Fraction Collisions, Divisibility, and Prime-Power Formulas for Integer Triple Counts

For positive integers k and 1 ≤ m ≤ k, consider the fractions fₖ(m)=(2k m)/m and the number $D(k)$ of distinct rational values occurring at least twice. We establish a divisor-pair characterization of their coincidences and classify the range D(k)≤ 5, obtaining uniform descriptions of five related OEIS entries A387426, A389863, A390100, A389221, and A390319. In particular, the level $D(k)=5$ consists of exactly eleven integers. We also examine the more general level sets Sᵣ=≥ 1:D(k)=r\ and derive a classification of their finiteness: S₀=\1\, |S₅|=11, and every other positive level is attained infinitely often. The latter statement uses a prime-residue stabilization principle, independently switchable fractional coincidences, and an explicit finite interval certificate. An exact criterion for a prescribed collision at a multiple of three relates OEIS A387319 to the existence sequence A343311. Finally, a Chinese-remainder-theorem factorization of the congruence uv(u+v)≡ 0 n, followed by an S₃ orbit count, yields prime-power formulas for the triple-partition counts A343270 and A343309. The arguments distinguish universal identities from finite certificates, with exact-arithmetic checking programs and data provided separately.