Categorical Refinement Dynamics and a Consistency Relation Among Standard Model Flavor Parameters

Abstract

I develop a framework in which physical configurations form a category 𝐶Ps - a possibility space of structural configurations - whose morphisms carry complex-valued weights encoding transformation cost and geometric phase. The single realized reality Sr traces a path through this possibility space - each instantaneous configuration a codimension-1 hypersurface in the spacetime sense, the trajectory in 𝐶Ps a path through such configurations - evolving by inconsistency minimization rather than wavefunction collapse. Composition of morphisms generates a path-integral structure in which quantum interference appears and the Born rule is recovered as a unique consistency condition. Assuming a Gaussian fixed point for the induced measure, motivated by Wilson–Kogut universality, yields quadratic effective dynamics as the leading-order approximation. Non-Gaussian corrections generate terms with the structural form of Standard Model (SM) interactions. Within the flavor sector, fermion generations are identified with eigenmodes of a refinement recursion operator (the framework is consistent with, but does not fully predict, the observed count of three generations); mixing angles are identified with Berry phases of inter-generation transitions; and CP-violating phases appear as loop holonomies in refinement space. A consistency relation connects quark and lepton CP-violating phases through mixing angles, mass ratios, and gauge coupling structure; the framework’s two calibrated parameters (𝜇₀ and 𝐾𝑞) are solved so that this relation holds with the lepton-sector phase set to 𝛿CP(lep) = 1.32 ± 0.05 (stat) ± 0.08 (scheme) rad, the value tested against DUNE. This consistency value aligns with T2K’s individual preference for large CP violation and is in tension with the NuFIT 6.0 global best fit (177–212 deg), which is near CP conservation for normal ordering; the consistency value lies outside the NuFIT 6.0 3-sigma range. The stat+scheme uncertainty of ±0.09 rad reflects parameter variation under the framework’s calibration held fixed; a broader Monte Carlo propagation over PDG input measurement uncertainties (see the footnote at its first statement and the Appendix for the reconstructed methodology) gives a 95% range of [0.93, 1.96] rad. The framework contains two calibrated parameters (𝜇₀ and 𝐾𝑞), fixed by the measured quark-sector CP phase together with the target lepton-sector value above, with 𝐾𝑙 determined from 𝐾𝑞 via the curvature ansatz Eq. (15) rather than independently fit; the joint-calibration 𝐾𝑞/𝐾𝑙 therefore equals the ansatz’s value by construction at the calibration ΛQCD, not as an independent cross-check; nor, as detailed in limitation L15, does the mere existence of a shared 𝜇₀ at a physically reasonable scale constitute additional non-trivial content, since 𝛿CP(lep) varies continuously and without bound as a function of 𝜇₀ near a pole in the calibration formula. The substantive, falsifiable content of the consistency relation is, in the ordinary sense, that the framework outputs a specific numerical value, 𝛿CP(lep) = 1.32 rad, directly testable against DUNE’s measurement, not a derivation of the lepton phase from quark data alone. DUNE and Hyper-Kamiokande will definitively resolve the T2K/NOvA tension and test this consistency relation. I explicitly label all results as Derived, Motivated, or Assumed, and identify fifteen specific open problems.