Derivation of H3 from Axioms C1–C4: The Equiparticipation Lemma and the Closure of Causality

The Indifference Lemma (Hypothesis H3) states that the natural measure on the 11,017 relations of the PDL proton closure, as seen by an external K₄ observer, is the uniform measure. Since D39, H3 has been the sole residual hypothesis preventing a fully axiomatic derivation of the gravitational engagement fraction κ = 310φ/11017 from the four PDL axioms C1–C4 alone. This document closes that gap. Three lemmas are established. The Equiparticipation Lemma (D42-L1) proves that on any complete signed graph satisfying C4-optimality (zero violated triangles), every edge participates in exactly (n−2) coherent triangles — a purely algebraic result verified exhaustively for n = 4, 5, 6, 7. The Cross-Triangle Blindness Lemma (D42-L2, from D39) establishes that the (A)∧(B) criterion cannot distinguish valence from sea relations. The S₄-Equivariance Lemma (D42-L3) proves that the count of coherent cross-triangles formed by any proton relation with K₄ is invariant under all 24 permutations of S₄ = Aut(K₄) — a consequence of the bijectivity of permutations on the set of unordered pairs, verified exhaustively over 24,576 cases with zero violations. Together, these three lemmas establish H3 as a theorem of C1–C4, without invoking D36 as an additional input. As a direct corollary, κ = 310φ/11017 ∈ ℚ(√5) is promoted from a conditional result to an unconditional theorem of the PDL axioms. The entire derivation chain — from C1–C4 to the gravitational coupling, the Hubble constant, and the Bekenstein–Hawking entropy — is now fully axiomatic with no residual free parameter. A corollary on causal closure is stated: within the PDL framework, nothing exists outside a closure, and the causal chain from axioms to physics is complete. A multi-scale correspondence is exhibited, showing that the same principle — indifference as the unique fixed point of constrained optimisation — governs signed graphs, nuclear stability, and Darwinian evolution.