Convergence Criteria for Dynamic Integer Systems

The paper claims to prove Collatz by three “equivalent” reductions culminating in the accelerated map on odd nonmultiples of 3 with a forced parameter choice ν > 1 so that n' = (3n+1)/2^ν < n, thereby invoking the old monotonicity criterion (Criterion 6). This ν > 1 enforcement is obtained by a third reduction that removes entire branches B(n) whenever ν = 1 gives n' > n, citing Block (4) as justification. But Block (4) presupposes that such a branch is a closed, convergent subsystem with root at its delegate n, which is not true for the Collatz graph and is not established anywhere; thus the reduction is not equivalence-preserving for “every n reaches 1” (the paper’s goal) and impermissibly deletes genuine trajectories. Without this deletion, ν = 1 occurs infinitely often (e.g., 11 → 17), so the asserted universal decrease on the reduced domain is false. The paper also asserts convergence from “self-similarity” because each knot has η = ∞ direct predecessors and further claims an isomorphism with an MP system n' = n/p(n); however, equality of infinite in-degree does not imply branch ≅ whole (self-similarity) nor a graph isomorphism, and the MP system is not a unique function unless a deterministic choice rule p(n) is fixed. These steps are therefore logically unsound. The candidate solution correctly identifies these flaws and that the problem remained open as of the paper’s date. Key paper claims and tools: Eq. (2) accelerated map; Block (4) removal of branches; Eq. (7) with ν > 1 “so that always n' < n”; use of Criteria 3/4 from η = ∞; and the purported isomorphism to MP—are all explicitly stated in the paper and are the basis of the errors noted here .