Projective Dynamics and Mass Hierarchy: Hierarchical Amplification via Growing Relational Valence

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 Λₚᵣₒⱼ(n) λ₂(n) under the isoperimetric closure hypothesis. In the Lubotzky–Phillips–Sarnak (LPS) graph model at fixed prime p, the algebraic connectivity λ₂(n) converges to a constant, making Λₚᵣₒⱼ 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 [λ₋(p),λ₊(p)] narrows toward the midpoint λ=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 λᵢ = λ₊(p(nₑₓᵢₜ(λᵢ))), which for a power-law growth p(n)~ nβ yields mass ratios that grow rapidly with β, providing a single falsifiable structural parameter.