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 K₄ and the gauge chain S₄ → S₃ → Dic₃ ⊂ 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 PDLSU3ₛcript1. py: (L1) S₄/V₄ ≅ S₃, the Weyl group of A₂, recalled from D57; (L2) the natural action of S₃ on the three non-identity elements of V₄, identified by an S₄-equivariant canonical bijection with the three V₄-orbits on the six edges of K₄ (new result of D58) ; (L3) A₄/V₄ ≅ Z₃, 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 Rₑ = 6 (D16a) ; (L5) identification of the A₂ root system in the trace-zero plane. By the Cartan–Killing–Weyl classification, the unique compact simply connected simple Lie group with Weyl group S₃, centre Z₃, and rank-2 Cartan with A₂ 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 H₁ (K₄; R) fails: V₄ acts faithfully, not trivially, on H₁ (K₄; R), clarifying why the natural S₃-set is V₄ \ 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 PDLSU3ₛcript1. py (deposited separately on Zenodo). The derivation uses no free parameters and introduces no auxiliary hypotheses beyond the four combinatorial axioms C1–C4.