A mirror barrier: without a lower bound, staged valuation certificates still cannot rule out cycles of the 3x+1 map
Year: 2026
Type: article
Abstract: H. Nashida proved that the shortcut 3x+1 map T has no generalized staged certificate of termination: finitely many stages, each with a function α ln x + Σ β vq(x − c) + g(x) with g bounded (Expanding cycles are not enough, Zenodo, doi:10.5281/zenodo.23081448, Definition 3.3 and Corollary 9.1). On stages with α < 0 his definition asks for a lower bound. To exclude cycles, as opposed to proving termination, decrease alone suffices. This note shows that, for the steps of T, dropping the lower bound does not help. The lower bound is replaced by the shadow of a positive rational cycle of T, such as 1/5 → 4/5 → 2/5: along integers that follow it for a long time, ln x decreases linearly while the valuations stay small. With several stages, words (EnCn)m put an expanding and a contracting block on one stage, and the valuations are read at the junctions and in the middles of the blocks. If the decrease by a fixed δ is dropped for bounded parts with infinitely many values, the question becomes a tautology. The result concerns one class of ranking functions. It says nothing about the Collatz conjecture itself, nor about certificates that are not functions along the orbits. The framework (Nashida's generalized staged certificates and their barrier) and most of the tools are Nashida's; the credits are in Remark 3.4 and Section 6 of the note. Files: the note (PDF and LaTeX source, 8 pages); miroir_checks.py, a numerical illustration of the construction (Python standard library; not part of the proof), with its expected output and a README. Use of AI: the argument, the script and the note were written by Claude Opus 5.5 (Anthropic), an AI system, in a research workflow directed by Eric Merle, who is responsible for the content. Adversarial reviews in separate AI sessions were made before release, until two consecutive reviews found no error (only, in the first of them, a minor remark on notation). No human expert has reviewed the note.
Author: Merle, Eric
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225581596