Valuation Without ℝ: A Category-Theoretic Foundation for Finite Measurement

Measurement in physics is always finite and approximate — yet its mathematical foundations are embedded in the Archimedean continuum of real numbers and the cumulative hierarchy of set theory. We invert this dependency: the act of measurement is axiomatized as a graded distinguishability map v: S^2 → N ∪ {∞} over a poset of states satisfying the ultrametric inequality, with no dependence on ℝ or set-theoretic foundations. The category Val of valuation spaces with non-expansive maps is self-contained; ℝ appears only as a limit idealization. The effective spatial dimension d emerges as the exponent in the asymptotic growth N(r) ∼ qd·r of the distinguishability graph at resolution r, constrained by the cohomological consistency of the refinement sheaf across resolutions. Falsifiable predictions are pre-registered at Planck-scale resolution.