D-exp-SP2: Topological Optimality Criterion for Coherent Photon-to-Electron Conversion — PDL Framework Exploratory Document

This exploratory document derives a topological optimality criterion for coherent photon-to-electron conversion from the axioms C1–C4 of the Projective Dynamic Logo (PDL) framework. The central result (Theorem PDL-SP2) establishes that zero geometric leakage ε_geom = 0 is achievable on a bond graph G if and only if G is bipartite, recovering the Harary balance theorem (1953) as a corollary of C1–C4 without presupposition. A four-criterion composite score derived from PDL selects six optimal inorganic candidates from 21 screened materials: BP, AlAs, GaP, AlP, GaN, and SiC, with GaN and BP identified as priority candidates for experimental validation. An extension to hierarchical graphs (Conjecture PDL-H) shows that global bipartition is the sole determining condition for zero leakage — local graph structure is irrelevant. This conjecture is verified exhaustively over 2²⁴ = 16,777,216 sign assignments. Biological connection: the Mg–Mg transfer network of Photosystem II is bipartite, explaining the two-scale architecture of natural photosynthesis within the PDL framework. Three falsifiable experimental predictions are formulated, testable by femtosecond pump-probe spectroscopy without free parameters.