sqrt(2) x ln(2): GEOMETRIC CONSTANTS FROM H_4 Complete Framework — Version 3.0

This project presents the complete K_AUD framework connecting H₄ geometry to information-theoretic constants, with the Binary Tower scaling and the gap scaling formula 400/11. Core Framework: The Floor (1/φ ≈ 0.618): Golden ratio self-similarity in H₄ vertex coordinates. The Ceiling K_AUD (√2 × ln(2) ≈ 0.980): Geometric embedding cost (√2) × information-theoretic binary distinction cost (ln(2), Shannon's fundamental unit). K_AUD exceeds Shannon's bound for a single binary decision by exactly the geometric factor √2. The factor √2 has multiple independent origins (H₄ circumradius, L2 norm, tesseract geometry, algebraic structure) — H₄ is one candidate among several. The Gap (G ≈ 2%): Irreducible mismatch between algebraic and transcendental values. Binary uniqueness: n = 2 is the only base producing a sub-unity ceiling. K_AUD can be constructed via four independent phi-free pathways without reference to H₄. Version 3.3 adds the Binary Tower: Master Formula: 2^k × G = 2^(k-6) × √φ × (1 − ε), with constant ε = −0.671% Epsilon constancy is algebraic (2^k cancels on both sides), not an empirical discovery — the entire tower reduces to the single relationship 64×G ≈ √φ. Tower target at k=5: √φ/2 ≈ 0.636 (nearby but distinct from 1/φ ≈ 0.618, differing by 2.9%). Pivot at k=6: 64×G ≈ √φ. Connection to 400/11 gap scaling formula: closure cycle 4² in ρ, 4³ = 64 as pivot. Literature status: What's established mathematics vs original to this framework. v3.3 corrections (February 2026): (a) Distinguished √φ/2 from 1/φ at k=5 (2.9% difference, separate coincidence). (b) Clarified epsilon constancy as algebraic tautology. (c) Acknowledged multiple origins for √2. (d) Corrected Table 11.2. (e) Corrected editorial oversight from v3.1 that incorrectly declared K_AUD unrelated to Shannon information theory. K_AUD exceeds Shannon's single binary decision bound by exactly √2. Update — 4 May 2026 (v3.3.1, link cleanup pass 5 May 2026): Two changes — no mathematical content changed. (a) Typography correction in §6.3 (φ-scaling ratio test). Displayed values of L₋₂ and L₋₃ corrected from 0.2273642... / 0.1405469... to 0.2273620... / 0.1405174... — the values that follow from the formula L_n = 1/(e × φ^(n-1)) given in §6.4 and used in §6.2 derivations. The φ-ratio claim now holds exactly with the displayed numbers (1.6180339887 = φ); previously the second ratio computed to 1.6176, which the paper labelled as φ. Formula and derivations were and remain correct. Flagged 2026-05-04 by Claude (Cowork) during systematic mpmath review of all published papers; verified by Claude (Chat) independently. (b) 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). The full UPDATE NOTICE appears at the top of the document; a v3.3.1 entry has been added to the version history block. The prior published version remains on OSF as an OLD- archived file. Mathematics verified across multiple AI architectures to machine precision. Empirical applications remain open for investigation. Available formats: PDF, Plain Text. 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