The Mathematics of Coexistence

A Formal Framework for Universal Ethics

Paper PDF: Download the latest revision from GitHub Releases or Zenodo.

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This repository contains the paper, mathematical derivations, verification scripts, and figure-generation code for The Mathematics of Coexistence.

What This Paper Does

We derive ethical principles as mathematical theorems from a single conditional axiom — the intent to persist — applied to agents modeled as open thermodynamic systems in a shared, finite-resource environment. This is applied mathematics, not speculative philosophy: every claim is either a theorem proven from axioms or a cited result from existing literature.

From one axiom, the framework proves:

• Rights as constraints. Using Lagrangian mechanics, rights emerge as inequality constraints whose shadow prices quantify the cost of violation.
• Cooperation as equilibrium. Using energy-denominated payoffs and repeated-game analysis, cooperation is the unique efficient Nash Equilibrium for agents with discount factor δ ≥ δ* = 0.363, resolving the Folk Theorem's equilibrium-selection problem.
• Deception as self-defeating. Using information thermodynamics, deliberate entropy injection degrades the network the deceiver depends on.
• Destruction as irrational. Accumulated negentropy — the integrated thermodynamic work embodied in complex systems — proves that destroying a biosphere to harvest its energy is irrational by a factor of ~10⁷.

The framework comprises 28 theorems, 46 definitions, 13 propositions, and 2 lemmas, all computationally verified (1,191 checks, 0 failures).

Keywords: mathematical ethics · thermodynamic game theory · AI alignment · accumulated negentropy · cooperative equilibrium · Hume's Is-Ought problem · constrained optimization

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Repository Structure

paper/                                Main manuscript (§0–§IX + references)
supplementary/
  A-mathematical-derivations/         Full proofs and derivations (5 files)
  E-conceptual-overview/              Conceptual overview for non-mathematical readers
scripts/
  simulations/                        Core verification scripts (6)
  simulations/applications/           Application verification scripts (6)
  simulations/case-studies/           Case study scripts (non-CORE)
  figures/                            Figure-rendering scripts
  build_paper.py                      Pandoc → PDF build script
  generate_figures.py                 Render all figures from pre-computed data
  run_verification.py                 Run all scripts + consolidated report
modules/                              Python computation library
output/
  figures/                            Generated figures (gitignored)
  data/                               Generated data (gitignored)

Paper Sections

§	File	Title
0	paper/00-abstract.md	Abstract
I	paper/01-introduction.md	Introduction
II	paper/02-literature-review.md	Literature Review
III	paper/03-axioms-definitions.md	Axioms & Definitions
IV	paper/04-rights-constraints.md	Rights as Constraints
V	paper/05-ethics-equilibrium.md	Ethics as Equilibrium
VI	paper/06-stress-testing.md	Stress-Testing
VII	paper/07-applications.md	Applications and Implications
VIII	paper/08-limitations-scope-future.md	Limitations, Scope & Future
IX	paper/09-conclusion.md	Conclusion
—	paper/references.bib	Bibliography

Supplementary Material

Appendix A — Mathematical Derivations. Complete proofs for:

File	Domain
lagrangian-constraints.md	Lagrangian formulation, shadow prices, KKT conditions
thermodynamic-friction.md	Entropy production bounds, friction costs
information-negentropy.md	Information-theoretic identities, deception costs, accumulated negentropy
game-theory-payoffs.md	Nash equilibrium existence, δ* threshold, cooperative surplus
value-dynamics.md	Value evolution equations, inequality dynamics

Getting Started

Requirements

• Python ≥ 3.11
• uv (recommended) or pip

For building the PDF:

• Pandoc ≥ 3.0
• A LaTeX distribution (TeX Live, MiKTeX, or TinyTeX)

Installation

git clone https://github.com/keithlostracco/ethics-theorem.git
cd ethics-theorem
uv venv
uv pip install -e .

Running Verification

All 1,191 computational checks can be run from the repository root:

python scripts/run_verification.py

This runs all 13 scripts and prints a consolidated report:

  FINAL REPORT
========================================================================
  Script                                            Passed  Failed    Time
  ------------------------------------------------  ------  ------  ------
  ✓ verify_lagrangian_constraints.py                    90       0    0.9s
  ✓ verify_thermodynamic_friction.py                   172       0    0.4s
  ✓ verify_information_entropy.py                      127       0    0.5s
  ✓ verify_game_theory.py                              143       0    0.4s
  ✓ verify_value_dynamics.py                           107       0    0.5s
  ✓ verify_accumulated_negentropy.py                   105       0    0.3s
  ✓ verify_misalignment_friction.py                    54       0    0.3s
  ✓ verify_deception_entropy.py                        105       0    0.6s
  ✓ verify_cooperative_equilibrium.py                   84       0    0.3s
  ✓ verify_resource_constraints.py                      23       0    0.6s
  ✓ verify_biosphere_preservation.py                    67       0    0.3s
  ✓ verify_foundation_collapse.py                      114       0    0.5s
  ------------------------------------------------  ------  ------  ------
  TOTAL                                               1191       0    5.7s

  ✓ ALL 1191/1191 CHECKS PASSED
========================================================================

Pass -v for verbose per-check output:

python scripts/run_verification.py -v

Verification Scripts

Core derivation scripts verify the mathematical results in the paper and Appendix A:

Script	Verifies
verify_lagrangian_constraints.py	Lagrangian formulation, shadow prices, KKT conditions (§IV)
verify_thermodynamic_friction.py	Entropy production bounds, friction costs (§IV)
verify_information_entropy.py	Information-theoretic identities, deception costs (§IV)
verify_game_theory.py	Nash equilibrium existence, δ* threshold, cooperative surplus (§V)
verify_value_dynamics.py	Value evolution equations, inequality dynamics (§V)
verify_accumulated_negentropy.py	Negentropy integrals, destruction-irrationality ratio (§V)

Application scripts verify the applied results in §VII and generate figure data:

Script	Verifies
verify_misalignment_friction.py	Reward-hacking friction landscape (§VII)
verify_deception_entropy.py	Deception-entropy cost surface (§VII)
verify_cooperative_equilibrium.py	Cooperative equilibrium basin of attraction (§VII)
verify_resource_constraints.py	Resource-allocation Pareto frontier (§VII)
verify_biosphere_preservation.py	Burning-library negentropy visualization (§VII)
verify_foundation_collapse.py	Foundation model ecosystem collapse (§VII)

Generating Figures

Figure scripts render from JSON data generated by the verification suite in output/data/:

python scripts/generate_figures.py   # render all figures

Figures are written to output/figures/.

Building the PDF

python scripts/build_paper.py

This runs verification, generates figures, then calls Pandoc with --citeproc to render all [@BibKey] citations from paper/references.bib and produces output/ethics-theorem.pdf. See python scripts/build_paper.py --help for options.

Computational Library

The modules/ package provides reusable implementations of the framework's core mathematics:

Module	Purpose
lagrangian.py	Constrained optimization and shadow-price computation
thermodynamics.py	Entropy production and thermodynamic friction
information.py	Information-theoretic measures and deception costs
game_theory.py	Nash equilibrium and cooperative surplus calculations
value_dynamics.py	Value evolution and inequality dynamics
negentropy.py	Accumulated negentropy and destruction-irrationality
plotting.py	Shared visualization utilities
figure_data.py	Save/load figure data (JSON) between verify and figure scripts
verify.py	Test harness (check() / section() helpers)

License

This work is licensed under the Creative Commons Attribution 4.0 International License — see LICENSE for details.

Citation

If you reference this work, please cite:

Lostracco, K. (2026). The Mathematics of Coexistence: A Formal Framework for Universal Ethics. Zenodo. https://doi.org/10.5281/zenodo.18900496