# The Thirteen-Bill Closure Pattern of Integer Factoring (2024-2026)

Kevin Russell, ProjectForty2

Pre-publication preprint, May 2026. IACR ePrint ID pending.

## Abstract

The factorization ledger audits 504 papers from 2024-2026 on integer-factoring claims spanning quantum, classical, and learning-style approaches. The harness consists of thirteen bills, six meta-costs, and three escape gates. Three bills are predicted empty: Bill 6, Bill 7, and Bill 8. Across the 504-paper batch, Bills 6 and 7 receive zero triggers, while Bill 8 receives one marginal candidate with zero clean frontier-gate crossings.

## Main Claim

Any frontier integer-factoring claim must pay at least one closure bill before it can count as a clean breach. The empty-space hypothesis predicts that the most important unclaimed space is not a missing implementation detail, but a structural closure gap: no paper in the audited corpus supplies the exact-point, cross-Frobenius, and cross-regime evidence required to move from suggestive construction to frontier breach.

## Method

The audit uses a public receipt layer:

- `data/factorization/bills_draft.md` defines the bill taxonomy.
- `data/factorization/corpus_synthesis.json` summarizes the audited corpus.
- `data/factorization/audit_scripts.md` documents reproducibility checks.
- `atlas.html` provides the public atlas overview.

## Status

This markdown export is the lightweight public receipt paired with `eprint.html`. The canonical public evidence remains the static ProjectForty2 ledger and data bundle.