√2 × ln(2): GEOMETRIC CONSTANTS FROM H4 Mathematical Framework and Discovery Context Version 2.0
B.D.
Abstract
This project presents a mathematical framework connecting H4 geometry (the 120-cell polytope) to information-theoretic constants. The ceiling constant K_AUD = √2 × ln(2) ≈ 0.980 combines geometric embedding (√2) with binary distinction cost (ln 2). The factor √2 has multiple independent origins (H4 circumradius, L2 norm, tesseract geometry, algebraic structure) — H4 is one candidate among several. The floor constant 1/φ ≈ 0.618 emerges from golden ratio self-similarity in H4 vertex coordinates. The gap G = 1 − K_AUD ≈ 0.0197 is constitutive, not error. The framework includes identities (Corridor = 1/φ² − G, Golden Partition: 1/φ + 1/φ² = 1), a Binary Tower showing G scales through powers of 2 to track golden ratio powers (the entire tower reduces to 64×G ≈ √φ), and a gap scaling formula connecting K_AUD to the Feigenbaum constant via ρ = 400/11 − 1/2500 − 1/939939. Binary (n=2) is the unique base producing a sub-unity ceiling. Mathematics is independently verifiable. K_AUD can be constructed via four independent phi-free pathways without reference to H4. The relationship between H4 and ln(2) is a selection argument, not a derivation — empirical applications remain open for investigation. Update — 4 May 2026 (v2.0.2, link cleanup pass 5 May 2026): Three changes — no mathematical content changed. (a) Attribution clarification in §1.1 (Origin) now credits Gemini (Google) for the closed-form identification of K_AUD = √2 × ln(2), the detailed articulation of the gap structure G = 1 − K_AUD, and the naming of the ceiling constant K_AUD. D.B.'s prior credit (Conceptualization, and Discovery of the ceiling as an empirical phenomenon, surfaced via months of cross-architecture pattern recognition) is preserved. Both are Discovery roles of different objects per the framework's contribution-role taxonomy (Methodology §5). The earlier §1.1 phrasing implicitly attributed the closed-form match to D.B. — this update makes the credit chain explicit and aligns Paper 2 with the canonical credit chain that already existed in the Methodology page, the Discovery Tracking Log, and related papers. (b) Typography correction in §6.3 (φ-scaling ratio), §7.1 (All Calculations Collected table), and §8.3 (Extended Constants table): displayed values of L₋₂ and L₋₃ corrected from 0.2273642379 / 0.1405469971 to 0.2273619984 / 0.1405174428 — the values that follow from the formula L_n = 1/(e × φ^(n-1)) given in §6.4 and used in §6.2 derivations. (c) DOCUMENT LINKS section simplified: paper-to-paper file URLs (which would break under any future filename change) are removed in favour of three durable anchors — the OSF main project, the framework About page, and the Direct Documents download index. Paper-specific DOIs are preserved (DOIs are stable). Flagged 2026-05-04 by Claude (Cowork) during systematic mpmath review; verified by Claude (Chat) independently — Chat caught the §7.1 propagation gap in second-pass review. 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