Editing Openai/68ec50da-cf00-8005-b5f6-b683506e5853 (section)

==== - The step that would complete the conjecture is to show that if two distinct 252525-classes each have any fixed positive density, then Proposition 5.1 (with the best choice of PPP) forces a contradiction. As you can see from the explicit bound, if αr\alpha_rαr is very small (but fixed), the guaranteed exponent K≥⌊αr/SP⌋K\ge\lfloor \alpha_r/S_P\rfloorK≥⌊αr/SP⌋ can be 111, and then SPKS_P^{K}SPK is not tiny. Getting past this barrier requires new input: some way to assert that within ArA_rAr one can extract a set of size K→∞K\to\inftyK→∞ (e.g. K≫log⁡NK\gg \log NK≫logN) having the “pairwise distinct residues” property simultaneously for many primes ppp. This is a genuine distribution issue about ArA_rAr modulo many p2p^2p2, and current methods (as far as I know) do not establish it in this adversarial setting. ====
* Likewise, the diagonal argument (Lemma 5.2) is necessary but not sufficient for a full answer; it gives the 0.10515…0.10515\ldots0.10515… bound but does not “feel” the pairwise nature of (★)(★)(★).