The Fermionic Matter Sub-Programme

The fermionic matter sub-programme of the Cosmochrony corpus asks where fermions, chirality, hypercharge, and three generations could come from in the admissible Weil fibre. The answer is layered, and each layer names what it needs. On a supplied Heisenberg carrier, given a real metaplectic model and a supplied doublet, the algebra is proved and uses no metric: mp(2,ℝ)_ℂ ≃ sl₂(ℂ), the symmetric square of the doublet is the adjoint, its exterior square the determinant line. The admissibility axioms do not select that carrier, and every result below inherits it. Two physical readings of this algebra are distinct hypotheses, supplied by no source and excluded by none. [H-Spin] identifies the SL(2,ℂ) of the doublet with the spin group of a four-dimensional Lorentzian co-metric on the effective base; under it Sym²(S_L) is the Lorentz sector, ∧²(S_L) a trivial line, and the projective endomorphism E_Π of the projected Dirac operator is left-admissible; left-admissibility is the definition of the orientation-compatible branch, an input, given the Born–Infeld datum (H1)–(H2) of O30. [H-Weak] supplies a distinct rank-two weak factor E_weak with structure group U(2): su(2)_L sits in Sym²(E_weak), the hypercharge line is L_Y ≔ ∧²(E_weak), and the V-A structure is its content. The anomaly constraints on the hypercharge weights need both hypotheses. Lorentz chirality and weak chiral selection are different statements; the second is not derived from the first. The rigidity of the hypercharge weights carries, beyond the two hypotheses, a supplied colour module and the admissible matter decomposition, and selecting the Standard Model pattern carries in addition the minimal integral normalisation of L_Y. The saturation invariant σ_c(n_3) = 3, a supplied rule, gives conditionally a gauge-singlet three-generation factor C³_gen; the generation sector is a set of model operators on a distinct supplied doublet, their reading as restrictions of E_Π², named [H-Res], is not supplied, the level-to-generation map is not established, and the split value is a matching value. The Schur form of the projective residue concerns a compression remainder, which is the projective endomorphism only under a symbol identity that no source states; a non-zero split, its Schur transport and its Born–Infeld genus are hypotheses, and no mechanism for such a split is derived. O31 is a withdrawal notice: the co-admissibility of the colour profiles, the uniqueness of SU(3), the arithmetic criterion for colour triplets and every gauge-factor derivation are withdrawn, so no admissible colour sector in that sense is established. A dedicated no-go note (NSR) excludes the two deposited candidates for an independent fermionic weak multiplicity M_L ≃ ℂ². NSR's finite-stratum obstructions are unconditional (not conditional on [H-Spin]); its Lorentz-lock input is imported from Q14, where the spinor reading of the carrier is under [H-Spin]. This note reads E_weak as the role such a multiplicity would play, so that E_weak remains an explicit architectural extension. Interpretive outlook (a reading, not a result). Chirality is where the internal algebra of the admissible fibre must meet spacetime geometry and the weak interaction at once; the two hypotheses locate that meeting point, and turn the question of why weak interactions are left-handed into the question of how the metaplectic symmetry of the fibre is joined to Lorentz frames and to a separate internal doublet.