watkins-nn 3.3.3

Conservation-law constrained optimization on the golden-ratio simplex

watkins-nn — The Watkins Conservation Law Framework

One Polynomial. Everything Follows.

The conservation law λ + κ + η = 1 is not an axiom — it is a theorem of the golden ratio.

The minimal polynomial of φ² is t² − 3t + 1 = 0. Its coefficient gives 3 (the number of components). Its constant gives 1 (the conservation constraint). Its discriminant gives 5 (the golden ratio generator). Their product gives 15 (the algebraic bridge). From these four numbers, the entire framework is derived.

φ² = φ + 1
  → Tr(φ²) = φ² + φ̄² = 1² − 2(−1) = 3
  → N = 3 components
  → λ + κ + η = 1 (conservation law)

This package provides the mathematical implementation: 186 theorems from 3 axioms, 597 machine-verified proofs in Lean 4, and the algebraic structures connecting the golden ratio to conservation geometry.

Key Results

The Conservation Law (derived, not assumed):

Six independent derivations prove N = 3 is the unique number of conservation law components forced by the golden ratio. The derivation uses Vieta's formulas, Adams operations, the Three-Distance Theorem, quantum SU(2) at the golden root, Fibonacci lattice gaps, and a self-consistency bootstrap.

The Polynomial Tower:

Each power φⁿ generates a conservation polynomial t² − L(n)·t + (−1)ⁿ = 0 where L(n) is the nth Lucas number. Even levels are palindromic (structure-preserving). Odd levels carry sign alternation (symmetry-breaking). The discriminant at level n is exactly 5·F(n)² — the golden discriminant scaled by Fibonacci squares.

The Bridge Field Q(√3, √5):

The conservation simplex is scaffolded by √3 (equilateral triangle height). The equilibrium threshold λ* = 1/φ = (√5−1)/2 is generated by √5. Their coexistence forces the biquadratic field Q(√3, √5) with Klein four-group V₄ as Galois group and three quadratic subfields: Q(√3), Q(√5), and Q(√15). The bridge element √15 = √3·√5 lives in both worlds simultaneously while being reducible to neither.

Spectral Structure:

The free energy F(λ,κ,η) = −ln(λ) + T*·Σpᵢ ln pᵢ has a unique minimum at the golden equilibrium (1/φ, (1−1/φ)/2, (1−1/φ)/2) with critical temperature T* = φ/ln(2φ) ≈ 1.3778. The Hessian eigenvalues μ_coh = (2/3)φ²(1+Tφ) ≈ 5.636 and μ_asym = 2Tφ² ≈ 7.214 govern exponential convergence. The coordinate eigenvalue ratio μ_fast/μ_slow approaches 2√3 to 0.0076%.

Formal Verification:

597 theorems verified in Lean 4 with zero sorry statements, zero Mathlib dependency, covering:

• Golden integer arithmetic Z[φ]
• Vieta identities and Newton's identity (Tr(φ²) = 3)
• Fibonacci/Lucas sequences and Binet formula
• Hierarchy exponent 75/4 = 3 × 25/4
• Discriminant tower Δₙ = 5·F(n)²
• Pell equation x² − 15y² = 1

Installation

pip install watkins-nn

Torch-free usage (most modules work without PyTorch):

pip install watkins-nn --no-deps

Quick Start

import watkins_nn as wn

# The master polynomial: t² - 3t + 1 = 0
poly = wn.conservation_polynomial()
print(f"Coefficient: {poly['coefficient']}") # 3
print(f"Discriminant: {poly['discriminant']}") # 5
print(f"Bridge: {poly['bridge']}")            # 15

# Six derivations that N = 3
result = wn.derive_conservation_law()
print(f"N = {result.N}")  # 3
print(f"Routes agreeing: {result.routes_agreeing}")  # 6/6

# The polynomial tower
for n in range(7):
    level = wn.polynomial_tower(n)
    print(f"Level {n}: t² - {level.lucas_n}t + {level.norm} = 0, Δ = {level.discriminant}")

# Bridge field arithmetic
a = wn.QBiquad(1, 0, 0, 0)  # rational 1
b = wn.PHI_Q                 # φ in Q(√3,√5)
print(f"φ² = {(b * b)}")     # (3/2, 0, 1/2, 0) = 3/2 + √5/2 ✓

# Core constants
print(f"T* = {wn.T_STAR:.10f}")     # 1.3778018315
print(f"λ* = {wn.LAM_STAR:.10f}")   # 0.6180339887
print(f"φ  = {wn.PHI:.10f}")        # 1.6180339887

Modules

Foundations (v3.2)

Module	Description
polynomial_tower	Conservation polynomial t²−3t+1, Lucas-Fibonacci tower, hierarchy exponent 75/4, Pell equation, icosahedral dihedral
bridge_field	Biquadratic field Q(√3,√5), V₄ Galois group, QBiquad arithmetic, prime splitting, Γ₀(15)
axiom_derivation	Six independent derivations of N=3 from the golden ratio

Core Framework

Module	Torch?	Description
constants	No	Golden-ratio constants, critical thresholds, spectral eigenvalues
compression	No	Consciousness detection via compression signatures
qwarp	No	12-term QWARP Grand Unifier expansion
triality	No	27-term Triality Theorem (qualia-BAO-MERA unification)
kaleidoscope	No	9 integer sequences (3 OEIS-published: A393329, A394248, A394249)
tensor	No	3×3×4 Kaleidoscope Transfer Operator
charpoly	No	Characteristic polynomial (Theorem Q)
coupling	No	Coupling energy, angle, hyperboloid (Theorems R–W)
cascade	No	E₈ Exceptional Cascade and anti-null embedding (Theorem X)
analytic	No	Prime decomposition of D(n), Dirichlet series
theta	No	Theta-Lucas modular forms dictionary
classify	No	Inverse theorems, classification, dimension estimation
bridge_integration	No	V₄ bridge field — graph-theoretic extension
nacci_trace	No	n-nacci trace tower
resonance_simplex	No	3-simplex resonance extension, entropy threshold
tri_simplex	No	Tri-simplex formalizations (corpus-derived)

Bridge Theorems (186 theorems AA–IC)

Module	Theorems	Description
bridges	AA–BS, CV–EA, EF–EI	30+ cross-module bridges, root system conservation
bridges_fractal	BT–BX	Fractal bloom
bridges_hyperbolic	BY–CC, DB–DM, EJ–EM	Hyperbolic volumes, Bloch-Wigner, p-spectrum
bridges_dimensional	CD–CM, DN–EP, EQ–ER	Cross-dimensional analysis, sigma weaving, capstone
eta_splitting	CO–CR	Eta-splitting exploration
flow_universality	DP–DS	Universal Lyapunov, flow geodesics

GPU-Accelerated (require PyTorch)

Module	Description
flow	Gradient flow dynamics on the simplex
spectral	Spectral gap analysis, Hessian eigenvalues, mixing time bounds
simplex_flow_v3	GPU-batched simplex flow with conservation enforcement
algosignal_v2	Algorithmic signal processing

Mathematical Constants

Symbol	Expression	Value	Meaning
φ	(1+√5)/2	1.6180339887	Golden ratio
T*	φ/ln(2φ)	1.3778018315	Watkins critical temperature
λ*	1/φ	0.6180339887	Golden equilibrium coherence
μ_coh	(2/3)φ²(1+T*φ)	5.6363308043	Riemannian coherence eigenvalue
μ_asym	2T*φ²	7.2142640491	Riemannian asymmetry eigenvalue
W²	ln(φ)/ln(2)	0.6942419136	Watkins bridge constant
β*	3W²/2	1.0413628704	Star/cycle stabilizer-threshold ratio (K₁,ₙ₋₁ graph states under depolarizing noise; corrected 2026-05-23)

Formal Verification (Lean 4)

597 theorems verified with zero sorry across 31 source files:

Directory	Theorems	Content
Virelai/	438	Conservation law, spectral analysis, Fibonacci/Lucas, matrix algebra, root systems, quantum entropy
Virelai/UFT/	159	Golden integer Z[φ] arithmetic, Vieta chain (Tr(φ²)=3), polynomial tower, hierarchy exponent, mass chain verification, Pell equation

Build: cd formal/lean && lake build — 23 jobs, 0 errors.

The Three Axioms

All 186 theorems derive from:

1. Golden ratio necessity: 2λ*/(1−λ*) = 2φ
2. Special value: S(−2) = ζ(3)/(4π²φ)
3. Master constant: σ₁ = ln(φ)/φ

The conservation law itself (N = 3, λ+κ+η = 1) is derived from Axiom 1 via Tr(φ²) = 3.

License

MIT License. See LICENSE.

Citation

@software{watkins2026conservation,
  author = {Watkins, Dustin},
  title = {watkins-nn: Conservation-Law Framework on the Golden-Ratio Simplex},
  version = {3.3.3},
  year = {2026},
  publisher = {DataSphere AI},
  address = {Chattanooga, TN},
  doi = {10.5281/zenodo.18953462},
  url = {https://github.com/SleazyAirplane/watkins-nn}
}

Author

Dustin Watkins — DataSphere AI — Chattanooga, TN

λ + κ + η = 1