The Fermionic Matter Sub-Programme

Structural chain

$$ \Pi \Rightarrow F_n \simeq V_\rho \;\Rightarrow\; \mathrm{Mp}(2,\mathbb{R}) \;\Rightarrow\; \mathrm{mp}(2,\mathbb{R})_\mathbb{C} \simeq \mathfrak{sl}_2(\mathbb{C}) \;\rightsquigarrow\; \mathrm{Spin}(3,1) \simeq \mathrm{SL}(2,\mathbb{C}) \;\Rightarrow\; \mathcal{S}_\Pi. $$

Then:

• $\mathrm{Sym}^2(\mathcal{S}_\Pi) \Rightarrow \mathrm{ad}_\mathbb{C}(P_\Pi)$ — the $\mathrm{SU}(2)_L$ sector.
• $\wedge^2(\mathcal{S}_\Pi) = L_Y$ — the $\mathrm{U}(1)_Y$ sector.
• $E_\Pi$ left-admissible — the $V-A$ chiral structure (established via the spinorial BI lift theorem of Q14).
• $\sigma_c(n_3) = 3 \to C^3_{\mathrm{gen}}$ — three generations as a gauge-singlet factor.

Core results (Q14)

• Theorem A (spinorial electroweak bundle, unconditional). The admissible Weil module $V_\rho$ induces, via the metaplectic lift and the Lorentzian complexification forced by $g^{\mu\nu} = 2\eta^{\mu\nu}$, an admissible spinor bundle $\mathcal{S}_\Pi$ whose tensor functors satisfy $\mathrm{Sym}^2(\mathcal{S}_\Pi) \simeq \mathrm{ad}_\mathbb{C}(P_\Pi)$ and $\wedge^2(\mathcal{S}_\Pi) = L_Y$. After compact real-form selection these define the $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ electroweak bundle data without additional input.
• Theorem B ($V-A$ and hypercharge, unconditional). The projected Dirac operator satisfies a Lichnerowicz-type identity $D^2_{\Pi,g,A} = -(\nabla^{S,A})^2 + R/4 + \tfrac{1}{2}\gamma^\mu\gamma^\nu F^a_{\mu\nu} T^a_R + E_\Pi$. The spinorial BI lift theorem of Q14 establishes that $E_\Pi$ is left-admissible ($P_R E_\Pi P_R = 0$), giving the $V-A$ structure. $\mathrm{U}(1)_Y$ invariance of the projected spectral functional imposes $\mathrm{Tr}_{\mathcal{S}_\Pi}(\gamma_5 Y A_\Pi(x)) = 0$ — the anomaly-cancellation trace condition. Hypercharge is a spectral coherence datum, not a free parameter.
• Theorem C (three generations, unconditional). The saturation invariant $\sigma_c(n_3) = 3$ admits two functorially distinct readings of the same rank-three admissible invariant: a geometric reading (three spatial directions) and a spinorial reading yielding a gauge-singlet generation space $C^3_{\mathrm{gen}} \subset \ker(\mathrm{ad}_{\mathrm{SU}(2)} \oplus Y)$. The identity $\mathbb{R}^3_{\mathrm{space}} \neq C^3_{\mathrm{gen}}$ is established explicitly: the spatial and generation spaces are distinct functorial images of the same invariant.
• Colour sector (unconditional at the pointwise level). The colour-coupled quark sector $\mathcal{S}_\Pi \otimes V_{\mathrm{color}}$ is obtained by tensoring the admissible spinor bundle with the colour module. $[\mathrm{H\text{-}color}]_{\mathrm{pointwise}}$ is proved analytically on the standard Cayley graph (O31, single-frequency BI fingerprint).
• Dynamic generation lifting (Q14 §6; qualitative). The static $J_\Pi$-protected degeneracy of $C^3_{\mathrm{gen}}$ is statically obstructed; the admissible lifts removing the degeneracy form a two-dimensional $J_\Pi$-odd sector (real-split direction $J_3$; mixing direction 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 fixed; its amplitude is deferred to the cascade normalisation.

Constituent papers

• Q14 — Fermionic matter and chirality from projective Dirac admissibility. Principal paper of the sub-programme: Theorems A–C and the qualitative mechanism for the inter-generation splitting (Q14 §6).
• PRS — The projective residue as a Schur complement. Operator-side companion note refining Q14 §6: Schur form $E_\Pi = -M^\dagger M$, localisation of $u$ in the antiunitary chiral equivariance defect $\Delta_\chi(P)$, A4 stratification, finite/Lorentzian separation, and reduction of the Schur transversality to a projected commutator with an exhaustive transverse/null dichotomy; the Lorentzian chiral curvature $\mu_\chi^2$ is then discharged electric in the Schur-transverse branch via the A4 companion.
• AAR — Reduction of the angular amplitude observable. Working note fixing, before any Weil–BFS computation, the precise observable for the absolute normalisation of the Q14 §6 angular split: accumulated oriented $J_3$-odd symplectic area normalised by the projected capacity, compared to the dictionary $\varepsilon = 1/10$ without fitting.
• Q11OF — The oriented frontier observable. Diagnostic note defining a frontier transfer observable on directed outgoing edges of the BFS cascade — the honest replacement for the identically-vanishing raw shell observable. Tests only whether the recursion produces a non-zero, bias-free oriented signal.
• A4-Note — The Born–Infeld saturation margin of the chiral modulus. Companion to PRS discharging its first open deliverable: antisymmetry lemma forcing the modulus variable to be an antisymmetric chiral two-form $F_{\chi,\mu\nu}(s)$; sign-locked Born–Infeld saturation functional $\mathcal{B}_{\mathrm{sat}}(s)$ (sign fixed by A4 admissibility, not by Maxwell weak-field matching); genus reduction $\mu_\chi = \tfrac{1}{2} P_{\chi,\mu\nu} P_\chi^{\mu\nu}$ of the second variation; Lorentzian genus determined electric in the Schur-transverse branch ($\mu_\chi^2 < 0$, $u \neq 0$); quartic radicand with vanishing Born–Infeld pseudoscalar on the electric locus and amplitude reduced to the cubic backreaction $P_\chi \cdot \mathcal{R}_\chi$; path-invariant Born–Infeld saturation on the corpus-derived real cascade (spin-2 obstruction to genus calibration, doubly conditional interior lock), magnitude $|u|$ dictionary-bound through the chiral-frontier normalisation $\mathcal{N}_A$.
• EBJ — The constrained jet of the Lorentzian eliminated block and the chiral generator. Companion taking up PRS's first open deliverable, the explicit $1 - P(s)$. The projector jet $1 - P(s) = Q_0 + sQ_1 + s^2Q_2 + O(s^3)$ is constrained ($Q_1 = [A, Q_0]$ a purely off-diagonal Grassmann tangent, $Q_2$ diagonal blocks pinned to $Q_1^2$ with opposite signs), giving $u(s) = s\,\langle J_3, \{E_0, E_1\}\rangle/\langle J_3, J_3\rangle$ with $v \equiv 0$ for phase-free data. The chiral generator is identified, not a new selector: $[\Pi_{J_\Pi\text{-odd}}(1-P)]_{LL} = \tfrac12\Delta_\chi(P)$ and $[\Pi_{J_\Pi\text{-odd}}\dot Q(0)]_{LL} = \tfrac12\partial_s\Delta_\chi(P)|_0$, reducing the value question to the single normalisation target $\partial_s\Delta_\chi(P)|_0$.
• CHO — Chiral orientation of the directed Heisenberg frontier. Companion note resolving the chiral weight $\sigma_L$ deferred by Q11OF. Parity argument closes the false route $\sigma_L = f(T(e))$ — edge inversion and the residual reflection $\varphi$ both send $T \mapsto -T$, so every function of $T$ has equal arrow- and $\varphi$-parity, whereas the admissible weight is arrow-even and $\varphi$-odd. The resolution: $\sigma_L$ is the Weyl representation type $\mathbf{2}$ versus $\bar{\mathbf{2}}$ (group inversion preserves it, complex conjugation exchanges it). In the inherited Heisenberg central-phase lift the edge phase is the linear cyclotomic character $\zeta_q^{\Delta A_c}$ (no Gauss, Maslov or $\sqrt{5}$ component), so the natural branch gives $\rho_\chi = 1$ and $\mathcal{N}_A = \theta_{\max} \neq 0$ — $q$-invariant on the verified range. The residual open point is an additional spin-Galois phase outside the central character (a $\sqrt{5}$ or ADE-spin factor orthogonal to $\zeta_q$), the same arithmetic orthogonality that governs the ADE gate.
• AOG — Spin-stratum type-rigidity: orthogonality of the ADE gate and closure of the chiral lift. Companion note isolating the exact logical status of the split value $\varepsilon = 1/10$: a theorem modulo the ADE case-selection gate (the generation-deficit normal form forces $\varepsilon = 1/10$ and $\kappa = 5/12$ once a case with external ratio $3{:}2$ is selected). The obstruction to deriving the gate is an arithmetic orthogonality between Born–Infeld parity (complex conjugation, fixing $\sqrt{5}$) and the Galois action $\sqrt{5} \mapsto -\sqrt{5}$ required to select the $\sqrt{5}$-locus. For the present corpus construction the negative theorem is an unconditional no-go: a bounded character-table audit of $2I \cong \mathrm{SL}(2,5)$ proves spin-stratum type-rigidity — every available operation fixes $\sqrt{5}$, so the generated group lies in the $\sqrt{5}$-fixing subgroup and by closure cannot manufacture the outer automorphism. The same rigidity cuts the opposite way for the Lorentzian chiral lift (chiral-lift corollary): the lift induces no action on $\sqrt{5}$, so it discharges the residual obstruction of CHO and closes $[\mathrm{H\text{-}orient}]$ with $N_A \neq 0$ unconditionally within the present spin stratum. Only the abstract full-tower type-rigidity remains open, broader than the gate requires.
• PYL — Projected Yukawa line and generation-level assignment. Companion note opening the mass-sector frontier and closing only its first stage. The determinant line $L_Y = \wedge^2(\mathcal{S}_\Pi)$ is the unique functorial line of the rank-two carrier, invisible to the $\mathrm{SU}(2)_L$ adjoint sector $\mathrm{Sym}^2(\mathcal{S}_\Pi)$ (its induced action is $\det = 1$) and the carrier of the abelian $\mathrm{U}(1)_Y$ weight, so the projected Yukawa sector is the determinant-line sector of the admissible spinor functorial closure, not an external bundle datum. On the gauge-singlet triplet the squared residue $E_\Pi^2|_{C^3_{\mathrm{gen}}} = \mathrm{diag}(1, \tfrac12 + u, \tfrac12 - u)$ orders the three generation levels by exit deficit ($e_0$ lightest, $e_-$ heaviest for $u > 0$). The method lock is explicit: $L_Y$ fixes the coupling line, $E_\Pi^2$ fixes the levels, and the mass comes after; the projected Yukawa operator $Y_\Pi : \mathcal{S}_{L,\Pi} \otimes L_Y \to \mathcal{S}_{R,\Pi}$, its norm, $\operatorname{sign}(u)$, and the mixing phase remain open, so no mass value is claimed.
• PYO — Projected Yukawa operator and chiral polar factor. Second mass-sector step, with a sharp partial no-go. The Schur-residue sector fixes the positive hermitian square $H_\Pi := Y_\Pi^\dagger Y_\Pi$, with $\mathrm{Spec}(H_\Pi)|_{C^3_{\mathrm{gen}}} = \lambda_Y^2\{1, \tfrac12 + u, \tfrac12 - u\}$, so $E_\Pi^2$ fixes the squared Yukawa levels, not the morphism. By polar decomposition $Y_\Pi = U_\Pi H_\Pi^{1/2}$ the unitary chiral factor $U_\Pi$ — carrying the chiral orientation and the complex mixing phase — is free, while the positive norm $\lambda_Y$ is a separate scale of $H_\Pi$; neither is visible to $E_\Pi^2$. The CP-real branch $U_\Pi = \mathbf{1}$ is the diagonal no-mixing choice. Front 3b is thereby closed as a squared-Yukawa-levels result, not a mixing result: $E_\Pi^2$ can close $H_\Pi = Y_\Pi^\dagger Y_\Pi$ but cannot close $Y_\Pi$ unless $U_\Pi$ is fixed.

Position within Cosmochrony

The fermionic matter sub-programme sits at the apex of Branch III. It is the first paper of the Q-series that depends simultaneously on all upstream layers of the programme:

• the admissible Weil fibre structure (Note 1 and Note 5);
• the closed Lorentzian geometry (Note 2);
• the gauge bundle (Note 3);
• the gauge–gravity synthesis (Q13, to be summarised in Presentation Note 8).

Open deliverables

• Explicit $E_\Pi$, splitting value and the Yukawa sector. The qualitative mechanism of the inter-generation splitting is closed in Q14 §6 (static obstruction, oriented $J_\Pi$-odd lift, non-zero cascade $J_3$ projection), and the amplitude mechanism is now derived: on the corpus-derived real cascade the A4 companion settles it as Born–Infeld saturation. The split value itself is dictionary-bound through the chiral-frontier normalisation $\mathcal{N}_A$ (angular analogue of the radial scale $\kappa = 5/12$). The genuine first-principles front is therefore upstream and structural: the ADE case selection and level-to-generation map, the derivation of the projective-resolution growth and the cascade exponent $\beta$, and the transfer constant $N_{\mathrm{casc}}$ that fixes $\mathcal{N}_A$. Downstream of these: the integral normalisation of $\mathcal{L}_Y$, the mass spectrum from the eigenvalues of $E_\Pi^2$, and generation mixing via the complex metaplectic phase. The Yukawa line and generation ordering are structurally closed (PYL): $L_Y = \wedge^2(\mathcal{S}_\Pi)$ and the level assignment from $E_\Pi^2|_{C^3_{\mathrm{gen}}} = \mathrm{diag}(1, \tfrac12+u, \tfrac12-u)$; the projected Yukawa operator, the mass normalisation, and the mixing remain open. The radial factor of the split modulus is fixed in form, $s_* = \beta/|E_P|$ in the affine electric gauge, with no numerical prediction.
• $[\mathrm{H\text{-}color}]_{\mathrm{pointwise}}$ beyond the standard Cayley graph. The colour-coupled sector is unconditional at the pointwise level on the standard Cayley graph (O31); extension to general admissible graphs is open.
• Full matter content in Lorentzian signature. $S^{\mathrm{matter}}_\Pi = (\mathcal{S}_\Pi \oplus (\mathcal{S}_\Pi \otimes V_{\mathrm{color}})) \otimes C^3_{\mathrm{gen}}$ with explicit gauge couplings and three-generation Yukawa structure.

References

Beau, J. The Fermionic Matter Sub-Programme: Presentation Note 6. Working paper, 2026. https://doi.org/10.5281/zenodo.20562665