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 V4e — 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 PDLD59ₛcript1. py (DOI: 10. 5281/zenodo. 20628926): L1 — S3 = S4/V4 acts faithfully and transitively on V4e (recalled from D58). L2 — The permutation representation on C³ decomposes as χ_π = χₜrivial + χₛtandard; 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 = nd − nᵤ = 4 > 0 (unconditional theorem of C4, D47) forces the canonical ordering k1 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 Δmᵢso = md − mᵤ. PDL corpus: D01–D59 + DS01 + DL01–DL02 + N01 + D-exp series. Zenodo community: pdl-framework.
Cédric Laubscher (Wed,) studied this question.