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

==== Fix a≡7 (mod 25)a\equiv 7\ (\mathrm{mod}\ 25)a≡7 (mod 25). Modulo 999 the congruence 9∣ab+19\mid ab+19∣ab+1 has exactly one solution class b≡−a−1 (mod 9)b\equiv -a^{-1}\ (\mathrm{mod}\ 9)b≡−a−1 (mod 9). Consequently: ====

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This can be fed into the main bound by simply removing p=3p=3p=3 from both SPS_PSP and TPT_PTP (for those bbb’s); formally, when you build B~(in),B~(out)\widetilde B^{(\text{in})},\widetilde B^{(\text{out})}B(in),B(out) you replace the universal set of allowable primes by the subset that is arithmetically compatible with the residue restrictions you know AsA_sAs satisfies. The same observation holds (with obvious changes) for any other fixed prime qqq: if AsA_sAs misses the unique residue class (mod qqq) that can yield q2∣ab+1q^2\mid ab+1q2∣ab+1, then qqq can be removed from the allowable list. In practice this strengthens the inequality (Main) once you can certify such absences (e.g. if one 333-class is missing from AsA_sAs). This is exactly the “stability bonus” you pointed out.