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Casey Race edited this page Jun 8, 2026 · 2 revisions

Welcome to the σ-Constant Verification Wiki

This repository contains the independent verification suite for the paper:

The σ-Constant: A Universal Algebraic Invariant for Energy Propagation in d-Dimensional Simplicial Lattices
Casey Lee Race · Calera Computing, Inc.
DOI: 10.5281/zenodo.20350425

These scripts reproduce the key empirical results from the paper using zero shared code with the production system. They are written in pure Python + NumPy/SciPy and construct all lattice geometry from mathematical definitions.

What Is the σ-Constant?

The $\sigma$-constant ($\sigma \approx 1.220744084605759$) is the unique positive real root of the quartic equation:

$$x^4 = x + 1$$

It is the 4-dimensional analog of the golden ratio ($\phi$) and the plastic constant ($\rho$) — the geometry-native organizing constant for energy propagation on the $A_4$ root lattice (the pentatope lattice in 4D).

The Dimensional Hierarchy

Each dimension $d$ has its own algebraic constant $c_d$, the positive real root of $x^d = x + 1$:

Dimension Equation Constant Name Geometry
2 $x^2 = x + 1$ $\phi \approx 1.6180$ Golden Ratio Triangle, Pentagon
3 $x^3 = x + 1$ $\rho \approx 1.3247$ Plastic Constant Tetrahedron, FCC
4 $x^4 = x + 1$ $\sigma \approx 1.2207$ $\sigma$-Constant Pentatope, $A_4$ Lattice
5 $x^5 = x + 1$ $\approx 1.1673$ — 5-Simplex
$\infty$ — $\to 1.0$ Isotropic Limit —

The corresponding per-hop decay base is $1/\sigma \approx 0.8192$, which determines how energy attenuates across each hop in the lattice.

Clean Room Verification Methodology

To ensure that the results are not artifacts of the production system, all verification scripts adhere to strict independence guarantees:

  • ✅ No imports from any proprietary codebase
  • ✅ No shared code — lattices are built from the mathematical definition ($\mathbb{Z}^{d+1}$ zero-sum constraint via BFS)
  • ✅ No hardcoded target values — algebraic constants are computed at runtime from their defining polynomials
  • ✅ Pure Python 3 + NumPy + SciPy — universally reproducible, MIT-licensed
  • ✅ Random seeds logged for full reproducibility

Verification Experiments

The suite consists of three independent verification scripts:

Simulation 1: Hopfield Associative Memory on $A_4$

File: sim1_hopfield_a4.py

Builds a Hopfield associative memory on the $A_4$ root lattice from scratch. Tests four decay constants ($\phi$-decay, empirical, $\rho$-decay, $\sigma$-decay) and reports recall accuracy at each hop depth. A grid search over $\alpha \in [0.50, 0.95]$ maps the recall landscape to find the unconstrained optimum.

  • Protocol: 50 independent random seeds $\times$ 20 trials $\times$ 8 patterns = 8,000 recall tests per decay constant.
  • Key Results:
    • $\sigma$-decay ($\alpha \approx 0.819$) achieves the highest 4-hop recall among the four named constants.
    • Grid search reveals a broad optimal plateau: all $\alpha \in [0.67, 0.95]$ perform within 1% of the grid-search best.
    • $\sigma$-decay lies within the optimal plateau, only 0.17% below the grid-search maximum.
    • Mean optimal $\alpha$ across 50 seeds is $0.857 \pm 0.089$.
  • Output: sim1_results.json

Simulation 2: Spectral Laplacian Analysis

File: sim2_spectral_a4.py

Builds the normalized graph Laplacian of $A_2$, $A_3$, and $A_4$ lattices ($N=125$). Computes the heat kernel at the mixing timescale $t^* = 1/\lambda_2$ and extracts the per-hop energy decay profile (alpha_raw) to check for geometric emergence.

  • Protocol:
    1. Construct adjacency and degree matrices for $A_2$, $A_3$, and $A_4$ lattices.
    2. Compute normalized graph Laplacian: $L = I - D^{-1/2}AD^{-1/2}$.
    3. Compute spectral gaps ($\lambda_2$) and evaluate heat kernel $H(t^) = \exp(-t^ L)$ for a point source.
    4. Extract the per-hop decay profile and verify matching growth of shell sizes with kissing numbers.
  • Key Results:
    • $A_d$ coordination numbers verified: 6 ($A_2$), 12 ($A_3$), 20 ($A_4$).
    • Spectral gaps are positive for all lattices, proving connectivity.
    • Heat-kernel profiles show monotonic distance decay.
    • Shell sizes match $A_d$ kissing number growth.
    • The raw decay rate alpha_raw is reported without arbitrary correction factors.
  • Output: sim2_results.json
Lattice Coord Interior $\lambda_2$ alpha_raw $1/\sigma_d$ (ref)
$A_2$ 6 71.2% 0.0271 0.973 0.618 ($1/\phi$)
$A_3$ 12 32.0% 0.0867 0.822 0.755 ($1/\rho$)
$A_4$ 20 12.8% 0.1968 0.592 0.819 ($1/\sigma$)

Simulation 3: Tetradic Sequence Verification

File: sim3_tetradic_sequence.py

Verifies the algebraic and combinatorial properties of the Tetradic sequence $T(n) = T(n-3) + T(n-4)$, which governs energy propagation on the $A_4$ lattice.

  • Protocol:
    1. Compute the Tetradic sequence up to $n=200$.
    2. Check ratio convergence $T(n)/T(n-1) \to \sigma$ as $n \to \infty$.
    3. Validate sequence terms against the 30 known OEIS A017817 values.
    4. Verify combinatorial equivalence by counting brute-force $(3,4)$-compositions of $n$.
    5. Validate the Semantic Octave identity $\sigma^{-3} + \sigma^{-4} = 1$ to 15 decimal places.
  • Key Results:
    • $T(n)/T(n-1) \to \sigma$ with error $< 3 \times 10^{-12}$ by $n=200$ ✅
    • All 30 OEIS A017817 terms match exactly ✅
    • Brute-force $(3,4)$-composition counts match $T(n)$ exactly ✅
    • Semantic Octave identity $\sigma^{-3} + \sigma^{-4} = 1$ verified to 15 digits of precision ✅
  • Output: sim3_results.json

Quick Start

# Clone the repository
git clone https://github.com/Calera-Computing-Inc/sigma-constant-verification.git
cd sigma-constant-verification

# Create a virtual environment
python3 -m venv .venv
source .venv/bin/activate
pip install -r requirements.txt

# Run the verification scripts
python3 sim1_hopfield_a4.py          # Hopfield recall (~30 seconds)
python3 sim2_spectral_a4.py          # Spectral analysis (~10 seconds)
python3 sim3_tetradic_sequence.py    # Tetradic sequence verification (~instant)

All scripts exit with code 0 on success, 1 on failure. Results are written to JSON files.

Repository Structure

sigma-constant-verification/
├── .gitignore
├── .zenodo.json                      # Zenodo publication metadata
├── CITATION.cff                      # Citation information
├── LICENSE                           # MIT License
├── README.md                         # Project overview
├── requirements.txt                  # Python dependencies (numpy, scipy)
├── sim1_hopfield_a4.py               # Simulation 1: Hopfield associative memory
├── sim1_results.json                 # Simulation 1 outputs
├── sim2_spectral_a4.py               # Simulation 2: Spectral Laplacian analysis
├── sim2_results.json                 # Simulation 2 outputs
├── sim3_tetradic_sequence.py         # Simulation 3: Tetradic sequence verification
└── sim3_results.json                 # Simulation 3 outputs

Links & Resources

Resource Link
Paper (Open Access) Zenodo DOI: 10.5281/zenodo.20350425
Organization Calera Computing, Inc.
License MIT (verification scripts only — no proprietary code)

Calera Computing, Inc. · Delaware C-Corp · Founded 2026
Lead with the physics. The industries will follow.

Calera Computing, Inc. · 2026 · All Glory to the Most High God!

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