Hercher 2023 no mcycles 91

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theorem_23: Hercher's 2023 main theorem that the shortcut Collatz map has no nontrivial cycle with at most 91 local minima, extending the Simons-de Weger exclusion from 75 by continued-fraction bounds and the verification bound 704 times 2^60; the cycle-exclusion frontier recorded on Problem 1135.

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Christian Hercher, There are no Collatz m-cycles with m <= 91, J. Integer Seq. 26 (2023), Article 23.3.5 (the journal reference the arXiv record carries; the journal's articles carry no DOI and no Crossref record was found). The retained folder-name PDF is arXiv:2201.00406v3 [math.NT] (4 April 2023), 22 pages with a complete text layer; the journal text is not held and was not compared; the locators below are the preprint's pages. The statement pages were read on the rendered page images of pp. 1--3 and 15--16. Source: PDF. The arXiv record (https://arxiv.org/abs/2201.00406, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Read status: claims checked for the abstract, Definition 1 (the map), Conjecture 2, Remark 3, Definition 4 (the verification bound $X_0$ used), Theorem 23 with the first lines of its proof, Corollary 24 with Table 1 and Remark 25, read clause by clause on the page images of pp. 1--3 and 15--16; the proof (Sections 2--3, Lemmas 5--22) was read for structure only and not checked.

Works with the shortcut map T -- the map convention is pinned from this primary text: Definition 1 defines the Collatz operator $C(n)=n/2$ for even $n$ and $(3n+1)/2$ for odd $n$ (the problem page's $f$). Extends the Simons-de Weger ladder from m <= 75 to m <= 91: sharpened bounds on the ratio (K+L)/K of a cycle's member counts, turned into lower bounds on K by continued fractions and the verification bound X_0, exclude every m-cycle with at most 91 local minima; the computations took a few minutes in a Sagemath worksheet (pp. 3--4). Relevance: Excludes Collatz m-cycles for all m <= 91, the current cycle-exclusion frontier for problem 1135.

Contents

• Abstract and Section 1 (pp. 1--3): the conjecture asserts that every starting value reaches the trivial cycle $1,2,1,2,\ldots$ (Conjecture 2, p. 2); an $m$-cycle is a nontrivial cycle with exactly $m$ local minima (Definition 5, p. 2); Simons and de Weger proved $m\ge76$, newer verification bounds give $m\ge83$, "In this paper, we prove $m\ge92$." (p. 1). Remark 3: the two ways the conjecture could fail (an unbounded trajectory; a nontrivial cycle), with Terras's and Tao's almost-all results quoted; Definition 4: $X_0$, the largest number up to which convergence is known, taken from Barina's project as $X_0=704\cdot2^{60}$ (p. 2; $\approx8.1\cdot10^{20}$, p. 15). The last part of the paper shows that raising the odd-member bound to $K\ge1.375\cdot10^{11}$ would follow from verification up to $1536\cdot2^{60}=3\cdot2^{69}$.
• Sections 2--3 (pp. 3--16): sums $T(n_i)$ of reciprocals of the run of odd members starting with each local minimum $n_i$, the bounds on $(K+L)/K$ they give (Theorem 16, Corollary 17, Theorem 21), continued fractions (Lemma 22, p. 15), the iterative improvement of the lower bound on the number $K$ of odd members against the Simons--de Weger upper bound $K<1.4784\,m\delta^m<2.2\cdot10^{20}$; Theorem 23 (Main Theorem) (p. 15): "There is no $m$-cycle with $m\le91$." Corollary 24 with Table 1 (p. 16): lower bounds on $K$ for $m$-cycles with $m\ge92$ (for instance $K>7.76\cdot10^{19}$ if $m\le98$; $K>7.20\cdot10^{10}$ for all $m$).
• Section 4 (pp. 16--22): cycles without knowing $m$; what must be proved for the next bound on $K$.

Compiled scope

The statements were read; Theorem 23 is compiled as a statement with the paper's proof pointer. No step was checked, the computation was not rerun, and nothing here is independently reviewed.

Bears on. #1135: the cycle-exclusion frontier for the page's map $f$: no nontrivial cycle with at most $91$ local minima exists; this rules out one of the two ways the conjecture could fail only for cycles of that shape and says nothing about divergent trajectories, as the paper's Remark 3 states.

Results.

• Theorem 23 (Main Theorem) (p. 15): there is no $m$-cycle with $m\le91$.
• Corollary 24 (p. 16): an $m$-cycle whose $m$ is at most a value in Table 1 has at least the $K$ paired with that value as its count of odd members (statement read, not compiled as a page).