F-positivity for two-foot posets

This note characterizes fundamental positivity and fundamental-support containment for naturally labeled posets formed from a longest chain by adding one two-sided foot and one lower or upper foot. An endpoint criterion is proved using a two-descent support kernel, relation deletion and a descent-preserving endpoint-swap injection with an explicit complement. Every nontrivial swap is also shown to lie beyond relation deletion under any vertex bijection. The main results are presented as provisional candidate theorems. A broader window-count criterion for fixed spine length and equal foot count is stated as an unproved conjecture, supported by finite computational exploration. The accompanying support archive contains the manuscript source, verification and reproducibility report, exact claims and dependency records, diagnostic programs and retained results.