A Search for the C5 Metric Candidate: Logical Geometry, the (A) ∧ (B) Standard, and the Parity Obstruction of K4

This preprint documents a single extended research session investigating whether a logical metric — a candidate for the missing axiom C5 of the Projective Dynamic Logo (PDL) framework — can be constructed from its combinatorial architecture (K4, K24, K28, the proton/neutron quintuplet). Main result: an exact parity obstruction is proved for the simultaneous (A)∧(B) stability criterion across all six edges of K4: with classical (±1-valued) cross-signs, joint stability is unconditionally impossible (zero solutions, verified exhaustively). A resolution is then given via a spin-type algebraic structure already present in the framework's Dirac-algebra construction (τ32=+I, T2=−I), yielding an exact joint-stability fraction of 41−n and a second, independent proof that n=4 is the unique non-trivial value for which this exponent equals the edge count of Kn. The document also reports, with the same rigour as the positive result, an extended series of exploratory constructions (geometric, thermodynamic, and mechanical models) tested against internal consistency and against external nucleon-structure data (magnetic radius, polarisabilities, diquark mixing parameters, the Weinberg angle) — the large majority of which failed and are documented as negative results, following the negative-results-as-first-class convention of this research programme. Four independent, self-contained verification scripts (Python, exhaustive enumeration and symbolic computation) are deposited alongside this paper under the same DOI. Keywords: signed graphs; Harary balance; combinatorial physics; parity obstruction; Clifford algebra; K4; nucleon structure; diquark models; Projective Dynamic Logo (PDL)