One axiom. One operation. Zero free parameters. Three Clay Millennium problems — the Riemann hypothesis, the Yang–Mills mass gap, and Navier–Stokes regularity — share a hidden assumption: the continuum. This paper shows the assumption is the obstruction. With no continuum (the line replaced by the fold lattice, which has a smallest scale and no zero), each has a direct structural answer. RIEMANN: the critical line Re(s)=1/2 is the unique self-antipodal axis of the fold reflection x ↔ 1−x — the involution of the zeta functional equation — so the self-dual zeros are forced onto the half-One. YANG–MILLS: the spectrum is discrete with a positive least element (the floor 1/2^d), so a mass gap exists by construction and equals the floor. NAVIER–STOKES: the energy cascade is repeated halving that reaches the floor in finitely many folds and stops — no sub-floor scale, no blow-up. Each is the fold removing the continuum and reading off what remains (the fold-native structural answer/mechanism, not the continuum-formulation Clay proof). Machine-checked; reproduces from one command. A standalone result within the Smithian Fold Theory of Everything (SFTOE). Full corpus, code, and the run-it-yourself VERIFY.md protocol: https://github.com/MettaMazza/Smithian-Fold-Theory