Abstract

We develop a finite-mode framework for fermion flavor built on a previously established

projector-resolved charged-cell construction. The parent theory supplies a localized body with complementary longitudinal and transverse response channels, together with an independent spacetime spinor factor. Flavor is represented on the conditional three-mode Hilbert space H3 = span{|00⟩, |10⟩, |20⟩}.  On this space, a finite operator basis spans every Hermitian 3 × 3 flavor kernel, allowing explicit construction of sector Yukawa matrices and the mixing matrices VCKM = U†uLUdL and UPMNS = U†eLUνL. An exact reconstruction of the frozen real CKM benchmark shows that rank-≤ 2 operator content is insufficient and that a rank-three transition response is required. Minimal one-shell and arbitrary-rank heavy completions are excluded within the fixed projected Legendre-tensor class. Positive local direct-source realizations are classified by a positive-semidefinite source Gram matrix. This establishes complete realizability while leaving the mixed normalization unselected in the absence of a parent-derived flavor selector. Generation counting is likewise conditional because the localized parent does not derive a self-adjoint flavor operator with an isolated, perturbatively stable rank-three spectral projector. A shared implementation verifies that the frozen curvature factors propagate through the canonical CKM benchmark path. The quark-mass interface separates perturbative transport from flavor matching through FQCD = Zmatch/R, and flavor-universal matching is excluded as the source of the observed hierarchy. The resulting framework provides exact finite-mode flavor mechanics, reproducible benchmark mixing reconstructions, no-go theorems for several minimal completions, and explicit parent interfaces for generation selection, electromagnetic charge readout, source normalization, and quark-mass matching.

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Keywords: Standard Model, particle physics, flavor physics, fermion masses, quark masses, lepton masses, CKM matrix, PMNS matrix, neutrino mixing, Yukawa matrices, CP violation, flavor puzzle, effective field theory, flavor symmetry, geometric flavor model, Axis Model, projector methods, reproducible research