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

This work tests a concrete and falsifiable hypothesis: whether an anomaly-consistent preon model with confining gauge group \(\mathrm{SU}(15)_p\) an account for a simple pattern in the spectrum of the nine charged fermions. For each charged sector \(s\in\{\ell,d,u\}\)—charged leptons, down-type quarks, and up-type quarks—we introduce the dimensionless logarithmic curvature \(A_s=\frac14\bigl(\ln y_{s,1}-2\ln y_{s,2}+\ln y_{s,3}\bigr),\) where \(y_{s,g}\) is the renormalized Yukawa-matrix eigenvalue of generation \(g=1,2,3\), with all quantities evaluated in a common renormalization scheme and at a common scale. We study \(\Delta_\Pi\equiv A_\ell+2A_d-A_u.\) For current central running inputs this combination is close to zero, but a source-informed covariance treatment supports only a weak diagnostic significance of about \(1.56 \sigma\). Accordingly, \(\Delta_\Pi\simeq0\) is treated here not as an established empirical law but as a phenomenological clue that requires an independent microscopic explanation. In the four-dimensional operator description we derive the exact identity \(\Delta_\Pi=C_{21}-C_{12},\) where \(C_{12}\) and \(C_{21}\) are coefficients of two independent crossed composite operators that differ by the order of two internal binding routes. Hence the exact plane \(\Delta_\Pi=0\) is equivalent to route reciprocity, \(C_{12}=C_{21}\). Within the unchanged fundamental field content of \(\mathrm{SU}(15)_p\), an explicit gauge-invariant point-split 24-preon source is constructed. The number 24 refers to the structure of the interpolating Wilson-network operator and is not identified with a proven number of physical constituents of a bound state. For the involution \(\tau\), defined as the reflection that exchanges the two routes and obeys \(\tau^2=1\), the explicit antisymmetric 24-preon source is an eigenoperator with eigenvalue -1. Thus the required one-dimensional sign sector already exists in the four-dimensional microscopic operator space. The dynamical problem is formulated by Schur–Feshbach reduction. Hidden composite states generate a positive-semidefinite matrix susceptibility that subtracts from the visible two-route kernel. For a general Hermitian two-route kernel, the existence of a route-exchange involution is equivalent, after an allowed relative phase choice, to equality of the two diagonal entries; the phase of the off-diagonal entry is not by itself a basis-invariant obstruction. For the known reduced kernel, the required diagonal compensation is 0.049406060606061 in the adopted dimensionless normalization. We prove the existence of positive-semidefinite hidden sectors that realize this compensation, and of exact Hermitian extensions in which the route-odd state is a nondegenerate ground state. This establishes the absence of an algebraic or naturalness-size obstruction, but it does not replace a nonperturbative calculation of the actual spectrum of the confining theory. To connect the construction directly to Euclidean calculations, the hidden susceptibility is represented by a positive-semidefinite Euclidean memory kernel. Route reciprocity and its energy stability become moment sum rules for the difference of the two diagonal memory-kernel components. Consequently, testing higher-order route locking does not require prior reconstruction of individual hidden poles. For the physical route-odd correlator we derive finite-time moment criteria for subthreshold existence, total subthreshold weight, and minimal multiplicity. We further prove an exact flat-extension criterion for a finite block-Krylov space: vanishing of the next-layer Schur complement means that the finite Krylov space has closed and equals the full cyclic subspace accessible to the chosen sources. Only under this closure condition does the number of subthreshold Ritz values become the exact physical number of source-supported bound states. If closure has not occurred, finitely many positive moments cannot impose a universal upper bound on arbitrarily weak additional bound states; their total spectral weight in a specified region can nevertheless be bounded rigorously by the flat-extension residual. The five-dimensional construction is treated as a possible effective composite description rather than as a mandatory fundamental spatial dimension. In the minimal one-band Jacobi–Robin class, a single physical scalar spectral measure determines the asymptotic bulk Jacobi coefficients, the boundary defect, and the canonically normalized Robin parameter. At the same time, we prove a fundamental limitation of scalar spectral data: masses and a positive scalar spectral measure do not determine the route parity of the lowest state. That information requires the matrix two-point function of two microscopically defined route sources. For this 2\(\times\)2 correlator we obtain finite-time inertia criteria for the number of source-supported subthreshold directions and a three-slice lower bound on route-odd subthreshold spectral weight, with no pole fit and no \(t\to\infty\) limit. Two further qualifications prevent a circular symmetry argument. First, the exact continuous flavor transformations of the published four-dimensional field content that commute with the gauged Pati–Salam embedding act separately on its inequivalent spectator blocks and cannot exchange the two routes. Second, after whitening, any full-rank 2\(\times\)2 correlator exactly saturated by two nondegenerate states possesses a spectral reflection. Thus a clean two-pole generalized-eigenvalue signal is necessary but is not, by itself, evidence for a microscopic route symmetry. The common reflection must remain stable when further states and continuum contributions are resolved, and an independently renormalized matching vertex must identify its even covector with the mass-plane direction. For a literal six-dimensional ultraviolet completion we analyze a smooth gravitating Einstein–Abelian–Higgs vortex with vectorlike six-dimensional parent fermions. Under the stated assumptions it supports exactly one four-dimensional Weyl zero mode per parent multiplet and satisfies the corresponding local and direct-product global anomaly tests. The minimal smooth two-derivative construction nevertheless has a structural limitation for a generic sourced radial "sausage" response: standard bulk localization of a constant four-dimensional gauge zero mode forces the angular radius to possess an interior turning point, and the standard scalar, vector, and tensor additions considered here do not remove this obstruction in a controlled way. A healthy \(R+\beta R^2\) scalaron supplies a genuine local scalar degree of freedom but does not generically eliminate the turning condition. We therefore examine a qualitatively different nonminimal possibility inspired by bulk-confinement localization. The smooth no-turn geometry \(L(r)=R_c\tanh(r/R_c),\qquad M(r)=\cosh^{-\kappa}(r/R_c)\) has a regular axis, no finite-radius turning point, a finite four-dimensional Planck norm, and a finite confinement-weighted gauge norm for every positive confinement exponent. It can be promoted to an exact effective background: a local two-field nonlinear sigma model with positive kinetic metric supports the same cylinder and solves the Einstein–matter equations analytically. A separate numerical boundary-value calculation finds a nonempty order-one-backreaction window in which a unit-winding Abelian–Higgs vortex coexists with the stabilizing sector. The compact phase is not an independent scalar zero mode but the Stueckelberg coordinate of angular diffeomorphisms. In the physical \(\theta\)-independent scalar/radion sector the exact Einstein-frame potential is a fake-supergravity potential plus a nonnegative square, and the coupled spin-zero Hamiltonian factorizes as a sum of positive operators. Smooth-cap boundary conditions exclude both \(m_4^2<0\) and a normalizable \(m_4^2=0\) eigenmode. The first non-axisymmetric sector can now be stated more sharply. Every regular real dipole perturbation of the sigma fields is a globally admissible scalar-clock deformation. Diffeomorphism invariance then implies that the stationary metric completion of any such clock direction has exactly zero Schur curvature. In particular, the previously found fixed-metric eigenvalue \(-0.347963/R_c^2\) is not a physical mass eigenvalue and must not be carried into the final stability test. After quotienting the scalar clocks and eliminating the constraints, any remaining \(m_4^2<0\) mode, if it exists, must be a discrete normalizable finite-core metric state. Its existence is exactly equivalent to a Birman–Schwinger threshold criterion: for a nonnegative reference operator \(H_+\) and the negative finite-core part \(G_-\) of the physical perturbation, \(B(\mu)=G_-^{1/2}(H_++\mu)^{-1}G_-^{1/2},\qquad \beta_0=\lim_{\mu\downarrow0}\lambda_{\max}B(\mu),\) with \(\beta_0<1\) excluding physical tachyons and \(\beta_0>1\) implying at least one. The marginal case \(\beta_0=1\) is a threshold problem. Because the essential spectrum begins at \(m_4^2=0\), no universal argument based only on a small gravitational correction is sufficient. The explicit constrained Einstein operator, and hence the physical value of \(\beta_0\), remain to be calculated; the six-dimensional branch is therefore prospective rather than complete. An ordinary finite-positive-coupling gauging of the winding does not support the infinite asymptotic cylinder, but a finite-radius interface provides a conditional exact escape. Israel and Maxwell matching fix its effective stress and the remaining bulk integration constant. The interface equation-of-state ratio \(T_b/Q_b\) determines its radius uniquely, while one common exponential scalar coupling closes the background scalar junction. The required interface is necessarily flux dominated, \(Q_b>\rho_b=T_b+Q_b/2\). This excludes its realization as the thin limit of a canonical positive-potential scalar wall, for which \(Q_{\rm can}\le\rho_{\rm can}\). A topological flux interface, a strongly backreacted thick defect, or genuinely nonperturbative confining stress remains possible. The finite-interface fluctuation problem is not covered automatically by the smooth infinite-background factorization. The microscopic confining origin, the admissible fermion representation, the full anomaly/inflow problem, and the finite-interface spectrum remain open. Finally, the previously considered seven-dimensional all-preon construction is re-evaluated. A genuine seven-dimensional Dirac spinor contains an additional twofold four-dimensional Weyl multiplicity under the stated inversion projector, so the earlier count of 405 selected zero-mode components becomes 810. Thus that particular 7D realization does not reproduce the desired spectrum without additional structure. The central conclusion is therefore deliberately limited. The operator, topological, Schur–Feshbach, spectral, and finite-data analyses show that the proposed route-odd mechanism is mathematically viable and sharply testable, and they identify explicit classes of higher-dimensional completions together with their stopping rules. They do not yet constitute a first-principles derivation of the observed charged-fermion mass plane. The decisive remaining four-dimensional calculation is the renormalized route-resolved nonperturbative correlator and three-point matching vertex of the explicit 24-preon sources.