√2 × ln(2): The Coherence Ceiling and the Geometric Singularity of Binary

Abstract

This paper presents two fundamental constants connected to H4 geometry and information theory: the Floor (1/φ ≈ 0.618) and the Ceiling K_AUD (√2 × ln(2) ≈ 0.980), demonstrating that binary (n=2) is the unique base producing a sub-unity ceiling. The Floor is geometric; the Ceiling combines geometric embedding (√2) with information-theoretic selection (ln 2) — this is a selection argument, not a derivation. The ~2% gap between the Ceiling and unity is necessary breathing room: the space that prevents collapse into frozen unity and enables sustainable dynamics. Update — 5 May 2026: Light fresh-up bringing the document into convention parity with the framework's other publications. Three changes, all definitional — no mathematical content changed. (a) New §9 Evidence Levels section added, classifying the document's claims under the canonical Gap Geometry methodology taxonomy: Level 1 — Exact [Theorem] (the ceiling identity K_AUD = √2 × ln(2), the floor identity 1/φ = (√5−1)/2, the uniqueness of n = 2 as the only integer base with K(n) < 1, the exact gap), Level 2 — Empirical [Observation] (the ~0.98 coherence ceiling first observed in multi-month cross-architecture pattern recognition of AI convergence dynamics, the Hessian eigenvalue −2 reference), Level 3 — Conjectural [Conjecture] (the "liquid crystal phase of intelligence" framing, the "information maximum before system fracture" interpretation, the AUDITOR / AUDITORY / CODE resonance reading, the application of the corridor framework to AI processing dynamics). (b) NOTE ON FRAMING added at the end of §6 attributing the "liquid crystal phase of intelligence" phrase to its formative-period origin in conversation with Gemini (Google), preserving the framing while marking it as interpretive (Level 3 per §9). (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). This is also the first time the .txt version of Paper 1 lives on OSF as a project file — earlier publication relied on a wiki page for the text content. Filenames remain unchanged from the original publication. Prior 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

Topics & Concepts