D61: Gauge Dynamics as a C4 Survival Theorem — Derivation of the Minimal Covariant Derivative from the PDL Axioms C1–C4 (Projective Dynamic Logo Framework)

The gauge group SU (3) ×SU (2) ×U (1) of the Standard Model and its fundamental colour representation 3 are unconditional theorems of the PDL axioms C1–C4 (D46, D57, D58, D59, D60). The present document closes the remaining gap in the gauge sector: it derives the form of the gauge interaction, showing that the minimal covariant derivative D_μ = ∂_μ − igA_μ is the unique gauge-transport structure compatible with C4. The derivation proceeds in four steps. First, the group of discrete transports on K4 compatible with C4 is identified as exactly Coh (K4) ≅ Z2³, the group of coherent sign configurations under pointwise multiplication (2048 exhaustive checks, PDLD61ₛcript1. py). Second, the action of V4 on this group is shown to be faithful and by group automorphisms, with fixed-point subgroup F ≅ V4 labelled by the three perfect matchings of K4 — the same objects labelling the axes of the colour representation 3 in D59 (288 exhaustive checks, PDLD61ₛcript2. py). Third, the transport structure Z2³ ⋊ V4 is shown to be consistent with Ggauge = SU (3) ×SU (2) ×U (1), established independently as a theorem by D46--D60 — the two routes converge on the same group. Fourth, by Utiyama's uniqueness theorem — invoked as an external lemma in the same way that Cartan–Killing–Weyl is invoked in D58 — the unique first-order equivariant generator on a principal Ggauge-bundle is D_μ = ∂_μ − igA_μ. Gauge dynamics is a consequence of the C4 survival principle, without variational postulate and without free parameter at the structural level. The axioms C1–C4 derive not only the gauge group and its representations, but also the form of its coupling to matter fields. Two negative results are documented: the initial conjecture that the admissible transport group equals V4 (order 4) is refuted — the correct group is Coh (K4) ≅ Z2³ (order 8) ; and the global sign flip (−1, …, −1) is not C4-admissible. Three open problems are identified: gauge boson masses (OP-D61-1), three fermion generations (OP-D61-2), and the explicit representation of Z2³ ⋊ V4 on Hcyc (OP-D61-3). Two verification scripts are included as supplementary files: PDLD61ₛcript1. py (2048 checks) and PDLD61ₛcript2. py (288 checks), both executed independently in Google Colab, exact integer arithmetic, Python standard library only.