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

==== Let A⊆{1,…,N}A\subseteq\{1,\dots,N\}A⊆{1,…,N} satisfy (★)(★)(★) and write A=⨆u (mod 25)AuA=\bigsqcup_{u\ (\mathrm{mod}\ 25)}A_uA=⨆u (mod 25)Au with densities αu:=∣Au∣/∣Su∣\alpha_u:=|A_u|/|S_u|αu:=∣Au∣/∣Su∣. ====
# (Diagonal bound) ∣A∣≤(0.10515+o(1))N|A|\le (0.10515+o(1))N∣A∣≤(0.10515+o(1))N with the constant written as an Euler product, and the proof detailed in §5.3.
# (Cross–class covering, Proposition 5.1) For any two classes r≠sr\ne sr=s with s≢−r−1 (mod 25)s\not\equiv -r^{-1}\ (\mathrm{mod}\ 25)s≡−r−1 (mod 25), αs ≤ inf⁡P[ SP ⌊αr/SP⌋+TP ]+o(1).\alpha_s\ \le\ \inf_{P}\Bigl[\, S_P^{\,\lfloor \alpha_r/S_P\rfloor}+T_P\,\Bigr]+o(1).αs ≤ Pinf[SP⌊αr/SP⌋+TP]+o(1). This is fully proved via Lemmas 3.1–3.2 and 4.1.
# (Stability bonus from missing moduli, §5.5) If, for some prime qqq, the set AsA_sAs misses the unique residue class modulo qqq (equivalently modulo q2q^2q2) that can yield q2∣ab+1q^2\mid ab+1q2∣ab+1 for a fixed aaa, then qqq can be deleted from the sums SP,TPS_P,T_PSP,TP in (Main) for those cross constraints, sharpening the bound.

These statements are mathematically correct and can be used as a framework for any refined optimization you would like to try (e.g. selecting PPP from primes ≡1 (mod 4)\equiv1\ (\mathrm{mod}\ 4)≡1 (mod 4) only; mixing in the “mod 333” restriction when available; or allowing PPP to depend on the pair (r,s)(r,s)(r,s)).

===== The hard barrier, now clearly isolated, is to manufacture a long list of elements of ArA_rAr with prescribed distinct residues modulo many squares p2p^2p2—long enough that the factor SPKS_P^{K}SPK becomes negligible even when SPS_PSP is close to its maximal value ∑p≠51/p2≈0.412 \sum_{p\ne5}1/p^2\approx 0.412∑p=51/p2≈0.412. Your “missing a residue class modulo 333” observation is exactly the sort of extra rigidity that reduces SPS_PSP and can move the needle. If you can systematically harvest such missing classes for several small primes (e.g. for 3,13,17,29,⋯≡1 (mod 4)3,13,17,29,\dots\equiv1\ (\mathrm{mod}\ 4)3,13,17,29,⋯≡1 (mod 4) in the cross terms), the inequality (Main) becomes strong enough that the conjectural N/25N/25N/25 bound comes within reach. =====

If you’d like, I can plug in explicit numerical choices of PPP (for instance P={13,17,29,37,41,… }P=\{13,17,29,37,41,\dots\}P={13,17,29,37,41,…}) together with a concrete hypothesis like “AsA_sAs misses one class modulo 333” and show exactly what quantitative αr↦αs\alpha_r\mapsto\alpha_sαr↦αs bound you get.