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_ρ induces, through its metaplectic lift and the Lorentzian complexification mp(2,R)C sl₂(C), an admissible spinor bundle S_Π whose symmetric and determinant tensor sectors reproduce the SU(2)L × U(1)Y electroweak bundle structure. Second, the projected Dirac operator DΠ,g,A contains a canonical zero-order endomorphism E_Π, the internal spectral residue of non-injective projection. Under the spinorial lift of BI parity (Theorem thm:spinorial-bi-lift), E_Π is left-admissible and produces the V-A chiral structure. The γ₅-weighted a₄ coefficient of DΠ,g,A² 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 σc(n₃) = 3, a supplied selection rule (O23 proves the three-dimensionality of the neutral sector of a supplied spinor carrier; the carrier selection and the identification of Σc with that dimension are open), admits a spinorial multiplicity reading, distinct from its geometric reading as three spatial directions, yielding conditionally a gauge-singlet three-generation factor C³gen ⊂ (ad_SU(2) ⊕ Y). The colour-coupled quark sector is obtained by tensoring with the colour module Vcolor, unconditional at the pointwise level ([H-color]pointwise established in ). Finally, we identify a qualitative mechanism by which the oriented cascade lifts the static J_Π-protected degeneracy of the generation factor; the amplitude of this splitting remains deferred to the cascade normalisation.