Cosmochrony

This repository contains the source of the Cosmochrony white paper.

Cosmochrony: A Non-Injective Projection Framework for Emergent Physics Quantum Mechanics, Spacetime Geometry, and Gauge-Matter Structure from a Static Relational Substrate

Cosmochrony is a theoretical framework in which quantum structure, spacetime geometry, and matter emerge from the non-injective projection of a single static relational substrate, denoted $\chi$, onto an observable space through a fixed map $\Pi$.

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 relational structure.

Core Thesis

Cosmochrony is built around the following central statements.

The substrate is static

The relational substrate $\chi$ is atemporally fixed. It carries no intrinsic dynamics, no temporal ordering, and no spatial localization. All apparent dynamics belong exclusively to the effective reprojection sequence $U_{t+1} = \Pi \circ \sigma(U_t)$, not to $\chi$ itself.

Non-injectivity is a structural necessity

Any framework that genuinely distinguishes an infra-physical level from an observable level must admit a non-injective projection $\Pi$. This is not an assumption but a theorem: a surjective projection that is both informationally faithful (injective) and genuinely emergent cannot exist. One of the two conditions must be abandoned; genuine emergence requires non-injectivity.

Observables are projected quantities

Physical observables belong to an effective descriptor obtained through a generally non-injective projection $\Pi : \Omega \to O$ from $\chi$. A given observable $o \in O$ carries an irreducible fiber $\Pi^{-1}(o)$ of indistinguishable pre-images. Effective descriptions underdetermine the underlying configuration, imposing intrinsic limits on reconstruction and factorization.

Quantum structure is an emergent representation

The Hilbert space framework is not a fundamental ontology. It is the minimal representational closure compatible with non-injective projection: the effective description must represent, without access to $\chi$, the coexistence of projectively admissible contributions from the same fiber. This forces an additive, phase-sensitive, norm-preserving structure — a complex Hilbert space. In the regime where the Born–Infeld bound keeps fluctuations subcritical and the background remains quasi-stable, a unique effective dynamics of Schrödinger type follows as a proved result.

Spacetime is emergent

Spacetime geometry arises as an effective continuum encoding of relational connectivity. The same non-injectivity that forces Hilbert structure on the quantum side forces an emergent metric on the gravitational side: the projection $\Pi$ cannot be globally injective, so the effective metric $g_{\mu\nu}$ is a descriptive construct encoding variations in the local projective ordering rate. The Born–Infeld bound selects the unique infrared dynamics compatible with descriptive invariance.

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
• static and relational: atemporally fixed, 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 (static)
• $\chi_{\mathrm{eff}}$, an effective scalar descriptor used only once a stable geometric regime exists

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

Causality, Saturation, and Bounds

Cosmochrony distinguishes between:

• $c_{\chi}$: an invariant structural bound governing admissible relational 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. 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

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, leading 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 underlies the emergence of mass hierarchies and connects directly to the spectral admissibility programme (O-series, O1–O26), which investigates the growth and stability of Weil representations of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ and their connection to the charged-lepton mass hierarchy.

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 $\mathrm{U}(1)$ 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

Quantum-like Phenomena

Quantum behavior is not postulated as fundamental. Instead:

• the Hilbert space is the minimal representational closure of a non-injective projection (structurally motivated, not yet a fully derived theorem)
• quantization arises only at the effective level
• quantum superposition is the observable trace of fiber multiplicity: a superposed state encodes the coexistence of projectively admissible contributions from $\Pi^{-1}(o)$, not a literal simultaneous occupation of distinct ontic states
• 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, not an ontological object
• chirality and CP asymmetry 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 arises as a cumulative relaxation effect, without dark energy as a fundamental input
• the Hubble tension may emerge from projective and relaxation effects
• flat galactic rotation curves are 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 behaviour is reinterpreted in terms of relaxation saturation and reprojection dynamics

Falsifiable Predictions

The framework yields concrete, falsifiable predictions:

• a ~5% enhancement of $H(z)$ at $z \sim 1$ relative to $\Lambda$CDM
• environment-dependent deviations in kinematic inference in cosmic voids, testable with current surveys
• a structural upper bound on the capacity exponent governing the fermion mass hierarchy

Status of the Framework

Cosmochrony is:

• structurally constrained: every effective description is derived under explicit hypotheses, with proved results, structurally motivated consequences, and open problems clearly distinguished
• phenomenologically incomplete: a research programme, not a finished theory

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
• proved emergence of Schrödinger dynamics under explicit structural hypotheses
• well-defined effective limits connecting to GR, QM, and the Standard Model structure
• quantitative directions and falsifiable predictions

Companion Papers

The white paper is supported by a series of companion papers:

Key	Title
BornInfeld	Born–Infeld admissibility and bounded flux
Gravity	Emergent spacetime geometry from bounded relational relaxation
Bell	Non-injective projection and Bell inequality violations
Spectral	Spectral admissibility and spinorial structure
TopologicalInvariants	Topological invariants of the projection fiber
ENI	Non-Injectivity as a Structural Necessity of Genuine Emergence
O1–O26	Spectral admissibility programme (Weil representations, capacity exponents)

Repository Contents

paper/
├── out/        # Compiled Cosmochrony PDF
├── tex/        # LaTeX sources (multi-file, one file per section)
├── 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 Non-Injective Projection Framework for Emergent Physics, 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.