Height and chain signs from P-partition support

For finite posets with arbitrary bijective labelings, we give a proof that containment of the fundamental supports of their P-partition enumerators forces reverse monotonicity of height. This answers a question of Lesnevich and McNamara and implies that equality of the enumerators preserves height, as asked by McNamara and Ward. The proof orders antichain levels against the successive label comparisons of a longer chain. We also characterize the binary sign subsequences occurring in every linear extension: they are exactly those occurring along saturated chains. The constructive converse produces an avoiding extension whenever no chain contains the prescribed subsequence. The resulting support invariant recovers height and both strict- and weak-edge maxima, and is strictly finer than these three numbers. The results are presented as provisional candidate results with complete combinatorial arguments.