AI referee report

The sharp-length question is internally incompatible with the object it proposes to calibrate. The manuscript explicitly explains that the radius is unweighted, omits retained-mean and atom-mass uncertainty, and has its natural local-regression scale h_n^β at zero threshold; nevertheless the remark requires its length and 2W_{n,B_n} to match r_n for every regime, including δ_n=0 through the q=0 case. At zero and regular/critical thresholds, the missing atom weight generally makes this component radius much larger than the complete clamp frontier, so the displayed matching limits cannot describe the defined interval. The requirement of one constant c across all phase limits is also unsupported and generally implausible when the relative regular and local contributions vary with the phase limit.

The manuscript states that the checks were run at commit 6f11a039165a75921bbb6499f3d9bf40f3f404f8, while the authoritative current verification contract identifies commit 84c32d60cb53b2f1cdef917448a45b96b08296ad. The reproduction and archive narrative therefore does not identify the source state supporting the submitted statements.

The paper proposes a nonstandard split-sample estimator, total-Gram stabilization rule, fallback branch, and honest interval, yet supplies only a one-cell exponent calculation. The manuscript itself identifies fallback frequency, realized coverage, and expected length as potentially sensitive to constants and moderate sample size. Without numerical evidence, readers cannot assess whether the proposed procedure is usable in the regimes for which it is advocated.

The prose repeatedly describes the density restriction as “lower-tail” thinning or as holding “near the lower endpoint,” but the stated model imposes c_-a^κ ≤ π_x(a) ≤ c_+a^κ almost everywhere on the entire interval [0,1]. This understates a material global restriction and can lead readers to believe the frontier has been proved under a merely local boundary condition.

The displayed minimax theorem writes the estimator risk as an expectation E_P even though the estimator is a function of an n-observation sample and the verified criterion uses the product experiment. As written, the measure on that expectation has the wrong sample space.

The presentation claims that each object has one canonical definition but then repeats the estimator and interval as synth_4 and synth_12, references synth_4 before defining it, and says synth_4 is the version used for both risk and length even though synth_4 defines only the estimator. This obscures which displayed procedure each theorem evaluates.

The zero-threshold bullet says stabilized risk and length are “both asymptotic to n^{-1/2},” while the verified conclusion is order comparability (AsympSeq/$\asymp ),$ not ratio convergence with constant one. Ordinary mathematical usage makes “asymptotic to” stronger than the delivered result.

The novelty boundary relative to existing LMTP threshold interventions remains somewhat blurred. The paper acknowledges prior threshold implementations but later describes “the present estimand” as having a different post-policy law without clearly separating the already available exact clamp policy/identification from the new minimax analysis of its moving atom.

The sentences saying split-indexed aliases “add no new content” and that “nothing in [the verification supplement] is needed to read the main results” use contribution-by-negation outside a designated limitations discussion.

The symbol h_n first appears in the frontier display without the required plain-language first-use gloss.