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 that they are not postulated as external fields: they arise as the spinorial face of the admissible Weil module ${V_\rho}$, once its metaplectic Lie-algebraic structure is complexified by the Lorentzian metric established in the geometric branch. The structural chain 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 the admissible spinor bundle ${\mathcal{S}_\Pi}$ produce the gauge-sector data of ${\mathrm{SU}}(2)_L \times {\mathrm{U}}(1)_Y$ — the complex adjoint from $\mathrm{Sym}^2({\mathcal{S}_\Pi})$ and the hypercharge line from $\wedge^2({\mathcal{S}_\Pi})$ — these being Lorentz-typed bundle data, not an independent fermionic weak doublet carrier; (B) the projected Dirac operator ${D_{\Pi,g,A}}$ contains a canonical projective endomorphism ${E_\Pi}$ that enforces the $V-A$ chiral structure (established via the spinorial BI lift theorem of Q14) and whose $\gamma_5$-weighted $a_4$ coefficient imposes anomaly-cancellation constraints making hypercharge a spectral datum; (C) the saturation invariant $\sigma_c(n_3) = 3$ has a spinorial multiplicity reading yielding a gauge-singlet three-generation factor ${C^3_{\mathrm{gen}}}$. The colour-coupled quark sector is unconditional at the pointwise level ($[\mathrm{H\text{-}color}]_{\mathrm{pointwise}}$ established in O31). The structural form of ${E_\Pi}$ is now fixed as a Schur complement of the eliminated spinorial block, and the generation-split amplitude mechanism is derived as Born–Infeld saturation in the companion note; the open deliverables are its explicit Lorentzian block and the Yukawa sector, while the split value itself is dictionary-bound, its first-principles derivation relocated upstream to the ADE case selection and the cascade exponent. A dedicated no-go note now closes the two deposited candidates for the independent fermionic weak multiplicity $M_L \simeq \mathbb{C}^2$ (the internal operator doublets and the erased history fibres) through exact obstructions culminating in the no-sequential-re-entry theorem; the matter-sector weak doublet is therefore an open frontier requiring a parallel substrate channel, stated as an explicit extension rather than a derivation.