Overview
This article extends the spectral relaxation analysis by removing the fixed-valence assumption imposed on the Lubotzky--Phillips--Sarnak relaxation graphs. The central question is whether the asymptotic saturation of the projective threshold $\Lambda_{\mathrm{proj}}(n)$ is a structural feature of the substrate or only an artefact of holding the graph valence fixed along the cascade.
The answer is clear: once the effective valence is allowed to grow with cascade depth, the Kesten–McKay spectral support contracts toward its midpoint. This produces an ordered exit of the ADE eigenvalue levels and restores a genuinely dynamical projective regime.
In this variable-valence setting, the fixed-valence inversion identified in the previous paper is corrected by support geometry alone. The result is a restored hierarchy ordering without introducing any additional dynamical hypothesis beyond increasing relational valence along the cascade.
Core contributions
- Variable-valence regime: the paper studies LPS relaxation sequences in which the effective valence parameter grows along the cascade, $p(n)\to\infty$.
- Kesten–McKay contraction: as $p(n)$ increases, the spectral support shrinks toward $\lambda=1$, replacing the fixed-valence saturation plateau with a moving spectral window.
- Ordered ADE exit: the ADE spectral levels leave the support in a fixed geometric order determined by their distance from the midpoint.
- Restored projective dynamics: the lower-bound dynamics of $\Lambda_{\mathrm{proj}}(n)$ becomes non-trivial again once the valence is allowed to vary.
- Ordering correction: the fixed-valence inversion is removed and the effective mass ordering is restored by support-exit geometry itself.
Interpretation
The article shows that the main failure of the fixed-valence relaxation model was not a defect of spectral stratigraphy, but a defect of the graph regime used to realise the relaxation cascade.
- Representation structure still determines the discrete ADE levels made available by spectral stratigraphy.
- Support geometry now determines when those levels cease to remain inside the admissible spectral window.
- Variable relational valence provides the minimal structural ingredient needed to restore non-static threshold dynamics.
Within this perspective, O1 is the correction paper to Spectral Relaxation: it shows that the asymptotically static threshold was not fundamental, and that a growing-valence cascade naturally recovers the physically expected ordering of generations.
Relation to the Cosmochrony program
O1 follows directly from the limitation identified in Spectral Relaxation. Spectral admissibility determines which sectors can survive bounded flux, spectral stratigraphy determines the discrete level structure, and spectral relaxation showed that a fixed-valence expander regime cannot generate a viable hierarchy.
The present paper removes that obstruction. It establishes that allowing relational valence to grow along the cascade restores a genuine projective dynamics and a correct ordering of stabilisation events. The remaining open problem is no longer the sign of the hierarchy, but its quantitative amplitude.
References
Jérôme Beau. Projective Resolution Dynamics on Variable-Valence Relaxation Graphs. Preprint, Zenodo.