Projective Dynamic Logo (PDL) — Global Mapping of Structures, Results, and Open Problems (Version 27)

Version 27 of the PDL Global Mapping document. Supersedes v26 (DOI: 10.5281/zenodo.20625504).This version adds Documents D59 and its supplementary verification script PDL_D59_script1.py, completing the gauge programme initiated in D46–D58.New in v27 — D59 (DOI: 10.5281/zenodo.20629282) constructs the fundamental representation 3 of SU(3) from the PDL axioms 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}, with axes canonically labelled by V4{e}. Five lemmas verified by exhaustive exact-integer arithmetic in PDL_D59_script1.py (DOI: 10.5281/zenodo.20628926): (L1) S3-action on V4{e} faithful and transitive; (L2) character decomposition χ_π = χ_trivial + χ_standard; (L3) W the unique S3-invariant complement of C(1,1,1); (L4) A2 root system in W, Cartan matrix [[2,−1],[−1,2]]; (L5) isospin asymmetry Δn = n_d − n_u = 4 > 0 (theorem of C4, D47) selects orientation 3 over 3-bar, identifying the matter sector from the axioms. Exact identity: k3/(k1+k2) = R_e = 6. Negative result documented: bijection (T2)↔(T3) physically motivated but not S4-equivariant.Open problems resolved in v27: OP-D58-1 (physical identification of SU(3)_c) and OP-D58-3 (fundamental representation 3 from C1–C4).New open problems introduced: OP-D59-1 (quark mass spectrum from C1–C4, priority medium) and OP-D59-2 (minimal covariant derivative D_μ = ∂_μ − igA_μ as the unique C4-compatible gauge transport — gauge dynamics as a consequence of the survival principle, priority high).Combined with D46 (U(1)), D57 (SU(2), Weinberg angle), D58 (SU(3) algebraic), and D59 (fundamental representation 3, matter orientation), the PDL programme has now derived both the gauge group structure and the first gauge representation from purely combinatorial axioms, without free parameters beyond Δm_iso = m_d − m_u.PDL corpus covered: D01–D59 + DS01 + DL01–DL02 + N01 + D-exp series. Zenodo community: pdl-framework.