Overview
This article investigates the dynamical side of spectral stratigraphy: how the projective threshold $\Lambda_{\mathrm{proj}}$ evolves along the relaxation cascade, and whether this evolution can amplify small spectral separations into physically meaningful mass hierarchies.
The analysis derives the projective resolution from global coherence properties of the relaxation graph. In expander families with spectral-isoperimetric saturation, the admissibility scale becomes controlled by the algebraic connectivity of the relaxation graph itself.
Applying this framework to the Lubotzky--Phillips--Sarnak graph family shows that, at fixed prime $p$, the spectral gap converges and the projective threshold becomes asymptotically static. In that regime, the resulting hierarchy is too weak and the ordering of levels is incorrect.
Core contributions
- Derived projective threshold: the projective resolution is related to the isoperimetric capacity of the relaxation graph, with $\Lambda_{\mathrm{proj}} \asymp h(G)^2$.
- Expander reduction: for spectrally saturated expanders, this yields $\Lambda_{\mathrm{proj}} \asymp \lambda_2$.
- LPS relaxation model: in the fixed-$p$ LPS family, the algebraic connectivity approaches a constant, making the threshold asymptotically static.
- Negative numerical result: the resulting one-level mechanism produces only order-unity hierarchies and gives the wrong level ordering.
- Structural diagnosis: the absence of a dynamical threshold $\Lambda_{\mathrm{proj}}(n)$ is identified as the reason large hierarchies do not emerge in the static regime.
Interpretation
The article clarifies which part of the hierarchy problem is already solved by spectral stratigraphy and which part still requires a dynamical mechanism.
- Representation structure determines the discrete spectral levels available for stabilisation.
- Relaxation-graph coherence determines the projective threshold governing when those levels become inaccessible.
- Static expander thresholds explain why hierarchy amplification fails in the present regime.
Within this perspective, the paper is not a final hierarchy model but a diagnosis paper: it shows why a static expander-based threshold cannot by itself account for the observed mass spectrum, and why a genuinely dynamical $\Lambda_{\mathrm{proj}}(n)$ is the next necessary step.
Relation to the Cosmochrony program
Spectral relaxation extends the spectral admissibility programme by addressing the missing dynamical link between admissible sectors and observed mass hierarchies. Spectral admissibility determines which sectors survive bounded flux, and spectral stratigraphy determines the discrete level structure of stabilisable modes.
The present paper asks whether the relaxation cascade itself can amplify these discrete levels into large physical hierarchies. Its conclusion is precise: not in the asymptotically static expander regime. A dynamical projective threshold remains the key open ingredient.
References
Jérôme Beau. Asymptotic Saturation of Projective Resolution: Expander Relaxation Graphs. Preprint, Zenodo.