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.
Cédric Laubscher (Mon,) studied this question.