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 only remaining non-numerical open problem in the gauge sector 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 5.1) 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 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. A falsifiable numerical test for O32 is formulated.