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

Papers O27–O30 complete the analytical account of the ${\mathrm{SU}}(2)$ sector of the admissibility programme, closing the chain $\text{emergence} \Rightarrow \mathfrak{su}(2) \Rightarrow d_\rho = 2 \Rightarrow \lambda_1/\lambda_2 = 2$. The remaining gauge-sector problem addressed here is the identification of ${\mathrm{SU}}(3)$ as an admissibility fixed point, and thereby the derivation of the full Standard Model gauge group ${\mathrm{SU}}(3) \times {\mathrm{SU}}(2) \times \mathrm{U}(1)$ as a composite. The present paper formulates this problem precisely and develops the structural framework for its resolution. Three results are obtained. First, we show that the O23 carrier route cannot directly yield $d_\rho = 3$: under the supplied spinor carrier of O23, the admissible algebra is ${\mathbb{H}}$, with $\dim_{\mathbb{R}} \operatorname{Im}{\mathbb{H}} = 3$ spectral directions and effective sector dimension ${d_{\rho}} = 2$. A new input is therefore required. Second, we define the colour triplet as an unordered set $\{c_1, c_2, c_3\} \subset ({\mathbb{Z}}/q{\mathbb{Z}})^*$ satisfying $c_1 + c_2 + c_3 \equiv 0 \pmod{q}$ and pairwise non-conjugacy $c_i \not\equiv q - c_j$ for all $i \neq j$. We state Hypothesis ${\mathbf{[H\text{-}color]}}$: the BFS fingerprint observables satisfy $\sigma_{c_1}(n) = \sigma_{c_2}(n) = \sigma_{c_3}(n)$ for all shells $n$ (triplet co-admissibility). Third, we prove (Theorem thm:su3-from-triplet) that if ${\mathbf{[H\text{-}color]}}$ holds, the group of Born–Infeld-compatible fibre-preserving transformations of the combined effective space $V_{\mathrm{color}} = V_{c_1} \oplus V_{c_2} \oplus V_{c_3} \cong {\mathbb{C}}^3$ is ${\mathrm{SU}}(3)$. Colour triplets exist if and only if $q \equiv 1 \pmod{3}$. We also prove (Theorem thm:generator-twist-isospectrality) that the Weil-sector Markov operator is isospectral under the generating-set twist ${\mathcal{S}_q} \mapsto \phi_\omega({\mathcal{S}_q})$, i.e.\ $\operatorname{spec}(\rho_c(P_{{\mathcal{S}_q}})) = \operatorname{spec}(\rho_c(P_{\phi_\omega({\mathcal{S}_q})}))$ for all $c$. A direct numerical computation (all primes $q \equiv 1 \pmod{3}$, $q \leq 157$) shows that this isospectrality does not extend across distinct central characters: $\operatorname{spec}(\rho_c(P_{{\mathcal{S}_q}})) \neq \operatorname{spec}(\rho_{\omega c}(P_{{\mathcal{S}_q}}))$ in general, establishing that ${\mathbf{[H\text{-}color]}}$ for the standard graph cannot be proved through global Markov-spectrum equality and that the BFS capacity is a strictly finer observable. We identify the correct algebraic mechanism: the irreducibility of the Weil representation (Schur's lemma) forces the Heisenberg centre to act as a scalar, so every horizontal coset $(a,b)$ contributes exactly one independent direction to each colour summand $V_{c_i}$; the Vandermonde invertibility of the triplet phase matrix $(\lambda_j^k)_{j,k}$, with $\lambda_j = e^{2\pi i c_j/q}$ and $(c_1,c_2,c_3) = (c,\omega c,\omega^2 c)$, then guarantees exact colour sector rank equality $r_c(n) = r_{\omega c}(n) = r_{\omega^2 c}(n)$ for all BFS depths, unconditionally (Lemma lem:vandermonde-coset and Proposition prop:colour-sector-rank). We further show (Lemma lem:single-frequency and Proposition prop:hcolor-frequency) that 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 $\sigma_c(n)$ is a count of distinct frequencies and the orbit dilation $B \mapsto \omega B$ preserves it exactly. This establishes ${\mathbf{[H\text{-}color]}}$ pointwise on the standard graph at finite $q$, so that the ${\mathrm{SU}}(3)$ identification, and with it $G_{\mathrm{SM}} = {\mathrm{SU}}(3) \times {\mathrm{SU}}(2) \times \mathrm{U}(1)$, holds unconditionally for the standard graph. We identify two candidate discrete groups for the analogue of the $Q_8 \to 2I \to {\mathrm{SU}}(2)$ chain: $\Delta(27) = {\mathrm{Heis}}_3(\mathbb{F}_3)$ as the minimal SU(3)-type group and $\Sigma(168) = \mathrm{PSL}(2,7)$ as the LPS candidate. The O32 numerical campaign has been conducted for $q \in \{61, 151, 211\}$ (test, $q \equiv 1 \pmod{3}$) and $q \in \{29, 101\}$ (control, $q \equiv 2 \pmod{3}$): ${\mathbf{[H\text{-}color]}}$ is confirmed for all three test primes, with variance ratios $R \leq 1$ relative to the control noise floor (Section ssec:o32-results). $q=307$: $R=0.7$, 4/4 triplets. The full dataset $q\in\{61,151,211,307\}$ uniformly satisfies $R\leq 3$ (ctrl\_ref $=1.657\times10^{-3}$).