Toward Fermionic Matter from Projective Dirac Admissibility: Chirality and Electroweak Structure Rest on a Lorentzian Spin Solder and a Distinct Weak Factor

The gauge–gravity synthesis of the Cosmochrony programme reads gravity and Yang–Mills dynamics, conditionally, as the $a_2$ and $a_4$ Seeley–DeWitt responses of one admissible spectral functional. This paper asks what the fermionic sector requires when fermions are sought in the Weil module of the admissible fibre on a supplied Heisenberg carrier, which every result below inherits. The finite carrier acts through ${\operatorname{SL}}(2,\mathbb{Z}/q\mathbb{Z})$, so a real metaplectic model and a doublet carrying its Lie algebra are supplied model data; given them, the algebra is proved: $\mathfrak{mp}(2,\mathbb{R})_{\mathbb{C}} \simeq \mathfrak{sl}_2(\mathbb{C})$, the symmetric square of the doublet is the adjoint and its exterior square the trivial line. Given the supplied Born–Infeld parity datum, the internal parity $HK$ satisfies $(HK)^2 = -1$ without any metric. The physical readings require two distinct hypotheses, stated in the paper and supplied by no source. [H-Spin] identifies the ${\operatorname{SL}}(2,\mathbb{C})$ acting on the doublet with the spin group of a four-dimensional Lorentzian co-metric on the effective base; the geometric branch supplies such a co-metric only conditionally, and nothing relates the metaplectic action to its frames. [H-Weak] supplies a distinct rank-two weak factor: by Schur's lemma a weak action commuting with Lorentz transformations cannot live on the same copy of the doublet, whose symmetric square is then a Lorentz sector and whose exterior square carries no hypercharge. Under [H-Spin], the projected Dirac operator contains a canonical zero-order endomorphism $E_\Pi$, the spectral residue of non-injective projection; the spinorial lift of the parity is unique up to a phase and reverses chirality, and $E_\Pi$ is left-admissible on the orientation-compatible branch, an input. The chiral selection of the weak interaction, the $V{-}A$ structure, and the anomaly-cancellation constraints on the hypercharge weights require [H-Weak] as well, and left-admissibility does not select the chiral assignment. Both hypotheses are missing identifications, not refutations. The supplied rank-three selection rule $\sigma_c(n_3) = 3$ (O23) admits a spinorial multiplicity reading, giving conditionally a gauge-singlet three-generation factor $\mathbb{C}^3_{\mathrm{gen}} \subset \ker(\operatorname{ad}_{{\operatorname{SU}}(2)} \oplus Y)$. The quark sector uses a supplied colour module; no $\mathrm{SU}(3)$ is derived. Finally, in a metaplectic step model the ordered step generator $\log g$ has a non-zero $J_3$ component in the sector that can lift the static degeneracy of the generation factor; its identification with an emergent ordering derivative is not supplied, and the amplitude is open. 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 paper locates that meeting point in two explicit hypotheses, which turns 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.