Eigenvalue-spacing statistics — universality across project Laplacians (2026-05-12)¶
Origin: User question following Hodge spike: "are eigenvalue spacings of these different Laplacians drawn from the same universal statistical distribution?" and follow-up "Let's do the Maass forms next!"
Framing: this is the testable proxy for the Maass-form universality question. Maass-form spectra famously belong to the GUE (Gaussian Unitary Ensemble) universality class per the Bohigas-Giannoni-Schmit conjecture for quantum chaotic systems. Asking "are our Laplacians Maass-like in the statistical sense?" is equivalent to asking "are they in the Wigner-Dyson universality classes (GOE / GUE / GSE)?"
The Wigner-Dyson taxonomy:
| Class | Wigner surmise P(s) | When it applies |
|---|---|---|
| Poisson | exp(-s) |
Integrable / non-interacting systems |
| GOE (real symmetric) | (π/2) s exp(-π s²/4) |
Time-reversal-symmetric quantum chaos |
| GUE (complex Hermitian) | (32/π²) s² exp(-4 s²/π) |
Broken-time-reversal quantum chaos — Maass forms live here |
| GSE (symplectic) | (higher-order) | Spin-½ with strong spin-orbit |
Reproduce: python -X utf8 docs/srmech/notes/eigenvalue_spacing_statistics_script.py — ~30s. Deterministic seed 20260512. Reuses cached 1UBQ.pdb from PR #333.
Pipeline¶
For each Laplacian:
1. Compute full spectrum via scipy.linalg.eigvalsh.
2. Unfold — fit polynomial smoothed cumulative density N(E), then map eigenvalues to N(eigenvalues). This normalizes so mean spacing = 1 across the spectrum.
3. Compute nearest-neighbor spacings (NNS) — differences of consecutive sorted unfolded eigenvalues.
4. KS-test the NNS empirical distribution against the three Wigner-Dyson theoretical PDFs + Poisson.
5. Classify by smallest KS distance.
Results¶
| Laplacian | n | Best fit | KS-Poisson | KS-GOE | KS-GUE |
|---|---|---|---|---|---|
| synthetic GOE (control) | 500 | goe ✓ | 0.203 | 0.048 | 0.101 |
| synthetic GUE (control) | 500 | gue ✓ | 0.278 | 0.078 | 0.018 |
| synthetic Poisson (control) | 500 | poisson ✓ | 0.032 | 0.230 | 0.296 |
| chess 8×8 king-move L₀ | 64 | poisson | 0.260 | 0.369 | 0.427 |
| cycle C₁₀₀ (integrable) | 100 | poisson | 0.506 | 0.506 | 0.506 |
| magnetic torus 8×8 flux=1 | 64 | poisson | 0.702 | 0.702 | 0.702 |
| magnetic torus 16×16 golden flux | 256 | poisson | 0.450 | 0.460 | 0.485 |
| protein 1UBQ L₀ | 76 | goe | 0.190 | 0.094 | 0.139 |
| protein 1UBQ L₁ Hodge | 326 | goe | 0.146 | 0.098 | 0.167 |
Synthetic controls hit expected class 3/3 — pipeline validated.
Headline finding — proteins ARE Maass-class (modulo time-reversal)¶
Protein contact graphs (both L₀ and L₁) belong to the GOE universality class with KS distance 0.09-0.10, comparable to the synthetic GOE control (0.05). This is the Wigner-Dyson universality for time-reversal-symmetric chaotic systems.
Maass-form spectra are GUE-universal (broken time-reversal — complex Hermitian operators on hyperbolic surfaces with no time-reversal symmetry). Proteins are GOE-universal (real-symmetric graph Laplacians have time-reversal symmetry). Same universality framework, sibling classes. The Bohigas-Giannoni-Schmit conjecture connects classical chaos to Wigner-Dyson statistics; the protein contact graph's spectrum is quantum chaotic in this precise sense.
Biological interpretation: protein contact networks are irregular enough that their Laplacian spectra cannot be distinguished statistically from a random symmetric matrix. The contact graph is "complex" in the random-matrix sense — there's no integrable substructure (or it's drowned by noise). This is consistent with known protein-dynamics chaos at long timescales and with the difficulty of analytically predicting fold dynamics.
Negative finding — highly-symmetric Laplacians are NOT in any Wigner-Dyson class¶
The three structured Laplacians tested all fail to belong to any standard universality class:
- Chess 8×8 king-move L₀ (KS = 0.26 / 0.37 / 0.43 vs Poisson/GOE/GUE): the king-move graph on a regular 8×8 board has many degenerate eigenvalues from the D₄ symmetry of the board. After unfolding, NNS distribution is bimodal (zero degeneracies + large jumps).
- Cycle C₁₀₀ (KS = 0.51 for ALL classes): eigenvalues are
2 − 2cos(2πk/N), a deterministic sequence. NNS is deterministic, not random — equally far from every random-matrix universality class. - Magnetic torus 8×8 at flux φ=1 (KS = 0.70 for ALL): at rational flux p/q, Hofstadter band structure produces q clusters of degenerate eigenvalues. Spectrum has structure no random-matrix class predicts.
- Magnetic torus 16×16 at golden-mean flux (KS = 0.45-0.49): irrational flux breaks the rational-band structure but at N=16 the residual symmetries of the toroidal lattice still dominate over disorder. Closer to GUE than rational-flux case, but not within statistical reach.
Why this is informative: random matrix theory predicts universality when the underlying system is generic chaotic. Highly-symmetric or integrable graphs (regular lattices, cycles, perfect tori) have non-generic spectra and require either disorder, larger N, or genuinely chaotic dynamics to enter Wigner-Dyson. The boundary is sharp and we can see it.
What it would take to see GUE in our framework¶
The magnetic torus is the natural candidate for GUE (complex Hermitian, broken time-reversal). To reach GUE statistics:
- Disordered magnetic torus — random link strengths or random vector potential on each edge. Breaks remaining lattice symmetries.
- Larger system size — at N=100+ the symmetries of finite tori become irrelevant; spectrum approaches universal limit.
- Random-matrix-like ensembles of project graphs — could test "average" universality over a graph ensemble.
The Hopf-fibration framework (MFO §VII.4.1.1 / PR #332) gives a clean route: a discrete principal U(1)-bundle over a random base graph would have complex-Hermitian Laplacian with broken time-reversal — natural setting for GUE statistics.
Connection to Maass forms — what the result actually says¶
We did NOT compute Maass forms. We tested whether the statistical universality class that Maass forms occupy (GUE) is reached by our Laplacians.
| Question | Answer |
|---|---|
| Do any of our Laplacians literally equal Maass forms? | No — wrong manifold, no arithmetic structure |
| Do any reach GUE statistics? | Not at tested sizes; magnetic torus is the natural route but needs disorder/larger N |
| Do any reach GOE statistics (sibling class)? | YES — protein L₀ and L₁ both pass with KS ≈ 0.10 |
| Are our project Laplacians universal across the board? | No — highly-symmetric ones are non-Wigner-Dyson; chaotic-enough ones are |
The universality question has a real answer: the Wigner-Dyson framework is the right universality framework when it applies, but applicability is gated on the underlying system being chaotic enough. Proteins are. Regular graphs are not. The discriminator is whether the spectrum has residual structure from symmetries the random-matrix ensembles average over.
Connection to MFO §VII.4.1.1 (Hopf bundle, PR #332)¶
Yesterday's Hopf-bundle Hodge framework predicts complex-Hermitian Laplacian on a principal U(1)-bundle should exhibit GUE statistics in the random-base limit. We didn't reach the random-base limit (our magnetic torus is regular), so the prediction is consistent but un-tested. Future spike: build a graph principal U(1)-bundle over a random connected base graph; verify the spectrum lands in GUE.
The fibre-harmonic content from MFO §VII.4.1.1 corresponds to the kernel of L₁ in Hodge theory (PR #333 result); the non-kernel part of L₁ is what carries the Wigner-Dyson statistical content. So the framework distinguishes: - Topological content (harmonic forms; Betti numbers; integer, categorical) - Statistical content (NNS distribution of non-harmonic eigenvalues; continuous, universal)
Both are real; both are testable; they live at different levels.
Architectural lesson for srmech catalog¶
This spike establishes a spectral-statistical-class column for any project Laplacian:
| Laplacian property | Catalog column | Diagnostic |
|---|---|---|
| Integrable / regular | Poisson | NNS exponential decay |
| Time-reversal-sym chaotic | GOE | Wigner-Dyson linear-onset |
| Broken-time-rev chaotic | GUE (Maass-class) | Wigner-Dyson quadratic-onset |
| Highly symmetric / band structure | Non-universal | KS large vs all three |
Two real value-adds: 1. Cross-domain comparison of project Laplacians can now check universality-class agreement before similarity scoring (proteins compared to proteins = both GOE; chess compared to chess = both non-universal in this respect). 2. Encoding lesson generalization: per the PR #333 / Hodge-spike rule ("encoding matches data's mathematical type"), the spectral-statistics class is a structural property the catalog should record. A GOE-class Laplacian's spectrum is literally a random matrix sample in distribution; encoding accordingly differs from a band-structured spectrum.
Open follow-ups¶
- Ephemerides 52-body resonance graph Laplacian — likely GOE (chaotic, disordered); would be a fourth Wigner-Dyson confirmation.
- Larger disordered magnetic graphs — to reach GUE in the discrete setting; tests MFO §VII.4.1.1 Hopf-bundle prediction concretely.
- Persistent spectral universality across cutoffs — vary protein contact cutoff in [5, 12] Å; check whether GOE class is preserved across the persistence range.
- Mixed-class graphs — chess king-move at very large N might transition from non-universal to GOE; testable.
- Selberg trace formula connection — the Maass-form trace formula links eigenvalue density to closed geodesic lengths on hyperbolic surfaces. The discrete analog: Ihara zeta function / cycle structure of graphs. Could be a future spike to compute Ihara zeta of project graphs and check the spectral / cycle-length duality empirically.
Files¶
eigenvalue_spacing_statistics_script.py— reproducible spike (~30s)eigenvalue-spacing-statistics-per-laplacian-2026-05-12.ndjson— per-Laplacian KS distanceseigenvalue-spacing-histograms-2026-05-12.png— 8-panel histogram comparison vs Wigner-Dyson curves
Citations¶
- Bohigas, Giannoni, Schmit 1984: "Characterization of chaotic quantum spectra and universality of level fluctuation laws" — the conjecture connecting classical chaos to GUE statistics
- Wigner 1955 surmise; Dyson 1962 three classes (orthogonal/unitary/symplectic)
- Selberg 1956 trace formula; Sarnak (multiple papers) on Maass-form GUE universality
- Mehta 2004 Random Matrices — standard reference for Wigner-Dyson statistics