Dirac Operator, Generation Count, Chirality Structure, and the m₃ Integer

Paper #48 identifies the fermion sector of the Standard Model with the low-lying spectrum of S = ψ† LT ψ built from the face Laplacian LT of the truncated octahedron and a 14-component field ψ. Section 5. 2 of Paper #48 quotes the heaviest neutrino mass m₃ = mₑ · exp− (11 + 13√17) /4 as a "sector-specific identification. " This paper upgrades that identification to derivation at the Dirac-operator and multiplicity level, addressing four questions: (i) does a canonical foam-cell Dirac operator exist? (ii) does the three-generation count follow from the cell's symmetry? (iii) does the chirality-mixing mass operator on the T₁u sector take a specific 2×2 Pauli-matrix form forced by the cell's geometric structure? (iv) is the integer pair (11, 13) in the m₃ exponent geometrically selected rather than fitted? Results: T72. 1 (existence of a canonical foam-cell Dirac operator DF with DF² = LT, self-adjoint, commuting with the permutation representation of Oₕ) — theorem with proof. T72. 2 (three-generation count = T₁u multiplicity, derived by Oₕ character theory) — theorem with proof. T72. 3a (chirality-mixing operator on the T₁u sector = −2 σₓ in the canonical face-type basis, via the LT = Ldiag + Lₒff orbit splitting) — theorem with proof, uniform across all three T₁u generation copies to machine precision. T72. 3b (physical chirality identification hex ↔ L, sq ↔ R) — conjecture with V10 geometric necessary condition: T₂g hosts only on the hex-orbit, forcing hex-T₁u doublets and sq-T₁u singlets under the standard SU (2) L ↔ T₂g embedding. T72. 4 (integer triple (11, 13, 4) in the m₃ exponent) — best-match primitive triple in a principled 12, 800-triple search; the unique primitive (gcd = 1) match within 1σ of PDG 2024 m₃ = 49. 50 meV, ranked #1 by rel-err accuracy (0. 019%) across the full 4σ window. Three of five sub-claims are theorems. T72. 3b rests on a geometric necessary condition pending substantiation of the SU (2) L ↔ T₂g embedding. T72. 4 has a characterised best-match identification pending a closed-form counting rule and resolution of the PDG-vs-NuFIT convention dependence. Both open items are well-defined closed-form problems with documented candidate paths. A companion verification script reproduces every numerical claim from raw cell integers F = 14, Fₕx = 8, Fₛq = 6, V = 24, E = 36, CA = 3 and the master equation λ² − 9λ + 16 = 0 in under ten seconds.