The Fermionic Matter Sub-Programme

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.

Structural chain

$$ F_n \overset{\mathrm{model}}{\simeq} \mathcal{H}_c \;\overset{\text{model}}{\rightsquigarrow}\; \mathrm{Mp}(2,\mathbb{R}) \;\Rightarrow\; \mathrm{mp}(2,\mathbb{R})_\mathbb{C} \simeq \mathfrak{sl}_2(\mathbb{C}) \;\overset{[\text{H-Spin}]}{\rightsquigarrow}\; \mathrm{Spin}(3,1) \simeq \mathrm{SL}(2,\mathbb{C}) \;\Rightarrow\; \mathcal{S}_\Pi. $$

The carrier dictionary and the real metaplectic model are supplied, and the arrow marked [H-Spin] is not an implication. Then:

• $\mathrm{Sym}^2(S_L) \simeq \mathrm{ad}(P_\Pi)$ and $\wedge^2(S_L)$ trivial — the Lorentz sector, under [H-Spin].
• $\mathrm{Sym}^2(E_{\mathrm{weak}}) \supset \mathrm{ad}_\mathbb{C}(\mathfrak{su}(2)_L)$ — the $\mathrm{SU}(2)_L$ sector, under [H-Weak].
• $L_Y := \wedge^2(E_{\mathrm{weak}})$ — the $\mathrm{U}(1)_Y$ sector, under [H-Weak].
• $E_\Pi$ left-admissible — Lorentz chirality, under [H-Spin] and the Born–Infeld datum of O30; left-admissibility is the definition of the orientation-compatible branch, an input.
• The $V-A$ chiral structure — the representation content of [H-Weak], not derived from left-admissibility.
• supplied rank-three rule $\sigma_c(n_3) = 3 \to C^3_{\mathrm{gen}}$ — three generations as a gauge-singlet factor, conditional on that rule.

Core results (Q14 3.3, as organised in FM-Note 3.3)

• Theorem A (algebra proved on supplied model data; two conditional readings). On the supplied carrier, given a real metaplectic model and a supplied doublet $V$, $\mathrm{Sym}^2(V) \simeq \mathfrak{sl}_2(\mathbb{C})$ and $\wedge^2(V)$ is the determinant line: this is algebraic, proved without either hypothesis, and independent of any metric. Under [H-Spin], $\mathcal{S}_\Pi = S_L \oplus S_R$ is a spinor bundle, $\mathrm{Sym}^2(S_L)$ is its Lorentz sector and $\wedge^2(S_L)$ is trivial. Under [H-Weak], the same identities applied to $E_{\mathrm{weak}}$ give the weak adjoint, whose compact real form is $\mathfrak{su}(2)_L$, and $L_Y = \wedge^2(E_{\mathrm{weak}})$ carrying $\mathrm{U}(1)_Y$ only because the structure group is $\mathrm{U}(2)$. $E_{\mathrm{weak}}$ is not constructed.
• Theorem B (chirality and hypercharge, conditional; the parts carry different hypotheses). Under [H-Spin], the projected Dirac operator satisfies a Lichnerowicz-type identity. Under [H-Spin] and the Born–Infeld datum, the spinorial lift of the parity is, up to a phase, the unique covariant antilinear lift and reverses chirality; on the orientation-compatible branch, an input, $E_\Pi$ is left-admissible ($P_R E_\Pi P_R = 0$). This is Lorentz chirality. The $V-A$ structure is a different statement, the representation content of [H-Weak], which Q14 does not derive from left-admissibility. Under both hypotheses, $\mathrm{U}(1)_Y$ invariance imposes the anomaly-cancellation trace condition $\mathrm{Tr}_{\mathcal{S}_\Pi}(\gamma_5 Y A_\Pi(x)) = 0$, in its gauge-curvature part only. Hypercharge stands on two further steps: the rigidity of the weights carries a supplied three-dimensional colour module, and selecting the Standard Model pattern carries in addition the minimal integral normalisation of $L_Y$, holds only up to an overall sign, rescaling and the exchange of $u_R$ and $d_R$, and rests on a connection Q14 leaves open.
• Theorem C (three generations, conditional). The selection rule $\sigma_c(n_3) = 3$, a supplied rule, admits a spinorial multiplicity reading giving a gauge-singlet generation space $C^3_{\mathrm{gen}} \subset \ker(\mathrm{ad}_{\mathrm{SU}(2)} \oplus Y)$, on a supplied rank-three carrier and, through the bundle it multiplies, under [H-Spin] and [H-Weak]. The generation space is identified with neither the weak adjoint nor the Lorentz sector, and not with the spatial triplet: $\dim\mathbb{R}^3_{\mathrm{space}} = \dim C^3_{\mathrm{gen}}$ does not identify the two spaces.
• Colour sector (open input). A colour-coupled quark sector would tensor the admissible spinor bundle with a separately supplied colour module $V_{\mathrm{color}}$. O31 version 2.0 withdraws the claimed pointwise co-admissibility and group identification, so this module is not constructed by the present chain.
• Dynamic generation lifting (Q14 §6; model operators). 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). The operators of Q14 §6 are model operators on a supplied doublet distinct from the soldered one; their reading as restrictions of $E_\Pi^2$ is the named hypothesis [H-Res], which is not supplied, and the level-to-generation map is not established. Within the metaplectic step model, the external block $R_{\mathrm{mix}}$ is not reached by the image of $\mathfrak{sl}_2(\mathbb{C})$ (Remark 6.4, a statement on $R_{\mathrm{mix}}$ only).
• Mass-sector frontier (PYL, PYO; no mass value). The line note types the Yukawa line as the conditional weak determinant line $L_Y = \wedge^2(E_{\mathrm{weak}})$ under [H-Weak] and orders the levels of the model operator $\mathrm{diag}(1, \tfrac12+u, \tfrac12-u)$ under [H-Res]. The operator note shows that $H_\Pi = Y^\dagger Y$ fixes the squared levels, not the polar factor of the morphism, and separates the polar generator of a morphism family from the projection $A_\Pi^{\mathrm{odd}}$ of the $\mathfrak{sl}_2$ lift, which vanishes, by representation theory, for the projection as defined, and does not determine the polar factor. The physical polar class is open, and the choice of the $J_\Pi$-odd part is a modelling choice that no source justifies.

Constituent papers

• Q14 — Toward fermionic matter from projective Dirac admissibility: chirality and electroweak structure rest on a Lorentzian spin solder and a distinct weak factor. Principal paper of the sub-programme (version 3.3): the metaplectic spinor algebra on supplied model data, the two hypotheses [H-Spin] and [H-Weak] on which the Lorentz and electroweak readings rest, and a model of the inter-generation splitting (Q14 §6).
• PRS — The projective residue as a Schur complement (version 3.0). Operator-side companion note refining Q14 §6. For a locking operator the Schur form $E_\Pi[P] = -N_P^\dagger N_P \preceq 0$ is unconditional; for the reference operator it is the compression remainder of the projected Dirac square, which is the projective endomorphism of Q14 only under the symbol identity $\Pi_S D^2 \Pi_S^{*} = \mathrm{Lich}$, which no source states. Under the generation-reading hypothesis [H-Gen] the reference operator gives $u = 0$ and a non-zero $u$ requires a locking operator that breaks $J_\Pi$-equivariance; the multiplicative reading [H-Mult] is a separate hypothesis. The finite sector is conditional. The Schur-null and Schur-transverse loci are defined and a dichotomy is proved under explicit hypotheses, which locus holds being open; the exact $\mathfrak{sl}_2$ opening is produced by $[E,F] = H$, and its transport to the locking operator is the hypothesis [H-Tr]. Neither the sign of the Born–Infeld genus nor the non-vanishing of $u$ follows, and no mechanism for a non-zero split is derived.
• AAR — Angular Amplitude Reduction: the Oriented Shear Product of the $\mathfrak{sl}(2)$ Step and the Central-Phase Observable of the Weil Cascade. Internal note fixing the observable for the absolute normalisation of the Q14 §6 angular split and reporting the Weil–BFS computations of it. The per-step $J_3$ component of the metaplectic cascade generator is the oriented shear product $ts$ produced by the $\mathfrak{sl}_2$ commutator $[E,F] = H$ (in the $\mathrm{SL}(2)$-on-$V$ model of Q14, no Heisenberg bracket); the proposed observable is the accumulated oriented central-phase increment of the Heisenberg lift, normalised by the projected capacity and compared to the dictionary $\varepsilon = 1/10$ without fitting. That this increment carries the $J_3$ signal is an input of the note, not a consequence of the reduction.
• 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: antisymmetric structure and Lorentzian genus of the generation split (version 3.0). Companion to PRS examining the Lorentzian saturation functional. The symmetric square of the $J_\Pi$-odd modulus is $J_\Pi$-even and cannot carry the orientation sign, so the carrier is taken, by hypothesis, to be an odd antisymmetric chiral two-form $F_{\chi,\mu\nu}(s)$; $\mathcal{B}_{\mathrm{sat}}(s)$ is the Born–Infeld determinantal saturation margin of such a supplied two-form, and its second variation $\mu_\chi^2 = \tfrac{1}{2} P_{\chi,\mu\nu} P_\chi^{\mu\nu}$ is the Lorentzian genus of the chiral polarisation. The genus is not determined: the central commutator supplies no electric assignment, the central direction not being temporal in the co-metrics of the corpus. The quartic coefficient of the radicand depends on the parametrisation and selects no lock; for an exactly collinear purely electric ray the bound is reached at $t_* = \beta/|\mathbf{E}_{F_0}|$, and the first contact is chart independent when it exists. The Front D factorisation of the contact split is formal and conditional, and the magnitude $|u|$ is not predicted.
• EBJ — The constrained jet of the Lorentzian eliminated block and the chiral defect-rate identity. Companion taking up PRS's first open deliverable, the explicit $1 - P(s)$. For every smooth family 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), with alternating parity under $J_\Pi$-equivariance. With a moving spinorial embedding, antiunitary parity makes every odd-order generation-mixing coefficient vanish, and an explicit covariant moving-Schur family realises $E_0^2 = \mathrm{diag}(1, \tfrac12, \tfrac12)$ with a non-zero split rate $u'(0) = 11 \cdot 2^{1/4}/9$. This finite witness has $J^2 = +1$ and lies outside [H-Gen](i) of PRS: it illustrates a moving-embedding family without being an instance of [H-Gen]. The identity $[\Pi_{J_\Pi\text{-odd}}\dot Q(0)]_{LL} = \tfrac12\partial_s\Delta_\chi(P)|_0$ fixes one block of the odd tangent, not a unique generator; the selection of the family and the normalisation of $|u|$ remain open.
• CHO — Chiral orientation of the directed Heisenberg frontier. Companion note resolving the chiral weight $\sigma_L$ deferred by Q11OF. Under [H-WS] (the residual reflection $R_b$ acts on the chiral carrier as the scalar $-1$; supplied by no source), a 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 $\mathcal{N}_A = \theta_{\max} \neq 0$ (by this definition) — $q$-stable to $0.4\,\%$ on the verified range, not $q$-invariant, in that lift and under [H-WS], not unconditionally. The residual open point is an additional spin-Galois phase outside the central character (a $\sqrt{5}$ or ADE-spin factor orthogonal to $\zeta_q$), an open point examined, under stated inputs, by the AOG-Note below.
• AOG — Spin-stratum type-rigidity: an obstruction to the Galois action on the binary icosahedral stratum, with conditional consequences for 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), whose first-principles status the note does not settle. The proved result is an obstruction concerning the Galois automorphism $\sigma : \sqrt{5} \mapsto -\sqrt{5}$: a character-table audit of $2I \cong \mathrm{SL}(2,5)$ shows that inner conjugations, $-I$ and $J_\Pi$ act trivially on $\mathrm{Irr}(2I)$, whereas $\sigma$ induces $(2\,2')(3\,3')$. Modulo [H-WS] for $R_b$, the A4 hypotheses for the locking, modelling assertions for the tensor functors and the Schur projection, and the modelling premise [M] that the generator list is complete (which implies carrier-preservation), the base-fibre Born–Infeld parity (a candidate hypothesis of the foundations, taken as a premise) fixing $\sqrt{5}$, no listed operation realises $\sigma$. No link between the gate and $\sigma$ is established (the outer automorphism fixes the order-four class and preserves each order-five locus), so the gate stays open. The Lorentzian chiral lift is an input that the note does not construct, the spin-solder hypothesis [H-Spin] of Q14; under [H-Spin], [H-WS], [M], the boost premise [B] and the inherited central-phase convention of CHO (with $\rho_\chi = 1$ by definition of $\sigma_L$) it induces no action on $\sqrt{5}$, so the residual caveat of CHO does not arise within the present spin stratum.
• PYL — Toward projected Yukawa couplings: a conditional weak determinant line and generation-level ordering (version 3.0.1). Companion note opening the mass-sector frontier; it derives no mass, Yukawa coupling or mixing. Algebra proved on supplied model data: on a rank-two bundle, $\wedge^2$ is the unique one-dimensional functor among $\mathrm{Sym}^k$ and $\wedge^k$, and it is trivial when the structure group lies in $\mathrm{SL}(2,\mathbb{C})$, so $\wedge^2(S_L)$ carries no abelian weight of the spin group under [H-Spin]. Under [H-Weak], the line $L_Y := \wedge^2(E_{\mathrm{weak}})$ with structure group $\mathrm{U}(2)$ carries the abelian weight: a conditional weak determinant line, not a line of the spinor bundle. On the generation triplet, the model operator $\mathrm{diag}(1, \tfrac12+u, \tfrac12-u)$ has exit deficits that order three levels for $0 < u < \tfrac12$ with a scale-independent ratio; reading it as $E_\Pi^2|_{C^3_{\mathrm{gen}}}$ is the named hypothesis [H-Res], not supplied, and the level-to-generation map is not established. The Yukawa operator needs a weak linking carrier supplied by no source; its norm, the sign of $u$ and the mixing remain open, so no mass value is claimed.
• PYO — Projected Yukawa operator and chiral polar factor (version 2.1). Second mass-sector step, with a partial no-go, stated on the generation block of the Yukawa morphism under the named premises [H-Res], [H-Sq], [H-Fac] and [H-Grad], none supplied. $H_\Pi = Y^\dagger Y$ fixes the squared levels $\lambda_Y^2\{1, \tfrac12+u, \tfrac12-u\}$ and not the polar factor $U_\Pi$, which carries the chiral orientation and the mixing; its rephasing class has three angles and one Jarlskog phase. The projection $A_\Pi^{\mathrm{odd}}$ of the $\mathfrak{sl}_2$ lift vanishes, by representation theory, for the projection as defined, and does not determine the polar factor: the polar generator of a morphism family solves a Sylvester equation and, at the identity, is the generally non-zero $J_\Pi$-even anti-hermitian part of the lift; an exact counterexample (the polar factor of $\mathrm{Sym}^2 g$, external entry $1/17$ at $(t, s) = (2, 1)$, not a Yukawa morphism) shows that a vanishing tangent does not exclude a finite external entry. The corpus supplies no family of Yukawa morphisms, so the physical polar class is open, and the choice of the $J_\Pi$-odd part is a modelling choice that no source justifies. The depth hierarchy of the generations is not derived. On the orientation-compatible branch of Q14, an input, in the framework of PRS (under its generation-reading hypothesis) a non-zero diagonal split requires a locking operator that breaks $J_\Pi$-equivariance, and no mechanism for it is derived.
• KUD — Charged-lepton Koide relation as a unit-dispersion constraint. A source note fixing the mass-sector target as a single scale-free invariant: $Q_\ell = (1 + \mathrm{CV}(r_\ell)^2)/3$, so Koide $\iff \mathrm{CV}(\sqrt m) = 1$ (standard deviation equal to mean). It separates the dispersion constraint from the concrete hierarchy (the phase placing one generation near a node of the square-root profile), and shows that the ladder $(1, \tfrac12 + u, \tfrac12 - u)$, taken by hypothesis as a normal form of the projected levels, under-disperses ($\mathrm{CV} \le 1/\sqrt2$), so it cannot reach $Q = 2/3$. A maximum-entropy reading is recorded as an open test, and a scoped no-go now excludes the linear self-adjoint determinant realisation of the singlet–doublet route (the equipartition mechanism otherwise remaining open); a target invariant, not a mass formula.
• TPC — Ternary phase carrier: colour, generations, and Koide as one carrier. A speculative bridge relating three facts through one complex three-phase carrier $z_g = S + A\,e^{i(\delta + 2\pi g/3)}$: its pure-phase part sums to zero ($1+\omega+\omega^2=0$, a colour-neutral class), its real part is the square-root mass profile of KUD. The singlet amplitude $S$ is the master dial — $S=0$ colour (confined), $S\neq0$ masses, $A=\sqrt2\,S$ equipartition (Koide, $\mathrm{CV}=1$), $\delta$ near a node the hierarchy — with a candidate capacity-balance filter saturated at $\mathrm{CV}=1$ (the selection of the saturated edge by projection locking being a further hypothesis that the cited sources do not supply). Explicitly not a derivation: it reorganises the open questions and states the construction charge sheet.
• NPI — Neutrino mixing and projective particle identity. A structural companion note on the ontology of fermionic identity: the misalignment between the weak flavour basis and the mass propagation basis is the generic situation (neutral meson systems exhibit the same structure; the neutrino is the elementary case), while alignment requires an identity-stabilising projection channel. The large PMNS mixing is then the unfixed default, consistent with the anarchy hypothesis, and the near-diagonality of CKM becomes the explanandum, conditional on a common projective frame. No mixing angles or masses are derived.
• SRN — Spin as a transformation class of rotating wave modes. Companion conceptual audit of the rotating-wave reading of spin: the relative phase is a measurable coordinate within a sector, the weight $m$ an axial winding degree, $j$ the $SU(2)$ representation label, and univalence $(-1)^{2j}$ a rule on the observable algebra. For Cosmochrony it records the exact central grading on $V_\rho^{\otimes n}$, the $e_0$ correction (spin-weight $m = 0$ within the $j = 1$ module), the Lorentz/weak factor separation, and types the open front: constructing the full $\operatorname{Spin}(3)$ action (representational problem) versus its dynamical realisation by the cascade, with axis and angle relation both to be derived (dynamical problem).
• NSR — Operator doublets, erased history fibres, and the no-sequential-re-entry theorem. No-go note closing the two deposited candidates for the independent fermionic weak multiplicity $M_L \simeq \mathbb{C}^2$: the commutant of the finite Weil module is spanned by the two parity projectors, so the canonical Hecke $M_2(\mathbb{C})$ doublets of $\mathrm{End}(V_c)$ admit no fermionic lift; the erased history fibres are structurally abelian ($C_4$ frame stabiliser, canonical $(\mathbb{Z}_2)^r$ torsor per $b$-shadow); the two odd Hermitian axes $i(A_X \pm A_Y)$ are exchanged by a semilinear automorphism of the cocycle-enriched datum; and the no-sequential-re-entry theorem ($\pi v = 0 \Rightarrow Q\pi v = 0$, for the record quotient and for the conditional expectations of every non-degenerate measure) excludes every projection tower. Corollary: the history-fibre route requires a parallel oriented substrate channel — an extension architecture, not a derivation.

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 Weil fibre structure on a supplied Heisenberg carrier (Note 1 and Note 5);
• a four-dimensional Lorentzian co-metric on the effective base, supplied and entering the fermionic sector only through [H-Spin] (Note 2);
• the gauge bundle for a supplied compact structure group (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 mass spectrum and the mixing require the explicit spectral construction of $E_\Pi$. The Schur form of the compression remainder, $-N^\dagger N \preceq 0$, is that of $E_\Pi$ only under the identity $\Pi_S D^2 \Pi_S^{*} = \mathrm{Lich}$; the open part is its explicit Lorentzian block $1 - P(s)$ along the $J_\Pi$-odd modulus. The Schur-null and Schur-transverse loci of the transported $J_\Pi$-odd tilt of the locking operator are defined in PRS, where a dichotomy is proved under explicit hypotheses; which locus holds is open. The exact $\mathfrak{sl}_2$ opening of the oriented metaplectic step is produced by $[E,F] = H$ (Q14, Proposition 6.5), but its transport to the locking operator is the hypothesis [H-Tr] of PRS, supplied by no source. The A4 companion does not determine the genus: the central commutator supplies no electric assignment, the central direction not being temporal in the co-metrics of the corpus, and no map from the generation operator to a two-form is available. Neither amplitude nor selection is settled there: the quartic coefficient of the radicand depends on the parametrisation of the modulus and selects no lock, and for an exactly collinear purely electric ray the bound is reached at $t_* = \beta/|\mathbf{E}_{F_0}|$; no choice between a boundary contact and an interior stationary point is established. What remains open is the existence of a non-zero split, its value $|u|$ and the Yukawa sector downstream of $u$. The value $|u|$ is not a parameter-free prediction: it 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, and the generation mixing, whose physical polar class is open. The Yukawa line and the generation ordering are recorded conditionally (PYL): $L_Y = \wedge^2(E_{\mathrm{weak}})$ under [H-Weak] and the ordering of the levels of $\mathrm{diag}(1, \tfrac12+u, \tfrac12-u)$ under [H-Res]; the Yukawa operator needs a weak linking carrier supplied by no source, and its norm, the sign of $u$, the polar class (PYO) and the mixing remain open.
• Spin solder and weak factor. Construct the principal $\mathrm{Spin}(3,1)$-bundle of [H-Spin] from the metaplectic action, and the distinct rank-two weak factor $E_{\mathrm{weak}}$ of [H-Weak]; neither is supplied, and neither is excluded.
• Colour carrier and group. Construct a colour module and prove its admissibility and compact symmetry group independently of the withdrawn O31 claims.
• 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). Version 3.3, 2026. https://doi.org/10.5281/zenodo.20562665