The Fermionic Matter Sub-Programme

Abstract

The fermionic matter sub-programme of the Cosmochrony corpus addresses a single central question: where do fermions, chirality, hypercharge, and three generations come from? The answer is stratified and conditional. They need not be postulated as external fields: they arise as the spinorial face of the Weil module ${V_\rho}$ carried by the admissible fibre on a supplied Heisenberg carrier, once its metaplectic structure is complexified by the Lorentzian metric of the geometric branch. The admissibility axioms do not select that carrier, and every result below inherits it. The structural chain, read on it, is: \[ \Pi \Rightarrow F_n \simeq {V_\rho} \Rightarrow {\mathrm{Mp}}(2,\mathbb{R}) \Rightarrow \mathrm{mp}(2,\mathbb{R})_\mathbb{C} \simeq \mathfrak{sl}_2(\mathbb{C}) \rightsquigarrow {\mathrm{Spin}}(3,1) \simeq {\mathrm{SL}}(2,\mathbb{C}) \Rightarrow {\mathcal{S}_\Pi}. \] Three modular structural results are established in Q14: (A) the tensor functors of ${\mathcal{S}_\Pi}$ produce the gauge-sector data of ${\mathrm{SU}}(2)_L \times {\mathrm{U}}(1)_Y$, Lorentz-typed adjoint and determinant data rather than an independent fermionic weak doublet carrier; (B) the projective endomorphism ${E_\Pi}$ of the projected Dirac operator enforces the $V-A$ chiral structure, and its $\gamma_5$-weighted $a_4$ coefficient imposes anomaly-cancellation constraints on the hypercharge weights; (C) the saturation invariant $\sigma_c(n_3) = 3$ yields a gauge-singlet three-generation factor ${C^3_{\mathrm{gen}}}$, conditionally on the supplied rank-three carrier. Their hypotheses differ, and (B) is itself stratified. The $V-A$ structure carries nothing beyond the supplied carrier. The rigidity of the hypercharge weights carries a supplied three-dimensional colour module, the anomaly sums being weighted by the representation dimension. Selecting the Standard Model pattern carries, beyond that module, the minimal integral normalisation of $L_Y = \wedge^2({\mathcal{S}_\Pi})$, holds only up to an overall sign and rescaling, and rests on a connection Q14 leaves open; the quark sector carries the module but not that normalisation. O31 is a withdrawal notice: no colour triplet is established as admissible, ${\mathrm{SU}}(3)$ is not identified, and no gauge factor is derived. The structural form of ${E_\Pi}$ is fixed as a Schur complement of the eliminated spinorial block and the generation-split mechanism is Born–Infeld saturation, but the split value is dictionary-bound, its first-principles front lying upstream in the ADE case selection and the cascade exponent. A dedicated no-go note closes the two deposited candidates for the independent fermionic weak multiplicity $M_L \simeq \mathbb{C}^2$ through the no-sequential-re-entry theorem, leaving the matter-sector weak doublet an explicit architectural extension rather than a derivation.