A reproducible re-examination of Beale ciphers B1 and B3: systematically closing the composite-cipher hypotheses, with methodological cautions

Abstract The three Beale ciphers (published by J. B. Ward in 1885) remain a touchstone problem in historical cryptanalysis: cipher B2 decrypts against the United States Declaration of Independence to a coherent description of a buried treasure, while B1 and B3 have resisted 140 years of attack. Building on Gillogly’s observation that B1 decoded against the Declaration produces an anomalous alphabetic run, we report a fully reproducible, open-source re-examination that (i) re-establishes the B2 positive control under a single explicit scoring framework, (ii) systematically forecloses the principal composite-cipher hypotheses for B1 and B3—Caesar, short- and long-keyword Vigenère, autokey, Beaufort, columnar transposition over all periods to 25, and the complete enumeration of all 157,248 invertible Hill 2 × 2 matrices—none of which yields English plaintext, and (iii) quantifies the Gillogly anomaly with a generative “alphabet-walking” construction model that yields a likelihood ratio of at least 100:1 in favour of deliberate construction over encryption. We additionally document two methodological pitfalls that recur in book-cipher analysis: the index-of-coincidence baseline inflation produced by a skewed key’s first-letter distribution, and the dependence of significance on null-distribution construction. Our conclusions agree with the prevailing hoax verdict for B1 and B3; the contribution is the rigour, completeness, and reproducibility of the negative result, together with a calibrated statement of residual uncertainty. All code, data, and a continuous-integration test suite are publicly archived. Acknowledgements The author thanks the maintainers of Project Gutenberg, Wikisource, and Documenting the American South for the public-domain key texts, and the cryptologic-history community—in particular the published work of J. Gillogly, L. Kruh, C. Hammer, L. Campanelli, R. Wassmer, and the analyses of N. Pelling—without which this re-examination would have no foundation. Disclosure of AI assistance The analysis pipeline, statistical computations, figure generation, and a draft of this manuscript were produced with the assistance of an AI coding agent operating under the direction of the human author, using MATLAB and Python tooling. All methods are fully specified and reproducible from the archived code; no result in this paper depends on an unverifiable AI judgement. The AI system is a tool and is not an author. Disclosure statement No potential conflict of interest was reported by the author(s). Data availability statement All code, cipher data, the key-text corpus, figure-generating scripts, phase-by-phase analysis reports, and a continuous-integration test suite are archived in the project repository at https://github.com/myhsieh1002/beale-cipher-analysis. A permanent archival snapshot of the version corresponding to this publication has been deposited on Zenodo: https://doi.org/10.5281/zenodo.21193924. Additional information Funding Notes on contributors Ming-Yu Hsieh Ming-Yu Hsieh is a physician and clinical researcher who directs the Center for Evidence-Based Medicine at Chung Shan Medical University Hospital and serves as Vice Director of the School of Medicine at Chung Shan Medical University, Taichung, Taiwan. This study is a cross-disciplinary application of clinical-research methodology—pre-specified analysis, reproducibility, and adversarial review—to a classic problem in cryptologic history.