Discrete Derivation of the Fano Plane from the Non-Freezing Principle

Abstract

This paper establishes, within the discrete regime of the Triniton framework, that the non-freezing filter MP(III) — the third constitutive component of the Meta-Principle — together with the selection filters N1–N4, P-admissibility, Class A, and a lexicographic CPE refinement, forces every CPE-minimal Class A finite core to be isomorphic to the Fano plane Ω₇, the unique 2-(7,3,1) combinatorial design. The derivation proceeds constructively through five stages: (i) finite core extraction from viable systems; (ii) the uniformization maillons M1′–M4; (iii) Fisher's inequality; (iv) lexicographic CPE selection; (v) exhaustive classification PCC-Ω₇ (6,724,520 candidate families verified). No free parameter, no ad hoc adjustment, and no external combinatorial structure is injected. The host structure Ω₇ is derived, not postulated. Key clarifications with respect to earlier versions: MP(III) is treated throughout as a structural selection filter (not a derived axiom); the observable σ₃ used in this document is the count of local equivalence classes in a ball (σ₃^obs, Chapitre III convention), not the cyclomatic invariant of Partie I; the unboundedness of σ₃^obs is derived directly from Théorème 10 (sufficiency proof) without circularity through Proposition 13; and the embedding of Ω₇ as a finite core of a viable system is declared as an open program. The combinatorial automorphism group Aut(Ω₇) ≅ PSL(2,7) is identified as a declared mathematical import. The identification G₂ = Aut(𝕆) belongs to a later regime and is not derived here. A technical companion paper (SQ-R and Non-Degeneracy: Technical Companion to D1) provides the complete proof of the SQ-R P-admissibility transfer theorem and the