Survival Frontiers in Balanced 2-Gap Companion Processes

This article examines companion processes that reproduce the exact global growth of 2-gaps but change where each filter removes them. Every parent produces r copies and exactly two are removed, as in the sieve sequence. The companion may place those two removals randomly, protect a chosen target, or direct them toward it. This construction separates the number of surviving 2-gaps from their location near the head. A filter's local destruction fraction is compared with the random rate through a normalized hazard. Under the stated spatial, availability, and mixing premises, the article derives square-window and head-recurrence survival frontiers for balanced and exact-quota companion processes. None of those premises is claimed for the real sieve; the resulting phase diagrams identify the additional arithmetic information a transfer theorem would require.