Thresholded Measure: Why Discrete Stability Is Unavoidable

Any nontrivial evaluation of magnitude forces thresholds—and with them, discreteness. This paper derives discrete stability from purely structural constraints, without invoking physics, probability, or observers. Building on prior results that establish irreversibility, ordered asymmetry, and quantitative measure as a forced refinement of ordinal order, the paper introduces a minimal viability evaluation mapping states into finitely many descriptive statuses (e.g., viable vs.\ non-viable). It is shown that any magnitude-consistent, non-constant evaluation induces a threshold partition of the range of the measure, and that thresholding generates discrete stable categories as equivalence classes under evaluation. Discreteness is therefore not assumed as fundamental; it emerges as a quotient structure created by thresholded evaluation of forced measure. The paper closes by identifying the next logical pressure: multiple closure criteria can induce incompatible evaluations, motivating a subsequent derivation of projection structure and duality.