This repository contains the source of the paper on bounded relaxation, Born–Infeld saturation, and the dynamical selection of spacetime geometry (paper C).

This work studies a class of relational systems whose effective continuum descriptions admit a finite maximal propagation or relaxation flux. We show that this minimal constraint uniquely enforces a Born–Infeld–type effective structure, and that this structure dynamically selects a restricted class of admissible operators whose continuum limits define physically meaningful spacetime geometries.

In the revised formulation, flux saturation is not only a dynamical constraint but also a limitation of projectability. In saturation regimes, unresolved relational structure is not lost but re-encoded into effective geometric and thermodynamic observables, such as curvature or horizon temperature. Born–Infeld saturation thus plays a dual role: it bounds admissible dynamics and governs how information is redistributed at the level of effective descriptions.

In particular, we demonstrate that:

• flat Minkowski spacetime emerges as the unique homogeneous and isotropic fixed point of bounded relaxation,
• Schwarzschild geometry arises universally as the effective exterior description of a localized and stationary obstruction,
• horizons correspond to flux saturation and loss of projectability, rather than to physical singularities.

The analysis is entirely operator-based and does not postulate independent metric dynamics or gravitational field equations.

Core Claims

The paper is organized around the following statements:

1. Bounded propagation excludes purely quadratic effective actions
   Any local continuum description admitting a finite maximal flux cannot be governed by a purely quadratic functional. Under mild assumptions, bounded propagation uniquely enforces a Born–Infeld–type structure.

2. Effective geometry is selected dynamically, not postulated
   The bounded-flux condition restricts the class of admissible relational Laplacians whose continuum limits admit a geometric interpretation. The effective metric arises from the principal symbol of the selected operator.

3. Minkowski spacetime emerges as a homogeneous relaxation fixed point
   In the absence of localized obstructions, homogeneous and isotropic relaxation uniquely selects a flat spacetime with pseudo-Riemannian signature ((- + + +)), without assuming Lorentz invariance at the microscopic level.

4. Schwarzschild geometry is the universal response to a localized obstruction
   For stationary and isotropic perturbations, flux conservation alone leads to a (1/r) profile and induces the Schwarzschild metric as the effective exterior geometry, independently of the microscopic details.

5. Horizons mark loss of projectability, not physical singularities
   At flux saturation, the effective operator becomes degenerate and the geometric description ceases to apply. Horizons are interpreted as boundaries of the projectable regime, consistent with structural analyses of non-injective mappings.

6. Born–Infeld saturation governs the re-encoding of unresolved structure
   In regimes where bounded relaxation saturates, effective geometric and thermodynamic quantities act as compensatory parameters, encoding relational degrees of freedom that cannot be resolved within a smooth spacetime description.

Relation to Other Papers

• This paper builds on results concerning spectral and relational emergence of geometry developed in paper A.
• Its interpretation of horizons and strong-field regimes is conceptually aligned with the analysis of non-injective effective descriptions developed in paper B.

Each paper is logically autonomous, but together they form a coherent analysis of emergent geometry, its limitations, and its operational meaning.

What This Paper Does Not Assume

To avoid conflating structural and ontological claims, the paper does not assume:

• a fundamental spacetime manifold
• independent metric degrees of freedom
• Einstein field equations
• a specific microscopic ontology
• modifications of quantum mechanics

All results are derived at the level of effective operators, symmetry constraints, and bounded relaxation dynamics.

Keywords

Emergent spacetime, Born–Infeld structure, bounded propagation, relational Laplacians, Minkowski spacetime, Schwarzschild geometry, horizons, effective operators

Repository Contents

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

Links

• 📄 Paper PDF: https://github.com/Cosmochrony/born-infeld-paper
• 🌐 Website: https://cosmochrony.org
• 💻 GitHub organization: https://github.com/Cosmochrony

Citation

If you reference this work, please cite:

J. Beau, Bounded Relaxation and the Dynamical Selection of Spacetime Geometry, 2026.

(Replace with DOI / venue when available.)

Acknowledgements

Portions of the editorial refinement benefited from iterative interactions with large language models. These tools were used as analytical assistants for exploring alternative formulations, checking internal consistency, and improving clarity. All claims and final formulations remain the sole responsibility of the author.

Contributions

This repository is intended as a research reference.

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