Endpoint Tangents and a New Universal Lower Bound for the Ropelength of Nontrivial Knots

Abstract

Preprint; not peer reviewed. This work sharpens the optimized universal diameter-normalized ropelength bound of Denne, Diao and Sullivan. Obstacle tangent cones, endpoint deficits, circular-turn comparisons, two-moving-point constraints, evolving-polar recovery, and a common-continuation estimate produce an auxiliary joint functional. Directed interval certificates prove 0.09548 ≤ Δ* < 0.095675 and hence Rop(K) > 15.75608926 for every nontrivial knot K. Read the manuscript (PDF). Version 5.0.1 is a reproducibility packaging correction to version 5.0.0. The preprint, theorem, numerical bound, and generative-AI disclosure are unchanged. The complete replay supplement restores 27 omitted upper-audit dependencies, each matched to its previously recorded hash, and adds a replay guide and verification notes. All 24,106 members of the original release archive are preserved byte-for-byte in the complete supplement. For reproduction, use schoen_ropelength_jktr_v5.0.0_supplement.zip as the complete package, or apply schoen_ropelength_v5.0.0_replay_addendum.zip to the original v5.0.0 release archive. Consult referee-guide.pdf and PACKAGING-UPDATE-v5.0.1.md for the file map and verification scope. The original files are retained for provenance.