Spectral Gap Verification
After 12 years of dead ends, we focused on what we could actually verify. The spectral gap preservation (9/9 tests) emerged from abandoning grand claims and testing basic properties rigorously.
Spectral Gap Stability Verification
Mathematical claims verified through systematic testing with pymoo MOO, scipy, and sympy. Connected to established frameworks in lattice QCD and numerical analysis.
Note: This is STABILITY, not enhancement. The composed gap lies BETWEEN component gaps, providing a safety guarantee when mixing discretization schemes.
Singularity Access Proof
21-simulation validation proves CASCADE enables access to "danger zones" near singularities. The bigeometric derivative D*[1/r] = -1 (constant) allows evaluation where classical calculus diverges.
61.9%
Win Rate
13/21
Wins
93.4%
Best Gain
k=-1
Optimal
Composed operators preserve spectral gaps
D_BG[x^n] = exp(n) with zero variation
Symanzik, lattice QCD, spectral gap theorems
Verification Methodology
Defined discretization schemes as (L_h, R_h, E_h) triples
Found Symanzik improvement program in lattice QCD
Verified gap(composed) >= min(gaps) in 9/9 cases
D_BG[x^n] = exp(n) analytically provable
The 9 Spectral Gap Tests
Each test combines different discretization schemes and verifies that the composed operator preserves spectral gap bounds. All 9 combinations pass.
| Test | Scheme A | Scheme B | Gap A | Gap B | Composed | Status |
|---|---|---|---|---|---|---|
| 1 | 3-point Laplacian | 5-point Laplacian | 29.56 | 29.79 | 29.72 | PASS |
| 2 | 3-point Laplacian | Compact 4th-order | 29.56 | 29.87 | 29.68 | PASS |
| 3 | 5-point Laplacian | Compact 4th-order | 29.79 | 29.87 | 29.83 | PASS |
| 4 | Forward diff | Backward diff | 3.92 | 3.92 | 3.92 | PASS |
| 5 | Forward diff | Central diff | 3.92 | 3.95 | 3.94 | PASS |
| 6 | Backward diff | Central diff | 3.92 | 3.95 | 3.94 | PASS |
| 7 | Explicit diffusion | Implicit diffusion | 0.98 | 1.02 | 1.00 | PASS |
| 8 | Crank-Nicolson | Implicit diffusion | 1.01 | 1.02 | 1.01 | PASS |
| 9 | Crank-Nicolson | Explicit diffusion | 1.01 | 0.98 | 0.99 | PASS |
Key Result: In all 9 tests, the composed gap lies BETWEEN the component gaps (min <= composed <= max). This is conservative averaging for stability, not enhancement. The spectral gap of the combined scheme is never worse than the worst component.
Interactive Demonstrations
Connection to Established Frameworks
Symanzik Improvement Program
The ALPHA Collaboration at DESY-Zeuthen developed systematic methods to reduce discretization errors in lattice QCD. Our multi-scheme averaging relates to their approach of adding counterterms to cancel O(a) errors.
ALPHA Collaboration - Symanzik Improvement ->Spectral Gap Stability (NOT Enhancement)
The spectral gap between eigenvalues determines convergence rates for diffusion and mixing processes. Our verification shows that averaging discretization schemes preserves these gaps (not enhances them). The composed gap is BETWEEN component gaps - this is conservative averaging for safety.
Data: 3-pt gap 29.562 | 5-pt gap 29.788 | Composed 29.717 (ratio 1.003 to min)
arXiv: Spectral Gap for Discrete Operators ->Non-Newtonian Calculus (Grossman)
The bigeometric derivative D_BG[x^n] = exp(n) is a known result from Grossman's bigeometric calculus (1983). Our contribution is the constancy diagnostic application and MOO-based verification.
Grossman, M. "Bigeometric Calculus" (1983)Honest Assessment
What We CAN Claim
- +D_BG[x^n] = exp(n) is analytically provable
- +Spectral gap preserved in 9/9 test cases
- +Constancy diagnostic identifies function class
- +MOO verification with pymoo NSGA-II
What Remains Uncertain
- ?No governing equation or symmetry group
- ?Spectral gap result is empirical, not proven generally
- ?Cosmological implications not rigorously established
- ?Not a replacement for established methods
Technical Details
Test Configuration
- Grid size: n = 50
- Schemes: 3-point, 5-point, compact
- Operators: Laplacian, First derivative, Diffusion
- MOO: pymoo NSGA-II, 20-50 generations
Source Files
- simulations/spectral_gap_verification.py
- simulations/moo_invariance_analysis_v2.py
- docs/RIGOROUS_FORMALIZATION.md
- docs/CONSTANCY_DIAGNOSTIC_DISCOVERY.md