Derivation of the SU(3) Gauge Structure from the PDL Axioms C1–C4: Completion of the Standard Model Gauge Group SU(3) × SU(2) × U(1) (Projective Dynamic Logo Framework — Document D58)

Documents D46 and D57 established U(1) and SU(2) as unconditional theorems of the PDL (Projective Dynamic Logo) axioms C1–C4, via the Hopf fibration of K_4 and the gauge chain S_4 → S_3 → Dic_3 ⊂ SU(2) at tree level. The present document closes this programme by establishing SU(3) as the third and final compact simply connected simple Lie factor of the Standard Model gauge group, derivable as an unconditional theorem of C1–C4 modulo the classical Cartan–Killing–Weyl classification of compact simple Lie groups.

The derivation rests on five lemmas, each verified by exhaustive integer computation in the supplementary script PDL_SU3_script1.py: (L1) S_4/V_4 ≅ S_3, the Weyl group of A_2, recalled from D57; (L2) the natural action of S_3 on the three non-identity elements of V_4, identified by an S_4-equivariant canonical bijection with the three V_4-orbits on the six edges of K_4 (new result of D58); (L3) A_4/V_4 ≅ Z_3, the centre of SU(3), distinguishing SU(3) from PSU(3); (L4) reduction to a rank-2 Cartan subalgebra forced by the invariance of the relational budget R_e = 6 (D16a); (L5) identification of the A_2 root system in the trace-zero plane.

By the Cartan–Killing–Weyl classification, the unique compact simply connected simple Lie group with Weyl group S_3, centre Z_3, and rank-2 Cartan with A_2 root system is SU(3). Combined with D46 and D57, this establishes the algebraic structure SU(3) × SU(2) × U(1) of the Standard Model gauge group as an unconditional theorem of C1–C4.

As a documented negative result, the script establishes that the alternative strategy via the cycle homology H_1(K_4; R) fails: V_4 acts faithfully, not trivially, on H_1(K_4; R), clarifying why the natural S_3-set is V_4 \ {e} and not the cycle homology.

The physical identification of SU(3) with the colour gauge group SU(3)_c acting on the proton triplet (u, u, d) in the fundamental representation 3 is identified as Open Problem OP-D58-1.

All claims are exhaustively verified by the supplementary Python script PDL_SU3_script1.py (deposited separately on Zenodo). The derivation uses no free parameters and introduces no auxiliary hypotheses beyond the four combinatorial axioms C1–C4.