A Carleman window class for the height floor in Hua–Yang, arXiv:2608.16034: explicit cutoffs, the Fourier–Laplace estimate, and the modified Proposition 9.2

Technical note (14 pages) accompanying Hua-Yang, arXiv:2608.16034. In referee correspondence on that paper we proposed replacing its Gevrey window class by the Carleman class with weight sequence M_m = m!(log(m+2))^(2m), lowering the height floor of its main theorem to T >= (log log q)(log log log q)^(2+eta)/log q with all constants unchanged, including C_MT. At the authors' request, this note supplies the complete details: the explicit box-convolution cutoffs with computed constants (c_0 = 0.5296605), the Fourier-Laplace estimate with decay exp[-cD/log^2(2+D)], the modified proof of their Proposition 9.2 with the interlock closing uniformly at H >= (log Q)(log log Q)^(2+eta), the two changed rows of their Section 12 error table, and a status section recording what is proved, what is machine-verified in Lean 4, and where we judge the argument least robust. The note has been through three adversarial checking passes on our side but is not refereed. Provenance: the improvement originated in our referee reading of Hua-Yang v2 (24 August 2026) and is deposited as a separate citable companion result by agreement with the authors, who are checking it independently. Related work: our polylogarithmic-height family theorem, doi:10.5281/zenodo.21980224.