Physical Identification of SU(3)_c and the Fundamental Representation 3 from the PDL Axioms C1–C4 (Projective Dynamic Logo Framework — Document D59)

Document D59 of the Projective Dynamic Logo (PDL) framework. Building on D58 (which established SU(3) as an unconditional algebraic theorem of the PDL axioms C1–C4), the present document constructs the fundamental representation 3 of SU(3) explicitly from C1–C4 and identifies the physical colour gauge group SU(3)_c. The carrier space is the trace-zero plane W = {(a,b,c) ∈ C³ : a+b+c = 0}, whose three natural axes are canonically labelled by the three elements of V4{e} — the structural triplet of K4 established in D58 Lemma L2. Five lemmas are established as unconditional theorems of C1–C4 and verified by exhaustive exact-integer computation in the supplementary script PDL_D59_script1.py (DOI: 10.5281/zenodo.20628926): L1 — S3 = S4/V4 acts faithfully and transitively on V4{e} (recalled from D58). L2 — The permutation representation on C³ decomposes as χ_π = χ_trivial + χ_standard; verified over all 24 elements of S4. L3 — The trace-zero plane W is the unique S3-invariant complement of C(1,1,1) in C³; the six S3-matrices on W are distinct 2×2 integer matrices with |det|=1. L4 — The A2 root system is realised in W with Cartan matrix [[2,−1],[−1,2]], confirmed by exact integer inner products. L5 — The isospin asymmetry Δn = n_d − n_u = 4 > 0 (unconditional theorem of C4, D47) forces the canonical ordering k1 < k2 < k3 and selects the orientation 3 over 3-bar, identifying the matter sector from the axioms. The ratio k3/(k1+k2) = R_e = 6 holds exactly. A documented negative result shows that no S4-equivariant bijection exists between the leakage-cycle triplet (T2) = {9, 19, 168} and the orbit triplet (T3) = V4{e}; the canonical bijection is physically motivated by the quintuplet, not forced by equivariance. Main theorem: SU(3)_c acting on W in the fundamental representation 3 is an unconditional theorem of C1–C4 modulo the classical Cartan–Killing–Weyl classification. The matter/antimatter orientation is derived from Δn = 4 > 0 without additional input. Open problems resolved: OP-D58-1 (physical identification of SU(3)_c) and OP-D58-3 (fundamental representation 3 from C1–C4). Combined with D46 (U(1)), D57 (SU(2), Weinberg angle), and D58 (SU(3) algebraic), D59 completes the derivation of the colour gauge group with its fundamental representation from purely combinatorial axioms, without free parameters beyond Δm_iso = m_d − m_u. PDL corpus: D01–D59 + DS01 + DL01–DL02 + N01 + D-exp series. Zenodo community: pdl-framework.