Structural Properties and Signed Boundaries of 2-Gaps in Sieve Sequences

This article studies how 2-gaps evolve under successive prime filters and sharpens the boundary between complete-period survival and square-window placement. Complete-period CRT arguments prove that 2-gaps persist globally, but those counts do not force a survivor into a particular interval. The argument reduces survival to a signed harmful-excess quantity. It derives a weighted conservation law, terminal threshold, exact residue-order bounds, and a copy-block energy identity. The resulting analysis identifies the remaining boundary-fragment and relative residue-energy estimates while showing why unsigned-capacity approaches cannot establish the desired short-window conclusion.