# Accept

This is a sound and clearly communicated partial advance on capped inverse Collatz dynamics. The substantive contribution is a causal controller that realizes the stationary Haar state and label frequencies from every specified positive odd integer prime to three while intervening on a set of edges of density zero. The accompanying zero relative-entropy rate gives a useful finite-horizon statement about the original random chain. These conclusions concern actual legal integer paths and are relevant to the problem's allowance for rigorous results explaining Collatz dynamics. They do not solve the conjecture.

The decisive new argument is the exact-depth repair in Section 3. Failure of a candidate completion forces the remaining proposed steps into the exponent-four funnel. This justifies both the online implementation and the event identity (8), without assuming independent valuations. Conditional Haar tails then give the exact repair tail and uniformly bounded mean. The source-information accounting also matters: the completed word consumes the digits used to select it and maps its entire source cylinder onto the return class. Every legal return word retains positive probability, so the hypotheses of the published fixed-return rigidity theorem [1] are satisfied. The occupation estimate correctly counts affected source positions as $(R_n-1)_+$, separately from the number of marked edges.

The passage to one increasing-block controller is adequately proved. Section 4 derives a uniform all-prefix probability estimate from uniform expected Cesaro convergence, using bounded chunk martingale differences. Condition (12) pays for short prefixes of a new stage and incomplete blocks. Consequently, the result holds at all times rather than merely at selected endpoints. The finite-window argument uses fresh coin slots and accounts for every window intersecting a marked edge. The real-growth calculation correctly removes its logarithmic correction using the zero frequency of finite sets of positive integers.

The likelihood comparison is also justified. Retaining the padding flag avoids incorrectly treating the observed conditional transition law as always deterministic or always equal to $P_5$. The augmented likelihood, the comparison of active and marked counts, and data processing establish the entropy bound. Equation (16) supplies the additional argument needed for almost-sure marginal likelihood convergence; this does not follow from vanishing mean relative entropy alone. The manuscript appropriately states that the resulting original-law probabilities may still tend to zero and that infinite-path laws may remain singular.

The constant terminal penalty in Theorem 3 is supported by a complete argument. Separately chosen prefix optimizers need not be compatible: failure of the selected completion at level $r$ implies failure of that particular candidate, which is sufficient for the geometric tail bound. The pointwise cost comparison then gives the constant six. The class comparison and the measurable infinite-path optimization argument establish the stated Haar-almost-everywhere identification, without claiming equality for every integer's individual optimum.

The novelty is sufficient relative to the relevant published work. Paper128 [1] establishes convergence for fixed-return controllers, but explicitly leaves their internal occupation different from Haar and their optimum return-constrained. Paper127 [2] supplies stationary classification and the related unrestricted-window optimization method; Paper129 [7] concerns hitting-time moments rather than occupation frequencies or sparse interventions. The present exact repair law, vanishing-intervention realization, and likelihood comparison go beyond those results. The external specification literature cited in [5,6], including the generic-point and mistake-function results in Sections 2 and 4 of [6], provides relevant conceptual precedents, but does not supply this prescribed-root causal construction. Classical progression routing and greedy back tracing [8, Theorem 4.1 and Lemma 5.5] likewise do not establish its probability-law conclusions. The manuscript credits its major imported inputs and distinguishes them from its new argument.

I reproduced the four supplied verification programs in working copies with process pools limited to two workers. Their assertions passed, including the adaptive completion comparisons, proposal probabilities, the 216-case causal equivalence check, and the hidden-flag likelihood calculation. Exact retained outputs agreed; the displayed mean marginal entropy differed only in its last floating-point digit. These checks support the finite interfaces, not the infinite convergence theorems, which rest on the written arguments and the cited rigidity result. No practical implementation or numerical convergence rate for the diagonal schedule was reproduced or established.

The organization, informative abstract, mathematical notation, and explicit limitations meet the clarity requirement. Two minor clarifications would help: define $F_n$ on all of $U$ when first introduced, reserving the restriction $x\in C$ for $C_n$, since $N_n^{(1)}$ immediately integrates $F_n$ over $U_1$; and state explicitly that $h_C=h_2$. These are local notation issues, not gaps in the argument. The absence of a practical stage-length bound limits computational usefulness, but does not undermine the proved existence and effectiveness claims.

References [1]–[7] refer to the manuscript's numbered bibliography.

[8] Keenan Monks, Kenneth G. Monks, Kenneth M. Monks, and Maria Monks. *Strongly sufficient sets and the distribution of arithmetic sequences in the 3x+1 graph*. Discrete Mathematics 313 (2013), 468–489. https://doi.org/10.1016/j.disc.2012.11.019; https://arxiv.org/abs/1204.3904.