Three Pillars of 3D Incompressible Navier–Stokes Theory: Kato Local Existence, Fujita–Kato Small-Data Globality, and the Beale–Kato–Majda Continuation Criterion (While the Roof Remains Open)

Expository classical compilation presenting, with expanded proof sketches and references to the original works, the three standard pillars of 3D incompressible Navier–Stokes theory on the torus: (i) Kato local existence and lifespan bound for arbitrary data in Hs (s > 5/2), with full Kato–Ponce both terms + absorb, mollified/Galerkin + Aubin–Lions, and L² uniqueness with ∇v; (ii) Fujita–Kato global regularity for small data in the critical space Ḣ1/2 with smallness linear in ν (‖u0‖Ḣ1/2 ≤ ε ν); (iii) the Beale–Kato–Majda continuation criterion in one-directional form (∫ ‖ω‖∞ dt < ∞ ⇒ T* > T), with log bound including +‖ω‖L². Smoothness stated for t > 0. Status matrix: Pillar I Unconditional Local / Pillar II Unconditional Small-Data Global / Pillar III One-directional / Clay Statement (B) Open. No unconditional large-data claim. Supplementary note (Littlewood_Paley_Conditional_and_SmallData.tex/.pdf): the critical-Besov small-data global regularity theorem and Kozono–Taniuchi continuation in matching one-directional / C/ν form. Companion geometric draft: 10.5281/zenodo.24079623. License: Apache-2.0. Creator: Frohman, Benjamin Stanley (ORCID 0009-0006-7068-3718). Keep unpublished until the author explicitly confirms.