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 $\Lambda_{\mathrm{proj}}$, we derive its dependence on the isoperimetric capacity of the cascade graph and show that $\Lambda_{\mathrm{proj}} \asymp h(G)^2$. For expander families satisfying spectral-isoperimetric saturation, this implies $\Lambda_{\mathrm{proj}} \asymp \lambda_2$.In the Lubotzky--Phillips--Sarnak (LPS) graph model with fixed prime $p$, the spectral gap converges to a constant, making $\Lambda_{\mathrm{proj}}$ 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 $\Lambda_{\mathrm{proj}}$.Combining this counting mechanism with the representation structure of ADE substrates yields a factorised prediction for mass ratios,\[\frac{\mathcal{M}_i}{\mathcal{M}_j}\;\propto\;\frac{F_{\mathrm{KM}}(\lambda_i)}{F_{\mathrm{KM}}(\lambda_j)}\cdot\frac{\dim\rho_{\lambda_j}}{\dim\rho_{\lambda_i}},\]where $F_{\mathrm{KM}}$ 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 $\Lambda_{\mathrm{proj}}$ 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.