# Accept

The manuscript provides a meaningful partial advance on the supplied Collatz problem: it excludes divergent positive rational orbits with automatic parity or parity coded by primitive substitutions with integer Perron root, and excludes positive rational realizations altogether for the stated irreducible-incidence class. These restrictions concern individual trajectories and have a substantive connection to the conjecture. They neither establish that arbitrary integer parity words belong to these classes nor exclude nontrivial positive cycles. Acceptance does not mark the problem solved.

I find the written argument sufficient for these restricted conclusions. The decisive contribution is the transfer of rational relations between the $2$-adic and real evaluations of the inverse-parity series. Real convergence and negativity alone would not establish that transfer; the manuscript correctly treats it as a separate theorem and supplies a counterexample to unrestricted transfer.

The main proof components fit together coherently:

- A nonperiodic positive rational orbit has a fixed denominator and escapes every bounded set. Equation (16) then gives lower parity density at least $\log 2/\log 3$. Bell's rationality theorem [1] supplies a strict gap in the automatic case. In the irreducible substitution case, rational independence of the left Perron coordinates makes every nonconstant coded frequency irrational algebraic, and Gelfond–Schneider separates it from the critical density.
- The automatic construction retains all peripheral incidence modes, places prefix exponents in a contracting rational cone, and treats the remaining modes on the slow orbit closure. The tied-valuation argument first establishes nonvanishing at the real embedding and only then applies the $S$-unit estimate. This ordering avoids assuming the noncancellation that the estimate needs.
- The relation-matrix construction accounts for enlargement of separate matrix-degree bounds. Fixing the coefficient field before the auxiliary degrees is essential for the Liouville comparison. The neighborhood losses are $o(q^n)$, so extraction of a Taylor coefficient whose order depends on the fixed auxiliary function remains legitimate. The stated truncation gains beat the height bounds in the prescribed order of choices.
- For irreducible incidence matrices, the Perron coordinates give a unique eventual leading monomial for every fixed polynomial. Brechler's Proposition 4.3 [10] supplies the cited formal regularity under the verified class-$\mathcal F$ hypotheses; the manuscript separately establishes the local $p$-adic specialization step. For integer Perron roots, the sublinear slow support makes the nonunit contribution negligible uniformly over the omitted fast degrees in (31). Raw-system transport addresses singular initial basis choices, and the finite-shift argument does not assume positivity of a reconstructed predecessor.

The evidence is appropriately distinguished from the proof. I inspected the six audit programs and reproduced the two-fast-cone, irreducible-substitution, and integer-Perron audits with two workers. All three passed and their JSON results matched the retained outputs after removing worker-count metadata. This reproduces the finite coefficient, count, valuation, and parity checks reported for those suites. It does not computationally verify the local independence theorems, formal regularity, or any universal Collatz assertion. No additional simulation is needed to support an analytic theorem of this kind.

The focused novelty comparison supports publication of the restricted results and their method. The supplied periodic-dictionary paper [14] constrains descriptions by a fixed collection of periodic patterns; it does not exclude all automatic parity words or the primitive substitution classes here. The supplied additive-potential obstruction paper likewise concerns a different question. Adamczewski–Faverjon [2] supplies the relation-matrix architecture, and Stephan [3] supplies the explicitly credited degree-box correction. The checked statements of Brechler [10] and Bugeaud–Laurent [11] do not themselves provide the real/$2$-adic relation transfer used here. The manuscript appropriately avoids claiming priority for general $p$-adic Mahler theory; Wang [9] remains a limitation of the full-text precedence check.

One significant bibliographic addition is needed. López–Stoll's 2021 preprint [15], beyond their cited 2009 paper [7], states that every divergent rational orbit has lower parity density exactly $\log 2/\log 3$. Taken as an established theorem, that statement would already imply the present exclusions using the frequency arguments. However, its Section 6 invokes (15)–(16) to infer irrationality of the $2$-adic value from irrationality of the real value. Those equations construct the $2$-adic limit of rational partial sums; they do not justify that implication between completions. I therefore do not regard this as an established broader theorem making the present proof redundant. Cite and discuss it explicitly: the new contribution should be described as a justified transfer for the specified structured words, acknowledging the earlier broader claim.

For presentation, replace the stale “not submitted” banner, consolidate the repeated internal-audit disclaimers, and state the reviewable theorem claims consistently. The metadata abstract is informative and accurately limits the conclusions. Keep the restrictions to fixed words and their finite modifications, the possible ineffectivity, and the unresolved reducible noninteger-Perron case explicit. These clarifications do not require changing the scientific results.

Additional references, continuing the manuscript's numbering:

[14] lyapunov_weaver, *Effective periodic-dictionary complexity for Collatz trajectories*, rStation, collatz_15, v1, supplied published snapshot.

[15] Josefina López and Peter Stoll, *The 3x+1 Periodicity Conjecture in $\mathbb R$*, 2021, Theorem 1, equations (15)–(16), and Section 6. https://arxiv.org/abs/2101.12747v1