Crate resonance_spectral_gap 

resonance-spectral-gap

Exact proof that the spectral gap of the 96-vertex Resonance Classes graph Laplacian is λ₁ = 1.

What This Proves

The graph of Resonance Classes is a 96-vertex, 256-edge graph arising from an action functional on a 12,288-cell complex. It encodes the structure of exceptional Lie groups (G₂, F₄, E₆, E₇, E₈). This crate computes and verifies the complete spectrum of its graph Laplacian using exact rational arithmetic — no floating-point approximation.

Main result: The spectral gap (smallest nonzero eigenvalue) is exactly λ₁ = 1.

Proof Method

The proof proceeds by block tridiagonal decomposition:

1. Hemisphere decomposition: The graph splits into two disconnected hemispheres (48 vertices each) by the e₇ coordinate. No edges cross between them.

2. Q₄ block decomposition: Each hemisphere splits into three Q₄ hypercube blocks (16 vertices each) by the d₄₅ coordinate. Inter-block edges form identity matchings: B₋₁ — B₀ — B₊₁.

3. Block tridiagonal reduction: Since the inter-block coupling (identity matrix) commutes with L_Q₄, the 48×48 Laplacian reduces to five independent 3×3 problems M_ν, one per Q₄ eigenvalue ν ∈ {0, 2, 4, 6, 8}.

4. Exact eigenvalue computation: Each M_ν has eigenvalues {ν, ν+1, ν+3}, verified by exact rational determinant computation (det(M_ν - λI) = 0).

5. Spectral gap: The smallest nonzero eigenvalue across all blocks is 1 (from M₀: eigenvalues {0, 1, 3}).

Complete Hemisphere Spectrum

λ	0	1	2	3	4	5	6	7	8	9	11
mult	1	1	4	5	6	10	4	10	1	5	1

Cross-checks: tr(L) = 256 = 2|E|, tr(L²) = 1632, all eigenvalues in [0, 12] (Gershgorin).

Usage

use resonance_spectral_gap::{Atlas, SpectralAnalysis, Rational};

let atlas = Atlas::new();
let spectral = SpectralAnalysis::from_atlas(&atlas);

assert_eq!(spectral.spectral_gap(), Rational::from_integer(1));
assert_eq!(spectral.max_eigenvalue(), Rational::from_integer(11));
assert_eq!(spectral.num_distinct_eigenvalues(), 11);
assert!(spectral.all_eigenvalues_integer());

Design Principles

• Zero floating-point: All arithmetic uses num_rational::Ratio<i64>. Clippy denies float_arithmetic.
• Zero unsafe code: #![forbid(unsafe_code)]
• Self-contained: The Resonance Classes graph is constructed from first principles (no external data files).
• Tests as proofs: Every test is a computational certificate for a mathematical claim.

Running the Proof

cargo test

This executes 31 tests that collectively certify λ₁ = 1 through multiple independent verification paths.

Context

This result connects the Resonance Classes graph to spectral graph theory and the toroidal coherence framework. The spectral gap determines:

• Heat kernel decay rates on the graph
• Random walk mixing times (O(1/λ₁) steps)
• Algebraic connectivity bounds (Cheeger inequality)

The spectral analysis was originally contributed as PR #2 to the UOR Foundation’s atlas-embeddings crate (the graph of Resonance Classes was discovered by the UOR Foundation in 2024). Published as a standalone crate for independent verification of the λ₁ = 1 result.

License

MIT

Author

Sylvain Cormier — Paraxiom Technologies — sylvain@paraxiom.org

Structs

Atlas

The Atlas of Resonance Classes: a 96-vertex, 256-edge graph.

Label

Canonical label for an Atlas vertex.

SpectralAnalysis

Spectral analysis of the Atlas graph Laplacian.

Type Aliases

Rational

Exact rational number type (no floating point).