Fermionic Matter and Chirality from Projective Dirac Admissibility

The preceding gauge–gravity synthesis of the Cosmochrony programme showed that gravity and Yang–Mills dynamics arise from the a₂ and a₄ Seeley–DeWitt responses of the same admissible spectral functional. The present paper extends this principle to the fermionic sector. Fermions are not introduced as external matter fields. They arise as the spinorial face of the Weil module already forced by non-injective admissibility, once its metaplectic Lie-algebraic structure is complexified by the Lorentzian metric established in the geometric branch of the programme. Three structural results are proved. First, the admissible Weil module V_ induces, through its metaplectic lift and the Lorentzian complexification mp (2, R) ₂ sl₂ (C), an admissible spinor bundle S_ whose symmetric and determinant tensor sectors reproduce the SU (2) L U (1) Y electroweak bundle structure. Second, the projected Dirac operator D, ₆, ₀ contains a canonical zero-order endomorphism E_, the internal spectral residue of non-injective projection. Under the spinorial lift of BI parity (Hypothesis hyp: spinorial-bi-lift), E_ is left-admissible and produces the V-A chiral structure. The ₅-weighted a₄ coefficient of D, ₆, ₀² then imposes anomaly-cancellation trace constraints on the hypercharge weights, which are thereby not free parameters but spectral coherence data. Third, the admissible saturation invariant c (n₃) = 3 admits a spinorial multiplicity reading, distinct from its geometric reading as three spatial directions, yielding a gauge-singlet three-generation factor C³₆₄₍ (adₒₔ (₂) Y). The colour-coupled quark sector is obtained by tensoring with the colour module V₂₎₋₎ₑ, unconditional at the effective level (H-color₄₅₅ established in).