Overview
This paper asks what the fermionic sector of the spectral approach rests on when fermions are sought in the Weil module of the admissible fibre on a supplied Heisenberg carrier, rather than added as external matter fields. The finite carrier acts through $\operatorname{SL}(2,\mathbb{Z}/q\mathbb{Z})$, which has no Lie algebra, so a real metaplectic model and a doublet carrying $\mathfrak{sl}_2(\mathbb{C})$ are supplied model data. On them the algebra is proved.
The physical readings rest on two hypotheses that no source supplies. [H-Spin] solders the $\operatorname{SL}(2,\mathbb{C})$ acting on the doublet to the spin group of a four-dimensional Lorentzian co-metric. [H-Weak] supplies a distinct weak factor: by Schur's lemma, a weak action commuting with Lorentz transformations cannot act on the same copy of the doublet. Both are missing identifications, not refutations.
The dictionary $F_n \overset{\mathrm{model}}{\simeq} \mathcal{H}_c$ is supplied. The Heisenberg carrier $\mathcal{H}_c$, its associated Weil block $V_c \simeq \mathbb{C}^q$, and the supplied spinor doublet $V_\rho \simeq \mathbb{C}^2$ are distinct; no map selecting the doublet is supplied. Connection-dependent admissible readings retain Q6a’s two unproved continuum hypotheses: a smooth effective fibre bundle and a differentiable spatial continuum, the latter open in Q5a.
Core contributions
- Algebra (proved on supplied model data): $\mathfrak{mp}(2,\mathbb{R})_\mathbb{C} \simeq \mathfrak{sl}_2(\mathbb{C})$; on the supplied doublet the symmetric square is the adjoint and the exterior square the trivial line, and a Hermitian form selects the compact real form $\mathfrak{su}(2)$.
- Two readings, two hypotheses: under [H-Spin], the symmetric square of the spinor factor is a Lorentz sector and its exterior square carries no hypercharge; the $\operatorname{SU}(2)_L \times U(1)_Y$ data come from the distinct $U(2)$ weak factor of [H-Weak].
- Lorentz chirality: under [H-Spin] and the supplied Born–Infeld datum, the projected Dirac operator $\mathcal{D}_{\Pi,g,A}$ carries, for a symbol-compatible lift, a canonical zero-order endomorphism $E_\Pi$, the spinorial lift of the parity is unique up to a phase and reverses chirality, and on an orientation branch taken as an input $E_\Pi$ is left-admissible; under the explicit hypothesis $E_\Pi \preceq 0$, which is not inherited from the Schur form of PRS (the compression remainder of PRS is $E_\Pi$ only under a symbol identity that no source states), a non-zero left-admissible $E_\Pi$ breaks that parity.
- Electroweak chiral selection: the $V-A$ structure of weak couplings is the representation content of [H-Weak]; left-admissibility is compatible with it but does not select it.
- Hypercharge: under both hypotheses, the gauge-curvature $\gamma_5$-weighted $a_4$ terms impose anomaly constraints that select the hypercharge twists up to sign, rescaling and the exchange of $u_R$ and $d_R$, with a supplied three-dimensional colour module, on the Standard Model content without $\nu_R$ and with the minimal integral normalisation of $L_Y$, using the known classification; the $E_\Pi$-dependent traces are not analysed.
- Three generations: the supplied selection rule $\sigma_c(n_3) = 3$ gives a gauge-singlet factor $\mathbb{C}^3_{\mathrm{gen}} \subset \ker(\operatorname{ad}_{\operatorname{SU}(2)} \oplus Y)$, conditional on that supplied selection rule and on the two hypotheses.
- Colour sector: a colour-coupled quark sector is written by tensoring with a supplied module $V_{\mathrm{color}}$; O31 version 2.0 withdraws the purported construction and group identification.
- Generation splitting (model): on a distinct supplied generation doublet, a static $J_\Pi$-real, weight-preserving model operator has a degenerate outer pair; the step generator $\mathcal{G}_g = \log g$ of an ordered metaplectic step carries the exact $J_3$ component $\alpha = ts\,\theta/\sinh\theta$, $\cosh\theta = 1 + ts/2$, non-zero for $ts > 0$, and, in its $J_\Pi$-odd projection, no $R_{\mathrm{mix}}$ component ($\mu = 0$).
From the Weil module to a spinor algebra
Q14 uses a supplied dictionary between the admissible fibre and the Heisenberg carrier $\mathcal{H}_c$ at fixed non-trivial central character; the associated Weil block is $V_c$, distinct from the supplied doublet. On the finite carrier the symmetries act through the finite group $\operatorname{SL}(2,\mathbb{Z}/q\mathbb{Z})$, while the metaplectic group of the real model acts on an infinite-dimensional representation with no finite-dimensional invariant subspace. The passage to a doublet carrying $\mathfrak{sl}_2(\mathbb{C})$ is therefore a modelling step, and the doublet is supplied model data. The tensor identities on that doublet are proved.
Spinors and weak doublets are two different factors
The same algebra $\mathfrak{sl}_2(\mathbb{C})$ is the algebra of Lorentz spin and the complexified algebra of weak isospin, but one copy of the doublet cannot carry both. On an irreducible Weyl fibre, every complex-linear map commuting with Lorentz transformations is a scalar (Schur's lemma), so a non-trivial weak action needs a second factor: the field carries a spinor index and a separate weak index, as in the Standard Model. Under [H-Spin], a compact real form of the Lorentz algebra is the stabiliser of a time-like direction and generates spatial rotations in that frame, not an internal symmetry. Q14 therefore states the Lorentz reading as [H-Spin] and the electroweak reading as [H-Weak], and constructs neither the solder nor the weak factor.
Spectral residue and chirality
Under [H-Spin], the endomorphism $E_\Pi$ entering the projected Dirac operator is the part of its square not resolved by the effective metric and gauge field: the spectral residue of non-injective projection. With the Born–Infeld parity datum supplied by O30's hypotheses, the spinorial lift of the parity is unique up to a phase and exchanges the two chiralities. On an orientation branch taken as an input, $E_\Pi$ has no right-handed self-coupling, and a non-zero such residue must break the parity.
That is a statement about Lorentz chirality. The chiral selection of the weak interaction, left-handed doublets and right-handed singlets, is the representation content of [H-Weak], and no argument in the paper derives it from left-admissibility.
Hypercharge from anomaly cancellation
Under both hypotheses, gauge invariance of the projected spectral functional imposes anomaly constraints through the gauge-curvature part of the $\gamma_5$-weighted $a_4$ coefficient. The hypercharge twists left free by [H-Weak] are then selected up to sign, rescaling and the exchange of $u_R$ and $d_R$, with a supplied three-dimensional colour module, on the Standard Model content without $\nu_R$ and with the minimal integral normalisation of $L_Y$; the paper uses the known classification of one-generation solutions and does not reprove it. Terms of $a_4$ that depend on $E_\Pi$ are not analysed.
Three generations from $\sigma_c = 3$
The supplied selection rule $\sigma_c(n_3) = 3$ admits a spinorial reading as the multiplicity of a gauge-singlet subspace, giving, conditionally on that rule, the factor $\mathbb{C}^3_{\mathrm{gen}}$. O23 proves three-dimensionality only for the neutral sector of a supplied spinor carrier; the carrier selection and the identification of $\Sigma_c$ with that dimension remain open, and no source reads this rule as three spatial directions.
A model of the generation splitting
Section 6 works on a supplied generation doublet distinct from the spinor factor, with the rank-three sector modelled as its symmetric square. A static $J_\Pi$-real, weight-preserving model operator cannot split the outer pair. The generator $\log g$ of an ordered metaplectic step carries an exact non-zero $J_3$ component, whose sign reverses with the order of the step, and no component along the mixing direction: no element of $\mathfrak{sl}_2(\mathbb{C})$ connects the two outer weights, so a complex metaplectic phase cannot produce mixing in this model.
Reading these operators as restrictions of $E_\Pi^2$ is not supplied, and neither is the identification of $\log g$ with an emergent ordering derivative: Q11 defines that derivative only on functions of the ordering parameter. The value $\varepsilon = 1/10$ is a matching value, and the amplitude is open.
Relation to the Cosmochrony program
Q14 follows the gauge–gravity synthesis of Q12–Q13, which organises gravity and Yang–Mills dynamics, conditionally, as Seeley–DeWitt responses of one admissible spectral functional. Its result is to locate what the fermionic sector needs beyond the admissible Weil fibre: a Lorentzian spin solder and a distinct weak factor. The geometric branch supplies a Lorentzian co-metric only conditionally: Q11 (Corollary 6.1) gives, under its stated hypotheses, a Lorentzian signature, which is all Q14 uses; Q11 supplies neither coefficient values, nor a Lorentzian spin soldering, nor an order generator, so [H-Spin] remains an additional hypothesis.
FM-Note 3.3, deposited on 2026-10-27, is aligned with this version. PYL 3.0.1, PYO 2.1, PRS 3.0, A4-Note 3.0, EBJ 2.1, KUD 1.5, TPC 1.2 and NPI 1.0.1, all deposited on 2026-10-07, predate this version and have not been checked against it. CHO 1.1 cites Proposition 6.3, Remark 6.4 and Proposition 6.5 of Q14, and Q11OF 1.1 cites Proposition 6.5 and section 6 of Q14; both cite the $\mathfrak{sl}_2$ commutator $[E,F]=H$ (Proposition 6.5); neither constructs the Lorentzian chiral lift, which AOG 1.8 names as the hypothesis [H-Spin] of Q14. Their use of $[\gamma_5,R_b]=0$ is carried by the explicit hypothesis [H-WS] of Q11OF. TempProj and EG-Note were also written before it and are still to be aligned. Q14 does not derive the Heisenberg carrier, the colour module, the gauge group, the rank-three selection rule, or either of its two hypotheses from the admissibility axioms.
References
Jérôme Beau. Toward Fermionic Matter from Projective Dirac Admissibility: Chirality and Electroweak Structure Rest on a Lorentzian Spin Solder and a Distinct Weak Factor. 10.5281/zenodo.20218409