Spectral Gap Verification
After 12 years of dead ends, we focused on what we could actually verify. The spectral gap study began with a small passing fixture, then expanded into a boundary map with explicit counterexamples.
Spectral-Gap Boundary Audit
Reproducible local testing with a normal-operator guard, including passing cases and counterexamples.
Scope: 3 distinct baseline compositions pass, while the guarded grid audit finds 9 counterexamples. No general safety guarantee is claimed.
Earlier wording said 5/5 non-trivial; two of the five repeat another case (see table)
912 non-normal cases were refused as out of scope
5 Dirichlet and 4 periodic Schrodinger/Laplacian mixes; worst ratio 0.21
Verification Methodology
Defined discretization schemes as (L_h, R_h, E_h) triples
Found Symanzik improvement program in lattice QCD
Characterized the passing subclass and reproduced 9 counterexamples
D_BG[x^n] = exp(n) analytically provable
Scoped Spectral-Gap Evidence
The original result file contains five composed-gap checks, but in that file the "compact" operator has exactly the 3-point gap, so "3-point + compact" is a self-composition and "5-point + compact" repeats "3-point + 5-point". That leaves 3 distinct checks. Four first-derivative entries have zero real-part spacing and are marked as not being spectral gaps, so they are not counted either. Gaps are in units of 1/h^2 with n = 50.
| Composition | Minimum component | Composed | Status |
|---|---|---|---|
| 3-point + 5-point Laplacian | 29.562 | 29.678 | PASS |
| 3-point + compact Laplacianself-composition (compact == 3-point here) | 29.562 | 29.562 | PASS (not distinct) |
| 5-point + compact Laplacianrepeats 3-point + 5-point | 29.562 | 29.678 | PASS (not distinct) |
| All three Laplacians | 29.562 | 29.640 | PASS |
| 3-point + 5-point diffusion | 29.601 | 29.717 | PASS |
Boundary result: The guarded grid rerun accepted 88 normal-operator compositions: 79 passed and 9 counterexamples failed the proposed bound. Another 912 non-normal compositions were refused because this diagnostic is not valid for them. This is an empirical subclass observation and boundary map, not a theorem or a general solver guarantee.
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 Boundary (NOT Enhancement)
The spectral gap between eigenvalues determines convergence rates for diffusion and mixing processes. The baseline shows averaging for selected Laplacian-family compositions, while the guarded grid audit contains 9 counterexamples.
Data: diffusion composed gap 29.717 vs minimum component 29.601 (ratio 1.004)
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
- +3 distinct baseline cases pass; the broader normal-operator claim has 9 counterexamples
- +Constancy diagnostic identifies function class
- +The boundary map is a reproducible 1000-case grid search
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
- Boundary map: 1000-case grid, normal-operator guard
Source Files
- simulations/spectral_gap_verification.py
- simulations/moo_invariance_analysis_v2.py
- docs/RIGOROUS_FORMALIZATION.md
- docs/CONSTANCY_DIAGNOSTIC_DISCOVERY.md