Machine Learning for Modular Forms: GNNs, Spectral Methods, and L-Function Phenomenology — Comprehensive Preprint

May 30, 2026

ResearchAI & Machine Learning

graph-neural-networksnumber-theorylmfdbmodular-formssato-taterandom-matrix-theoryconnes-spectral-triplesfunsearchgat-attentionriemann-hypothesispreprint

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

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

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.

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Machine Learning for Modular Forms: GNNs, Spectral Methods, and L-Function Phenomenology — Comprehensive Preprint

What the Preprint Covers

Negative Results (Thread A — Cayley Graphs of SL(2,Fₚ))

Positive Results (Thread B — LMFDB-Scale Learning)

Structural Discoveries (Thread C — CvS, GUE, Sato-Tate)

New in the Preprint (Not Previously Published)

Key Experimental Constants

How to Cite

Repository

Never miss a deep-dive

What the Preprint Covers

Negative Results (Thread A — Cayley Graphs of SL(2,Fₚ))

Positive Results (Thread B — LMFDB-Scale Learning)

Structural Discoveries (Thread C — CvS, GUE, Sato-Tate)

New in the Preprint (Not Previously Published)

Key Experimental Constants

How to Cite

Repository