Q14 — Fermionic Matter and Chirality from Projective Dirac Admissibility

J. Beau, Independent Researcher, France

Status

Preprint. DOI: 10.5281/zenodo.20218409

Abstract

The preceding gauge–gravity synthesis of the Cosmochrony programme showed that gravity and Yang–Mills dynamics arise from the $a_2$ and $a_4$ Seeley–DeWitt responses of the same admissible spectral functional. The present paper extends this principle to the fermionic sector. Fermions are not introduced as external matter fields. They arise as the spinorial face of the Weil module already forced by non-injective admissibility, once its metaplectic Lie-algebraic structure is complexified by the Lorentzian metric established in the geometric branch of the programme.

Three structural results are proved:

1. Electroweak spinor bundle: The admissible Weil module $V_\rho$ induces, through its metaplectic lift and the Lorentzian complexification $\mathfrak{mp}(2,\mathbb{R})_\mathbb{C} \simeq \mathfrak{sl}2(\mathbb{C})$, an admissible spinor bundle $S\Pi$ whose symmetric and determinant tensor sectors reproduce the $\operatorname{SU}(2)_L \times U(1)_Y$ electroweak bundle structure.

2. Chiral structure and hypercharge rigidity: The projected Dirac operator $\mathcal{D}{\Pi,g,A}$ contains a canonical zero-order endomorphism $E\Pi$, the internal spectral residue of non-injective projection. Under the spinorial lift of BI parity, $E_\Pi$ is left-admissible and produces the $V-A$ chiral structure. The $\gamma_5$-weighted $a_4$ coefficient then imposes anomaly-cancellation trace constraints on the hypercharge weights, which are thereby not free parameters but spectral coherence data.

3. Three-generation multiplicity: The admissible saturation invariant $\sigma_c(n_3) = 3$ admits a spinorial multiplicity reading, yielding a gauge-singlet three-generation factor $\mathbb{C}^3_{\mathrm{gen}} \subset \ker(\operatorname{ad}{\operatorname{SU}(2)} \oplus Y)$. The colour-coupled quark sector is obtained by tensoring with the colour module $V{\mathrm{color}}$, unconditional at the pointwise level ($[H\text{-color}]_{\mathrm{pointwise}}$ proved in O31).

4. Dynamic lifting of the generation degeneracy: A qualitative mechanism is identified by which the oriented spectral cascade lifts the static $J_\Pi$-protected degeneracy of the generation factor $\mathbb{C}^3_{\mathrm{gen}}$. The static obstruction is established explicitly; the oriented odd sector and the cascade bridge provide the lifting channel. The amplitude of the generational splitting remains deferred to the cascade normalisation.

Position in the programme

Q14 is the fermionic extension of the gauge–gravity synthesis established in Q12–Q13. Together, Q12–Q14 constitute the matter–geometry unification arc of the Q-series: Yang–Mills dynamics (Q12), joint Einstein–Yang–Mills equations (Q13), and fermionic matter with chirality and generation structure (Q14).

Compilation

cd q14
bash compile.sh
# or manually:
pdflatex -output-directory=out tex/q14.tex