Asymptotic Saturation of Projective Resolution: Expander Relaxation Graphs

We investigate the spectral mechanism governing mode stabilisation in the Cosmochrony projective cascade framework. Starting from the admissibility condition defining the projective resolution ₏ₑ₎₉, we derive its dependence on the isoperimetric capacity of the cascade graph and show that ₏ₑ₎₉ h (G) ². For expander families satisfying spectral-isoperimetric saturation, this implies ₏ₑ₎₉ ₂. In the Lubotzky--Phillips--Sarnak (LPS) graph model with fixed prime p, the spectral gap converges to a constant, making ₏ₑ₎₉ asymptotically static. In this regime, the stabilisation of modes is governed not by the admissibility threshold but by saturation of the cumulative spectral count below ₏ₑ₎₉. Combining this counting mechanism with the representation structure of ADE substrates yields a factorised prediction for mass ratios, \MᵢMⱼ\;\;F₊₌ (㶁) F₊₌ (ⱼ) 䲛㶁, F₊₌ is the Kesten--McKay cumulative distribution function. Numerical evaluation shows that this single-level mechanism produces mass ratios of order unity and inverts the ordering expected from the admissibility envelope. We identify the asymptotic constancy of ₏ₑ₎₉ as the origin of both limitations, and establish that this structural obstacle historically motivated the O-series adoption of the Heisenberg BFS cascade. We further identify a two-regime growth law for the projectable mode count --- polynomial in the BFS shell regime, exponential in the trajectory-branching regime --- as a structural candidate for a cosmological model of decelerated then accelerated expansion, compatible with recent DESI evidence for dynamical dark energy.