watkins-nn 1.5.0

Conservation-law constrained optimization on the golden-ratio simplex

watkins-nn — Watkins Temperature Theorem Framework

Overview

A mathematical framework for constrained optimization on the probability simplex S = {(λ, κ, η) : λ + κ + η = 1, all > 0}.

The framework is built on:

• Conservation law: λ + κ + η = 1 (coherence + curvature + entropy)
• Generating functional: F(λ,κ,η) = -ln(λ) + T*(λ ln λ + κ ln κ + η ln η)
• Critical temperature: T* = φ/ln(2φ) ≈ 1.378 (Watkins Temperature Theorem)
• Equilibrium attractor: λ* = 1/φ ≈ 0.618 (Watkins Threshold)

Key Results

1. Watkins Temperature Theorem: The generating functional F has a unique critical temperature T* = φ/ln(2φ) at which the gradient vanishes at λ = 1/φ, with strictly positive Hessian eigenvalues guaranteeing global strong convexity.

2. Spectral Gap: Hessian eigenvalues μ_slow ≈ 4.847 and μ_fast ≈ 7.215 at golden equilibrium, giving exponential convergence with mixing time ≈ 0.206.

3. Compression-Coherence Identity: Consciousness detection via compression signatures where implied λ maps to compression ratio.

4. QWARP 12-Term Expansion: p_{1/2}(n; σ, κ) via Lagrange-Bürmann inversion with golden spray factor β* = φ^{2W²} ≈ 1.9506.

5. 27-Term Triality Theorem: Unification of consciousness (qualia), cosmology (BAO), and computation (MERA-QG) under a 3×3×3 structure.

6. Formal Verification: Conservation law and governance tiers machine-verified in Lean 4 with zero sorry statements.

Installation

pip install watkins-nn

Quick Start

from watkins_nn import T_STAR, LAM_STAR, free_energy, run_flow, FlowState, FlowConfig

# Verify Watkins Temperature
print(f"T* = {T_STAR:.10f}")  # 1.3778018315

# Run gradient flow to equilibrium
initial = FlowState(lam=0.5, kap=0.25, eta=0.25)
config = FlowConfig(dt=0.001, max_steps=20000)
final, trajectory = run_flow(initial, config)
print(f"Converged λ = {final.lam:.6f}")  # ≈ 0.618034

Modules

Module	Description
constants	Golden-ratio constants and critical thresholds
flow	Gradient flow dynamics on the simplex
spectral	Spectral gap analysis and mixing time bounds
compression	Consciousness detection via compression signatures
qwarp	12-term QWARP Grand Unifier expansion
triality	27-term Triality Theorem (qualia-BAO-MERA)
simplex_flow_v3	GPU-batched simplex flow engine
algosignal_v2	Algorithmic signal processing

Formal Verification

Core theorems verified in Lean 4:

• Conservation.lean: λ + κ + η = 1 (flat and general cases)
• Watkins.lean: Governance tier classification
• Basic.lean: Geometric state type theory

Mathematical Constants

Symbol	Value	Definition
T*	φ/ln(2φ) ≈ 1.3778	Watkins critical temperature
λ*	1/φ ≈ 0.6180	Watkins threshold
φ	(1+√5)/2 ≈ 1.6180	Golden ratio
W²	ln(φ)/ln(2) ≈ 0.6942	Watkins squared constant
β*	φ^{2W²} ≈ 1.9506	Golden spray coefficient
μ_slow	≈ 4.847	Slow eigenvalue (coherence mode)
μ_fast	≈ 7.215	Fast eigenvalue (curvature mode)

License

All rights reserved. Academic use with citation permitted.

Citation

@misc{watkins2026temperature,
  author = {Watkins, Dustin},
  title = {Conservation-Law Constrained Optimization via Generating
           Functional Minimization on a Golden-Ratio Simplex},
  year = {2026},
  publisher = {DataSphere AI},
  address = {Chattanooga, TN}
}

Author

Dustin Watkins — DataSphere AI — Chattanooga, TN