Fermionic Matter and Chirality from Projective Dirac Admissibility

Core contributions

• Admissible spinor bundle: The Weil module $V_\rho$, through its metaplectic lift and the complexification $\mathfrak{mp}(2,\mathbb{R})_\mathbb{C} \simeq \mathfrak{sl}_2(\mathbb{C})$, induces 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.
• V−A chirality from $E_\Pi$: 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 enforces the $V-A$ chiral structure observed in weak interactions.
• Hypercharge rigidity: The $\gamma_5$-weighted $a_4$ Seeley–DeWitt coefficient of $\mathcal{D}_{\Pi,g,A}^2$ imposes anomaly-cancellation trace constraints on the hypercharge weights. Hypercharge assignments are therefore not free parameters but spectral coherence data.
• Three-generation multiplicity: The admissible saturation invariant $\sigma_c(n_3) = 3$ admits a spinorial multiplicity reading, distinct from its geometric reading as three spatial directions, yielding a gauge-singlet factor $\mathbb{C}^3_{\mathrm{gen}} \subset \ker(\operatorname{ad}_{\operatorname{SU}(2)} \oplus Y)$.
• Colour sector: The colour-coupled quark sector is obtained by tensoring with a colour module $V_{\mathrm{color}}$, unconditional on the standard graph via $[H\text{-color}]_{\mathrm{pointwise}}$ proved analytically in O31 v1.5 (single-frequency BI fingerprint, Proposition 4.23).
• Dynamic lifting of the generation degeneracy: The static $J_\Pi$-protected degeneracy of $\mathbb{C}^3_{\mathrm{gen}}$ is statically obstructed; admissible lifts form a two-dimensional $J_\Pi$-odd sector. In a diagnostic model of the Q11 cascade generator $\partial_\tau$, the $J_\Pi$-odd projection carries a non-zero weight component ($\alpha \neq 0$) whose sign reverses under cascade reversal. The qualitative mechanism of the inter-generation splitting is thereby fixed; its amplitude is deferred to the cascade normalisation.

Electroweak bundle from the Weil module

The Weil module $V_\rho$ is not an auxiliary construction: it is the canonical representation forced by non-injective admissibility. Its metaplectic Lie-algebraic structure, once complexified by the Lorentzian metric, provides the data for a spinor bundle $S_\Pi$ whose tensor decomposition contains an $\operatorname{SU}(2)$ doublet sector and a $U(1)$ determinant line, matching the $\operatorname{SU}(2)_L \times U(1)_Y$ electroweak gauge structure without an additional hypothesis.

Spectral residue and chiral selection

The endomorphism $E_\Pi$ entering the projected Dirac operator is a direct consequence of non-injectivity. When the projection $\Pi$ is non-injective, the fibre degeneracy generates an internal zero-order term in the lifted Dirac operator. This term, $E_\Pi$, is the spectral residue of projection and cannot be removed without breaking admissibility.

Under the spinorial lift of BI parity — the discrete symmetry selected by the Born–Infeld admissibility bound — $E_\Pi$ is left-admissible. The resulting Dirac operator couples differently to left- and right-handed spinors, producing the $V-A$ structure of the Standard Model weak sector as a structural consequence, not as an input.

Hypercharge from anomaly cancellation

The $\gamma_5$-weighted Seeley–DeWitt coefficient $a_4$ of the squared projected Dirac operator encodes the gravitational anomaly. Requiring this coefficient to vanish — spectral coherence of the fermionic sector — imposes linear trace constraints on the hypercharge assignments of the fermion multiplets. These constraints reproduce the standard anomaly-cancellation conditions of the Standard Model and determine hypercharge up to an overall normalization.

Hypercharge is therefore not a free parameter of the theory but a spectral coherence datum constrained by the same admissibility principle that governs the geometric and gauge sectors.

Three generations from $\sigma_c = 3$

The saturation invariant $\sigma_c(n_3) = 3$ was previously identified as a structural bound on the number of independent spatial directions in the admissible projective geometry. The same integer admits an independent spinorial reading as the multiplicity of a gauge-singlet subspace within the fermionic sector. This yields a gauge-singlet three-generation factor $\mathbb{C}^3_{\mathrm{gen}}$ without introducing a new integer parameter. The triple generational structure of the Standard Model is a consequence of the same saturation invariant that fixes the spatial dimension.

Dynamic lifting of the static generation degeneracy

A $J_\Pi$-real, weight-preserving restriction of $E_\Pi^2$ has a degenerate outer pair: the two light generations cannot be split statically. This static obstruction is established explicitly (Q14 §6). The admissible lifts removing the degeneracy form a two-dimensional $J_\Pi$-odd sector, with a real-split direction along the weight generator $J_3$ and a mixing direction along a doublet rotation.

In a diagnostic model of the Q11 cascade generator $\partial_\tau$, the $J_\Pi$-odd projection carries a non-zero $J_3$ component ($\alpha \neq 0$), whose sign reverses under cascade reversal. The qualitative mechanism of the inter-generation splitting is thereby fixed. Its amplitude is deferred to the cascade normalisation, and generation mixing requires the complex metaplectic phase. These items are the primary open quantitative deliverables of the fermionic sector.

Relation to the Cosmochrony program

Q14 closes the matter–geometry arc of the Q-series. Q12 derived Yang–Mills dynamics from the vertical heat-kernel variation of the admissible spectral functional. Q13 synthesized these with the gravitational sector, establishing joint Einstein–Yang–Mills equations and the hierarchy ratio. Q14 adds the fermionic matter content: spinor bundle, chirality, hypercharge rigidity, and generation multiplicity — all from the same admissibility framework.

Together, Q12–Q14 demonstrate that the bosonic gauge and gravitational fields and the fermionic matter fields of the Standard Model arise from a single structural source: the non-injective projection admissibility constraint on the Weil module.

References

Jérôme Beau. Fermionic Matter and Chirality from Projective Dirac Admissibility. 10.5281/zenodo.20218409