SU(3) from Colour Triplet Co-admissibility: Towards the Complete Standard Model Gauge Group O31 — Spectral Admissibility Sub-programme

Papers O27–O30 complete the analytical account of the SU(2) sector of the admissibility programme, closing the chain emergence ⇒ su(2) ⇒ d_ρ = 2 ⇒ λ₁/λ₂ = 2. The only remaining non-numerical open problem in the gauge sector is the identification of SU(3) as an admissibility fixed point, and thereby the derivation of the full Standard Model gauge group SU(3) × SU(2) × 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_ρ = 3: the Born–Infeld parity forces the unique minimal admissible algebra to be H, with R ImH = 3 spectral directions and effective sector dimension dρ = 2. A new input is therefore required. Second, we define the colour triplet as an unordered set ₁, c₂, c₃\ (Z/qZ)^* satisfying c₁ + c₂ + c₃ 0 q and pairwise non-conjugacy cᵢ ≡ q - cⱼ for all i ≠ j. We state Hypothesis [H-color]: the BFS fingerprint observables satisfy σc₁(n) = σc₂(n) = σc₃(n) for all shells n (triplet co-admissibility). Third, we prove (Theorem thm:su3-from-triplet) that if [H-color] holds, the group of Born–Infeld-compatible fibre-preserving transformations of the combined effective space Vcolor = Vc₁ ⊕ Vc₂ ⊕ Vc₃ C³ is SU(3). Colour triplets exist if and only if q ≡ 1 3. We also prove (Theorem thm:generator-twist-isospectrality) that the Weil-sector Markov operator is isospectral under the generating-set twist Sq ↦ φ_ω(Sq), i.e.\ spec(ρc(PSq)) = spec(ρc(Pφ_ω(Sq))) for all c. A direct numerical computation (all primes q ≡ 1 3, q ≤ 157) shows that this isospectrality does not extend across distinct central characters: spec(ρc(PSq)) ≠ spec(ρω c(PSq)) in general, establishing that [H-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 two candidate discrete groups for the analogue of the Q₈ → 2I → SU(2) chain: Δ(27) = Heis₃(F₃) as the minimal SU(3)-type group and Σ(168) = PSL(2,7) as the LPS candidate. The O32 numerical campaign has been conducted for q ∈ \61, 151, 211\ (test, q ≡ 1 3) and q ∈ \29, 101\ (control, q ≡ 2 3): [H-color] is confirmed for all three test primes, with variance ratios R ≤ 1 relative to the control noise floor (Section ssec:o32-results). Extension to $q = 307$ is in progress.