Derivation of the Tree-Level Weinberg Angle and the SU(2) Gauge Structure from the PDL Axioms C1–C4 (Projective Dynamic Logo Framework — Document D57)

We derive, as an unconditional theorem of the four PDL axioms C1–C4, the tree-level value sin²θ_W = sin²(π/R_e) = 1/4, where R_e = 6 is the relational budget of the minimal admissible closure K₄ (the number of its edges), itself a theorem of C1–C4 (D16a). The derivation proceeds via two independent routes that are shown to be equivalent: the combinatorial route (fraction of (A)∧(B)-stable configurations, D29) and the orbit route (ratio of the trivial to the dynamic orbit under the action of the symmetry group S₄ = Aut(K₄)). Both routes identify the factor 1/4 with |V₄|/16 = 4/16, where V₄ ⊂ S₄ is the Klein four-group. We further establish, as a well-motivated conjecture (Hypothesis H_SU2), that axiom C1 (binary pulsation indistinguishability) forces V₄ to act trivially on the physical configuration space, yielding the effective symmetry group S₃ = S₄/V₄ whose double cover Dic₃ ⊂ SU(2) embeds the gauge structure of the weak interaction. The connection to the exact value θ_W = 19π/119 (D55) is identified as the Berry-phase correction of the tree-level angle π/R_e = π/6. All numerical results are verified exhaustively by companion Python scripts (Scripts 1–7, Session 50). The connection A₄/V₄ ≅ ℤ₃ ↔ β₁ = 3 provides a partial resolution of Open Problem OP-OFN-1 posed in the joint PDL–OFN investigation (N01, Laubscher & Evdokimov 2026). Document D56 is the logical predecessor; document D55 provides the exact value; document D58 (forthcoming) will treat SU(3).