Lower bounds for the unknotting number over a corpus of built curves

VERSION 3 (17 October 2026) - STATUS UPDATE, READ FIRST: measured against B. A. Burton's census (up to 19 crossings) and Maguire's census (up to 17) and the corrected novelty boundary (all prime knots enumerated up to 20 crossings, alternating ones up to 24), 8 of the 13 knots with u determined are table knots (calibration, not a result), 4 have a proven crossing number of only c >= 20 (inside the enumerated range), and only 1 lies beyond every existing enumeration (crossing number 24, proven, non-alternating, u = 3). The '5 beyond any enumeration' below counted crossings of our diagrams, not proven crossing numbers. Of the 22 gap-one knots, 15 are table knots. Per-knot status in status_20261017.json; data unchanged. ---- Text of version 2, corrected: Six lower-bound quantities for the classical unknotting number, computed over a corpus of 479 distinct knots (788 records, diagram crossing numbers 3 to 151), built mostly outside the standard tables. This is a deposit, not a publication claim: NO THEOREM IS ASSERTED, and none of the mathematics is ours. The bounds are textbook - signature (Murasugi), Rasmussen s, Ozsvath-Szabo tau, slice genus, the Nakanishi index (minimal number of generators of the Alexander module), and the branched-cover generator bound. What is deposited is the corpus and the values. Most of the knots come from a symmetry-driven construction (atomic module + C_n/S_n symmetry + closure constraint); 15 of the distinct knots are realised by the centre line of one of the sculptures (of the nineteen sculptures, five exist in welded stainless steel, one is cut but not welded, and thirteen are rendered designs). That these knots are absent from the tables is a statement about the tables' range, NOT a claim of new mathematics. Where a lower bound meets an upper bound found by explicit crossing changes, the unknotting number is determined; that happened for 63 distinct knots, and we then removed most of them on purpose - 41 are table knots after all (there the bounds are calibration, not a result), 3 are a connected sum, a satellite or a Whitehead double, and 6 are not hyperbolic, which by our own rule never counts as new. Thirteen remain, with diagram crossing numbers 15 to 30 and u in {1,...,4}. Of those 13 we owe the reader one further subtraction: the prime knots with twenty crossings were tabulated in 2018 by Thistlethwaite and, independently, by Burton (account: M. B. Thistlethwaite, Algebraic & Geometric Topology 25 (2025) 329-344) and WE NEVER CHECKED AGAINST THAT, so 8 of the 13 could in principle sit in those tables; only 5 are beyond any enumeration that exists. A further 22 knots are open by a gap of exactly one, listed as an attack surface, not a result. Ten cross-checks ran over the corpus, each able to fail, and their verbatim counts are in README.md; one value was retracted on 11 August 2026 (the sculpture double-6, where a stored upper bound belonged to a different diagram) and the retraction is kept visible rather than removed. Two negative results are recorded because someone else would otherwise repeat them: the Ma-Qiu index is not computable as a lower bound in practice, and the X-torsion order from Alishahi's theorem came out as 1 on all 12 cases it was run on (the non-fibered knots among the 22 gap-one knots). Determining unknotting numbers at scale is done far better elsewhere (Applebaum, Blackwell, Davies, Edlich, Juhasz, Lackenby, Tomasev and Zheng determined the unknotting number of 57k knots with reinforcement learning, arXiv:2409.09032, to appear in Experimental Mathematics), and computing a knot type from a physical object is not new either (Raymer and Smith, PNAS 2007). The material is motivated by the degenerate case that Myfanwy Evans singles out in her work on the untangling number of 3-periodic tangles, where the untangling number coincides with the classical unknotting number; the Nakanishi index, her reference [29], is among the deciding bounds for 11 of the 13. HONESTY NOTE: no invariant in this deposit is computed by an AI. Every number comes from established, open, reproducible tools - SnapPy/spherogram, Regina, SageMath, SymPy, KnotJob - and can be recomputed from the planar diagrams included here; in identification mode our pipeline looks nothing up in a knot table. This work was done by Werner Alois Stanggassinger in collaboration with Claude Code (Anthropic): the geometry, the questions and the hardware are Stanggassinger's; the pipeline that drives the tools and the machines that generate the space curves are the joint contribution. Contents: u_bounds_data.json (479 knots, all six quantities, the sieve sets), pd_codes.json (planar diagrams of the 13 determined and the 22 gap-one knots), reproduce.py (pure Python, no dependencies), NOTE.md and README.md. Separate from, and not part of, our Kawauchi deposit chain (10.5281/zenodo.21590760).