Monograph synthesis of a 42-paper program in constructive reverse mathematics of mathematical physics. We prove that the logical resources required for every empirical prediction in known physics are exactly BISH + LPO: Bishop's constructive mathematics plus the Limited Principle of Omniscience. Three principal theorems anchor the result:(I) Fekete's subadditive lemma is equivalent to LPO over BISH, establishing that LPO is physically instantiated through phase transitions — not a mathematical convenience.(II) The Fan Theorem is dispensable: every empirical prediction derived using compactness arguments (Heine–Borel, Bolzano–Weierstrass, Arzelà–Ascoli) can be recovered at BISH + LPO.(III) Dependent Choice is dispensable: iterative constructions (mean ergodic theorem, martingale convergence, Picard iteration) are scaffolding over finite-precision predictions. Approximately 60 calibrations across 12 physical domains — statistical mechanics, quantum mechanics, quantum field theory, general relativity, quantum information, thermodynamics, classical mechanics, electrodynamics, particle physics, quantum gravity, AdS/CFT, and vacuum energy — yield zero exceptions. A conservation metatheorem explains the pattern: physical predictions are finite compositions of computable functions at computable inputs (BISH), with LPO entering only through completed limits without computable modulus of convergence. Eleven principal findings include: the BISH/LPO ceiling, Fan Theorem and Dependent Choice dispensability, Fekete universality, dissolution of the cosmological constant discrepancy as a non-prediction, duality as axiom-preserving map, physical undecidability as thermodynamic (spectral gap ≡ Ising), Bell ≡ Kochen–Specker at LLPO, and the structural alignment of the perturbative/non-perturbative boundary with the BISH/LPO boundary. The characterisation is explicitly falsifiable: any empirical phenomenon requiring Σ₂⁰ reasoning refutes the thesis.
Paul Chun-Kit Lee (Mon,) studied this question.