This repository contains the source of the Cosmochrony white paper.

Cosmochrony is a pre-geometric theoretical framework in which time ordering, spacetime geometry, dynamics, and matter emerge from the irreversible relaxation of a single relational substrate, denoted $\chi$.

The framework does not assume a pre-existing spacetime manifold, metric, fundamental fields, or quantization postulates. Instead, familiar physical structures arise as regime-dependent effective descriptions, valid only in projectable regimes, obtained through a generally non-injective projection of the underlying $\chi$ dynamics.

Core Thesis

Cosmochrony is built around the following central statements.

Time is not fundamental

Physical time corresponds to the intrinsically directed (irreversible) ordering of $\chi$ relaxation, rather than to an external parameter or coordinate.

Spacetime is emergent

Spacetime geometry arises as an effective continuum encoding of relational connectivity within $\chi$. In projectable regimes, it can be reconstructed from spectral and correlation properties of the underlying relational structure.

Observables are projected quantities

Physical observables belong to an effective descriptor $\chi_{\mathrm{eff}}$, obtained through a generally non-injective projection from $\chi$. As a result, effective descriptions underdetermine the underlying configuration, imposing intrinsic limits on reconstruction and factorization.

Dynamics are derived, not postulated

The effective dynamical law governing admissible regimes is obtained ab initio from structural constraints on relaxation and causal saturation of fluxes. The resulting Born–Infeld–like action is not a fundamental postulate, but the unique effective encoding compatible with bounded relaxation and projectability.

The $\chi$ Substrate

The substrate $\chi$ is:

• not a field defined on spacetime
• pre-geometric and relational
• devoid of intrinsic spatial or temporal localization
• defined prior to any metric, coordinate, or causal structure

Spacetime notions arise only after projection.

A fundamental distinction is made between:

• $\chi$, the infra-physical relational substrate
• $\chi_{\mathrm{eff}}$, an effective scalar descriptor used only once a stable geometric regime exists

The mapping $\chi \rightarrow \chi_{\mathrm{eff}}$ is mediated by a projection operator $\Pi$ whose fibres $\Pi^{-1}(O)$ encode equivalence classes of relational configurations producing the same effective observable state.

This projection is generically non-injective. The resulting structural indeterminacy constrains reconstruction, underlies effective indeterminacy, and plays a central role in the emergence of quantum-like behavior.

Causality, Saturation, and Bounds

Cosmochrony distinguishes between:

• $c_{\chi}$: an invariant structural bound governing pre-temporal relaxation fluxes in the $\chi$ substrate
• $c$: the emergent causal constraint appearing within effective spacetime descriptions

Relativistic causality is recovered as a projective limit of more primitive saturation constraints, avoiding circular dependence between geometry and dynamics.

Planck’s constant $\hbar$ arises analogously as a bound on the minimal resolvable granularity of reprojection, placing relativistic and quantum limits on a common structural footing.

Spectral Structure and Admissibility

A central structural result of the framework is that admissible relational configurations are constrained by the spectrum of a relational operator $L_\chi$ acting on the configuration space of the substrate.

Under bounded relaxation fluxes, each spectral mode admits a maximal effective amplitude

[ A_n^{\max} = \frac{c_\chi}{\sqrt{\lambda_n}} ]

which defines an admissibility envelope for the projected dynamics.

The geometry of admissible generator sets is further constrained by character relations and Gram identities. These constraints lead to a classification of finite groups capable of supporting neutral spectral sectors.

The resulting admissible families are

[ Q_8, \qquad \mathrm{Dic}_n \ (n \ge 3), \qquad 2O, \qquad 2I ]

while the binary tetrahedral group $2T$ is structurally excluded.

This spectral structure also underlies the emergence of mass hierarchies and connects directly to the spectral admissibility programme (O1–O17), which investigates the growth and stability of admissible configurations on discrete relational models.

Matter, Charge, Gravitation, and Spectral Structure

Within Cosmochrony:

• localized, stable configurations of $\chi$ correspond to matter-like excitations
• stability is characterized by spectral and topological invariants rather than fundamental couplings
• mass emerges from the energetic cost of maintaining non-contractible spectral and topological configurations against global relaxation
• electric charge is identified as a $\pi_1$ winding invariant of the $U(1)$ projection fiber
• the absence of magnetic monopoles follows from the admissibility-induced triviality of $\pi_2$ on the canonical admissible base
• gravitation arises from sustained inhibition of relaxation, recovering general-relativistic behaviour in appropriate regimes.

Spacetime curvature, time dilation, and gravitational dynamics emerge as continuum and thermodynamic limits of relational structure.

Quantum-like Phenomena

Quantum behavior is not postulated as fundamental.

Instead:

• quantization arises only at the effective level
• quantum correlations emerge from shared underlying $\chi$ configurations and non-factorizable projection
• violations of Bell inequalities follow from non-injective projection, without invoking superluminal influence or dynamical nonlocality
• the wavefunction is an effective descriptor rather than an ontological object
• chirality and CP asymmetry can arise as structural consequences of non-injective projection

Classical behavior is recovered when the effective projection becomes approximately injective.

Cosmology and Strong-Gravity Regimes

Cosmological and strong-gravity phenomena follow from the same bounded relaxation dynamics:

• large-scale expansion reflects global relaxation ordering
• apparent acceleration can arise as a cumulative relaxation effect
• the Hubble tension may emerge from projective and relaxation effects without requiring dark energy
• flat galactic rotation curves can be reproduced from saturation of relaxation dynamics without invoking dark matter particles
• horizon-like thresholds correspond to deprojection regimes where effective spacetime descriptions lose validity
• black-hole-like behaviour can be reinterpreted in terms of relaxation saturation and reprojection dynamics

These regimes provide concrete avenues for phenomenological tests.

Status of the Framework

Cosmochrony is:

• foundational and pre-geometric
• structurally constrained
• phenomenologically incomplete

It does not claim:

• experimental validation
• full derivation of the Standard Model
• final unification

It does provide:

• a uniquely derived dynamical core
• a coherent and explicitly stated minimal ontology
• well-defined effective limits
• quantitative directions for further development

Repository Contents

paper/
├── pdf/ # Compiled Cosmochrony PDF
├── tex/ # LaTeX sources
├── figures/ # Diagrams and illustrations
└── README.md

Links

• 📄 Paper PDF
  https://github.com/Cosmochrony/white-paper/blob/main/out/Cosmochrony.pdf

• 🌐 Website
  https://cosmochrony.org

• 💻 GitHub organization
  https://github.com/Cosmochrony

Citation

If you reference this work, please cite:

J. Beau,
Cosmochrony: A Pre-geometric Framework for Emergent Spacetime,
Dynamics, and Matter,
Zenodo, 2026.

Acknowledgements

Portions of the conceptual development, formal clarification, and editorial refinement of Cosmochrony benefited from iterative interactions with large language models used as analytical assistants. All theoretical claims, structural choices, and final formulations remain the sole responsibility of the author.

Contributions

This repository is intended as a research reference.

Critical feedback, independent analyses, and theoretical scrutiny are welcome. Please open an issue to discuss conceptual points, technical details, or potential extensions.