Triprime Contributions and Signed Remainders in Square-Root Intervals

Let N = 2n, x = N², J_n = [x - N/2, x + N/2] ∩ Z and u = N / (2 log N). We study the two- and three-prime components arising from a decomposition at the lower roughness cutoff N^{1/4}. An exact largest-cofactor orientation, with its repeated-prime corrections, reduces their centered sum to a specific truncated-Möbius continuation and a prime-cofactor core. We prove that the positive rough triprime core is absorbed, up to o(u), by negative continuation on the same integers. We also construct a bounded lower weight for a coupled rough degree pool, normalize its four-factor detector expansion, and prove a uniform O(N² / (PR log⁴ N)) variance bound on its first-pair dyadic blocks. The corresponding diagonal is negligible. A function-space example shows why an average over all centers does not by itself give the required saving on the sparse prime-pair sample. The resulting sharp one-sided comparison is stated explicitly and remains open. This work complements the semiprime-tail bound of the preceding paper. It proves neither its rough-number hypothesis nor a positive prime lower bound in J_n.