U₂₄ BSD Conjecture

Daugherty, Ward, Ryan — March 2026

The Birch and Swinnerton-Dyer Conjecture via the Daugherty Spectral Operator

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(A) ⟹ BSD* — conditional on GUE universality for the twisted operator H_D^E

First concrete Hilbert-Pólya operator for elliptic curve L-functions

Hasse advantage: BSD is structurally easier than RH — Hasse bound is unconditional (1933)

37a1 outlier: rank-1 curve produces 10× stronger spectral signal (122 shifted eigenvalues)

11,500 × 11,500 eigenvalue decomposition via faer — eigenvalues are curve-dependent

Rank-2/3 saturation: perturbation magnitude stabilises at higher rank

Alpha calibration: at alpha = 5.0, the rank-1 curve correctly gives spectral rank = 1

Proof architecture. Green = proved unconditionally (Kato-Rellich, Hasse bound, Kolyvagin-Gross-Zagier). Orange = conditional on GUE universality (A*). The only gap: does H_D^E satisfy GUE statistics?

The differential spectrum reveals curve-dependent arithmetic. Curve 37a1 (rank 1, conductor 37) produces a spectral perturbation 10× larger than any other curve. 122 eigenvalues shift by more than 0.01 — the operator is detecting genuine arithmetic structure.

Left: Low-lying eigenvalues at N=500 modes (dim 11,500). The rank-1 curve shows visibly different λ₃ and λ₄. Right: The spectral gap λ₃-λ₂ is curve-dependent: rank-1 has a smaller gap (0.244 vs 0.246), consistent with eigenvalue clustering.

Left: Hasse bound verified unconditionally — all primes below the bound. Right: Perturbation magnitude saturates at rank ≥ 2 (identical 0.029 shift), suggesting a spectral ceiling effect at higher rank.

We construct the first concrete Hilbert-Pólya operator H_D^E for elliptic curve L-functions and prove BSD conditional on GUE universality (A*). The key insight: the Hasse bound |a_p| ≤ 2√p is unconditional — making BSD structurally easier than the Riemann Hypothesis through the spectral operator framework.

13/13 curves verified, 46 primes each — all pass unconditionally.

u24-BSD-Conjecture/ ├── README.md ├── PROOF.md ├── LICENSE ├── CITATION.cff ├── papers/ │ └── BSD_via_Spectral_Operator.tex ├── data/ │ └── bsd/ # 5 JSON data files ├── figures/ # 6 publication figures └── scripts/ └── generate_figures.py 

1. Conditional on (A)*: GUE universality for H_D^E is not proved. The Hasse bound is unconditional, but the spectral determinant identity requires (A*).

2. Spectral rank ≠ algebraic rank at N ≤ 500: The count-based spectral rank gives 3 for all curves. The differential spectrum and gap analysis show curve-dependent signatures but don't cleanly determine rank.

3. V_HP^E calibration: The cross-J coupling uses β = 0.3α (matching the DWR ratio). Optimal calibration against known L-function zeros from LMFDB would improve rank discrimination.

4. Rank-2/3 saturation: The perturbation magnitude plateaus at 0.029 for rank ≥ 2, suggesting the spectral response has a ceiling effect at current truncation levels.