Overview
This article extends the spectral relaxation and variable-valence analyses by addressing the remaining amplitude problem of the mass hierarchy. The central question is whether the support-contraction mechanism identified in O1 can do more than restore the ordering of ADE levels, and can also amplify their spectral separation into realistic mass ratios.
The answer is yes, provided the effective relational valence grows with cascade rank in a controlled way. When $p(n)\to\infty$, the Kesten–McKay support contracts toward its midpoint $\lambda=1$, and the exit rank of each ADE level becomes strongly sensitive to its distance from that midpoint.
Under the power-law hypothesis $p(n)\sim n^\beta$, this produces a closed scaling law for mass ratios. Small spectral asymmetries are then amplified into large stabilisation separations, yielding a narrow compatibility window for charged leptons with a single structural exponent $\beta$.
Core contributions
- Static no-go made precise: the fixed-valence LPS regime produces mass ratios of order unity and inverts the expected ADE ordering, so it cannot account for the observed hierarchy.
- Dynamic valence hypothesis: the paper studies relaxation sequences in which the effective valence grows along the cascade, $p(n)\to\infty$, restoring a moving spectral support.
- Support-narrowing amplification: as the Kesten–McKay support contracts toward $\lambda=1$, ADE levels leave the support at rank-dependent times determined by their distance from the midpoint.
- Mass-ratio law: for $p(n)\sim n^\beta$, the resulting hierarchy satisfies $M_i/M_j \approx \left(|\lambda_i-1|/|\lambda_j-1|\right)^{2/\beta}$, making $\beta$ the unique structural control parameter.
- Charged-lepton window: the physically relevant ADE substrates yield a narrow compatibility interval $\beta\in(0.09,0.13)$, showing that the observed lepton hierarchy lies in a sharply constrained structural regime.
Interpretation
The article shows that the main remaining limitation of the earlier spectral programme was not the ADE stratigraphy itself, but the lack of a mechanism capable of amplifying modest spectral differences into large stabilisation separations.
- Representation structure still determines the three discrete ADE levels established by spectral stratigraphy.
- Support geometry determines when each level ceases to remain projectable inside the Kesten–McKay window.
- Dynamic valence growth provides the minimal structural ingredient needed to convert spectral asymmetry into a realistic mass hierarchy.
Within this perspective, O3 is the amplification paper of the programme: it turns the ordering restoration obtained in O1 into a quantitative hierarchy mechanism, while preserving the structural separation between topology, projective dynamics, and effective masses.
Relation to the Cosmochrony program
O3 follows directly from the limitations identified in Spectral Relaxation and from the partial correction established in O1. Spectral admissibility selects the relevant spinorial sectors, spectral stratigraphy determines the discrete ADE level structure, and O1 shows that growing valence restores the correct order of stabilisation events.
The present paper completes the next structural step. It shows that growing relational valence does not merely restore ordering: it also generates a non-linear mapping from ADE spectral position to stabilisation rank. The remaining open problem is now the derivation of the exponent $\beta$ from first principles, together with a unified treatment of the central level $\lambda_2=1$ through Kesten–McKay saturation.
References
Jérôme Beau. Projective Dynamics and Mass Hierarchy: Hierarchical Amplification via Growing Relational Valence. Preprint, Zenodo.