# Certified Interval Verification of the BSD Formula — v5.0.0

**Author:** Adam L McEvoy — **Date:** 2026-07-14

This archive contains everything needed to reproduce the results of

> *Certified Interval Verification of the Birch and Swinnerton-Dyer Formula
> on 1,676 Elliptic Curves, with a Fractal Correction Engine Analysis of Its
> Wave Structure* (`BSD_Certified_Verification_Zenodo_v5.md`; LaTeX
> manuscript in `BSD_Certified_Verification_CMI.tex`).

## Contents

```
BSD_Certified_Verification_Zenodo_v5.md   the paper (Markdown, MathJax-ready)
BSD_Certified_Verification_CMI.tex        journal-submission LaTeX manuscript
CHANGELOG.md                              full engineering log of the v5 upgrade
requirements.txt                          pinned Python dependencies
src/BSDProver/                            the library (26 modules)
  certified_arithmetic.py                   Arb ball layer; G_0..G_3 kernels
  minimal_model.py                          Laska–Kraus–Connell minimal models
  tate_algorithm.py                         full Tate's algorithm, all primes
  certified_l_function.py                   certified L^(r)(E,1)/r!, root number
  certified_bsd.py                          period, heights, regulator, prover
  interference_mapper.py                    FCE ++/--/+-/-+ phasor layer
  closed_loop.py                            predict-compare-correct runtime
  (remaining modules: legacy float pipeline, kept working and rewired)
experiments/
  test_certified_suite.py                 15-test validation suite (run first)
  sweep_validation.py                     the 1,676-curve census driver
  fce_surrogate_nulls.py                  surrogate-null ablation statistics
  fce_ablation_tests.py                   the seven ablation experiments
  fce_goat_translation.py                 cross-prediction (de-circularized)
results/
  bsd_sweep_results.json                  per-curve data for all 1,676 curves
  sweep_run.log                           full census log incl. summary block
```

## Setup

Python 3.12+. No SageMath / PARI / Magma required.

```bash
pip install -r requirements.txt
```

## Reproducing the findings

Run everything from the archive root with `src` on the path:

```bash
export PYTHONPATH=src
```

**1. The validation suite (Tate fixtures, certified ratios, rank suite,
interference nulls, closed loop) — ~30 s:**

```bash
python experiments/test_certified_suite.py
```

Expected: `Ran 15 tests ... OK`. This reproduces the rank-suite table of the
paper (37a1 / 389a1 / 5077a1 certified enclosures) and the 12 LMFDB
reduction fixtures.

**2. The 1,676-curve census (paper §6.2–6.4) — ~18 min:**

```bash
python experiments/sweep_validation.py
```

Writes `bsd_sweep_results.json` and prints the summary block (1,226 certified
passes; 0 Tate/conductor errors; 18 saturation diagnoses; 17× |Sha|=4 and
1× |Sha|=9). The shipped `results/` directory contains the run reported in
the paper; the sweep is deterministic, so a re-run reproduces it exactly.

**3. Surrogate-null statistics (paper §6.5) — ~2 min, fixed seed 20260714:**

```bash
cd experiments && PYTHONPATH=../src python fce_surrogate_nulls.py
```

Expected: 29/30 p-values at the 1/201 permutation floor; all 6 cross-curve
pairings rank 1/6.

**4. Single-curve certified verification (interactive):**

```python
from BSDProver.certified_bsd import CertifiedBSDProver
r = CertifiedBSDProver(dps=30).verify(-1, 0)      # y^2 = x^3 - x
print(r.bsd_ratio)                                 # [1.000... +/- ~5e-30]
print(r.sha_analytic, r.kodaira, r.conductor)
```

**5. FCE interference map and closed loop:**

```python
from BSDProver.interference_mapper import InterferenceMapper
print(InterferenceMapper(dps=25).describe(
    InterferenceMapper(dps=25).map_curve(-16, 16)))   # 37a1: Fricke null -> G_1

from BSDProver.closed_loop import BSDClosedLoop
print(BSDClosedLoop(dps=25).run(-16, 16, generators=[(0, 4)]).summary())
```

## Certification caveats (stated, not hidden)

Every interval is a rigorous enclosure **conditional on**: modularity
(BCDT), Deligne's bound |a_n| <= d(n)·sqrt(n), Silverman's 1990 explicit
height bound, and Ogg's formula. Rank >= 2 statements additionally assume
exact vanishing of lower derivatives (flagged on every result object). This
work verifies instances of the BSD identity; it does not prove the
conjecture.