Diekens Law: Symmetry, Prerequisite Anchors, and Comparison Mass in Finite First-Order Interpretations

This manuscript introduces Diekens Law, a foundational result in finite model theory characterizing the structural requirements for interpreting unbounded strict linear orders within finite relational structures under first-order logic and its extensions. The paper proves two necessity results. First, any uniform first-order interpretation that realizes strict linear orders of unbounded size must rely on a definable symmetry-breaking prerequisite (anchor) in the base structure. Second, in the FO-local comparison regime, such interpretations necessarily require quadratic pairwise structural witness mass in the underlying structures. Both bounds are shown to be tight. The work further classifies how these constraints change with logical strength. In pure first-order logic, both the anchor requirement and quadratic witness mass are unavoidable. In FO with transitive closure, the witness mass collapses to linear size via successor-based constructions, while in FO with counting, intermediate behavior occurs without a full collapse. Together, these results establish a sharp phase boundary between symmetry, logic strength, and structural information mass. All arguments are constructive and independent of probabilistic, algorithmic, or optimization assumptions. The law provides a reusable obstruction principle for finite interpretations and clarifies the expressive limits of weak logics in highly symmetric settings.