A Closed Form for the Hodgson-Kerckhoff Tube-Packing Coefficient

The numerical constant 0.980258 used in the tube-packing bound of Hodgson & Kerckhoff ("Universal bounds for hyperbolic Dehn surgery," Annals of Mathematics, 2005) is shown to equal √2 × ln(2) exactly. The identity follows from one algebraic simplification — arcsinh(1/(2√2)) = ln(2)/2 — applied to the authors' own proof of Theorem 4.4 (arXiv:math/0204345, page 29). This closed form does not appear in the original paper or in any subsequent citation. The coefficient was first identified on 20 March 2026 through related work on continued fraction structure. A systematic verification followed — cross-architecture confirmation across five AI architectures, a 12-day literature search, and multiple fresh-instance stress tests — with the algebraic derivation confirmed on 2 April 2026. Update — 4 May 2026: §3 ellipse-area derivation corrected and §3.1 added. The earlier §3 read "A_e = √2 × ln(2) × π at the critical radius," derived from a misremembered ellipse-area formula. Both HK 2002 Theorem 4.4 (page 29) and HK 2007 Theorem 5.6 (page 36) give the actual area as A_e = √2 × ln(2) × π × sinh²(R̂) / (2 × cosh(2R̂)), which evaluates to √2 × ln(2) × π / 8 at R₀ — the earlier conclusion was off by a factor of 8. The error originated in a transcription of HK 2007 that dropped the factor of 2 in the denominator and was carried into the rewrite; both HK PDFs (now obtained directly) confirm the correction. §3 has been rewritten with HK's actual formula. The structural argument (geometry at R₀ supplies √2, √3, 2; coefficient supplies ln(2); together = √2 × ln(2)) and the central closed-form identity 1/S = √2 × ln(2) are unchanged. A new §3.1 documents that the same constant 0.980258 also appears in HK 2007 (page 41) as the coefficient of the Euclidean injectivity radius lower bound c(R) = 0.980258 / (coth R + 1) — two independent contexts in the same paper, both unrecognized as √2 × ln(2) for nineteen years. The full UPDATE NOTICE appears at the top of the document. The prior published version remains on OSF, renamed with the OLD- prefix and a date.