frac{(1+t)(1+256t)}{{(1+16t)}^2}},{\left.{\displaystyle \frac{d{\Delta }_{\Pi }}{dt}}\right|}_{t=0}={\displaystyle \frac{225}{4}}.$$

(133)

One unresolved real matching coordinate can therefore change the observable. Equation (133) does not prove that only one coefficient is physically present; it proves that common normalization cancellation and covariance alone do not close the shape problem.

More generally, let $d\left(\theta \right)$ denote all independently measured responses at a parameter point and let $F\left(\theta \right)={\Delta }_{\Pi }$. Exact identifiability on a fibre requires

$$d\left(\theta \right)=d\left({\theta }^{\prime }\right)⟹F\left(\theta \right)=F\left({\theta }^{\prime }\right).$$

(134)

At a regular point, a necessary local condition is

$$kerDd\subseteq kerDF,$$

(135)

or equivalently $DF$ lies in the row space of $Dd$. The current analytic reductions establish the finite shape quotient and the exact estimator, but the physical response map d for the complete chiral vacuum has not yet been calculated. The physical value of ${\Delta }_{\Pi }$ is therefore not claimed as a first-principles result.

The algebra in this section has been checked independently with exact rational matrices on the generic patch and with random complex matrices away from degeneracies. The generic basis in Eq. (123) has rank nine, the single-spurion basis has rank three, Cayley–Hamilton reconstruction is exact, and the determinant, median-projector, cubic-root, and singular-value forms agree to double-precision residuals below $4.1\times {10}^{-15}$ in the tested samples. These are algebraic and synthetic numerical checks; no physical $\mathrm{SU}{\left(15\right)}_p$ inversion enters them.

14. Physical Chiral Measure and Regulator Status

The regulator is specified more sharply than a merely local feasibility argument, although the physical dynamical measure remains incomplete. The construction uses fundamental and matrix-free $\overline{\mathbf{120}}$ Wilson–overlap operators on a gauge-invariant admissible component ${\mathfrak{A}}_{{ϵ}_0}^0$. Gauge covariance, Ginsparg–Wilson convergence, the local $\Omega FF$ singlet, the 24-field source count, and the cubic anomaly cancellation have been verified directly. The admissibility condition

$${ϵ}_0<{\displaystyle \frac{1}{60}}$$

(136)

gives volume-independent lower bounds

$${\Delta }_{15}\ge \sqrt{1-30{ϵ}_0},{\Delta }_{\overline{120}}\ge \sqrt{1-60{ϵ}_0}$$

(137)

for the two overlap kernels. At ${ϵ}_0=1/120$ these become ${\Delta }_{15}\ge \sqrt{3}/2$ and ${\Delta }_{\overline{120}}\ge 1/\sqrt{2}$. Every fixed smooth continuum connection belongs to this admissible domain once the lattice spacing is sufficiently small, so the domain is not a finite-volume curiosity.

On a gauge-covariant principal-log chart, Kato transport and Poincaré homotopy solve the local determinant-line coboundary problem. Uniform kernel gaps and resolvent bounds imply exponential locality of the overlap projector, its link derivatives, the curvature kernel, and the compensated connection. On overlaps of the controlled charts the remaining transition phases are field-independent constants. For six representative configurations on the controlled core,

$$\underset{e}{max}{\parallel X_e\parallel }_2=0.0314118652<0.05<r_{*}=0.0625101770,{ϵ}_{max}=0.00861453613<{\displaystyle \frac{1}{60}},$$

(138)

while the smallest measured kernel gaps were $0.8611410548$ in the fundamental and $0.6950739762$ in the $\overline{\mathbf{120}}$ representation. The determinant-line phase can therefore be anchored globally on this controlled core. An independent plaquette-loop check is consistent with the same statement: compensation reduced the total-multiplet phase from $7.5\times {10}^{-7}$ to $3.2\times {10}^{-22}$, with gauge invariance at $2.7\times {10}^{-14}$.

Topology is also explicit at the sector level. For continuum topological charge Q, the overlap indices in the conventions used here are

$$ind{D}_{15}=Q,ind{D}_{\overline{120}}=17Q.$$

(139)

Thus the left-handed zero-mode count of the $19\mathbf{15}oplus\overline{\mathbf{120}}$ multiplet is $36Q$ in this convention. This index count must not be confused with the cubic gauge anomaly coefficient, for which $19A\left(\mathbf{15}\right)+A\left(\overline{\mathbf{120}}\right)=19-19=0$.

The specialized local overlap operator and a phase convention are defined on the controlled admissible core. The uncomputed object is the modulus of the physical Weyl determinant (or its equivalent Pfaffian representation) in the complete scalar and Pati–Salam background, together with extension to a physically sufficient admissible domain, the selected broken state, nonperturbative universality and the confining dynamical ensemble. Consequently the gauge-only and valence-weighted ensembles used in controls are not the physical measure. The broader literature likewise distinguishes perturbative anomaly-free Ginsparg–Wilson constructions from unresolved global and locality questions in proposed nonperturbative completions [22,23,24,25,26,27]. The present paper does not use an extra-dimensional regulator as evidence for route reciprocity.

15. Canonical Route Correlator as an Alternative Reciprocity Certificate

The local source algebra establishes that both crossed routes exist, but it does not determine their propagation through the confining vacuum. The physical object is the connected, renormalized Euclidean correlator

$$C_{ij}^{AB}(t;a_{\mathrm{src}},\mu )={\langle J_{Ai}^R\left(t\right)J_{Bj}^{R†}\left(0\right)\rangle }_c,A,B\in \{12,21\},$$

where $i,j$ label smearing or translation profiles constructed with the same point-split source geometry. A multi-profile basis is essential: a small odd pole can otherwise be hidden by a large source contaminant; the corresponding generalized-eigenvalue strategy is standard in finite-volume spectroscopy [16]. As an algebraic conditioning control, one unnormalized full-source pair has

$$G_{\mathrm{src}}^{\mathrm{alg}}=\left(\begin{array}{cc}56& 24\\ 24& 56\end{array}\right),spec\left(G_{\mathrm{src}}^{\mathrm{alg}}\right)=\{80,32\},$$

so its condition number is $2.5$. This full-source normalization is distinct from the normalized internal three-factor overlap, whose exact value is $1/64$ and whose even/odd Gram eigenvalues are $65/64$ and $63/64$; neither matrix is a renormalized vacuum correlator.

15.1. Source Metric, Whitening, and Basis Covariance

Choose a positive reference time $t_0$ after contact subtraction and define the source metric and canonical correlator by

$$G=C\left(t_0\right)\succ 0,\widehat{C}\left(t\right)=G^{-1/2}C\left(t\right)G^{-1/2},\widehat{C}\left(t_0\right)=I.$$

(140)

Under any nonsingular recombination of the renormalized source basis, $\widehat{C}$ changes by unitary similarity. Generalized eigenvalues, pole energies, and all norms below are therefore independent of arbitrary source normalizations. The route-exchange operator must be transformed by the same unitary similarity; fixing $U={\sigma }_x$ after an unrelated basis rotation would test a different exchange.

In the canonical two-route basis let $U={\sigma }_x$ and write a Hermitian kernel as

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

Its exchange-breaking component is

$$K_{\perp }={\displaystyle \frac{1}{2}}(K-UK{U}^{†})=k_y{\sigma }_y+k_z{\sigma }_z,\parallel K_{\perp }{\parallel }_2=\sqrt{{\displaystyle \frac{{(K_{11}-K_{22})}^2}{4}}+{(\Im K_{12})}^2}.$$

(141)

An exchange-odd pole proportional to $P_{-}=(I-{\sigma }_x)/2$ is compatible with $K_{\perp }=0$; route parity and route breaking are different notions.

If an isolated odd branch has relevant gap ${\Delta }_{-}^{\left(15\right)}$, the physical dimensionless propagation defect is

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

(142)

Both numerator and denominator must be measured at a continuously tracked pole $E_{*}$. Static tensor algebra fixes neither quantity. Equation (142) is therefore a nonperturbative diagnostic for the independent route-correlator certificate. It is not required to define the primary mass-only estimator of Section 11; the results below give rigorous sufficient tests of the stronger route realization before the entire spectrum is known.

15.2. Schur–Feshbach Reduction and Stability Contract

Let a visible two-route subspace with kernel $K_0$ couple through V to a self-adjoint hidden sector $H_H$. Below the hidden support, exact elimination gives

$$K_{\mathrm{eff}}\left(E\right)=K_0-V^{†}{(H_H-E)}^{-1}V,{(H_H-E)}^{-1}\succ 0.$$

(143)

This formula turns the unknown strong dynamics into a positive matrix-valued Stieltjes response. It also shows why evaluating the kernel at a fixed reference energy is unsafe: a moved hidden pole can amplify the resolvent even when the pole-tracked response is small.

If a mode decomposition gives $K_{\perp }={\sum }_i{c}_i$, define its cancellation conditioning by

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

(144)

Mode-level relative errors $|\delta c_i|\le \delta |c_i|$ then imply

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

A small finite-volume defect is accepted only if the pole, gap, residue, $K_{\perp }$, and ${\mathcal{C}}_{\mathrm{canc}}$ have controlled volume and regulator limits. This contract prevents a numerically delicate cancellation or a near-resolvent accident from being advertised as reciprocity.

16. Channel-Resolved Reduction for the Alternative Certificate

The exact crossed Pati–Salam source rows in the ordered irreducible channel basis $(03,150,153)$ are

$$B_{12}=(0,-4,4),B_{21}=(-2,0,6).$$

For channel-preserving connected propagation,

$${\Pi }_{\mathrm{conn}}\left(E\right)=diag({\Pi }_{03},{\Pi }_{150},{\Pi }_{153}),$$

and direct multiplication gives

$${\Sigma }_{\mathrm{conn}}=B{\Pi }_{\mathrm{conn}}{B}^{†}=\left(\begin{array}{cc}16({\Pi }_{150}+{\Pi }_{153})& 24{\Pi }_{153}\\ 24{\Pi }_{153}& 4{\Pi }_{03}+36{\Pi }_{153}\end{array}\right).$$

With route exchange $U={\sigma }_x$,

$${\displaystyle \frac{1}{2}}({\Sigma }_{\mathrm{conn}}-U{\Sigma }_{\mathrm{conn}}U)=2{\mathcal{G}}_{\mathrm{conn}}{\sigma }_z,{\mathcal{G}}_{\mathrm{conn}}=4{\Pi }_{150}-{\Pi }_{03}-5{\Pi }_{153}.$$

(145)

Thus this connected block is route even if and only if ${\mathcal{G}}_{\mathrm{conn}}\left(E_{★}\right)=0$. If the three subthreshold responses are nonnegative and ${\Pi }_{153}>0$, define $r_{03}={\Pi }_{03}/{\Pi }_{153}$ and $r_{150}={\Pi }_{150}/{\Pi }_{153}$. Then

$$4r_{150}-r_{03}-5=0,r_{150}\ge {\displaystyle \frac{5}{4}}.$$

(146)

The inequality is a necessary possibility condition for this positive, channel-preserving branch. Equal responses do not work: they give the raw diagonal mismatch $1/9$.

For a general Hermitian $3\times 3$ channel response, the route-odd matrix has the two-coordinate form

$${\Sigma }_{\perp }=\left(\begin{array}{cc}d& iq\\ -iq& -d\end{array}\right),{\parallel {\Sigma }_{\perp }\parallel }_2=\sqrt{d^2+q^2},$$

where

$$\begin{array}{cc}\hfill d=& -2{\Pi }_{03,03}+8{\Pi }_{150,150}-10{\Pi }_{153,153}+12\Re {\Pi }_{03,153}-16\Re {\Pi }_{150,153},\hfill \end{array}$$

(147)

$$\begin{array}{cc}\hfill q=& 8\left(\Im {\Pi }_{03,153}-\Im {\Pi }_{03,150}-3\Im {\Pi }_{150,153}\right).\hfill \end{array}$$

(148)

Hence even after channel mixing, the connected decision is two real numbers, not an unrestricted operator search.

16.1. A Positivity-Compatible Order-One Control

To test existence rather than predict the physical response, consider the common-residue Casimir–resolvent control

$${\Pi }_{03}={\displaystyle \frac{1}{\mu +2r}},{\Pi }_{150}={\displaystyle \frac{1}{\mu +4}},{\Pi }_{153}={\displaystyle \frac{1}{\mu +4+2r}},\mu ,r>0.$$

Substitution into (145) yields the exact positive cancellation curve

$$r_{★}\left(\mu \right)={\displaystyle \frac{4-\mu +\sqrt{9{\mu }^2+40\mu +80}}{8}}.$$

(149)

It is monotone on $\mu \ge 0$ and

$$r_{★}\left(0\right)={\displaystyle \frac{1+\sqrt{5}}{2}},r_{★}\left(1\right)={\displaystyle \frac{3+\sqrt{129}}{8}}=1.794727086450.$$

Therefore the source algebra and positivity do not forbid the cancellation, and the control requires only an order-one stiffness ratio. This curve is not a coupling prediction. In particular, identifying r directly with an inverse gauge-coupling ratio is unjustified because a physical response also contains spectral gaps, residues, vertices and continuum weight.

17. Alternative Route-Correlator, Feynman–Hellmann, and Matching Certificate

The direct $G_2$ theorem of Section 11 is the primary route from the physical two-point function to ${\Delta }_{\Pi }$. The construction in this section asks a stronger and logically separate question: whether a tracked composite pole, its regular complement, and the full route-resolved matching each transform equivariantly. It is retained as an independent cross-check and as a possible diagnosis of why the mass-only result succeeds or fails; it is not a second definition of the running masses.

17.1. Initial Null, RG Preservation, and the Minimal Physical Data

The equality to be transported must first be identified in the physical renormalized theory. In this subsection write

$${\Delta }_C\left(\mu \right)\equiv C_{21}\left(\mu \right)-C_{12}\left(\mu \right)={\Delta }_{\Pi }\left(\mu \right).$$

(150)

The last equality is the exact route identity in the fixed normalization; it does not identify an arbitrary bare response with ${\Delta }_C$. At an initial matching scale ${\mu }_0$, the primary mass-only determination is

$${\widehat{\Delta }}_C\left({\mu }_0\right)={\displaystyle \frac{1}{4}}\left(ln{\widehat{\rho }}_{\ell }+2ln{\widehat{\rho }}_d-ln{\widehat{\rho }}_u\right),{\widehat{\rho }}_s={\displaystyle \frac{{\sigma }_{s,1}{\sigma }_{s,3}}{{\sigma }_{s,2}^2}},$$

(151)

where ${\sigma }_{s,i}$ are the ordered singular values of the matched matrix $W_s=F_{L,s}^{-1/2}{J}_s{F}_{R,s}^{-1/2}$. All three sectors, all matching momenta, and all nuisance fits must be resampled jointly. With $\mathbf{r}={(ln{\rho }_{\ell },ln{\rho }_d,ln{\rho }_u)}^T$ and $\mathbf{c}={(1,2,-1)}^T/4$,

$$Var\left({\widehat{\Delta }}_C\right)={\mathbf{c}}^{T}Cov\left(\mathbf{r}\right)\mathbf{c}.$$

(152)

Theorem 7

(Transport of a physical codimension-one null). Let a complete renormalized coefficient vector evolve between two scales by the affine map

$${\mathbf{a}}_{{\mu }^{\prime }}=T({\mu }^{\prime },\mu ){\mathbf{a}}_{\mu }+\mathbf{b}({\mu }^{\prime },\mu ),$$

(153)

and let the physical targets be ${\Delta }_{\mu }={\mathbf{q}}_{\mu }^T{\mathbf{a}}_{\mu }$ and ${\Delta }_{{\mu }^{\prime }}={\mathbf{q}}_{{\mu }^{\prime }}^T{\mathbf{a}}_{{\mu }^{\prime }}$, with both covectors defined in their stated schemes. The complete hyperplane ${\Delta }_{\mu }=0$ is mapped into ${\Delta }_{{\mu }^{\prime }}=0$ if and only if there exists a scalar $\zeta ({\mu }^{\prime },\mu )$ such that

$${\mathbf{q}}_{{\mu }^{\prime }}^{T}T=\zeta {\mathbf{q}}_{\mu }^T,{\mathbf{q}}_{{\mu }^{\prime }}^T\mathbf{b}=0.$$

(154)

Proof. 

The functional ${\mathbf{q}}_{{\mu }^{\prime }}^{T}T$ must vanish on $ker{\mathbf{q}}_{\mu }^T$. The annihilator of this codimension-one hyperplane is $span\left\{{\mathbf{q}}_{\mu }^T\right\}$, proving the first condition. The image of the origin gives the second. Conversely, Eq. (154) gives ${\Delta }_{{\mu }^{\prime }}=\zeta {\Delta }_{\mu }$ directly. □

Suppose now that a complete closed response has been reduced to physical coordinates $\mathbf{u}={(u_{+},u_{-})}^T$ with

$$u_{-}\left(\mu \right)=\mathcal{N}\left(\mu \right){\Delta }_C\left(\mu \right),\mathcal{N}\left(\mu \right)\ne 0,$$

(155)

and that its finite step is exactly homogeneous, $\mathbf{u}\left(s\mu \right)={\Sigma }_{\mathrm{rt}}(s,\mu )\mathbf{u}\left(\mu \right)$. Theorem 7 then reduces to

$${\left({\Sigma }_{\mathrm{rt}}\right)}_{-+}(s,\mu )=0.$$

(156)

Indeed, $u_{-}^{\prime }={\left({\Sigma }_{\mathrm{rt}}\right)}_{-+}{u}_{+}+{\left({\Sigma }_{\mathrm{rt}}\right)}_{--}{u}_{-}$, so ${\left({\Sigma }_{\mathrm{rt}}\right)}_{--}$ is irrelevant to preservation of an exact initial zero. It is needed only to quantify the evolution of a nonzero defect.

Equation (156) is minimal only after Eq. (155) and closure have been proved. In an m-dimensional complete space the general condition is

$${\mathbf{q}}_{{\mu }^{\prime }}^{T}T{P}_{ker{\mathbf{q}}_{\mu }^T}=0,$$

(157)

which contains $m-1$ independent tangent-to-target components generically. Thus a generic four-channel basis requires three such zero conditions, and a generic 36-complex-dimensional nonderivative flavour basis requires 35 complex conditions unless exact intertwiners reduce the problem. A displayed $2\times 2$ subblock cannot certify closure against an omitted operator.

There are three exact obstructions to extending a tangent theorem beyond its hypotheses. First, the triangular matrix $\left(\begin{array}{cc}2& 1\\ 0& 3\end{array}\right)$ has zero $-+$ entry, but an additive vector ${(0,1)}^T$ generates an odd output from an initially odd-null state. Second, the nonlinear map $u_{-}^{\prime }=u_{-}+u_{+}^2$ has zero tangent $-+$ entry at the origin but does not preserve the finite manifold $u_{-}=0$. Third, in

$$T_3=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 0& 0& 1\end{array}\right)$$

(158)

the visible $2\times 2$ block is triangular while the omitted third coordinate feeds the odd output. Therefore the relation

$$\mathbf{u}\left(sL\right)={\Sigma }_{\mathrm{rt}}(s,L)\mathbf{u}\left(L\right)+O\left(u^2\right)$$

(159)

is a tangent theorem. Exact finite-amplitude preservation additionally requires $F_{-}(u_{+},0)=0$ along the relevant manifold, or a uniform bound on the nonlinear remainder.

Proposition 5

(When one initial scalar is sufficient). Let $M:V\to W$ be the complete linear measurement map and let $f:V\to \mathbb{R}$ be the physical scalar target. The data $Mx$ determine $f\left(x\right)$ if and only if

$$kerM\subseteq kerf,$$

(160)

equivalently $f\in rowM$.

Proof. 

Two parameter points give the same data precisely when their difference lies in $kerM$. The target is constant on every such fibre precisely when f vanishes on that kernel. □

Accordingly, one real number is sufficient at ${\mu }_0$ only when it is the already matched physical scalar in Eq. (151), or equivalently

$${\widehat{\Delta }}_C={\mathbf{w}}_B^T{\widehat{\mathbf{a}}}_B+{\widehat{\delta }}_{\mathrm{add}},{\mathbf{w}}_B=Z^{-1}{\mathbf{q}}_R,$$

(161)

with the additive contact/regular term ${\delta }_{\mathrm{add}}$ fixed. A single unmatched bare scalar or a slope along ${\mathbf{n}}_{\mathrm{dir}}$ is insufficient. For an unknown generic target row in an m-dimensional closed basis, at least m independent projected matching conditions are required to determine that row. If momentum-dependent nuisances span a matrix $N_s$ of rank $r_s$ in sector s, a mass projector requires at least $r_s+1$ independent momentum responses and exists only if the mass coefficient vector is outside $span{N}_s$.

The resulting measurement contract has three logically independent gates.

• Initial-null gate. Measure the matched $F_{L,s}\left(p_k\right),F_{R,s}\left(p_k\right),J_s\left(p_k\right)$ for all three charged sectors and enough nonexceptional momenta to satisfy the nuisance-rank condition. Retain the complete joint covariance and evaluate Eqs. (151) and (152). A coefficient-based alternative must first determine ${\mathbf{w}}_B$ and ${\delta }_{\mathrm{add}}$ in the complete basis.
• RG-preservation gate. Prove Eq. (155). For a real closed two-coordinate linear response, the full baseline matrix $R_1=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ and only the odd row $(g,h)$ of the scaled matrix are needed:

  $${\left({\Sigma }_{\mathrm{rt}}\right)}_{-+}={\displaystyle \frac{gd-hc}{ad-bc}}.$$

  (162)
  The minimal raw dataset is therefore the six real responses $\{a,b,c,d,g,h\}$ with their full $6\times 6$ covariance and a lower bound on $|ad-bc|$ or, more robustly, on ${\sigma }_{min}\left(R_1\right)$. Complex responses require 12 real components, their joint covariance, and two real zero conditions. If the origin is not analytically fixed, the odd intercept is an additional datum; paired $\pm \delta u_{+}$ and zero deformation are needed to bound the first nonlinear remainder.
• Continuum/volume gate. Every preceding quantity must be followed on one line of constant physics. No universal finite number of cutoff or volume points proves a limit. With a rigorously fixed leading form and a remainder bound, two lattice spacings or two volumes can bound an intercept; without that bound, at least three of each are required to fit one leading correction and test stability. One pair $(L,sL)$ tests one scale step; two adjacent pairs additionally test composition. Exact all-scale preservation still requires an analytic flow/symmetry theorem or a uniform certified bound.

A finite numerical interval that merely contains zero is not a proof of exact reciprocity. A numerical acceptance statement must instead use predeclared equivalence bands $|{\Delta }_C|<{\epsilon }_{\mathrm{init}}$ and $|{\left({\Sigma }_{\mathrm{rt}}\right)}_{-+}|<{\epsilon }_{\mathrm{RG}}$ after continuum, volume, matching, and nonlinear-remainder uncertainties are included. A controlled nonzero limit excludes reciprocity. For an exact claim, the corresponding zero must follow from the analytic intertwining conditions.

This analysis also fixes a necessary precondition. A new large inversion for ${\left({\Sigma }_{\mathrm{rt}}\right)}_{-+}$ is not physically interpretable until the action, selected state, determinant/Pfaffian modulus, complete operator space, and bridge (155) are specified. Once those inputs exist, the first expensive observable is the matched mass-only scalar (151), which removes the initial-null uncertainty. Only after it and the bridge pass does the six-response quantity (162) remove the distinct RG-generation uncertainty.

Exact finite-dimensional checks. All rank, kernel, transpose, inverse-transpose, reflection, and projector identities in this subsection were evaluated with exact rational arithmetic. The calculation gives $det{\mathsf{K}}_{\mathrm{rt}}=64$, $rank{R}_{\mathrm{ch}}=3$, $R_{\mathrm{ch}}{(1,1,1,1)}^T=0$, ${\mathbf{q}}^T{\mathbf{n}}_{\mathrm{dir}}=1$, $S_a^2=I_4$, and ranks $(3,1)$ for $(I_4\pm S_a)/2$. A non-diagonal rational test matrix with determinant 219 gives identical values for ${\mathbf{q}}_R^T{Z}^{-T}{\mathbf{a}}_B$ and ${\left(Z^{-1}{\mathbf{q}}_R\right)}^T{\mathbf{a}}_B$, while the naive ${\mathbf{q}}_R^T{\mathbf{a}}_B$ differs. The same exact calculation reproduces $det{G}_B=9$, the determinant formula ${\left({\Sigma }_{\mathrm{rt}}\right)}_{-+}=(gd-hc)/(ad-bc)$, and $K_{\mathrm{rel},--}={\left({\chi }_{--}\right)}^{-1}$ on a nonsingular rational block control.

17.2. Pole-Resolved Matching Is a Separate Physical Gate

Define

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

After pole amputation and subtraction of operator mixing, the strong route-symmetric matching condition adopted here is

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

(163)

The first inequality rules out the trivial solution in which the reciprocal sector does not couple to the charged operator. The second condition removes the route-odd matrix element of the tracked pole, and the third removes odd regular spectral weight in the same decomposition. A numerical cancellation between a nonzero pole-odd term and a nonzero regular remainder would be a different, less predictive mechanism.

For every finite invertible parity-preserving scheme change,

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

with nonsingular $Z_{\pm }$. Thus the zero/nonzero statements in Eq. (163) are scheme stable within this class. If a scheme mixes parity blocks, the entire matching vector and projectors must be transformed; quoting $F_{-}$ alone is then meaningless.

17.3. Finite Channel Bridge and the Route-21 Reduction

In a complete renormalized Pati–Salam channel basis ordered as $a=(a_{00},a_{03},a_{150},a_{153})$, a microscopic deformation ${ϵ}_X$ produces $a_X^{\prime }=da/d{ϵ}_X$. Let ${\mathbf{j}}_{\Pi }=\partial {\Delta }_{\Pi }/\partial a$ be the flavour-response covector and let $s_{\Pi B}={(\partial {\Delta }_{\Pi }/\partial {ϵ}_B)}_0\ne 0$ normalize the canonical crossed-amplitude direction at the same matching point. The observable-equivalent response is

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

(164)

This is a finite derivative, not a free strong coefficient. It can be evaluated only after the physical running basis, thresholds, and matching prescription have been fixed.

The source algebra narrows one side further. In the irreducible basis,

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

Let ${\mathcal{P}}_{03}\left(0\right)$ and ${\mathcal{P}}_{153}\left(0\right)$ include the corresponding residues, amputated vertices and common-scheme renormalization factors, and set

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

After one common response and one overall route-21 amplitude ${\alpha }_{21}$ are factored out, the linear mass-plane response is

$$D_{21}\left(r\right)={\displaystyle \frac{-2j_{\Pi ,03}+6j_{\Pi ,153}r}{s_{\Pi B}}},\delta {\Delta }_{\Pi }={\alpha }_{21}{D}_{21}\left(r\right)+O\left({\alpha }_{21}^2\right).$$

(165)

Equal channel responses are not required: the two reduced responses enter with the fixed source coefficients $-2$ and 6, while the covector components encode the independently computed low-energy sensitivity. This remains a one-sided result: full reciprocity requires the route-12 response and common physical normalization in the same scheme.

The complete non-derivative operator class contains four Pati–Salam channels times nine complex flavour directions, i.e. 36 complex operators. The six-tensor flavour subset used in the benchmark is not closed under the allowed wave-function directions. A physical numerical value of Eq. (164) must therefore follow construction of the full renormalized basis, rather than assigning provisional anomalous dimensions to an incomplete subspace.

17.4. Paired Feynman–Hellmann Slopes as an Independent Cross-Check

Let the finite-volume Hamiltonian or action be deformed by renormalized even and odd matching operators,

$$H({\lambda }_{+},{\lambda }_{-})=H_0+{\lambda }_{+}{𝒪}_{+}^R+{\lambda }_{-}{𝒪}_{-}^R.$$

For an isolated, continuously tracked odd branch with energy gap ${\mathcal{E}}_{-}$,

$$F_{\pm }^R\left(0\right)={\left.{\displaystyle \frac{\partial {\mathcal{E}}_{-}}{\partial {\lambda }_{\pm }^R}}\right|}_0.$$

(166)

Thus the diagonal matching gate can be extracted from deformed two-point spectroscopy rather than a separate three-point tail fit, following a method already used extensively in lattice calculations [17,18]. If an exact route involution exists for this branch, ${\mathcal{E}}_{-}({\lambda }_{+},{\lambda }_{-})={\mathcal{E}}_{-}({\lambda }_{+},-{\lambda }_{-})$, so $F_{-}\left(0\right)=0$ exactly. In the weaker emergent branch, paired $\pm {\lambda }_{-}$ calculations and a controlled ${\lambda }_{-}\to 0$ extrapolation measure the odd slope. The required conditions remain

$$F_{+}\left(0\right)\ne 0,F_{-}\left(0\right)=0$$

in the exact limit.

The odd null does not require a full operator-renormalization matrix. After metric whitening, only the cross-parity intertwining defect is relevant:

$$P_{-}^R\widehat{Z}{P}_{+}^0={\displaystyle \frac{1}{2}}{P}_{-}^R(\widehat{Z}{\widehat{\Pi }}_0-{\widehat{\Pi }}_R\widehat{Z})P_{+}^0.$$

This identity gives a norm bound on the renormalized odd slope from the measured cross-parity block.

17.5. One-Sided Step Scaling and the Determinant Test

Let canonical response matrices at matched volumes/scales be

$$R_1=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right),R_2=\left(\begin{array}{cc}e& f\\ g& h\end{array}\right),{\Sigma }_{21}=R_2{R}_1^{-1}.$$

This is the route-resolved analogue of finite-volume step scaling [19,20]. Only generation of an odd output from an even input matters. Therefore the minimal invariant-manifold condition is

$${\left({\Sigma }_{21}\right)}_{-+}=0,$$

not the vanishing of the full parity commutator. Direct inversion gives

$${\left({\Sigma }_{21}\right)}_{-+}={\displaystyle \frac{gd-hc}{ad-bc}}.$$

(167)

Thus the full scaled matrix is unnecessary for this gate: the four entries of $R_1$ and the odd row $(g,h)$ of $R_2$ are the six real response components required by Eq. (162). Their covariance must be retained jointly. This reduction concerns data volume only; it has a physical interpretation for $C_{21}-C_{12}$ only after the bridge (155), basis closure, and the absence of an odd affine term have been established. If $u=(c,d)$, $w=(e,f)$, $v=(g,h)$ and $[x,y]=x_1{y}_2-x_2{y}_1$, the relative generated leakage is

$$r_{G,2|+}=\sqrt{{\displaystyle \frac{g_{-}}{g_{+}}}}{\displaystyle \frac{\left|\right[v,u\left]\right|}{\left|\right[w,u\left]\right|}},$$

(168)

where $g_{\pm }$ are the positive output parity-metric weights. In a whitened basis $g_{+}=g_{-}=1$, and the notation $r_{2|+}=\left|[v,u]\right|/\left|[w,u]\right|$ is used. A common nonsingular change of the two bare deformation coordinates multiplies both alternating products by the same determinant and therefore leaves their ratio unchanged. Independent changes of units in the output parity sectors are instead compensated by the metric factor in Eq. (168). The input determinant cancels, but ${\sigma }_{min}\left(R_1\right)$ must remain separated from zero for the step itself to be well conditioned; the denominator issue is the analogue of the standard ratio-inference obstruction [21].

The uncertainty can be propagated without linearizing the determinants. For the real, CP-preserving case stack the response rows as $z={(u_1,u_2,w_1,w_2,v_1,v_2)}^T$ and assume the simultaneous confidence ellipsoid

$${\mathcal{E}}_{\kappa }=\left\{z:{(z-\tilde{z})}^T{\mathcal{C}}^{-1}(z-\tilde{z})\le {\kappa }^2\right\},\mathcal{C}\succ 0.$$

(169)

Here $\tilde{z}$ is the central response vector, $\mathcal{C}$ is its joint covariance matrix, and $\kappa$ fixes the simultaneous coverage. Define $N\left(z\right)=[v,u]$ and $D\left(z\right)=[w,u]$, with gradients

$$\begin{array}{cc}\hfill {\ell }_N& ={(-{\tilde{v}}_2,{\tilde{v}}_1,0,0,{\tilde{u}}_2,-{\tilde{u}}_1)}^T,\hfill \end{array}$$

(170)

$$\begin{array}{cc}\hfill {\ell }_D& ={(-{\tilde{w}}_2,{\tilde{w}}_1,{\tilde{u}}_2,-{\tilde{u}}_1,0,0)}^T.\hfill \end{array}$$

(171)

Let the symmetric matrices $H_N$ and $H_D$ be fixed by

$$\begin{array}{cc}\hfill \delta z^T{H}_N\delta z& =\delta v_1\delta u_2-\delta v_2\delta u_1,\hfill \end{array}$$

(172)

$$\begin{array}{cc}\hfill \delta z^T{H}_D\delta z& =\delta w_1\delta u_2-\delta w_2\delta u_1,\hfill \end{array}$$

(173)

and set

$$\begin{array}{cc}\hfill {\eta }_N^{\mathrm{cov}}& =\kappa \sqrt{{\ell }_N^T\mathcal{C}{\ell }_N}+{\kappa }^2{\parallel {\mathcal{C}}^{1/2}{H}_N{\mathcal{C}}^{1/2}\parallel }_2,\hfill \end{array}$$

(174)

$$\begin{array}{cc}\hfill {\eta }_D^{\mathrm{cov}}& =\kappa \sqrt{{\ell }_D^T\mathcal{C}{\ell }_D}+{\kappa }^2{\parallel {\mathcal{C}}^{1/2}{H}_D{\mathcal{C}}^{1/2}\parallel }_2.\hfill \end{array}$$

(175)

Theorem 8

(correlated determinant interval). For every $z\in {\mathcal{E}}_{\kappa }$,

$$|N\left(z\right)-N\left(\tilde{z}\right)|\le {\eta }_N^{\mathrm{cov}},|D\left(z\right)-D\left(\tilde{z}\right)|\le {\eta }_D^{\mathrm{cov}}.$$

If $|D\left(\tilde{z}\right)|>{\eta }_D^{\mathrm{cov}}$, then

$$max\left(0,{\displaystyle \frac{|N\left(\tilde{z}\right)|-{\eta }_N^{\mathrm{cov}}}{|D\left(\tilde{z}\right)|+{\eta }_D^{\mathrm{cov}}}}\right)\le r_{2|+}\le {\displaystyle \frac{|N\left(\tilde{z}\right)|+{\eta }_N^{\mathrm{cov}}}{|D\left(\tilde{z}\right)|-{\eta }_D^{\mathrm{cov}}}}.$$

(176)

The bounds for $r_{G,2|+}$ follow by multiplying both endpoints by the simultaneously bounded factor $\sqrt{g_{-}/g_{+}}$.

Proof. 

Write $\delta z=z-\tilde{z}={\mathcal{C}}^{1/2}y$ with ${\parallel y\parallel }_2\le \kappa$. Since N and D are quadratic polynomials, their Taylor expansions terminate exactly:

$$N\left(z\right)-N\left(\tilde{z}\right)={\ell }_N^T\delta z+\delta z^T{H}_N\delta z,$$

and analogously for D. Cauchy–Schwarz bounds the linear term by $\kappa \sqrt{{\ell }_N^T\mathcal{C}{\ell }_N}$, and the spectral norm bounds the quadratic term by ${\kappa }^2{\parallel {\mathcal{C}}^{1/2}{H}_N{\mathcal{C}}^{1/2}\parallel }_2$. The same argument applies to D. Direct and reverse triangle inequalities then give the numerator and denominator intervals, and division is legitimate precisely when the lower denominator is positive. □

For several scales write ${\Sigma }_k=\left(\begin{array}{cc}{A}_k& B_k\\ C_k& D_k\end{array}\right)$ and $z_k=F_{-,k}/F_{+,k}$. Projective evolution is the exact Möbius map

$$z_{k+1}={\displaystyle \frac{C_k+D_k{z}_k}{A_k+B_k{z}_k}}.$$

(177)

Consequently, if $r_k\ge \left|z_k\right|$ and $|A_k|-|B_k|r_k>0$, then

$$|z_{k+1}|\le r_{k+1}\equiv {\displaystyle \frac{|C_k|+|D_k|r_k}{|A_k|-|B_k|r_k}}.$$

(178)

This follows directly from the triangle inequality in the numerator and the reverse triangle inequality in the denominator. Starting with $r_0=0$ therefore gives a rigorous accumulated leakage bound. Stepwise control is essential: for example,

$$\left(\begin{array}{cc}1& 0\\ -0.1& 1\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0.1& 1\end{array}\right)=I$$

although each factor generates a nonzero odd component from an even input. A small endpoint value alone can thus be a cancellation rather than evidence for an invariant route-even manifold.

18. A Computable Spectral Possibility Theorem

18.1. Exact Hidden-Susceptibility Criterion

Let a canonically normalized visible route-odd state of energy $E_{-}^{\left(0\right)}$ be coupled by a vector v to an arbitrary self-adjoint hidden odd sector $H_H$. Assume that the coupled hidden support lies above the lowest even pole:

$$E_{+,0}>0,H_H-E_{+,0}\equiv G_H\succ 0,{\Delta }_0\equiv E_{-}^{\left(0\right)}-E_{+,0}>0.$$

The odd block is

$$K_{-}=\left(\begin{array}{cc}{E}_{-}^{\left(0\right)}& v^{†}\\ v& H_H\end{array}\right).$$

Define

$${\chi }_H\left(E\right)=v^{†}{(H_H-E)}^{-1}v.$$

Theorem 9

(hidden-susceptibility threshold). The lowest eigenvalue $E_{-}$ of $K_{-}$ satisfies

$$E_{-}<E_{+,0}\iff {\chi }_H\left(E_{+,0}\right)>{\Delta }_0.$$

(179)

For every nonzero hidden spectral shape above $E_{+,0}$, there is a nonempty open interval of overall couplings for which (179) holds and $K_{-}$ remains strictly positive.

Proof. 

For E below the hidden support, the Schur complement is

$$s\left(E\right)=E_{-}^{\left(0\right)}-E-{\chi }_H\left(E\right).$$

Since $s^{\prime }\left(E\right)=-1-v^{†}{(H_H-E)}^{-2}v<0$, its lowest zero is below $E_{+,0}$ exactly when $s\left(E_{+,0}\right)={\Delta }_0-{\chi }_H\left(E_{+,0}\right)<0$. For the second statement write $v=\alpha v_0$ and define $A=v_0^{†}{H}_H^{-1}{v}_0$ and $B=v_0^{†}{(H_H-E_{+,0})}^{-1}{v}_0$. Because $B>A>0$ and ${\Delta }_0<E_{-}^{\left(0\right)}$, the interval

$${\displaystyle \frac{{\Delta }_0}{B}}<{\alpha }^2<{\displaystyle \frac{E_{-}^{\left(0\right)}}{A}}$$

is nonempty. Its lower inequality gives favorable ordering and its upper inequality is the Schur condition for $K_{-}\succ 0$. □

This theorem supplies a minimal existence proof for the spectral branch. It proves compatibility, not that the physical theory chooses a point inside the interval.

18.2. Finite Positive-Time Moments Can Certify the Required Ordering

Define the positive Euclidean memory kernel

$$f\left(t\right)=v^{†}{e}^{-t{G}_H}v\ge 0,{\chi }_H\left(E_{+,0}\right)={\int }_0^{\infty }f\left(t\right)dt.$$

At a reference time $\tau >0$, let

$$w_{\tau }=e^{-\tau G_H/2}v,m_k\left(\tau \right)={(-1)}^k{f}^{\left(k\right)}\left(\tau \right)=\langle w_{\tau },G_H^k{w}_{\tau }\rangle .$$

For $i,j=0,\dots ,N-1$ set

$${\left(T_N\right)}_{ij}=m_{i+j+1},{\left(b_N\right)}_i=m_i,{\chi }_N\left(\tau \right)=b_N^{†}{T}_N^{+}{b}_N.$$

Theorem 10

(gap-free finite-data certificate). For every N for which the positive moment problem is well defined,

$$0\le {\chi }_1\left(\tau \right)\le {\chi }_2\left(\tau \right)\le \dots \le \chi \left(\tau \right)\le {\chi }_H\left(E_{+,0}\right).$$

(180)

Consequently

$${\chi }_N\left(\tau \right)>{\Delta }_0⟹E_{-}<E_{+,0}.$$

(181)

Failure to cross the threshold before spectral closure is inconclusive, not a proof of absence.

Proof. 

For x in the Hilbert space use the variational identity

$$\langle w_{\tau },G_H^{-1}{w}_{\tau }\rangle =\underset{x}{sup}\{2\Re \langle w_{\tau },x\rangle -\langle x,G_{H}x\rangle \}.$$

Restricting x to the nested Krylov space ${\mathcal{K}}_N=span\{w_{\tau },G_H{w}_{\tau },\dots ,G_H^{N-1}{w}_{\tau }\}$ gives $b_N^{†}{T}_N^{+}{b}_N$. Nested restriction proves monotonicity and the lower bound. Positive time suppresses heavy weight, so $\chi \left(\tau \right)\le \chi \left(0\right)={\chi }_H\left(E_{+,0}\right)$. Equation (181) then follows from Theorem 9. □

At $N=1$, ${\chi }_1=f{\left(\tau \right)}^2/[-f^{\prime }\left(\tau \right)]$. Thus even one value and one local derivative may certify the required pole ordering. Alternatively, positive-time samples $f(\tau +na)$ form a Hausdorff moment sequence in $x=e^{-a{G}_H}\in (0,1)$. Exact flat extension reconstructs the full source-supported discrete spectrum; before flat closure, only one-sided claims are justified.

18.3. Metric Covariance, Larger Operator Bases, and Finite Errors

Let $\mathcal{G}\succ 0$ be a metric in an enlarged renormalized operator basis, ${\mathcal{M}}_k⪰0$ its hidden-memory moment matrices, and R the tracked embedding of the two physical route sources. Define

$$S=R^{†}\mathcal{G}R,N_k=R^{†}{\mathcal{M}}_{k}R,m_k^{\left(S\right)}=Tr\left[P_{-}{S}^{-1/2}{N}_k{S}^{-1/2}\right],$$

where $P_{-}=(I-{\sigma }_x)/2$. These induced quantities are unchanged under an arbitrary nonsingular change of the full operator basis if R is transformed contragrediently. Closure of the two-source subspace under renormalization is not required; tracking the full embedding is.

For noisy moments, let ${\mathcal{E}}_{\alpha }$ be a joint $1-\alpha$ confidence region for the moment vector and for ${\Delta }_0$ let ${\overline{\Delta }}_{0,\alpha }$ be a simultaneous upper confidence bound. Define the robust Krylov lower bound by the semidefinite problem

$${\underline{\chi }}_{N,\alpha }=\underset{m\in {\mathcal{E}}_{\alpha },t}{min}\left\{t:T_N\left(m\right)⪰0,\left(\begin{array}{cc}{T}_N\left(m\right)& b_N\left(m\right)\\ b_N{\left(m\right)}^{†}& t\end{array}\right)⪰0\right\}.$$

(182)

For nonsingular $T_N$, the Schur complement makes the minimizing t equal to $b_N^{†}{T}_N^{-1}{b}_N$ for each admissible moment vector. Therefore

$${\underline{\chi }}_{N,\alpha }>{\overline{\Delta }}_{0,\alpha }⟹E_{-}<E_{+,0}\mathrm{with}\mathrm{the}\mathrm{declared}\mathrm{simultaneous}\mathrm{coverage}.$$

(183)

This is a directly computable, covariance-aware possibility condition. It does not invert a Laplace transform and cannot turn an inconclusive finite lower bound into a negative conclusion.

18.4. Numerical Control

For the three-state control

$$\begin{array}{cc}\hfill g& =(4.0198205527,10.0198205527,17.0198205527),\hfill \\ \hfill w& =(8.3472459562,3.7098870916,1.4491746452),\hfill \end{array}$$

the first four moments at $\tau =0$ are

$$(m_0,m_1,m_2,m_3)=(13.5063076930,95.3915261926,927.1322364549,11418.9223649736).$$

They give

$${\chi }_1=1.9123328327,{\chi }_2=2.4469585356>2.3017483391={\Delta }_0,$$

so the second Krylov level proves favorable ordering. The strict margin is $0.1452101965$, or $6.31\%$ of the threshold. The exact susceptibility is $2.5319231731$. All of these values belong to a declared control spectrum.

19. A Finite-Resolution Feasibility Certificate for the Full Mechanism

No single scalar can prove the exact physical plane at finite regulator, volume and statistical precision. A useful scalar can, however, summarize a declared sufficient branch without mixing dimensions or hiding tolerances. Define the measurable ratios

$$\begin{array}{cc}\hfill r_{\mathrm{conn}}& ={\displaystyle \frac{|4{\Pi }_{150}-{\Pi }_{03}-5{\Pi }_{153}|}{4{\Pi }_{150}+{\Pi }_{03}+5{\Pi }_{153}}},\hfill \end{array}$$

(184)

$$\begin{array}{cc}\hfill r_F& ={\displaystyle \frac{|F_{-}\left(0\right)|}{|F_{+}\left(0\right)|}},\hfill \end{array}$$

(185)

$$\begin{array}{cc}\hfill r_{\mathrm{reg}}& ={\displaystyle \frac{\parallel K_{\perp }^{\mathrm{reg}}{\parallel }_2}{{\Delta }_{-}}},\hfill \end{array}$$

(186)

with positive denominators certified in the same scheme. Let ${\overline{r}}_{\mathrm{conn}}$, ${\overline{r}}_F$, ${\overline{r}}_{\mathrm{reg}}$ and ${\overline{r}}_{\mathrm{RG}}$ be simultaneous upper confidence bounds, and choose declared resolution targets ${ϵ}_{\mathrm{conn}}$, ${ϵ}_F$, ${ϵ}_{\mathrm{reg}}$ and ${ϵ}_{\mathrm{RG}}$. Set

$$\begin{array}{cc}\hfill m_H& ={\displaystyle \frac{{\underline{\chi }}_{N,\alpha }}{{\overline{\Delta }}_{0,\alpha }}}-1,\hfill \end{array}$$

(187)

$$\begin{array}{cc}\hfill m_X& =1-{\displaystyle \frac{{\overline{r}}_X}{{ϵ}_X}},X\in \{\mathrm{conn},F,\mathrm{reg},\mathrm{RG}\},\hfill \end{array}$$

(188)

$$\begin{array}{cc}\hfill {\mathfrak{F}}_N& =min\{m_H,m_{\mathrm{conn}},m_F,m_{\mathrm{reg}},m_{\mathrm{RG}}\}.\hfill \end{array}$$

(189)

Proposition 6

(minimal finite-resolution mechanism certificate). Assume the source metric and all denominators above have simultaneous positive lower confidence bounds. In the branch where the connected spectator block and the regular remainder are required to be separately route even,

$${\mathfrak{F}}_N>0$$

(190)

is sufficient to establish, at the declared regulator, volume, scale and coverage, all of the following: a positivity-compatible hidden completion with the required pole ordering; connected route leakage below ${ϵ}_{\mathrm{conn}}$; nonzero even matching with odd/even leakage below ${ϵ}_F$; a regular remainder below ${ϵ}_{\mathrm{reg}}$; and generated RG leakage below ${ϵ}_{\mathrm{RG}}$.

Proof. 

The first component follows from (183) and the nonempty positive coupling interval in Theorem 9. The other four components are their defining upper-bound inequalities. Simultaneous coverage permits their conjunction. All margins are dimensionless, so taking their minimum only records the weakest passed gate and does not compare quantities with different units. □

This proposition is intentionally modest. It proves finite-resolution feasibility, not the exact continuum equality. An exact mechanism requires a controlled sequence $a\to 0$, $L\to \infty$ and the stated infrared/matching limits in which the four leakage bounds tend to zero while the positive denominators and the hidden-susceptibility margin remain separated from zero. A cancellation between individually nonzero connected and regular blocks could also make the total coefficient vanish, but that would be a different, less-predictive cancellation mechanism and is not counted as route reciprocity in Proposition 6.

20. Numerical Controls and Excluded Shortcuts

Finite numerical controls test parts of the proposed mechanism and its variance reduction, but they do not measure the physical quantity $C_{21}-C_{12}$. Their role is to delimit which approximations can be used as evidence.

1.

A twelve-function hidden-profile source space is algebraically independent, but the associated correlator blocks fail positivity and the relevant Ward residual is large; generalized eigenvalues become complex. This excludes that discrete-profile GEVP as a physical spectrum. It does not exclude an orbital completion with a different positive operator basis.

2.

Five consecutive Euclidean time separations are available in the finite control for which the first flat-extension condition fails and the second holds. In the corresponding exploratory finite-volume calculation, however, the heavy-tailed weights reduce the effective sample size to approximately one dominant source. Algebraic availability of five slices therefore does not provide a statistically controlled physical certificate.

3.

Exact averaging over all 120 unordered pairs in each of sixteen conditioned source groups removes a median $99.1351\%$ of the conditional source-pair second moment. Nevertheless, the normalized route-odd ratios remain finite: the two separately selected groups of eight conditioned sources give medians $0.344711$ and $0.426782$, with respective ranges $0.039240$–$0.707019$ and $0.236649$–$0.520312$. The two groups have different selection designs, are not independent identically distributed physical samples, and are therefore reported separately.

4.

Stabilization of partial sums does not remove the route-odd component. For a Hermitian two-channel matrix, the exact Frobenius distance to a matrix with zero mixed entry is $\sqrt{2}$ times the normalized mixed-entry ratio, so the representative central matrices require order-one relative changes. A targeted degree ladder for the dominant $\Omega$-propagator contributions also shows substantial changes between the tested truncation degrees, so convergence and inverse-solve error remain unresolved. This conclusion applies only to the bare finite surrogate. It neither proves nor disproves equality of the renormalized coefficients, because the determinant modulus, selected scalar vacuum, operator mixing, and matching have not been included.

These results exclude a specific shortcut. Increasing the precision of the same bare screening sum cannot by itself establish or refute Eq. (56). Renormalization-group evolution preserves an already protected zero when the odd covector is a left eigenvector of the anomalous-dimension and threshold maps, but a regular finite evolution does not generically create an exact zero from a nonzero initial value. The next decisive calculation must therefore use either the complete symmetry theorem with all hypotheses verified, or the physical matched two-point response of Section 11 in the selected chiral measure of Section 12.

21. Supporting Same-Pole and Response Controls

This section gives supporting charged-branch results after the finite-error and vacuum-selection reductions. Equations derived from positivity, the stated Hamiltonian form, or the stated moment constraints are exact within those assumptions; displayed numerical values are finite representative controls. None is substituted for an uncomputed renormalized $3+1$ dimensional $\mathrm{SU}{\left(15\right)}_p$ observable.

21.1. Three-Channel Sign Theorem and the $B=(15,1)$ Exclusion

Use the orthonormal Pati–Salam channel basis

$$A=(15,3),B=(15,1),C=(1,3),$$

and define the diagonal contrast operator

$$D_B=diag\left(-{\displaystyle \frac{1}{2}},1,-{\displaystyle \frac{1}{2}}\right).$$

For a positive Schur–Feshbach correction

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

the induced shift of the B-channel coordinate is

$$\delta d_B\left(E\right)=-Tr\left[D_{B}M\left(E\right)\right].$$

For a rank-one contribution $M=mq{q}^{†}$, $m\ge 0$, $\parallel q\parallel =1$, let $p_B={\left|q_B\right|}^2$. Since $|q_A{|}^2+|q_B{|}^2+{\left|q_C\right|}^2=1$,

$$\begin{array}{cc}\hfill Tr\left(D_{B}q{q}^{†}\right)& =-{\displaystyle \frac{1}{2}}(1-p_B)+p_B\hfill \end{array}$$

(191)

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

(192)

and hence

$$\delta d_B=-{\displaystyle \frac{m}{2}}(3p_B-1).$$

(193)

Therefore a positive hidden state produces the required negative B-diagonal shift if and only if

$$p_B>{\displaystyle \frac{1}{3}}.$$

(194)

For the exact one-dimensional route-odd direction used by the canonical three-channel control,

$$q_{-}={\displaystyle \frac{1}{\sqrt{20-6\sqrt{5}}}}\left(\begin{array}{c}\sqrt{5}-3\\ -\sqrt{5}\\ 1\end{array}\right),$$

one finds

$$p_B^{(-)}={\displaystyle \frac{10+3\sqrt{5}}{22}}=0.7594638151\dots >{\displaystyle \frac{1}{3}},$$

and

$$c_{-}\equiv q_{-}^T{D}_B{q}_{-}={\displaystyle \frac{8+9\sqrt{5}}{44}}=0.6391957227\dots .$$

(195)

Consequently an odd-only positive susceptibility has

$$M_{-}\left(E\right)={\chi }_H\left(E\right)P_{-},\delta d_B\left(E\right)=-c_{-}{\chi }_H\left(E\right)<0.$$

(196)

This is the exact bridge between the scalar odd susceptibility and the three-channel B contrast in this one-dimensional odd branch.

The same formula gives an exact negative result for an older proposed rescue mechanism. If the positive hidden correction has support only in the $A=(15,3)$ and $C=(1,3)$ channels, then $M_{BB}=M_{AB}=M_{BC}=0$ and

$$\delta d_B={\displaystyle \frac{1}{2}}(M_{AA}+M_{CC})\ge 0.$$

Thus a positive $03/153$-only Schur correction cannot by itself generate the negative B shift required by the three-channel rescue. Such a sector may remain relevant to route-21 matching, but not to this diagonal rescue mechanism.

Exact control-level limitation.Within the positive Schur–Feshbach interpretation, a hidden contribution must carry more than one third of its normalized weight in the $B=(15,1)$ channel to lower $d_B$. Varying only the restricted $03/153$ positive sector cannot solve this sign problem.

21.2. Exact Translation Between Running and the Three-Channel Coordinate

Let

$$\eta \left(\mu \right)={\displaystyle \frac{{\alpha }_R\left(\mu \right)}{{\alpha }_4\left(\mu \right)}}$$

be the ratio used in the refined gauge-running control, and let d denote the additional symmetry-compatible B-diagonal coefficient. In the declared route-symmetric three-channel family the full kernel obeys the exact identity

$$K(\eta ,d)=K_{\mathrm{aff}}\left(d_{\mathrm{eff}}\right)+2(\eta -1)I_3,d_{\mathrm{eff}}=d+2(1-\eta ).$$

(197)

The second term is a common scalar shift. It changes absolute eigenvalues but not their differences or eigenvectors. Pole ordering in this control family is therefore controlled by $d_{\mathrm{eff}}$ alone.

At $\mu =2.6{\Lambda }_{\mathrm{pre}}$ the refined constant-b control gives

$$\eta =0.6744335583,2(1-\eta )=0.6511328835.$$

The exact odd-ground interval for the additional d coordinate is

$$-8.2570993799<d<-2.1842591755,$$

and imposing the minimal positive Schur realization narrows it to

$$-4.7746393488<d<-2.1842591755.$$

(198)

Thus the benchmark $d=-2$ lies just outside this positive-Schur window by $0.1842591755$, whereas $d=-2.5$ lies inside.

Equation (197) is an exact coordinate identity only inside this explicitly defined control family. It does not identify the physical renormalized B response with a perturbative Casimir ratio. The actual $3+1$ dimensional channel matrix must still be obtained from the same renormalized source and regulator scheme used for the route correlator.

21.3. One Common RG–Hidden–Portal Hamiltonian

At the same control scale, the visible spectrum at $d=0$ is

$$E_{+,0}=6.721093894892478,E_{-}^{\left(0\right)}=10.231869360122246,E_{+,1}=14.344770978004576,$$

so that

$${\Delta }_0\equiv E_{-}^{\left(0\right)}-E_{+,0}=3.510775465229768.$$

(199)

The positive hidden control has gaps and weights

$$\begin{array}{cc}\hfill ({\Delta }_1,{\Delta }_2,{\Delta }_3)& =(4.0198205527,10.0198205527,17.0198205527),\hfill \end{array}$$

(200)

$$\begin{array}{cc}\hfill (w_1,w_2,w_3)& =(8.3472459562,3.7098870916,1.4491746452),\hfill \end{array}$$

(201)

which give

$${\chi }_3={\displaystyle \sum _{a=1}^3}{\displaystyle \frac{w_a}{{\Delta }_a}}=2.531923173071781.$$

At this running point ${\chi }_3<{\Delta }_0$, so the three-state hidden sector by itself does not invert the visible ordering. Adding one odd-projected portal state with

$$g_{45}=0.555038,{\Delta }_{45}=0.25,{\displaystyle \frac{g_{45}^2}{{\Delta }_{45}}}=1.232268725776.$$

The threshold susceptibility becomes

$${\chi }_{\mathrm{tot}}=3.764191898848>{\Delta }_0.$$

The corresponding finite odd Hamiltonian has a lowest eigenvalue

$$E_{-}^{\mathrm{full}}=6.676962005127341<E_{+,0},$$

with binding

$$G_{\mathrm{b}}\equiv E_{+,0}-E_{-}^{\mathrm{full}}=0.044131889765137.$$

(202)

The visible probability of this same eigenstate is

$$Z_v=0.195774020761.$$

This construction is an explicit compatibility control. The values of $g_{45}$ and ${\Delta }_{45}$ are not claimed as physical $\mathrm{SU}{\left(15\right)}_p$ measurements.

21.4. Action-Insertion Response and Why It Was Not Yet the Charged Mass Operator

Differentiating the same Hamiltonian with respect to the portal coupling gives the even action insertion

$${𝒪}_{45,+}^{\mathrm{ctrl}}={\displaystyle \frac{\partial H}{\partial g_{45}}}=|v\rangle \langle 45|+|45\rangle \langle v|.$$

For the normalized rescued state $|X_{-}\rangle$, Feynman–Hellmann gives

$$F_{45,+}^{\mathrm{ctrl}}\left(0\right)={\displaystyle \frac{\partial E_{-}}{\partial g_{45}}}=2a_v{a}_{45}=-0.738865962625676\ne 0.$$

(203)

The exact parity-resolved control also has $F_{-}^{\mathrm{ctrl}}=0$ and $C_{-}^{\mathrm{reg},\mathrm{ctrl}}=0$. This establishes that the interaction that rescues the pole can have a nonzero even same-pole response. It is not yet the charged mass matching because $\partial H/\partial g_{45}$ is an action insertion rather than the low-energy operator carrying the required Pati–Salam and family tensor.

21.5. Same-Pole Charged Mass Matching

The minimal already-defined charged even insertion is taken to act on the visible odd core of the same rescued state,

$${𝒱}_{M,+}^{min}\left(x\right)=g_M{\widehat{F}}_{+}(0;x)P_v,{\widehat{F}}_{+}(0;x)=\sqrt{2}{\displaystyle \frac{x}{sinhx}},$$

(204)

with the full Pati–Salam and family image

$${𝒪}_{M,+}^{min}={𝒱}_{M,+}^{min}\otimes Q_{RT}\otimes D_{+},Q_{RT}=(B-L)T_3^R,D_{+}=diag(1,-2,1).$$

Let $P_v=|v\rangle \langle v|$. Feynman–Hellmann on the same tracked pole gives

$$F_{M,+}^{min}(0;x)=g_M{\widehat{F}}_{+}(0;x)Z_v.$$

(205)

For $x>0$ and nonzero $g_M$, this cannot vanish if $Z_v>0$. At $x=1$, $g_M=1$,

$${\widehat{F}}_{+}(0;1)=1.203379957425177,F_{M,+}^{min}=0.235590532768260.$$

The induced charged-curvature direction is

$$(\delta A_{\ell },\delta A_d,\delta A_u)=F_{M,+}^{min}\left({\displaystyle \frac{3}{4}},-{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}}\right),$$

so that

$$\delta A_{\ell }+2\delta A_d-\delta A_u=0$$

identically. The statement is a compatibility theorem for this minimal embedding: the dressing required for spectral rescue need not decouple the same pole from the charged even operator.

21.6. Two Hidden States Suffice in the Finite Benchmark

The third hidden pole of the original representative is not required. Keeping only $h_1$, $h_2$, and the portal gives

$${\chi }_{12+45}=3.679045604813,{\chi }_{12+45}-{\Delta }_0=0.168270139583>0,$$

and direct diagonalization yields

$$E_{-}=6.692956285484438,G_{\mathrm{b}}=0.028137609408041,$$

$$Z_v=0.180884593134,F_{M,+}(x=1,g_M=1)=0.217672893984.$$

No smaller attractive subset of the fixed benchmark passes. This is a minimality result only for that finite set of poles and couplings; it is not a statement that the physical hidden spectrum contains exactly two states.

For the full representative, standard eigenvalue perturbation bounds imply that the spectral ordering is preserved under Hermitian perturbations satisfying

$$\parallel \delta H_{-}{\parallel }_2+{\parallel \delta H_{+}\parallel }_2<G_{\mathrm{b}}=0.044131889765137.$$

In the symmetric allocation, each block may therefore vary by less than $0.022065944883$. Standard perturbation bounds give a conservative visible-residue floor $Z_v\ge 0.176462368249$ and, at $x=1$, $g_M=1$, $F_{M,+}\ge 0.212351277191$ in the stated perturbation class. These are finite representative robustness bounds; the stronger moment-class theorem below removes the fixed pole count.

21.7. Pole-Count-Free Moment Certificate

Let $d\mu (\Delta )\ge 0$ be an arbitrary positive hidden spectral measure, discrete, continuous, or mixed, with gaps $\Delta >0$ relative to $E_{+,0}$. Define

$$m_k={\int }_0^{\infty }{\Delta }^{k}d\mu (\Delta ),{\chi }_H\left(0\right)={\int }_0^{\infty }{\displaystyle \frac{d\mu (\Delta )}{\Delta }}.$$

(206)

For

$${\left(T_N\right)}_{ij}=m_{i+j+1},{\left(b_N\right)}_i=m_i,i,j=0,\dots ,N-1,$$

positivity gives

$${\chi }_N=b_N^T{T}_N^{-1}{b}_N\le {\chi }_H,$$

with the Moore–Penrose inverse on a rank-deficient Krylov space. For the representative positive measure, the moments are

$$\begin{array}{cccc}\hfill m_0& =13.506307693000,\hfill & \hfill m_1& =95.391526192612,\hfill \end{array}$$

(207)

$$\begin{array}{cccc}\hfill m_2& =927.132236454888,\hfill & \hfill m_3& =11418.922364973629,\hfill \end{array}$$

(208)

$$\begin{array}{cccc}\hfill m_4& =161175.4164342328,\hfill & \hfill m_5& =2453085.1947956034.\hfill \end{array}$$

(209)

It gives

$${\chi }_1=1.912332832684,{\chi }_2=2.446958535635,{\chi }_3=2.531923173072.$$

For $N=2$,

$${\chi }_2={\displaystyle \frac{m_0^2{m}_3-2m_0{m}_1{m}_2+m_1^3}{m_1{m}_3-m_2^2}}.$$

(210)

Hence the four-moment class condition

$$m_1{m}_3-m_2^2>0,{\chi }_2+{\displaystyle \frac{g_{45}^2}{{\Delta }_{45}}}>{\Delta }_0$$

(211)

is sufficient for every positive measure with those four moments. For the displayed moment sequence,

$${\chi }_2+{\displaystyle \frac{g_{45}^2}{{\Delta }_{45}}}-{\Delta }_0=0.168451796181>0.$$

The $N=1$ truncation does not satisfy the criterion, so $m_0,\dots ,m_3$ are the first moments in this sequence that certify the required inequality. The lower bound ${\chi }_2$ is optimal for these four moments: an explicit positive two-node Gaussian quadrature measure reproduces them and saturates the bound. Therefore adding more trial poles cannot improve what is logically guaranteed by these four moments alone.

Without a lower support floor, finitely many positive moments do not give a universal finite upper bound on ${\chi }_H$: arbitrarily weak weight may approach $\Delta =0$ while changing positive moments only slightly. The correct no-extra-assumption statement is one-sided, ${\chi }_H\ge {\chi }_N$. If an independent support condition

$$supp\mu \subset [\delta ,\infty ),\delta >0,$$

is supplied, localizing-matrix positivity gives a finite upper bound. For the stated $\delta =4$ class, the six-moment sequence yields

$$2.531923173072\le {\chi }_H\left(0\right)\le 2.534050669900.$$

(212)

The interval width is only $2.1275\times {10}^{-3}$.

21.8. Uniform Binding Theorem for the Same Positive-Measure Class

The previous threshold inequality proves the existence of a rescued pole but does not by itself give its binding. Let

$$G_{\mathrm{b}}=E_{+,0}-E_{-}>0$$

and define the energy-dependent susceptibility

$${\chi }_H\left(G_{\mathrm{b}}\right)={\int }_0^{\infty }{\displaystyle \frac{d\mu (\Delta )}{\Delta +G_{\mathrm{b}}}},p_{45}\left(G_{\mathrm{b}}\right)={\displaystyle \frac{g_{45}^2}{{\Delta }_{45}+G_{\mathrm{b}}}}.$$

The scalar star reduction obeys the exact secular equation

$${\Delta }_0+G_{\mathrm{b}}={\chi }_H\left(G_{\mathrm{b}}\right)+p_{45}\left(G_{\mathrm{b}}\right).$$

(213)

Equivalently,

$$f\left(G\right)={\chi }_H\left(G\right)+p_{45}\left(G\right)-{\Delta }_0-G.$$

For a positive measure,

$$f^{\prime }\left(G\right)=-\int {\displaystyle \frac{d\mu (\Delta )}{{(\Delta +G)}^2}}-{\displaystyle \frac{g_{45}^2}{{({\Delta }_{45}+G)}^2}}-1<0,$$

so there is at most one rescued root. A positive threshold surplus $f\left(0\right)>0$ and $f\left(G\right)\to -\infty$ establish exactly one positive root.

Using the lower and upper measure bounds propagated away from threshold for the six-moment $\Delta \ge 4$ class gives the class-wide interval

$$0.03741447436\le G_{\mathrm{b}}\le 0.05003749190.$$

(214)

The explicit three-pole representative, $G_{\mathrm{b}}=0.0441318897651$, lies strictly inside this interval.

21.9. Universal Visible-Residue Lower Bound

Normalize the same rescued eigenstate as

$$|X_{-}\rangle =a_v|v\rangle +|h\rangle +a_{45}|45\rangle ,Z_v\equiv {\left|a_v\right|}^2.$$

For the scalar star reduction,

$$Z_v^{-1}=1+{\int }_0^{\infty }{\displaystyle \frac{d\mu (\Delta )}{{(\Delta +G_{\mathrm{b}})}^2}}+{\displaystyle \frac{g_{45}^2}{{({\Delta }_{45}+G_{\mathrm{b}})}^2}}.$$

(215)

For every $\Delta >0$,

$${\displaystyle \frac{1}{{(\Delta +G)}^2}}\le {\displaystyle \frac{1}{G}}{\displaystyle \frac{1}{\Delta +G}},$$

and the same inequality holds for the portal term. Applying the secular equation (213),

$$\int {\displaystyle \frac{d\mu }{{(\Delta +G)}^2}}+{\displaystyle \frac{g_{45}^2}{{({\Delta }_{45}+G)}^2}}\le {\displaystyle \frac{{\Delta }_0+G}{G}}.$$

Therefore

$$Z_v\ge {\displaystyle \frac{G_{\mathrm{b}}}{{\Delta }_0+2G_{\mathrm{b}}}}>0.$$

(216)

Once the positive binding $G_{\mathrm{b}}$ has been established, this inequality is independent of both the hidden pole count and a lower support floor. It is weak when $G_{\mathrm{b}}\ll {\Delta }_0$, but it is completely general within the stated scalar star reduction.

The six-moment support-floor class is much tighter. Combining its moment information with the binding interval gives

$$Z_v\ge 0.179924761839236.$$

(217)

Thus the rescued pole cannot become arbitrarily invisible to the very visible core from which the Schur reduction is built.

21.10. Uniform Same-Pole Mass-Vertex Theorem

Deform the same Hamiltonian by the charged even visible insertion,

$$H\left({\lambda }_M\right)=H+{\lambda }_M{\kappa }_M{P}_v,{\kappa }_M=g_M{\widehat{F}}_{+}(0;x).$$

By Feynman–Hellmann,

$${\displaystyle \frac{d{E}_{-}}{d{\lambda }_M}}{|}_0=\langle X_{-}|{\kappa }_M{P}_v|X_{-}\rangle ={\kappa }_M{Z}_v,$$

(218)

so for an arbitrary positive hidden measure in the class,

$$F_{M,+}=g_M{\widehat{F}}_{+}(0;x)Z_v.$$

(219)

At $x=1$, $g_M=1$,

$${\widehat{F}}_{+}(0;1)=1.203379957425177,$$

and Eq. (217) gives

$$F_{M,+}\ge 0.216517852241835>0.$$

(220)

This converts the positive-measure moment class into a uniform same-pole non-decoupling certificate: one Hamiltonian class simultaneously supplies spectral rescue and a nonzero minimal even charged mass response.

A paired finite-difference check of the explicit representative, using ${\lambda }_M=\pm {10}^{-6}$, gives

$$F_{M,+}^{\mathrm{FD}}=0.235590532859931,$$

whereas Eq. (219) gives

$$F_{M,+}^{\mathrm{exact}}=0.235590532768261.$$

The absolute difference is $9.17\times {10}^{-11}$.

For an additional unknown parity-even mass insertion $\delta {𝒱}_{+}$, the class-wide sign is certainly preserved if

$$\parallel \delta {𝒱}_{+}{\parallel }_2<0.2165178522.$$

(221)

If the correction acts only in the hidden-plus-portal block and has no visible–hidden cross matrix elements, a sharper sufficient budget is

$$\parallel \delta {𝒱}_{+,H}{\parallel }_2<0.2640219362.$$

(222)

These are norm budgets in the declared normalization, not estimates of the actual physical omitted operator.

21.11. Exact Same-Pole Surrogate in the Bare Eight-State $\mathrm{SU}\left(3\right)$ Control and Its Ordering Obstruction

A separate check is useful because it prevents a misleading future proposal to “complete” the programme by rerunning only a three-point function in the bare eight-state confining control. In its route-odd sector,

$$H_{-}^{\left(8\right)}=\left(\begin{array}{cc}{\Delta }_F& -t\\ -t& {\Delta }_F\end{array}\right),|X_{-}^{\left(8\right)}\rangle ={\displaystyle \frac{|J_{-}\rangle +|K_{-}\rangle }{\sqrt{2}}},E_{-}^{\left(8\right)}={\Delta }_F-t.$$

The parity-even action insertion $\partial H_8/\partial {\Delta }_F=P_{\mathrm{split}}$ is the identity on this odd two-dimensional sector. Hence

$$\langle X_{-}^{\left(8\right)}\left|P_{\mathrm{split}}\right|X_{-}^{\left(8\right)}\rangle =1,$$

(223)

and the corresponding normalized three-point/two-point ratio is exactly one for all insertion times. A route-odd insertion changes sign under exchange and therefore has vanishing diagonal matrix element on the same parity eigenstate; its diagonal three-point function vanishes exactly. Thus the bare control has an exact same-pole surrogate pattern “even nonzero, odd zero” for these action insertions.

This positive surrogate does not solve the spectral gate. The lowest even eigenvalue is

$$E_{+,0}^{\left(8\right)}={\displaystyle \frac{{\Delta }_F-2t-\sqrt{{({\Delta }_F-2t)}^2+12t^2}}{2}},$$

so

$$E_{-}^{\left(8\right)}-E_{+,0}^{\left(8\right)}={\displaystyle \frac{{\Delta }_F+\sqrt{{({\Delta }_F-2t)}^2+12t^2}}{2}}>0({\Delta }_F>0).$$

(224)

For ${\Delta }_F=1$, $t=0.30$, this is $1.0567764363$. The bare eight-state $\mathrm{SU}\left(3\right)$ control therefore cannot itself realize the required odd-ground-state inversion. This is consistent with the Perron–Frobenius even-ground-state argument already used in the control hierarchy, but Eq. (224) is the explicit closed-form gap.

Exact control-level exclusion. A bare same-pole action-insertion result in the eight-state $\mathrm{SU}\left(3\right)$ model is not a physical mass-matching result. That control has the wrong pole ordering for every ${\Delta }_F>0$. A future $\mathrm{SU}\left(3\right)$ calculation is informative only if a faithful charged mass operator is embedded in a confining completion that independently satisfies the spectral-inversion condition; repeating the bare action-insertion test would not address the missing physical input.

21.12. What Is Now Closed, What Remains Open, and the Minimal Next Physical Test

The late-stage sequence changes the logical status of several proposed next steps.

Strictly established within the declared controls. The sign condition (193), the exact coordinate translation (197), the pole-count-free four-moment certificate (211), the monotone secular equation (213), the class-wide binding interval (214), the gap-only residue bound (216), the six-moment residue floor (217), and the same-pole Feynman–Hellmann identity (219) all follow from the explicitly stated positive-measure/star-reduction assumptions. The bare eight-state $\mathrm{SU}\left(3\right)$ ordering obstruction (224) is exact.

Finite representative checks. The common Hermitian completion has $G_{\mathrm{b}}=0.0441318897651$, $Z_v=0.195774020761$, and $F_{M,+}=0.235590532768$, lies inside the class-wide binding interval, and agrees with the paired finite-difference Feynman–Hellmann value to $9.17\times {10}^{-11}$. The two-hidden-plus-portal subset also passes. These numbers check the finite representative under the stated assumptions.

Still open in the physical theory. The renormalized moments of the explicit 24-preon route source, the physical 45-channel response, the physical pole binding and residue, the fully renormalized charged mass insertion, the continuum and infinite-volume limits, $F_{-}\left(0\right)$, and the pole-subtracted $C_{-}^{\mathrm{reg}}$ have not been computed in the full chiral $3+1$ dimensional $\mathrm{SU}{\left(15\right)}_p$ theory. The independent dynamical alignment of $D_{+}$ also remains open. Therefore the uniform control theorem must not be presented as a first-principles mass prediction.

The most economical future calculation should keep one tracked pole branch and one renormalization scheme and determine, on the same ensembles or regulated Hamiltonian family:

1.

a physical pole binding $G_{\mathrm{b}}$ or a positive moment lower certificate for the same branch;

2.

the paired $+{\lambda }_{+}$ Feynman–Hellmann slope of the renormalized charged even operator, which can replace a separate diagonal three-point plateau fit for $F_{+}\left(0\right)$;

3.

the paired $\pm {\lambda }_{-}$ slope for the odd operator, testing $F_{-}\left(0\right)=0$;

4.

the pole-subtracted regular odd remainder $C_{-}^{\mathrm{reg}}$, which is not determined by Feynman–Hellmann slopes alone;

5.

the one-sided finite-volume/step-scaling leakage ${\left({\Sigma }_{\mathrm{rt}}\right)}_{-+}$ with covariance and continuum control.

If the scalar star reduction applies to the tracked physical branch, any directly resolved positive $G_{\mathrm{b}}$ and residue should also satisfy the consistency inequality (216). Violation would exclude that reduction even if a pole were seen.

Scope after the uniform same-pole certificate. No further hidden poles need be added merely to prove possibility; no full hidden spectrum needs to be reconstructed if the moment certificate is available; no new finite Hamiltonian is needed merely to re-prove that the same rescued pole can have $F_{M,+}\ne 0$; and a bare eight-state $\mathrm{SU}\left(3\right)$ same-pole action-insertion calculation cannot settle the physical problem because its ordering obstruction is exact. New work should supply physical $3+1$ dimensional response data or a genuinely stronger theorem about those data.

22. Scope of the Higher-Dimensional Constructions

The higher-dimensional constructions are not premises of the four-dimensional reciprocity theorem. The “five-dimensional” description is an exact Jacobi representation of a selected positive four-dimensional spectral measure: if $c\left(\zeta \right)=\int d\mu \left(s\right)/(\zeta +s)$ with $d\mu \ge 0$, Lanczos recursion gives a semi-infinite Jacobi operator $H_J$ such that $c\left(\zeta \right)=\langle 1\left|{(\zeta I+H_J)}^{-1}\right|1\rangle$. This representation is effective and does not imply a fundamental fifth spacetime dimension.

The six-dimensional vortex construction is only a prospective environment for chirality and localization. Its physical metric-fluctuation threshold, confining transverse phase, full matter content, and anomaly/inflow completion have not been established, so it supplies no evidence for ${\Delta }_{\Pi }=C_{21}-C_{12}$. The tested seven-dimensional embedding is instead excluded as a realization of the intended spectrum: the projection produces two selected four-dimensional Weyl modes per parent and hence 810 rather than 405 modes. Further details of these auxiliary constructions appear in Ref. [10]; none is used in the four-dimensional proof.

23. Discussion: Central Results, Supporting Controls, and Open Physical Inputs

The results form a strict hierarchy. At the first level are statements that follow from finite-dimensional route algebra or from the displayed symmetry hypotheses. At the second are constructive existence theorems showing that the hypotheses are compatible with interacting dynamics. At the third are regulator and control calculations that test mechanisms or eliminate shortcuts. The final level consists of physical inputs that have not yet been calculated. Keeping these levels separate is essential because a strong result at one level cannot substitute for a missing premise at another.

Table 5. Logical hierarchy of the principal results.

Level	Principal result	Meaning and limitation
Exact	${\Delta }_{\Pi }=C_{21}-C_{12}$; conditional renormalized reciprocity; neutral no-prediction and W2 non-closure theorems	The charged target is exactly the crossed-route difference. A complete unbroken linear involution enforces its vanishing. The charged-invisible direction is an independent mixed neutral operator, so charged data alone cannot close the neutral sector.
Exact matching and transport	${\mathbf{a}}_R=Z^{-T}{\mathbf{a}}_B$; dual projector $\mathbf{q}\ne {\mathbf{n}}_{\mathrm{dir}}$; codimension-one null-transport theorem	A one-sided RG entry preserves the physical zero only in a closed homogeneous two-coordinate reduction whose odd coordinate is $\mathcal{N}{\Delta }_{\Pi }$. A generic m-dimensional basis requires $m-1$ tangent-to-target conditions.
Exact observable reduction	Mass-only identity and direct noncommuting $G_2$ theorem	The target can be extracted from matched running mass singular values or from the spectrum of $J{F}_R^{-1}{J}^{†}{F}_L^{-1}$. Raw composite correlators still require contact, regular, threshold, and scheme matching.
Exact state constraint	$U{\left(1\right)}_{\chi }$ Ward obstruction and connected six-preon theorem	The mixed mass block vanishes in the symmetric state. A nonzero mass requires an oriented broken state and a nonzero projection of the connected six-preon cumulant.
Constructive	Positive channel metric, interacting symmetric vacuum, and same-pole control classes	Exact route symmetry is compatible with nontrivial stable dynamics. These constructions prove nonemptiness of the sufficient conditions, not their realization by the physical $\mathrm{SU}{\left(15\right)}_p$ vacuum.
Numerical control	Representation-correct sources, exterior-power contractions, Rao–Blackwell reductions, and excluded shortcuts	The source tensors and contractions are computable, and major conditional noise components can be removed exactly. Fixed bare route residuals are not renormalized physical coefficients.
Open physical input	Chiral determinant/Pfaffian modulus, scalar parameter point, selected vacuum, complete matching, and continuum/volume limits	The specialized overlap regulator and determinant-line phase are defined on a controlled admissible core; the missing modulus and physical state determine whether that construction supplies the required ensemble and whether the two-point spectrum can be converted into the common running-mass scheme.

Table 5. Logical hierarchy of the principal results.

Level	Principal result	Meaning and limitation
Exact	${\Delta }_{\Pi }=C_{21}-C_{12}$; conditional renormalized reciprocity; neutral no-prediction and W2 non-closure theorems	The charged target is exactly the crossed-route difference. A complete unbroken linear involution enforces its vanishing. The charged-invisible direction is an independent mixed neutral operator, so charged data alone cannot close the neutral sector.
Exact matching and transport	${\mathbf{a}}_R=Z^{-T}{\mathbf{a}}_B$; dual projector $\mathbf{q}\ne {\mathbf{n}}_{\mathrm{dir}}$; codimension-one null-transport theorem	A one-sided RG entry preserves the physical zero only in a closed homogeneous two-coordinate reduction whose odd coordinate is $\mathcal{N}{\Delta }_{\Pi }$. A generic m-dimensional basis requires $m-1$ tangent-to-target conditions.
Exact observable reduction	Mass-only identity and direct noncommuting $G_2$ theorem	The target can be extracted from matched running mass singular values or from the spectrum of $J{F}_R^{-1}{J}^{†}{F}_L^{-1}$. Raw composite correlators still require contact, regular, threshold, and scheme matching.
Exact state constraint	$U{\left(1\right)}_{\chi }$ Ward obstruction and connected six-preon theorem	The mixed mass block vanishes in the symmetric state. A nonzero mass requires an oriented broken state and a nonzero projection of the connected six-preon cumulant.
Constructive	Positive channel metric, interacting symmetric vacuum, and same-pole control classes	Exact route symmetry is compatible with nontrivial stable dynamics. These constructions prove nonemptiness of the sufficient conditions, not their realization by the physical $\mathrm{SU}{\left(15\right)}_p$ vacuum.
Numerical control	Representation-correct sources, exterior-power contractions, Rao–Blackwell reductions, and excluded shortcuts	The source tensors and contractions are computable, and major conditional noise components can be removed exactly. Fixed bare route residuals are not renormalized physical coefficients.
Open physical input	Chiral determinant/Pfaffian modulus, scalar parameter point, selected vacuum, complete matching, and continuum/volume limits	The specialized overlap regulator and determinant-line phase are defined on a controlled admissible core; the missing modulus and physical state determine whether that construction supplies the required ensemble and whether the two-point spectrum can be converted into the common running-mass scheme.

The mass-only route is the most consequential practical simplification. The observable is a shape of three singular values per charged sector and is insensitive to a sector-wide scalar normalization. It does not require the separate reconstruction of every microscopic Higgs vertex. At the same time, the exact counterexample in Eq. (133) shows why this simplification is not a prediction: a single unfixed relative coefficient can move the target. The correct question is therefore not whether enough algebraic invariants can be written down, but whether the physical two-point response identifies the median logarithmic shape after all additive matching directions have been controlled.

The chiral analysis changes the interpretation of a vanishing mixed propagator. In a $U{\left(1\right)}_{\chi }$-invariant or vacuum-averaged state, $J_s=0$ is required even if massive oriented vacua exist. Conversely, a nonzero $J_s$ probes a connected six-preon channel rather than a mandatory elementary $\langle \Omega \Omega \rangle$ line. A physical calculation must therefore specify the action, select the state, and include the appropriate determinant weight (or a demonstrably equivalent Pfaffian formulation) before a zero or nonzero mixed block is interpreted as a mass statement.

The route question is now considerably more constrained than a generic search for a hidden symmetry. The unchanged chiral theory does not contain a local microscopic transformation that exchanges the two complete crossed routes: the route swap is neither an inner nor an outer $\mathrm{SU}{\left(15\right)}_p$ automorphism, and the two local Pati–Salam tensors have unequal ranks, 4 and 8, for the rank-one breaking direction used here. These facts close the local-symmetry branch under the stated assumptions. They do not close the composite branch. In the physical three-channel space the explicitly constructed involution $S_{\mathrm{ch}}$ exchanges the visible route images, preserves a positive metric, and admits positive spectra with distinct even and odd energies. The route symmetry is therefore allowed precisely where a confining composite theory would be expected to generate it: in a mixed channel space after confinement and Pati–Salam breaking, rather than as a local transformation of the microscopic fields.

The nonlinear two-node construction adds an important second layer. It shows that the allowed channel symmetry can coexist with nontrivial propagation, a strictly positive odd curvature, and a unique globally selected symmetric vacuum. Because the coefficients of that action are chosen within a positive control domain rather than derived from $\mathrm{SU}{\left(15\right)}_p$, this is a proof of compatibility and vacuum-selection possibility, not a prediction of the physical vacuum. The symmetry-breaking countercontrol is equally important. It keeps the vacuum on the plane and preserves positive odd curvature while making $C_{+-}$ nonzero. Therefore “the vacuum lies on the plane” and “the plane is locally stable” are not sufficient substitutes for a direct measurement of the mixed even–odd correlator. The remaining route-sector question has become operational: determine whether the renormalized physical correlator realizes the already identified channel-space symmetry.

The phenomenological input must retain its modest status. The charged-mass plane was identified from the data analyzed in Ref. [10], and the present covariance-aware significance is weak. The separate rank-and-scale study with Lytvynenko [11] is used only as a methodological comparison. What gives the curvature combination theoretical value here is not its numerical smallness but the independent route identity ${\Delta }_{\Pi }=C_{21}-C_{12}$ in the fixed Pati–Salam basis. The explicit source construction then proves that both crossed channels exist in the unchanged field content and that the required nontrivial color coefficient arises from fixed recoupling algebra. At the same time, the all-length closure analysis rules out an overly strong interpretation in which the two complete microscopic histories are related by literal reversal. The only viable explanation left within this framework is dynamical equality of the renormalized crossed amplitudes, supplemented by independent family alignment and low-energy matching.

Family alignment remains a distinct dynamical problem. A single antisymmetric three-family spurion cannot pick out a unique median direction. The symmetric spurion can reconstruct the median projector basis covariantly, and a gauge-neutral ${\overline{\mathbf{6}}}_F$ composite channel supplies a constructive nonlinear vacuum branch. This spurion projector $P_2\left(H_S\right)$ is not the physical mass projector $P_2\left(M_s{M}_s^{†}\right)$ without an additional alignment theorem. The restricted $U{\left(1\right)}_X\times SU{\left(4\right)}_F$ branch of the separate four-flavour parent embedding goes further by forbidding the parent cubic invariant that would induce the light-family determinant term. This selection rule is exact within that branch but is not an exact symmetry of the minimal three-$\psi$ Pati–Salam theory. Other EFT completions can generate the determinant operator, although the determinant-tolerant theorem still leaves a finite globally stable domain. The physical completion must therefore determine both the branch and the renormalized scalar coefficients rather than having them fixed by the route argument.

Matching remains an independent requirement, but the mass-only and route-certificate versions must be distinguished. The primary calculation must subtract additive source terms and identify the matched running-mass blocks before the singular-value estimator is applied. In the stronger route-resolved test, a small route-breaking kernel, or even exact route exchange in the two-point function, does not by itself produce the charged-fermion relation: the same pole branch must also satisfy $F_{+}\left(0\right)\ne 0$, $F_{-}\left(0\right)=0$, and $C_{-}^{\mathrm{reg}}=0$ after amputation, mixing subtraction, and a common pole/regular decomposition. These zero/nonzero statements are stable under finite invertible parity-preserving changes of source normalization and renormalization scheme when the decomposition is implemented consistently. The finite channel Jacobian makes the bridge explicit. For route 21 the source algebra fixes $B_{21}=-2E_{03}+6E_{153}$, so only one reduced response ratio $r_{153/03}$ and one overall amplitude remain on that side. This is genuine progress, but it is deliberately one-sided and auxiliary to the direct mass observable.

The dual matching analysis prevents a further shortcut. Operator matrix elements transform with Z, whereas their coefficients transform with $Z^{-T}$. Consequently the direct deformation ${\mathbf{n}}_{\mathrm{dir}}$ and the dual target projector $\mathbf{q}$ have distinct roles, and the bare row representing the target is ${\mathbf{w}}_B=Z^{-1}{\mathbf{q}}_R$. The null-transport theorem then separates the remaining computation into three decisions: establish the initial physical scalar, establish that the RG odd coordinate is a nonzero normalization of that same scalar in a closed basis, and only then test its one-sided RG generation. Computing ${\left({\Sigma }_{\mathrm{rt}}\right)}_{-+}$ for an unmatched bare response would decide none of these physical questions. Conversely, once the bridge is proved, only the six real response components in Eq. (162), rather than an entire second $2\times 2$ matrix, are needed for one linear scale step.

The lower-dimensional calculations supply mechanism tests rather than direct evidence for the four-dimensional numerical answer. Their collective value is nonetheless substantial. The confining $1+1$ dimensional $\mathrm{SU}\left(3\right)$ control shows that equal route-sensitive excitation costs can generate exact reciprocity dynamically and that unequal costs are suppressed by the flux gap. The representation-matched $\mathrm{SU}\left(N\right)$ open chain establishes an $O\left(N\right)$ odd gap uniformly in chain length, and the long-chain $\mathrm{SU}\left(15\right)$ calculation shows finite large-volume limits for the pole-tracked susceptibility and its cancellation conditioning. Point splitting, plaquette and cross-junction recoupling, string breaking, transverse $2+1/3+1$ dimensional dressing, and variational source decontamination were included precisely because each can invalidate a tempting shortcut. These controls demonstrate, for example, that large N does not suppress every numerator, that configuration-space heaviness is not the same as spectral heaviness, that transverse plaquettes can preserve a gap without filtering an additive odd source, and that unscreened string arguments fail once string breaking reorganizes the spectrum. The control hierarchy therefore contributes both positive mechanisms and explicit exclusions, while none of its numerical values is imported into the physical four-dimensional theory.

The higher-dimensional material has a deliberately limited role. Positivity and Lanczos reconstruction give an exact Jacobi representation of a selected four-dimensional cyclic correlator, without implying a fundamental fifth dimension. The six-dimensional construction remains incomplete, and the tested seven-dimensional embedding has the wrong zero-mode multiplicity. None of these statements is used to support the four-dimensional numerical relation.

The positive-measure analysis further narrows the alternative route certificate. No finite hidden-pole ansatz is needed, even at the control level. Four hidden moments plus the portal response suffice for a one-sided spectral certificate, while six moments and a lower support floor can make the statement two-sided. In that class the rescued binding, the visible residue, and the minimal charged even Feynman–Hellmann slope are all uniformly separated from zero. The central result is therefore a same-pole implication rather than another isolated existence example. The exact bare-$\mathrm{SU}\left(3\right)$ control supplies the complementary warning: an even nonzero/odd zero same-pole response is insufficient if the spectral ordering is wrong. These statements strengthen the independent route-resolved diagnostic but do not displace the direct $G_2$ mass-shape theorem.

The final first-principles calculation is technically difficult but conceptually ordered. A fully specified chiral/scalar measure and oriented state must first yield the jointly averaged and renormalized blocks $F_{L,s}$, $F_{R,s}$, and $J_s$ for all three charged sectors. Source contacts, heavy regular terms, condensate corrections, thresholds, and scheme conversion must then be removed or bounded before Eq. (107) is evaluated inside joint resamples. This initial-null estimator is the first expensive physical observable because it decides whether the target vanishes at the matching scale. Only after an operator-closure proof identifies $u_{-}=\mathcal{N}{\Delta }_{\Pi }$ does the one-sided step-scaling entry become the next minimal observable; it then decides whether RG evolution regenerates an odd defect. On the same physical branch, a parity-resolved multi-profile basis may additionally determine the route correlator $C_{ij}^{AB}\left(t\right)$, its tracked pole and threshold, $K_{\perp }$, $J_{-}$, $C_{+-}\left(t\right)$, and the paired Feynman–Hellmann or three-point responses. The latter quantities test the mechanism more deeply; the former directly decide the charged mass shape. Either outcome is scientifically decisive because the observable, matching requirements, and acceptance conditions are fixed before the physical data are known.

24. Conclusions

The charged-mass observation motivating this work is deliberately modest. For running Yukawa eigenvalues evaluated at one scale and in one scheme, ${\Delta }_{\Pi }=A_{\ell }+2A_d-A_u$ lies about $1.56\sigma$ from zero in the stated covariance reconstruction. The plane ${\Delta }_{\Pi }=0$ is therefore a falsifiable structural hypothesis, not an established empirical law.

Its operator content is exact. In the fixed four-route Pati–Salam basis,

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

(225)

The same route map also proves a sharp limitation: its charged projection has kernel $span(1,1,1,1)$, and that invisible direction changes the neutrino-Dirac curvature. At source level the sum of the four routes is the independent mixed operator $-{𝒪}_X$; its absence requires the separate renormalized closure row $e_X^T{\gamma }_c{P}_{\overline{X}}=0$. Charged data consequently give neither a sign nor a proper finite interval for the neutral quantity, even if $C_{12}=C_{21}$ is imposed.

The origin and scope of reciprocity are now separated. A local microscopic exchange is impossible under the stated field content and rank-one Pati–Salam breaking, because the crossed local tensors have ranks four and eight. The confined channel space nevertheless admits the exact positive- metric involution $S_{\mathrm{ch}}$. If the complete renormalized functional, the selected state and external input, and every stage of source renormalization, operator mixing, amputation, pole subtraction, regular completion, and finite counterterm matching intertwine this involution, then $C_{12}=C_{21}$ follows. An interacting strictly convex control proves that the sufficient hypotheses are realizable, while the countercontrol proves that a symmetric vacuum location and positive odd curvature alone do not establish reciprocity.

Renormalization fixes which null is being transported. Matrix elements obey ${\mathbf{p}}_R=Z{\mathbf{p}}_B$, but coefficient vectors obey ${\mathbf{a}}_R=Z^{-T}{\mathbf{a}}_B$. In the complete four-channel basis the dual projector $\mathbf{q}={(1,-2,1,0)}^T/4$ is not the direct deformation ${\mathbf{n}}_{\mathrm{dir}}={(0,-1,2,1)}^T$. The bare row for the physical target is therefore ${\mathbf{w}}_B=Z^{-1}{\mathbf{q}}_R$. The exact null-transport theorem shows that ${\left({\Sigma }_{\mathrm{rt}}\right)}_{-+}=0$ preserves ${\Delta }_{\Pi }=0$ only after the response is proved closed and its odd coordinate is $u_{-}=\mathcal{N}{\Delta }_{\Pi }$. Outside that two-coordinate reduction, a generic m-dimensional basis has $m-1$ tangent-to-target mixing conditions, and affine or nonlinear odd terms must also vanish.

The primary physical observable is the mass-only two-point estimator. With $F_{L,s}\succ 0$, $F_{R,s}\succ 0$, and a rank-three mixed block $J_s$, the singular values of $W_s=F_{L,s}^{-1/2}{J}_s{F}_{R,s}^{-1/2}$ give the sectoral mass shape and hence ${\Delta }_{\Pi }$ without a separate Higgs-vertex reconstruction. This algebraic reduction does not remove physical matching: additive contacts, heavy regular terms, condensate corrections, thresholds, and scheme conversion must still be controlled. Nor does symmetry alone fix the state. The global $U{\left(1\right)}_{\chi }$ is free of the relevant $\mathrm{SU}{\left(15\right)}_p^2-U{\left(1\right)}_{\chi }$ ABJ anomaly and forces $J_s=0$ in an invariant state; a nonzero raw mass block is a projection of the connected six-preon cumulant. The complete two-scalar action and the generic rank-nine mass-tensor basis make the remaining relative-shape freedom explicit rather than hiding it in an unspecified coefficient.

Four linked requirements remain. The scalar/flavour action point and oriented broken state must be specified independently of the desired value of ${\Delta }_{\Pi }$; the full chiral determinant modulus (or a demonstrably equivalent Pfaffian formulation) must enter the measure; the composite blocks must be matched to one running-mass convention; and the continuum, volume, and statistical limits must control the nonlinear singular-value estimator. The overlap regulator, admissible-domain gap bounds, sector indices, and determinant-line phase are already explicit on a controlled core; it is the physical modulus, complete background, selected state, and universality of the resulting dynamical ensemble that remain open. The route-resolved correlator, spectral moments, and paired Feynman–Hellmann or three-point responses are retained as an independent, stronger test of the symmetry and its matching, not as a competing definition of the primary mass observable.

Thus the article establishes a hierarchy rather than a first-principles mass prediction: an exact charged operator identity, a proven sufficient mechanism for renormalized reciprocity, constructive compatibility with an interacting symmetric vacuum, a direct physical two-point target, a minimal separation of initial-null, RG-preservation, and continuum requirements, and exact boundaries on what the charged sector cannot predict. The next expensive observable is the matched mass-only initial-null scalar; the one-sided RG block becomes meaningful only after the physical odd-coordinate bridge is established. Whether the physical renormalized $\mathrm{SU}{\left(15\right)}_p$ state realizes that mechanism remains a well-defined dynamical question.