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Source. Fornal–Sun, Remark 1, equations (32)–(35), p. 12 of arXiv v1.

Source sketch; full quantitative proof not reconstructed. A materially different proposed partition groups integers $n\le d$ by the numbers

$$\omega_i(n)=|\{p\text{ prime}:v_p(n)\ge i\}|.$$

Taking $1\le i\le\lfloor\log d/\log2\rfloor$ includes every possible nonzero exponent. Each integer has one such vector, so its nonempty profile classes indeed form a partition. The source writes $L=\lfloor\log d\rfloor$; for natural logarithms that length can omit the highest exponent of a power of 2. The base-2 length above is the appropriate complete-profile convention.

For these classes, the source asserts

$$T_i\ll\exp\!\left((2+o(1))\sqrt{\log d\log\log d}\right),$$

where $T_i$ is the normalized gcd row sum defined in Proposition 2.1. It points to the Hardy–Ramanujan partition estimate, a divisor-function mean estimate, and an elementary enumeration, without writing that argument. This page records the proposed method and the asserted bound; it does not certify the omitted enumeration or its leading constant.

The logarithms are multiplied under the square root in this remark. That is a weaker scale than the quotient in Proposition 2.1 and Theorem 1.1. The remark is not used anywhere in the compiled main proof chain. No claim about the current status of a separately proposed improvement follows from it.

Dependencies and limits. The partition itself is elementary. The quantitative estimate remains a source sketch, with the exact partition/divisor estimates and their application not supplied here.

Bears on. Problem 202, as a possible alternative route to related structural bounds.