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

==== 1. Single‑prime obstruction sets. Fix a prime ppp and aaa with p∤ap\nmid ap∤a. Then the congruence p2∣ab+1p^2\mid ab+1p2∣ab+1 forces ====

b≡−a−1(modp2).b\equiv -a^{-1}\pmod{p^2}.b≡−a−1(modp2).
Thus, within any fixed 252525-class StS_tSt with gcd⁡(25,p2)=1\gcd(25,p^2)=1gcd(25,p2)=1, the set

Cp(a;t):={b∈St: p2∣ab+1}C_p(a;t):=\{b\in S_t:\ p^2\mid ab+1\}Cp(a;t):={b∈St: p2∣ab+1}
is exactly one residue class modulo p2p^2p2, hence has relative density 1/p21/p^21/p2 inside StS_tSt. (If p∣ap\mid ap∣a then p2∤ab+1p^2\nmid ab+1p2∤ab+1 for all bbb, so Cp(a;t)=∅C_p(a;t)=\varnothingCp(a;t)=∅.)
# Independence across primes (CRT). For any finite set of distinct primes PPP and any choice of residue classes Rp ( mod  p2)R_p\ (\bmod\,p^2)Rp (modp2), the intersection

{b∈St: b≡Rp ( mod  p2) ∀p∈P}\{b\in S_t:\ b\equiv R_p\ (\bmod\,p^2)\ \forall p\in P\}{b∈St: b≡Rp (modp2) ∀p∈P}
has relative density ∏p∈P1p2\prod_{p\in P}\frac{1}{p^2}∏p∈Pp21 inside StS_tSt (up to O(1)O(1)O(1) boundary error). In particular, for a fixed aaa the sets {Cp(a;t)}p∈P\{C_p(a;t)\}_{p\in P}{Cp(a;t)}p∈P lift to pairwise disjoint unions of residue classes modulo ∏p∈Pp2\prod_{p\in P}p^2∏p∈Pp2; hence

∣⋃p∈PCp(a;t)∣∣St∣=∑p∈P1p2(exact for finite P, up to O(1/N)).\frac{\Bigl|\bigcup_{p\in P} C_p(a;t)\Bigr|}{|S_t|}=\sum_{p\in P}\frac{1}{p^2}\qquad\text{(exact for finite \(P\), up to \(O(1/N)\)).}∣St∣⋃p∈PCp(a;t)=p∈P∑p21(exact for finite P, up to O(1/N)).