The companion papers in the Admissibility Sub-Programme established two results: the representation-theoretic structure of ADE binary Cayley graphs fixes the number and ordering of stratigraphic levels, and the projective resolution satisfies ₑ₎₉ (n) ₂ (n) under the isoperimetric closure hypothesis. In the Lubotzky–Phillips–Sarnak (LPS) graph model at fixed prime p, the algebraic connectivity ₂ (n) converges to a constant, making ₑ₎₉ asymptotically static. This static regime produces mass ratios in 0. 10, \, 2. 2, far below the 10^-3–10^-5 range observed in the Standard Model, and inverts the mass ordering expected from the admissibility envelope. We show that both failures are resolved by allowing the effective relational valence p to grow with the cascade rank n. As p (n), the Kesten–McKay spectral support - (p), + (p) narrows toward the midpoint =1. The three fixed ADE eigenvalue levels exit the support in a rank-dependent order determined by their distance from 1: the level farthest from 1 exits first and therefore stabilises at the highest mass. For the physically relevant ADE substrates this restores the admissibility ordering M₃>M₂>M₁ without additional tuning. The exit ranks satisfy an implicit equation ₈ = + (p (n₄ₗ₈ₓ (₈) ) ), which for a power-law growth p (n) n^ yields mass ratios that grow rapidly with, providing a single falsifiable structural parameter.
Jérôme Beau (Sat,) studied this question.