Q14 — Toward Fermionic Matter from Projective Dirac Admissibility: Chirality and Electroweak Structure Rest on a Lorentzian Spin Solder and a Distinct Weak Factor
J. Beau, Independent Researcher, France
Preprint. DOI: 10.5281/zenodo.20218409
Version 3.0 was deposited on 2026-09-30 as Zenodo record 23071163.
The gauge–gravity synthesis of the Cosmochrony programme reads gravity and Yang–Mills dynamics,
conditionally, as the
-
Algebra (proved on supplied model data): 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. Given them,$\mathfrak{mp}(2,\mathbb{R})_\mathbb{C} \simeq \mathfrak{sl}_2(\mathbb{C})$ ; the symmetric square of the doublet is the adjoint and its exterior square the trivial line, and a Hermitian form selects the compact real form$\mathfrak{su}(2)$ . Given the supplied Born–Infeld datum (hypotheses (H1)–(H2) of O30), the internal parity$HK$ satisfies$(HK)^2 = -1$ without any metric. -
Two hypotheses, 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; the geometric branch supplies such a co-metric only conditionally (Q5b, Q8, Q11). [H-Weak] supplies a distinct rank-two weak factor$E_{\mathrm{weak}}$ : by Schur's lemma a weak action commuting with Lorentz transformations cannot act on the same copy of the doublet. Under [H-Spin],$\operatorname{Sym}^2(S_L)$ is a Lorentz sector and$\wedge^2(S_L)$ carries no hypercharge. Both are missing identifications, not refutations. -
Chirality and hypercharge (conditional): under [H-Spin], the projected Dirac operator $\mathcal{D}{\Pi,g,A}$ contains a canonical zero-order endomorphism $E\Pi$, the spinorial lift of the parity is unique up to a phase and reverses chirality, and, given the Born–Infeld datum and the orientation-compatible branch (an input),
$E_\Pi$ is left-admissible,$P_R E_\Pi P_R = 0$ ; a non-zero left-admissible$E_\Pi \preceq 0$ must break that parity. The chiral selection of the weak interaction ($V-A$ ) and the anomaly constraints on the hypercharge weights require [H-Weak] as well; left-admissibility does not select the chiral assignment. The hypercharge selection needs in addition$Y_e \neq 0$ , which excludes the known degenerate solution (the$U(2)$ structure of [H-Weak] already excludes it), and holds up to sign, rescaling and the exchange of$u_R$ and$d_R$ . -
Generation multiplicity (conditional): the supplied rank-three selection rule
$\sigma_c(n_3) = 3$ (O23) admits a spinorial multiplicity reading, giving a gauge-singlet three-generation factor$\mathbb{C}^3_{\mathrm{gen}} \subset \ker(\operatorname{ad}_{\operatorname{SU}(2)} \oplus Y)$ , conditional also on [H-Spin] and [H-Weak] through the bundle it multiplies. The quark sector uses a supplied colour module; O31 is a withdrawal notice and no$\operatorname{SU}(3)$ is derived. -
Generation splitting (qualitative): a static
$J_\Pi$ -real, weight-preserving restriction cannot split the outer pair; in a metaplectic step model on a distinct generation doublet, the ordered step generator$\mathcal{G}_g = \log g$ carries the exact$J_3$ component$\alpha = ts,\theta/\sinh\theta$ ($\cosh\theta = 1 + ts/2$ ) and no mixing component, for any$\mathfrak{sl}_2(\mathbb{C})$ generator. Its identification with an emergent ordering derivative is not supplied, and the amplitude is open.
Q14 is the fermionic step after the gauge–gravity synthesis of Q12–Q13. It isolates what the fermionic sector needs beyond the admissible Weil fibre: a Lorentzian spin solder ([H-Spin]) and a distinct weak factor ([H-Weak]).
cd q14
bash compile.sh
# or manually:
pdflatex -output-directory=out tex/q14.tex