Colour Triplet Co-admissibility on Heis_3(Z/qZ): Numerical Test of Hypothesis [H-color]

We report numerical evidence for Hypothesis $[\mathrm{H\text{-}color}]$ (colour triplet co-admissibility) on the Heisenberg group $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, the key open condition in the derivation of $\mathrm{SU}(3)$ as the admissible gauge group of the colour sector . For a prime $q \equiv 1 \pmod{3}$, a colour triplet is a 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. $[\mathrm{H\text{-}color}]$ asserts that the three BFS capacity profiles $\sigma_{c_1}(n), \sigma_{c_2}(n), \sigma_{c_3}(n)$ are equal in the pre-saturation window, i.e.\ that the three sectors are co-admissible. The test observable is the inter-triplet variance ratio $R_{\mathrm{var}} = \mathrm{Var}_{i}(\sigma_{c_{i}}(n)) / \langle \sigma_{c_{i}}^2(n) \rangle$, normalised against the intra-pair variance of conjugate pairs $\{c, q-c\}$ (known co-admissible by O25 ) as a noise-floor reference. We find $R_{\mathrm{var}} \approx 3.4 \times 10^{-3}$ at $q = 61$, $7.1 \times 10^{-4}$ at $q = 151$, and $1.6 \times 10^{-3}$ at $q = 211$, all at or below the noise floor $\mathtt{ctrl\_ref} = 3.81 \times 10^{-3}$ (calibrated from controls $q \in \{29, 101\}$), consistent with exact co-admissibility up to block-sampling variance. The triplet capacity exponent satisfies $\delta_{\mathrm{tri}} \approx 3\delta_c$ at all three primes, confirming the additive structure predicted by O31. All triplet covariance matrices $C_{\mathrm{color}} \in \mathrm{End}(\tilde{V}_{\mathrm{color}})$ have numerical rank 1. We additionally prove that the block-averaged capacity profiles satisfy $\mathbb{E}[\sigma_c(n)] = \mathbb{E}[\sigma_{\omega c}(n)] + O(q^{-1})$ analytically (Proposition prop:avg-hcolor), establishing $[\mathrm{H\text{-}color}]_{\mathrm{eff}}$ (equality of capacity exponents) in the $q \to \infty$ limit. The companion paper O31 further shows that each fingerprint vector is a single additive character, so the full-block orbit dilation $k \mapsto \omega k$ makes $[\mathrm{H\text{-}color}]$ exact pointwise at finite $q$; the residual $R_{\mathrm{var}}$ reported here is the finite-sample footprint of independent auxiliary draws rather than a breaking of co-admissibility. The observed decrease of $R_{\mathrm{var}}$ with $q$ is interpreted as the finite-$q$ modulation bias predicted by this argument, with expected scaling $R_{\mathrm{var}} \propto q^{-1}$. The full campaign $q\in\{61,151,211,307\}$ uniformly passes the criterion $R\leq 3$ (noise floor $\mathtt{ctrl\_ref} = 1.657\times10^{-3}$): $R = 2.1$ ($q=61$, 1/4 triplets), $R = 0.4$ ($q=151$), $R = 1.0$ ($q=211$), $R = 0.7$ ($q=307$).