Mathematics Without ℝ The Computable Foundations of a Discrete Universe

We argue that the only mathematics capable of describing physical reality is the mathematics of computable objects — objects for which a finite algorithm exists that generates their complete specification. The continuous real line ℝ, as classically constructed, contains overwhelmingly more non-computable than computable numbers; none of the non-computable ones has ever been measured, observed, or required by any successful physical theory. We diagnose the standard foundations of mathematics — Platonism, ZFC set theory with completed infinities, and the uncritical use of ℝ as a physical substrate — as a 2500-year-old case of Framework Lock: a formalism that has been confused with an ontology. We formalize the Computational Razor: no mathematical argument about physical reality may depend on objects for which no finite algorithm specifies them. We reinterpret Gödel's incompleteness not as evidence that mathematics exceeds computation, but as a diagnosis that purely linguistic axiomatic systems (L2) cannot close upon themselves without recourse to observation (L1). We register a falsifiable prediction: none of the active Millennium Problems requires non-computable objects in its solution. The present paper establishes the framework; six companion papers (P vs NP, Yang–Mills, Navier–Stokes, Riemann, Hodge, Birch–Swinnerton-Dyer) will apply it, and an appendix will test it against Perelman's proof of the Poincaré conjecture.