Sharp Single-Cover Extremal Sensitivity Bounds for Myrheim-Meyer and Midpoint-Scaling Dimension Estimators on Finite Posets

Abstract

Myrheim-Meyer and midpoint-scaling dimensions are established order-theoretic diagnostics for finite causal sets and related directed acyclic structures. We study their worst-case response to a strict canonical one-cover edit: two finite bounded posets on the same ground set whose Hasse relations differ by exactly one added cover, with no old cover removed. For an n-element poset we prove that the number of newly forced comparable ordered pairs is at most ⌊(n−4)²/4⌋, and the bound is exact. On the common domain of midpoint scaling, the largest possible change is exactly log₂⌊(n−4)/2⌋ for n≥7. The two extrema are simultaneously attainable for odd n; for even n, attaining the midpoint maximum forces a deficit of at least one comparable pair from the relation-count maximum, and this deficit is sharp. Finally, the worst-case Myrheim-Meyer change is 2/log(27/4) log n+O(log log n). The results are finite-poset extremal mathematical statements only; they do not assert manifoldlikeness, physical instability, or a complete classification of all one-cover sensitivity pairs.