Asymptotic Saturation of Projective Resolution: Expander Relaxation Graphs
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2026
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| _version_ | 1866901757597057024 |
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| author | Beau, Jérôme |
| author_facet | Beau, Jérôme |
| contents | <div> <p>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 $\Lambda_{\mathrm{proj}} \asymp \lambda_2$.<br><br>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 $\Lambda_{\mathrm{proj}}$.<br><br>Combining this counting mechanism with the representation structure of ADE substrates yields a factorised prediction for mass ratios,<br>\[<br>\frac{\mathcal{M}_i}{\mathcal{M}_j}<br>\;\propto\;<br>\frac{F_{\mathrm{KM}}(\lambda_i)}{F_{\mathrm{KM}}(\lambda_j)}<br>\cdot<br>\frac{\dim\rho_{\lambda_j}}{\dim\rho_{\lambda_i}},<br>\]<br>where $F_{\mathrm{KM}}$ is the Kesten--McKay cumulative distribution function.<br><br>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.</p> </div> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20147423 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Asymptotic Saturation of Projective Resolution: Expander Relaxation Graphs Beau, Jérôme Spectral graph theory Ramanujan graphs Expander graphs Cheeger inequality Kesten–McKay distribution Graph Laplacian spectrum Emergent mass hierarchy <div> <p>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 $\Lambda_{\mathrm{proj}} \asymp \lambda_2$.<br><br>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 $\Lambda_{\mathrm{proj}}$.<br><br>Combining this counting mechanism with the representation structure of ADE substrates yields a factorised prediction for mass ratios,<br>\[<br>\frac{\mathcal{M}_i}{\mathcal{M}_j}<br>\;\propto\;<br>\frac{F_{\mathrm{KM}}(\lambda_i)}{F_{\mathrm{KM}}(\lambda_j)}<br>\cdot<br>\frac{\dim\rho_{\lambda_j}}{\dim\rho_{\lambda_i}},<br>\]<br>where $F_{\mathrm{KM}}$ is the Kesten--McKay cumulative distribution function.<br><br>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.</p> </div> |
| title | Asymptotic Saturation of Projective Resolution: Expander Relaxation Graphs |
| topic | Spectral graph theory Ramanujan graphs Expander graphs Cheeger inequality Kesten–McKay distribution Graph Laplacian spectrum Emergent mass hierarchy |
| url | https://doi.org/10.5281/zenodo.20147423 |