Fermionic Matter and Chirality from Projective Dirac Admissibility

Core contributions

• Admissible spinor bundle under [H-L]: On the conditional Lorentzian geometry, 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, on the supplied carrier, 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 constraints and 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. Their rigidity is conditional on a supplied three-dimensional colour module; selection of the Standard Model pattern additionally uses the minimal integral normalisation of the determinant line.
• Three-generation multiplicity: The selection rule $\sigma_c(n_3) = 3$ yields a gauge-singlet factor $\mathbb{C}^3_{\mathrm{gen}} \subset \ker(\operatorname{ad}_{\operatorname{SU}(2)} \oplus Y)$, conditional on a supplied rank-three carrier.
• Colour sector: A colour-coupled quark sector can be written by tensoring with a supplied module $V_{\mathrm{color}}$. O31 version 2.0 withdraws the purported construction and group identification, so this remains a conditional input.
• 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

Q14 identifies the admissible fibre with the Weil module $V_\rho$ on a supplied finite-Heisenberg carrier and fixed non-trivial central character. The axioms alone force irreducibility and non-commutation, but do not select the finite Heisenberg group or this identification. The module's 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 bundle structure relative to that carrier.

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. Rigidity of the weights requires a supplied three-dimensional colour module, while selection of the Standard Model pattern additionally requires the minimal integral normalisation of the determinant line and remains up to overall sign and rescaling.

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 $\sigma_c = 3$

Conditional on a supplied rank-three carrier, the selection rule $\sigma_c(n_3) = 3$ admits a spinorial reading as the multiplicity of a gauge-singlet subspace within the fermionic sector. This yields the factor $\mathbb{C}^3_{\mathrm{gen}}$. O23 proves three-dimensionality only for the neutral sector of a supplied spinor carrier; carrier selection and its observable identification remain open.

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

Conditional on [H-L], Q14 connects fermionic matter to the geometry 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 for a supplied compact structure group. Q14 adds the fermionic matter architecture with its [H-L], carrier, colour, and rank-three hypotheses kept explicit.

The structural results therefore share one projective framework, but Q14 does not derive the Heisenberg carrier, a colour module, a gauge group, or the rank-three selection rule from the admissibility axioms.

References

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