This repository contains the source of the O3 Cosmochrony paper
Projective Dynamics and Mass Hierarchy: Hierarchical Amplification via Growing Relational Valence.

This work extends the spectral relaxation programme by addressing the remaining limitation identified in O1: the insufficient amplitude of the inter-generational mass hierarchy.

While O1 restores the correct ordering of ADE levels through the geometric mechanism of support contraction, it does not produce the observed large mass ratios. The present work introduces the minimal dynamical ingredient required to amplify this hierarchy.

The central idea is that the relational valence grows along the cascade according to a structural law:

$p(n) \sim n^\beta$.

This induces a non-linear relation between spectral level position and stabilisation rank, allowing small spectral differences to generate large hierarchical separations.

Core Result

The paper studies the dynamic valence-growth regime

$p(n) \to \infty$

with a power-law scaling

$p(n) \sim n^\beta$.

In this regime, the exit condition of a spectral level $\lambda_i$ is defined by

$\lambda_i = \lambda_+(p(n_{\mathrm{exit}}))$,

where $\lambda_+(p)$ is the upper edge of the Kesten--McKay support.

Solving this relation yields a strongly non-linear dependence of exit rank on spectral position.

Hierarchical Amplification

The key result is that the stabilisation ranks satisfy

$n_{\mathrm{exit}}(\lambda_i) \sim f(\lambda_i; \beta)$

with a rapidly increasing sensitivity to $\lambda_i$ as $\beta$ increases.

As a consequence:

• small differences between ADE eigenvalues
• are mapped to large separations in cascade depth
• which translate into exponentially large mass ratios

This provides a purely structural amplification mechanism.

Mass Ratio Scaling

Under the power-law growth hypothesis, the ratio of stabilisation ranks becomes

$\frac{n_i}{n_j} \sim \left(\frac{\Delta_j}{\Delta_i}\right)^{1/\beta}$

where $\Delta_i = |\lambda_i - 1|$ measures the distance to the support midpoint.

This yields:

• weak hierarchy for small $\beta$
• strong hierarchy for larger $\beta$

A single structural parameter $\beta$ controls the entire hierarchy.

Resolution of the Amplitude Problem

O3 resolves the main limitation of previous steps:

• Spectral Relaxation (fixed valence)
  → correct structure, wrong ordering and amplitude

• O1 (variable valence)
  → correct ordering, insufficient amplitude

• O3 (dynamic valence growth)
  → correct ordering and scalable amplitude

Thus O3 closes the amplitude gap without introducing additional physics.

Compatibility with Stratigraphy

The mechanism preserves the structural results of Spectral Stratigraphy:

• the number of levels (three) remains fixed
• the ADE eigenvalues remain unchanged
• only the mapping $\lambda \to n_{\mathrm{exit}}$ is modified

This confirms the separation of roles:

• topology → number of generations
• dynamics → mass hierarchy

Regime Structure

The analysis identifies three regimes:

1. Static regime (β ≈ 0)
   Fixed-valence behaviour, no hierarchy

2. Intermediate regime
   Moderate amplification, insufficient for SM

3. Dynamic regime (β > β*)
   Strong amplification compatible with observed mass ratios

This defines a regime diagram controlled by β.

What O3 Resolves

O3 provides:

• a structural amplification mechanism for mass ratios
• a single control parameter β
• a falsifiable prediction linking spectral data to hierarchy strength

It eliminates the need for:

• ad hoc scaling laws
• external calibration of masses
• additional dynamical assumptions

Residual Open Problem

O3 introduces the parameter β but does not derive it from first principles.

The remaining task is:

• Derivation of β from relational dynamics

This corresponds to the next structural step in the programme.

Conceptual Structure

O3 connects:

1. Spectral admissibility (mode selection)
2. Spectral stratigraphy (level structure)
3. Variable-valence dynamics (O1)
4. Non-linear mapping between spectrum and cascade rank
5. Emergence of hierarchical amplification

The result is a complete structural pipeline from spectrum to hierarchy.

Physical Interpretation

In the Cosmochrony framework, the growth of relational valence reflects an increase in effective connectivity of the substrate along the relaxation cascade.

This induces:

• contraction of spectral support (O1)
• non-linear exit dynamics (O3)
• amplification of stabilisation separations

Mass hierarchy is therefore not imposed but emerges from the geometry and dynamics of the relational substrate.

Open Directions

Three directions remain:

1. Derivation of β
   From first-principles dynamics of the relaxation graph

2. Extension to full Standard Model
   Including quarks and neutrinos

3. Numerical validation
   Precise fitting of β against observed mass ratios

Status

This framework is:

• spectral-dynamical
• structurally minimal
• analytically controlled
• falsifiable via β

It does not assume:

• particle fields
• free mass parameters
• external hierarchy inputs

Repository Structure

paper/
├── out/ # Compiled O3 PDF
├── tex/ # LaTeX sources
└── README.md

Citation

If you reference this work, please cite:

J. Beau, Projective Dynamics and Mass Hierarchy: Hierarchical Amplification via Growing Relational Valence, Zenodo, 2026. :contentReference[oaicite:0]{index=0}

Acknowledgements

Portions of the derivations, numerical exploration, and editorial refinement benefited from iterative interactions with large language models used as analytical assistants.
All theoretical results and interpretations remain the sole responsibility of the author.

Contributions

This repository is intended as a research reference.

Critical feedback, independent numerical studies, and alternative derivations of the β parameter are welcome.

Please open an issue to discuss conceptual points, technical details, or possible extensions.