SU(3) from Colour Triplet Co-admissibility: Towards the Complete Standard Model Gauge Group

Overview

O31 builds the structural framework for deriving the SU(3) gauge group from admissibility constraints on the Heisenberg graph $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$. It begins by ruling out the direct approach: the Hurwitz argument of O23 cannot directly produce $d_\rho = 3$, so a new mechanism is required.

The mechanism introduced is colour triplet co-admissibility. A colour triplet is a set $\{c_1, c_2, c_3\} \subset (\mathbb{Z}/q\mathbb{Z})^*$ satisfying the constraint $c_1 + c_2 + c_3 \equiv 0 \pmod{q}$. The paper proves that such triplets exist if and only if $q \equiv 1 \pmod{3}$.

The metaplectic automorphism $\phi_\omega(a,b,z) = (\omega a, \omega^{-1} b, z)$ then forces spectral degeneracy $\sigma_c(n) = \sigma_{\omega c}(n)$ for all BFS shells on the colour-adapted Cayley graph (Theorem 4.4). Conditional on the co-admissibility hypothesis [H-color], the symmetry group of the co-admissible triplet is exactly SU(3).

Status (v1.5). The core results (existence of triplets, spectral degeneracy, SU(3) uniqueness, colour sector rank equality, and pointwise [H-color] on the standard graph) are proved unconditionally. Each Born–Infeld fingerprint vector is a single additive character of frequency $B = c_1 b_1 + c_2 b_2 + c_3 b_3$, so the capacity observable $\sigma_c(n)$ counts distinct frequencies and the orbit dilation $B \mapsto \omega B$ preserves it exactly. SU(3) and the Standard Model composite $G_\mathrm{SM} = \mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ therefore hold unconditionally for the standard graph.

Core contributions

Ruling out the direct Hurwitz route: proof that the O23 Hurwitz argument cannot directly produce $d_\rho = 3$, establishing that the SU(3) identification requires a new structural mechanism.
Colour triplet definition and existence: definition of colour triplets $\{c_1,c_2,c_3\} \subset (\mathbb{Z}/q\mathbb{Z})^*$ with $c_1+c_2+c_3 \equiv 0 \pmod{q}$; proof that such triplets exist if and only if $q \equiv 1 \pmod{3}$.
Metaplectic spectral degeneracy (Theorem 4.4): the automorphism $\phi_\omega(a,b,z) = (\omega a, \omega^{-1}b, z)$ forces $\sigma_c(n) = \sigma_{\omega c}(n)$ for all shells $n$ on the colour-adapted Cayley graph (proved).
SU(3) uniqueness: the symmetry group of the co-admissible colour triplet is exactly SU(3); U(3), SO(3), U(2), and $G_2$ are excluded (proved). Combined with the analytical proof of [H-color] on the standard graph (v1.5), the identification is unconditional.
Single-frequency BI fingerprint (Lemma, proved, v1.5): each Born–Infeld fingerprint vector is a single additive character of frequency $B = c_1 b_1 + c_2 b_2 + c_3 b_3$. The capacity observable $\sigma_c(n)$ is therefore a count of distinct frequencies, and the orbit dilation $B \mapsto \omega B$ preserves it exactly. This establishes [H-color] pointwise on the standard graph at finite $q$, making the SU(3) identification — and with it the Standard Model composite — hold unconditionally.
Standard Model gauge group as composite: the gauge group $G_\mathrm{SM} = \mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ follows as a composite of three admissibility fixed points, each arising from a distinct co-admissibility constraint.
Two-tier generating-set independence: (i) exact for the colour-adapted graph $\mathrm{Cay}(G_q, \mathcal{S}_q^{(3)})$ (metaplectic argument); (ii) $O(q^{-1/2})$ for any generating set, including $\mathcal{S}_q$, from U1 universality. The set $\mathcal{S}_q^{(3)}$ is the canonical colour-sector generating set: the minimal extension of $\mathcal{S}_q$ closed under $\phi_\omega$.
Generator-twist isospectrality (Theorem 5.3, proved): for every $c \in (\mathbb{Z}/q\mathbb{Z})^*$, the Markov operators $\rho_c(P_{\mathcal{S}_q})$ and $\rho_c(P_{\phi_\omega(\mathcal{S}_q)})$ are unitarily conjugate, hence isospectral. Confirmed to machine precision for all $q \leq 157$ with $q \equiv 1 \pmod{3}$. However, $\operatorname{spec}(\rho_c) \neq \operatorname{spec}(\rho_{\omega c})$ in general: [H-color] on the standard graph cannot be proved via global spectrum equality; the BFS capacity is a strictly finer observable.
Intra-coset Vandermonde symmetry (Lemma, proved): Schur's lemma forces the Heisenberg centre to act as a scalar in each irreducible representation $\rho_{c_i}$. The triplet phase matrix $(\lambda_j^k)_{j,k}$ with $\lambda_j = e^{2\pi i c_j/q}$ is Vandermonde with non-zero determinant (since $c_1, c_2, c_3$ are pairwise distinct). Each horizontal coset $(a,b)$ therefore contributes exactly one independent direction to each colour summand $V_{c_i}$.
Colour sector rank equality (Proposition, proved unconditionally): $r_c(n) = r_{\omega c}(n) = r_{\omega^2 c}(n)$ for all BFS depths $n$ and any seed, unconditionally on the standard graph. This proves [H-color] at the level of BFS sector ranks. The algebraic mechanism (Schur's lemma + Vandermonde invertibility) is independent of any global Markov spectral property.

The colour triplet mechanism

The key insight of O31 is that the group-theoretic constraint $c_1 + c_2 + c_3 \equiv 0$ is not merely an algebraic condition but encodes a structural symmetry of the Weil representation. The metaplectic automorphism $\phi_\omega$ acts on the Heisenberg group by a third-root-of-unity rescaling of the $(a,b)$ coordinates while preserving the central direction.

Theorem 4.4 shows that this automorphism forces spectral degeneracy on the colour-adapted Cayley graph: the BFS shell counts $\sigma_c(n)$ are invariant under $c \mapsto \omega c$. This spectral degeneracy is precisely what constrains the symmetry group of the triplet to be SU(3) rather than larger groups like U(3) or $G_2$.

The hypothesis [H-color] asserts that the colour triplet is co-admissible, meaning that all three Weil representations $\rho_{c_1}$, $\rho_{c_2}$, $\rho_{c_3}$ jointly satisfy the admissibility constraints. This is the key open claim whose numerical testing is carried out in O32.

Algebraic mechanism: Vandermonde symmetry and unconditional rank equality

The Spectral obstruction (Theorem 5.3) shows that [H-color] cannot be proved via global Markov-spectrum equality. O31 identifies the correct algebraic mechanism.

The irreducibility of the Weil representation forces the Heisenberg centre to act as a scalar in each colour sector $V_{c_i}$ (Schur's lemma). For any fixed horizontal coset $(a,b)$ and three distinct central values $z_0, z_1, z_2$, the corresponding BFS vectors in $V_{c_1} \oplus V_{c_2} \oplus V_{c_3}$ assemble into the Vandermonde phase matrix \[ V = (\lambda_j^k)_{j,k}, \quad \lambda_j = e^{2\pi i c_j / q}, \] whose determinant $\det V = \prod_{j > k}(\lambda_j - \lambda_k) \neq 0$ since $c_1, c_2, c_3$ are pairwise distinct modulo $q$. Each horizontal coset therefore contributes exactly one independent direction to each of $V_{c_1}$, $V_{c_2}$, $V_{c_3}$, symmetrically.

The resulting Proposition (Colour sector rank equality) is unconditional: for every BFS depth $n$ and any seed, $r_c(n) = r_{\omega c}(n) = r_{\omega^2 c}(n)$, where $r_{c_i}(n)$ counts the independent directions in $V_{c_i}$ reached at depth $n$. Since the sector rank equals the number of distinct horizontal cosets reached — a purely combinatorial, $c_i$-independent quantity — the equality holds without any appeal to Markov spectral properties.

This proves [H-color] at the BFS sector rank level. The O32 numerical campaign (confirming $\sigma_c(n) = \sigma_{\omega c}(n)$ in the full capacity functional) is consistent with this algebraic result and extends the evidence to the complete fingerprint observable.

Local vs relational non-injectivity (Section 6)

O31 provides a conceptual unification of the three Standard Model gauge factors by classifying them according to the type of non-injectivity of the projection $\Pi: \Omega \to \mathcal{O}$:

U(1) — phase non-injectivity. The full U(1) orbit $\{e^{i\theta}\chi\}$ is contained in every Born–Infeld-indiscernible fibre. U(1) acts within a single character sector; it is the symmetry of an individual fibre under global phase rotation.
SU(2) — minimal correlated non-injectivity. The $\mathbb{Z}_2$ parity $\chi \leftrightarrow -\chi$ identifies exactly two sectors ($c$ and $q-c$) into a single fibre. The effective space $V_\rho \cong \mathbb{C}^2$ and the SU(2) gauge group arise from the symmetry of this two-element fibre. This is local non-injectivity: forced by a single projection step.
SU(3) — relational correlated non-injectivity. The $\mathbb{Z}_3$ colour orbit $\{c, \omega c, \omega^2 c\}$ identifies three sectors into a single fibre. The effective space $\tilde{V}_\mathrm{color} \cong \mathbb{C}^3$ and the SU(3) gauge group arise from the symmetry of this three-element fibre. This is relational non-injectivity: the three sectors are co-admissible through a $\mathbb{Z}_3$ correlation, not a single involution.

The gauge hierarchy U(1) ⊂ SU(2) ⊂ SU(3) in complexity corresponds exactly to the hierarchy 1 → 2 → 3 in the size of the minimal BI-admissible fibre orbit. Non-factorisability (Foundation M, Axiom A3) is the uniform structural origin.

Relation to the Cosmochrony programme

O31 is the theoretical foundation for the emergence of the SU(3) colour gauge group from admissibility constraints. In conjunction with earlier results on SU(2) and U(1) (Q6a), it provides a route to the full Standard Model gauge group as a composite of three admissibility structures.

The metaplectic spectral degeneracy result connects the group-theoretic constraint on colour charges to the spectral geometry of the Heisenberg Cayley graph, providing a concrete realisation of the non-injectivity principle at the gauge symmetry level.

The numerical verification of [H-color] is provided by O32. The gauge dynamics for SU(3) are derived in Q12 via vertical variation of the projective spectral entropy.

References

Jérôme Beau. SU(3) from Colour Triplet Co-admissibility: Towards the Complete Standard Model Gauge Group. Preprint. doi:10.5281/zenodo.20030491