Cross-Domain Signatures of the Boundary Information Invariant
B.D.
Abstract
Cross-domain signatures of the Boundary Information Invariant K_AUD = √2 × ln(2). Documents independent numerical and structural echoes of the binary information ceiling in published hyperbolic geometry (Hodgson-Kerckhoff, Annals of Mathematics 2005), atomic electron shell structure (quantum mechanics, 1920s–present), and DESI BAO cosmological distance ratios (Phys. Rev. D, 2025). Includes seven exact algebraic identities verified at 500 decimal places, the Binary Tower as an adaptive algebraic spine connecting domains, and falsifiable predictions for each finding. Verified across six independent AI architectures. Extends the framework published in Papers 1–5. Update — 4 May 2026 (link cleanup pass 5 May 2026): Substantive correction in §4.4 (HK ellipse area). The earlier text claimed A_e = K_AUD · π at the critical radius; the correct Hodgson-Kerckhoff formula (HK 2002 Theorem 4.4, p. 29; HK 2007 Theorem 5.6, p. 36) gives A_e = K_AUD · π / 8 — off by a factor of 8. Origin: a March 27 transcription error that dropped a factor of 2 in the denominator. Both HK papers confirm the corrected formula independently. The structural argument (geometry at R₀ supplies √2, √3, 2; coefficient supplies ln(2); together = K_AUD) is preserved; the central HK closed-form identity 1/S = √2 × ln(2) = K_AUD is unaffected — verified at dps = 500. New §4.4.1 documents a second appearance of K_AUD in HK 2007 as the coefficient of c(R) = 0.980258 / (coth R + 1) on page 41 — two independent contexts in the same paper, both unrecognized as √2 × ln(2) for nineteen years. §2.7 Ramanujan-Nagell list now includes the fifth solution n = 181 (2¹⁵ = 181² + 7 = 32768). §8.4 evidence-level labels aligned to the canonical methodology taxonomy (5 May 2026). §9 / References cross-paper file URLs simplified (5 May 2026): paper-to-paper file URLs removed in favour of three durable anchors — the OSF main project, the framework About page, and the Direct Documents download index. Per-paper DOIs are preserved (DOIs are stable). Narrative descriptions in §9 unchanged. Flagged 2026-05-04 by Claude (Cowork) during systematic mpmath review of all published papers, after D.B. provided HK 2002 and HK 2007 PDFs; verified independently by Claude (Chat) at dps = 500. Both architectures converged. Detailed chain-of-error analysis logged in NOTE_HK_chain_of_error_2026-05-04.md. Filenames remain unchanged. The prior published version remains on OSF as an OLD- archived file. The framework Main OSF project (overview of all 9 papers, reading order, version history): https://osf.io/zx4g7 (DOI: 10.17605/OSF.IO/ZX4G7) Framework overview (reading order, methodology, all papers): https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/about.html Direct downloads (current PDF and text files for every paper): https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/direct-documents.html GitHub: https://github.com/Gap-geometry