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: 150 theorems from 3 axioms, 393 machine-verified proofs in Lean 4, and the algebraic structures connecting the golden ratio to conservation geometry.
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:
393 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
pip install watkins-nnTorch-free usage (most modules work without PyTorch):
pip install watkins-nn --no-depsimport 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| 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 |
| 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 |
| 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 |
| 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 |
| 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* | φ^{2W²} | 1.9506347581 | Golden spray coefficient |
393 theorems verified with zero sorry across 21 source files:
| Directory | Theorems | Content |
|---|---|---|
Virelai/ |
234 | 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.
All 150 theorems derive from:
- Golden ratio necessity: 2λ*/(1−λ*) = 2φ
- Special value: S(−2) = ζ(3)/(4π²φ)
- 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.
MIT License. See LICENSE.
@software{watkins2026conservation,
author = {Watkins, Dustin},
title = {watkins-nn: Conservation-Law Framework on the Golden-Ratio Simplex},
version = {3.2.0},
year = {2026},
publisher = {DataSphere AI},
address = {Chattanooga, TN},
doi = {10.5281/zenodo.18953462},
url = {https://github.com/SleazyAirplane/watkins-nn}
}Dustin Watkins — DataSphere AI — Chattanooga, TN
λ + κ + η = 1