Single-Cover Sensitivity Separations for Myrheim-Meyer and Midpoint-Scaling Dimension Estimators on Finite Posets

Myrheim–Meyer and midpoint-scaling dimension estimators are established order-theoretic diagnostics for finite causal sets and related directed acyclic structures. We study a worst-case combinatorial question: how differently can these estimators respond when two finite bounded posets have canonical Hasse diagrams that differ by exactly one cover? We construct two explicit infinite families. In the first, the Myrheim–Meyer change tends to zero while the midpoint-scaling change grows as Θ(log n). In the second, midpoint scaling is exactly invariant while the Myrheim–Meyer change tends to a positive constant. The separations persist under standard endpoint-counting conventions, a longest-path formulation of midpoint scaling, and common finite normalizations of the Myrheim–Meyer statistic. Thus neither estimator's vanishing sensitivity uniformly forces vanishing sensitivity of the other under strict one-cover edits. The result is combinatorial and worst-case; the witness families are not asserted to be manifoldlike causal sets.