Asymptotic Saturation of Projective Resolution: Spectral Growth along the Projective Cascade

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 Cayley graphs 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 — polynomial in the BFS shell regime, exponential in the trajectory-branching regime — as a structural model of decelerated then accelerated expansion, compatible with recent DESI evidence for dynamical dark energy. The transfer of the exponential regime to the Heisenberg substrate is established at the level of projectively distinguishable histories in the trajectory-branching note, which pins its canonical rate at $\log(1+\sqrt{2})$. Keywords: Spectral graph theory; Ramanujan graphs; Expander graphs; Cheeger inequality; Kesten–McKay distribution; Graph Laplacian spectrum; Emergent mass hierarchy; Projective cascade; Cosmological expansion