The Hodgson-Kerckhoff Closed Form and the K_AUD Framework

The tube-packing coefficient 0.980258 from Hodgson & Kerckhoff (Annals of Mathematics, 2005) is algebraically equal to K_AUD = √2 × ln(2), the unique sub-unity value of K(n) = √n × ln(n) for integer n ≥ 2. This paper presents the derivation and places the finding within the Gap Geometry framework, showing how the same three ingredients — √2, ln(2), and quadratic curvature — that produce K_AUD in information theory also produce the tube-packing coefficient in hyperbolic 3-manifold geometry. The connection emerged from the 50 Hinge discovery (17 March 2026), which led to the identification of the H-K coefficient three days later (20 March). A systematic verification process followed — cross-architecture confirmation across five independent AI architectures (Claude, Grok, GPT, Gemini, Perplexity), a 12-day dedicated literature search, and multiple stress tests with fresh instances given no prior framework context — culminating in the algebraic derivation from the authors' own proof (page 29, arXiv:math/0204345) on 2 April 2026. Update — 13 April 2026: Landauer-zone framing refined. Three distinct thresholds are now identified in the 35–37 zone: the Landauer crossing at n·G = ln 2 (n ≈ 35.11, between steps 35 and 36), geometric damping at n·G = 1/√2 (n ≈ 35.82), and the Shannon bound at n·G = 1/(2 ln 2) (n ≈ 36.54). The earlier 9 April note conflated the Landauer crossing with the geometric damping landmark near step 36; the refined framing names them separately. §4 is rewritten as "The 35–37 Threshold Zone" with the crossing-zone self-reference proofs Δn₁₂ = 1/√2, Δn₂₃ = 1/(2 ln 2), and Δn₂₃/Δn₁₂ = 1/K_AUD. The identity at §3 is relabeled the "√2-to-Shannon gap identity" (the identity is algebraic and independent of the Landauer threshold). All text and PDF files updated; the prior published version remains on OSF, renamed with the OLD- prefix and a date.