Projective Dynamic Logo (PDL) — Global Mapping of Structures, Results, and Open Problems (Version 25)

This document provides a systematic and self-contained guide to the Projective Dynamic Logo (PDL) programme, intended for a scientific colleague who wishes to understand, verify, or continue the research. PDL derives fundamental physical constants and dynamical laws from four axioms on finite signed graphs, without presupposing spacetime, particles, or fields. The minimal admissible closure is K₄ (four vertices, six edges) ; the proton is characterised by the quintuplet (24, 28, 930, 10087, 11017). Version 25 incorporates two new documents. Document D57 derives sin²θW (tree) = sin² (π/Rₑ) = 1/4 as an unconditional theorem of C1–C4, where Rₑ = 6 is the relational budget of K₄ (D16a). Two independent routes are proved equivalent via the Klein four-group V₄ ⊂ S₄ = Aut (K₄): the combinatorial route (192/768 from the (A) ∧ (B) criterion, D29) and the orbit route (dim (trivial) /dim (orbit-4) = 1/4). Hypothesis HSU2 (well-motivated conjecture) identifies the effective symmetry group as S₄/V₄ ≅ S₃, whose double cover Dic₃ ⊂ SU (2) embeds the weak gauge structure. The quotient A₄/V₄ ≅ ℤ₃ is identified as the combinatorial origin of β₁ (K₄) = 3, providing a partial resolution of OP-OFN-1. The joint PDL–OFN bridge note N01 (Laubscher & Evdokimov) documents the convergence β₁ (K₄) = b₁ (Ω₂₁) = 3 and poses Open Problems OP-OFN-1 and OP-OFN-4. Four new open problems are registered: OP-OFN-1, OP-D57-1 (formal proof of Lemma C1-V₄), OP-D57-2 (Dic₃ as weak gauge group), and OP-SU3 (SU (3) from the proton hierarchy). The dependency diagram is updated with D57 and HSU2 nodes. All prior results (D01–D56, DL01–DL02) remain unchanged.