Two errata for the Collatz extension modulo 2ᵖ+2^q

Abstract

Bouhamidi has proposed an extension of the Collatz conjecture in which the classical map is replaced by the map of the triplet (2ᵖ+2^q, 2ᵖ+2^(q+1), 2ᵖ)₊, and has conjectured that every such triplet is strongly admissible, of order one outside an exceptional set E of eight pairs and of order two (three at (5,2)) on it. We correct two printed claims of that paper. First, its Table 1 of maximum total stopping times M_∞(p)=max{σ_∞(n): 1 ≤ n ≤ 10⁷, 0 ≤ q ≤ p} is refuted at seven of its twenty-six entries: a recomputation over exactly that range gives 429,324,433,459,465 at p=0,1,2,3,4 against the printed 246,213,268,374,349, while M_∞(6) ≥ 930>737 and M_∞(19) ≥ 524288>131091. The last of these needs no search: we prove that at q=0 the orbit of 2ᵖ⁺¹ first meets 2ᵖ after exactly 2ᵖ steps, whence M_∞(p) ≥ 2ᵖ for every p ≤ 22 with (p,0) ∉ E. Second, the paper's worked example for the triplet (12,14,10)₊ prints the cycle Ω(4) as 4 → 8 → 16 → 22 → 4 "of length 6"; the stated length is correct and the printed chain is short by two, the cycle being 4 → 8 → 16 → 22 → 34 → 48 → 4. We also give an unconditional proof, with the minimality of the return time that the original argument omits, of the trivial-cycle statement inside the conjecture — a statement the source proves in a wider setting — and record what an independent recomputation confirms about the exceptional set. All statements are machine-checked in Lean 4.

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