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 quaternionic minimality argument cannot directly yield $d_\rho = 3$: the Born–Infeld parity forces the unique minimal admissible algebra to be $\mathbb{H}$, 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 $=1.657\times10^{-3}$).