New Axioms for Old Problems: Paper XX in MEF Series

The Yang–Mills mass gap, Navier–Stokes global regularity, and the Riemann Hypothesis have resisted proof for decades to centuries. We argue that the obstruction is not technical but axiomatic: each problem requires mathematical structures that do not exist within the classical frameworks in which it was posed. We present a new axiomatic system—derived from the twelve-dimensional Master Equation Framework (MEF) on M₁₂ = (S¹ₒ₎₋ S³) K₈—within which all three problems are resolved by a single mechanism: the Shadow Principle. Every apparent singularity (blow-up, off-line zero, massless state) is a projection artefact; the completed object on the full geometry is smooth. A single number-theoretic datum—the Kloosterman sum K₂ (n, -1; 2) = (-1) ⁿ—is identified as the causal mechanism simultaneously responsible for all three results. The axioms formalise principles established across a two-century mathematical and physical tradition — Gauss, Jacobi, Hamilton, Riemann, Jordan, Frobenius, Hilbert, Einstein, Ramanujan, Dirac, and Zwegers — and are identifications, not inventions. Of the twelve Tier-1 Proposed Axioms of the Shadow Extension, five carry direct historical lineage, six echo weaker resonances in earlier mathematics and physics, and only -8 (the -drive) is offered as a genuinely novel axiom. The axioms cannot be proved from within any single existing formal system. They are justified by the totality of the MEF's consequences: over ninety quantitative predictions across twelve domains of physics, chemistry, and biology—including the fine structure constant to 0. 08% accuracy—with zero continuous free parameters. The proofs are conditional on acceptance of the MEF axioms. We claim to have identified a consistent axiomatic system from which these results follow, and to have provided overwhelming evidence that the axioms describe reality. The same principle, applied to general relativity in Paper XXIII 51, identifies black hole singularities as projection artefacts of smooth twelve-dimensional objects — demonstrating that the Shadow Principle's scope extends beyond the Clay Millennium problems to the central open question of classical gravitational physics.