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

The Cosmochrony gauge–gravity synthesis conditionally reads gravity and Yang–Mills dynamics as $a_2$ and $a_4$ responses of one spectral functional. This paper asks what its fermionic sector requires. A supplied dictionary identifies the admissible fibre with a Heisenberg carrier $\mathcal{H}_c$; its associated Weil block $V_c \simeq \mathbb{C}^q$ is distinct from the supplied spinor doublet $V_\rho \simeq \mathbb{C}^2$, and no map selecting that doublet is constructed. The finite symplectic group has no Lie algebra. A real metaplectic model and its Lie-algebra action on the doublet are independent model inputs. Given them, $\mathfrak{mp}(2,\mathbb{R})_\mathbb{C} \simeq \mathfrak{sl}_2(\mathbb{C})$; the symmetric square is the adjoint and the exterior square the trivial line. Given the supplied Born–Infeld parity datum, $(HK)^2=-1$ without a metric. The physical readings require two hypotheses supplied by no source. [H-Spin] identifies the doublet action with the spin group of a four-dimensional Lorentzian co-metric, which the geometric branch supplies only conditionally. [H-Weak] supplies a distinct rank-two weak factor: Schur's lemma prevents a commuting weak action on the same Lorentz doublet. Connection-dependent admissible readings also retain Q6a's two unproved continuum hypotheses: a smooth effective fibre bundle and a differentiable spatial continuum, the latter open in Q5a. Under these geometric assumptions the projected Dirac operator contains a zero-order endomorphism $E_\Pi$; the spinorial parity lift is unique up to phase and reverses chirality. Left-admissibility is an input orientation branch. Weak $V{-}A$ selection and hypercharge anomaly constraints additionally require [H-Weak]; left-admissibility alone does not select the weak assignment. The supplied rank-three rule $\sigma_c(n_3)=3$ gives a conditional three-generation factor. The quark sector uses a supplied colour module; no $\mathrm{SU}(3)$ is derived. An ordered metaplectic step model can lift the static generation degeneracy, but its identification with an ordering derivative and its amplitude remain open. Interpretive outlook (a reading, not a result). Chirality locates the missing meeting of internal algebra, Lorentz frames and a separate weak doublet; the paper identifies the assumptions needed to study why weak interactions are left-handed.