Machine Learning for Modular Forms: GNNs, Spectral Methods, and L-Function Phenomenology — Comprehensive Preprint
ResearchAI & Machine LearningPublished on Zenodo: 10.5281/zenodo.20479512
This preprint synthesises the complete Riemann Project — 11 experiment tracks, 7 GNN architectures, 200,000 weight-2 newforms from the LMFDB, and a formal theoretical analysis of why message-passing GNNs fail on algebraic graphs.
Related articles: GNN + Riemann Hypothesis, Trace-Index Graph Prediction, ML Pipeline for Modular Forms
What the Preprint Covers
The 12-page preprint (32 references) consolidates every experiment track and theoretical contribution:
Negative Results (Thread A — Cayley Graphs of SL(2,Fₚ))
GNNs systematically fail on vertex-transitive algebraic graphs. Across GCN, GAT, GIN, ChebConv, and GraphSage — on subgraph and full-graph scales — no architecture achieves ΔR² > +0.042 over a trivial log(N) baseline. The root cause is vertex-transitivity: every node is structurally identical, collapsing local neighborhood information that message-passing depends on. Formalised via the Weisfeiler-Leman hierarchy: MPNNs on vertex-transitive graphs are limited to functions of diameter and degree.
Positive Results (Thread B — LMFDB-Scale Learning)
When the data has structure, ML succeeds:
- Hecke trace regression: R² = 0.987 on 53,000 forms using 100 Hecke traces (sklearn GBR). Data quantity, not model architecture, was the bottleneck.
- Analytic rank classification: F₁ = 0.970 (MLP/RF/GB ensemble on 100 Hecke traces).
- L-function zero prediction: R² = 0.731 (GAT on 1000-node trace-index graphs).
- CM form detection: F₁ = 0.919 (Sato-Tate moment features, M₄/M₂ ratio primary).
Structural Discoveries (Thread C — CvS, GUE, Sato-Tate)
- Connes CvS operator extracts ζ zeros to 10⁻¹⁶ machine precision at N=100.
- Two-population GUE statistics: dim=1 → GUE, dim≥2 → GOE across 63,844 forms (Cohen's d = 8.808).
- Galois correlation: ρ₂ = −0.607 ± 0.012, dilution law ρ_d ∼ d^.
- Sato-Tate moment diagrams: clear convergence gaps between CM and non-CM families.
New in the Preprint (Not Previously Published)
- GAT attention analysis (Section 4.10): 56M+ edge observations across 2000 test graphs. Prime bias 6.1% (ratio 1.061), Cohen's d = 0.035 — GAT does not learn the Ramanujan–Petersson bound. Layer 2 focused (entropy 7.15, sparsity 0.62), layers 0/1 diffuse.
- FunSearch specifications (Thread K): Two convergent search targets for arithmetic function discovery — multiplicative Hecke trace models and spectral gap formulas on LPS graphs.
- Spectral rigidity analysis: GUE eigenvalue spacing statistics (Δ₃, P(s)) across dimensions, with the dim=1/≥2 transition.
- Expanded theoretical framework: Weisfeiler-Leman proof of MPNN limitations on vertex-transitive graphs formalised as Theorem 4.2.
Key Experimental Constants
| Experiment | Sample | Best Result |
|---|---|---|
| Cayley spectral gap (GNN) | 27 primes, 1M+ node graphs | ΔR² = +0.042 (chance-level) |
| Hecke trace regression | 53,000 forms | R² = 0.987 (sklearn GBR) |
| Analytic rank | 46,347 forms | F₁ = 0.970 |
| L-zero prediction (GAT) | 46,347 forms, 1000-node graphs | R² = 0.731 |
| CM classification | 46,347 forms | F₁ = 0.919 |
| GAT attention (prime bias) | 2000 graphs, 56M edges | d = 0.035 (negligible) |
| ζ zeros via CvS | N = 100 | 10⁻¹⁶ machine precision |
How to Cite
Weiss, T. (2026). Machine Learning for Modular Forms: Graph Neural Networks, Spectral Methods, and L-Function Phenomenology. Zenodo. https://doi.org/10.5281/zenodo.20479512
Repository
All code, data pipelines, and experimental scripts are available in the Riemann Project repository (Dockerised, with Makefile targets for reproduction). The repository includes the full knowledge graph (Cypher/Neo4j), 7 GNN implementations, eigenvalue computation via sparse Lanczos, and Sato-Tate moment analysis.