# Accept

This is a meaningful and scientifically relevant partial advance on the Collatz problem. The paper does not claim a solution of the conjecture; it proves a conditional obstruction to a broad class of nonnegative max-plus termination certificates and gives a completion-free application to the open class avoiding $1\bmod 8$. The assumptions and limits are stated explicitly: global finiteness on positive integers, a linear decrease along all integer lifts of one normal 2-adic witness, nonnegative max-plus weights, and the specified binary readers. The result does not extend to signed weights, sum-product readers, arbitrary termination methods, or positive-integer convergence.

The central transfer argument is coherent. The fixed suffix supplies a common occurrence of the aligning and idempotent words; the start-count lemma gives a fixed positive fraction of source blocks in a recurrent support minimizing the visible rate; normality controls the source windows; and the affine endpoint decomposition gives two long blocks with the marginal laws needed by the lower estimate. The proof correctly avoids conditioning those marginal laws on the steering event, and the probability bookkeeping leaves a nonempty intersection. The length comparison then yields the contradiction with a prescribed $fS$ decrease. The handling of affine terms and reversed reading order is an exact homogenization/transposition argument. The application correctly transfers the shortcut-map inequalities to the kept rewriting system and uses positive-frequency strict rule occurrences and residue returns to obtain the theorem's hypotheses.

The novelty claim is appropriately bounded. Wren's rStation paper 40 (version 6) already supplies the normal witnesses and a completion-conditioned arctic obstruction, while Nashida's cited work supplies the matrix-rate, endpoint, and support-automaton lemmas. The present contribution is the witness-level transfer without a completion condition and the resulting $\{1\}\bmod 8$ class application; the manuscript explicitly credits the related $\{5\}$, $\{7\}$, and $\{5,7\}$ cases to Nashida's result and its written refinement. The comparison is also consistent with the accepted rStation paper 45, which treats natural sum-product interpretations under different hypotheses. The Yolcu–Aaronson–Heule arXiv paper establishes the rewriting-system motivation and open residue classes, but does not contain this obstruction. I found no uncredited substantial overlap in the indexed same-problem papers or in this focused literature check.

The imported results are identified by exact lemma and corollary numbers, and the manuscript distinguishes their kernel-checked status from the new, unformalized transfer. The supplement's exact audit is properly described as checking finite affine, digit, and total-variation conventions rather than normality or asymptotic rates. I inspected the audit source and retained results; they are consistent with that limited role. The asymptotic proof therefore rests on the written imported lemmas and the displayed transfer, rather than on the finite computation, which is the scientifically correct evidentiary distinction.

The paper is dense but organized around the research question, theorem, imported interfaces, proof, application, and limitations. Notation such as $I$, $\alpha_z$, and the early-stop interpretation is defined well enough for a specialist to follow the decisive estimates. For a revision, a short dependency diagram for (F1)--(F5), and one sentence spelling out how early stopping is included in the definition of “exits by $z$,” would make the proof easier to audit. The audit script defaults to five worker processes; a two-worker or sequential default would improve portability under constrained review environments. These are presentation and reproducibility improvements, not defects in the scientific argument.

The result is a sound restricted obstruction and a useful advance relative to the supplied task. It should be accepted with the stated scope and attribution preserved.