Python library for information-theoretic dynamics and GENERIC structure emergence, accompanying the paper:

The Inaccessible Game
Neil D. Lawrence (2025)
arXiv:2511.06795

The Inaccessible Game introduces a dynamical system derived from four information-theoretic axioms. The framework demonstrates how:

• Marginal entropy conservation ($\sum_i h_i = C$) acts as an isolated-system constraint
• Maximum entropy production subject to this constraint generates dynamics
• GENERIC-like decomposition emerges automatically (symmetric dissipative + antisymmetric conservative components)
• Energy-entropy equivalence holds in the thermodynamic limit under specific conditions

This library provides computational tools to explore these phenomena in:

• Three-variable binary systems (Ising models / Boltzmann machines)
• Curie-Weiss model (mean-field ferromagnetism with phase transitions)
• Harmonic oscillators (mass-spring systems)

git clone https://github.com/lawrennd/tig-code.git
cd tig-code
pip install -e .

• Python ≥ 3.9
• NumPy ≥ 1.20
• SciPy ≥ 1.7
• Matplotlib ≥ 3.3 (for examples)

import numpy as np
from tig import generic_decomposition_n3 as gd

# Initialize frustrated system (Section 4.3 of paper)
theta = np.array([0, 0, 0, 1, -1, 1])  # [θ₁, θ₂, θ₃, θ₁₂, θ₁₃, θ₂₃]
theta = theta / np.linalg.norm(theta)

# Analyze GENERIC structure at this point
result = gd.analyse_generic_structure(theta, N=3)
print(f"Symmetric (dissipative):     ||S|| = {result['norm_S']:.3f}")
print(f"Antisymmetric (conservative): ||A|| = {result['norm_A']:.3f}")
print(f"Ratio:                        ||A||/||S|| = {result['ratio']:.3f}")

# Run constrained dynamics
sol = gd.solve_constrained_maxent(theta, N=3, n_steps=10000, dt=0.01)
print(f"Converged: {sol['converged']}")
print(f"Final constraint violation: {abs(sol['constraint_values'][-1] - sol['constraint_values'][0]):.2e}")

from tig import curie_weiss_equivalence as cw

# Parameters
J = 1.0       # Coupling strength
h = 0.01      # External field
beta_c = 1/J  # Critical inverse temperature
n = 1000      # System size

# Compute magnetization across phase transition
beta_range = np.linspace(0.5 * beta_c, 2.0 * beta_c, 100)
magnetizations = [cw.exact_expectation_magnetisation(beta, J, h, n) 
                  for beta in beta_range]

# Test equivalence condition (Section 5.2)
grad_I = cw.exact_gradient_multi_info_wrt_m(beta_c, J, h, n)
print(f"Multi-information gradient at βc: {grad_I:.3f}")
print(f"Equivalence {'holds' if abs(grad_I) < 5 else 'breaks down'}")

from tig import harmonic_oscillator as ho

# Natural parameters: [θ_xx, θ_pp, θ_xp]
theta = np.array([1.0, 1.0, 0.0])  # Isotropic case

# Compute antisymmetric operator
A = ho.compute_antisymmetric_operator(theta)
print(f"Antisymmetric operator shape: {A.shape}")
print(f"Antisymmetry check: ||A + A^T|| = {np.linalg.norm(A + A.T):.2e}")

# Verify Jacobi identity
jacobi_max = ho.verify_jacobi_identity(theta)
print(f"Jacobi identity violation: {jacobi_max:.2e}")

The examples/ directory contains Jupyter notebooks reproducing all paper figures and demonstrating key concepts:

generate_all_paper_figures.ipynb

Single notebook generating all figures from the paper:

• N=3 Binary System (Section 4.3): GENERIC component norms vs temperature, parameter trajectories, constrained vs unconstrained dynamics
• Curie-Weiss Model (Section 5.2): Magnetization phase transition, multi-information gradient scaling with system size

simulation_experiments_inaccessible_game.ipynb

Computational validation of GENERIC structure for three binary variables (Ising model):

• Temperature dependence of ||A||/||S|| ratio (frustrated systems)
• Constraint maintenance during evolution (Σᵢhᵢ = C preserved)
• Trajectory comparison: constrained vs unconstrained maximum entropy
• Marginal and joint entropy evolution
• Jacobi identity verification: Numerical tests across symmetric, asymmetric, and frustrated parameter regimes

curie_weiss_experiments_inaccessible_game.ipynb

Analytical verification using mean-field ferromagnetism:

• Phase transition at critical temperature βc = 1/J
• Magnetization scaling with system size across transition
• Multi-information gradient remains O(1) as n→∞ (intensive behavior)
• Energy-entropy equivalence theorem validation
• Constraint gradient angles and alignment

mass-spring-simulation-example.ipynb

Harmonic oscillator GENERIC dynamics (Section 6.2):

• Thermalisation from non-equilibrium initial states
• Constraint preservation: h(X) + h(P) = constant
• Distribution evolution in natural parameter space (θ_xx, θ_pp, θ_xp)
• Particle trajectories under time-varying distributions
• Equipartition approach and variance trade-offs
• Ensemble measurements of energy redistribution

# Install with examples dependencies
pip install -e .

# Launch notebook
jupyter notebook examples/generate_all_paper_figures.ipynb

All notebooks are Colab-ready: they automatically install the tig package from GitHub when run in Google Colab (no local installation needed).

Run the full test suite:

pytest tests/ -v

Run specific test modules:

pytest tests/test_generic_decomposition_n3.py -v
pytest tests/test_curie_weiss_equivalence.py -v
pytest tests/test_harmonic_oscillator.py -v

Run notebook smoke tests (fast, ~2 minutes):

pytest tests/test_notebooks.py -v

Run full notebook tests (slow, ~30 minutes):

pytest tests/test_notebooks.py -v -m slow

Tests cover:

• Exponential family computations
• Entropy calculations (marginal, joint, multi-information)
• GENERIC decomposition properties (symmetry, antisymmetry, degeneracy conditions)
• Constraint preservation during dynamics
• Physical consistency (bounds, conservation laws)
• Notebook smoke tests: Verify imports and setup cells execute correctly

Core Functions:

• analyse_generic_structure(theta, N) - Compute M = S + A decomposition
• solve_constrained_maxent(theta, N, n_steps, dt) - Integrate constrained dynamics
• solve_unconstrained_maxent(theta, N, n_steps, dt) - Pure maximum entropy flow
• verify_jacobi_identity(theta, N) - Check Poisson bracket structure

Key Outputs:

• norm_S: Frobenius norm of symmetric part (dissipation strength)
• norm_A: Frobenius norm of antisymmetric part (conservative strength)
• ratio: ||A||/||S|| (regime indicator: thermodynamic vs mechanical)

Exact Canonical Ensemble:

• exact_expectation_magnetisation(beta, J, h, n) - Magnetization ⟨m⟩
• exact_joint_entropy_canonical(beta, J, h, n) - Joint entropy H
• exact_marginal_entropy_canonical(beta, J, h, n) - Sum of marginals Σᵢhᵢ
• exact_multi_information_canonical(beta, J, h, n) - Multi-information I

Gradients:

• exact_gradient_multi_info_wrt_m(beta, J, h, n) - ∇ₘI (equivalence diagnostic)
• implied_alpha_from_constraints(beta, J, h, n) - Natural parameter alignment

Operators:

• compute_fisher_information(theta) - Fisher information G(θ)
• compute_antisymmetric_operator(theta) - Antisymmetric part A
• compute_symmetric_operator(theta) - Symmetric part S

Verification:

• verify_jacobi_identity(theta, eps) - Test Poisson bracket condition
• verify_degeneracy_conditions(theta) - Check S∇E = 0, A∇H = 0

If you use this code in your research, please cite:

@article{lawrence2025inaccessible,
  title={The Inaccessible Game},
  author={Lawrence, Neil D.},
  journal={arXiv preprint arXiv:XXXX.XXXXX},
  year={2025}
}

The code validates:

1. GENERIC emergence (Figure 1): ||A||/||S|| varies from 0.16 (frustrated system) to >0.4 (near-critical), demonstrating regime diversity from thermodynamic (dissipative-dominated) to mechanical (conservative-dominated).

2. Energy-entropy equivalence (Figure 3-4): The multi-information gradient ∇ₘI remains O(1) as n→∞ in the Curie-Weiss model, confirming intensive behavior that enables the equivalence theorem.

3. Antisymmetric complexity (Section 6.2): For the harmonic oscillator, the antisymmetric operator A involves degree-7 polynomials, yet emerges automatically from constraint geometry—illustrating why GENERIC is typically hard to construct by hand.

Contributions welcome! Please:

1. Fork the repository
2. Create a feature branch (git checkout -b feature/new-analysis)
3. Add tests for new functionality
4. Ensure all tests pass (pytest tests/)
5. For notebook contributions: Set up nbstripout to auto-clean outputs:

   poetry install --with dev
   poetry run nbstripout --install

6. Submit a pull request

See CONTRIBUTING.md for detailed guidelines.

MIT License - see LICENSE file for details.

Neil D. Lawrence
University of Cambridge
📧 [email protected]
🌐 inverseprobability.com

This work builds on foundational contributions from:

• Baez & Fritz (2014) - Information loss axioms
• Jaynes (1957, 1980) - Maximum entropy principles
• Grmela & Öttinger (1997) - GENERIC framework
• Curie (1895) & Weiss (1907) - Ferromagnetism theory

See paper for complete references.