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Alpöge and Buckmaster’s separately credited forced-Euler work is not this Navier–Stokes theorem.","independent_review":"pending"},"schema_version":"research-record-v3","claims":[{"statement":"For every ν > 0 and every smooth divergence-free initial velocity u₀ on ℝ³ whose spatial derivatives decay faster than every inverse power, the unforced incompressible Navier–Stokes equations admit global smooth velocity and pressure with uniformly bounded kinetic energy.","id":"claim-navier-stokes-unforced-r3","slug":"navier-stokes-unforced-r3","version":1,"kind":"conjecture","title":"Unforced global regularity in the whole space","assumptions":["Three space dimensions; whole-space domain ℝ³; viscosity ν > 0.","No external force: f = 0.","Every spatial derivative of u₀ is rapidly decreasing."],"scope":"The official alternative A is not answered by the selected forced C/D constructions.","source_entry_ids":["navier-stokes:entry:clay:v1","navier-stokes:entry:paper:v1"],"formal_declarations":[]},{"statement":"For every ν > 0 and every smooth divergence-free periodic initial velocity, the unforced incompressible Navier–Stokes equations admit global smooth periodic velocity and pressure on ℝ³/ℤ³.","id":"claim-navier-stokes-unforced-periodic","slug":"navier-stokes-unforced-periodic","version":1,"kind":"conjecture","title":"Unforced global regularity on the three-torus","assumptions":["Three-dimensional unit-periodic domain.","Positive viscosity and no external force."],"scope":"The official alternative B; a separate domain from whole-space regularity.","source_entry_ids":["navier-stokes:entry:clay:v1","navier-stokes:entry:paper:v1"],"formal_declarations":[]},{"statement":"For every ν > 0 there exist smooth divergence-free rapidly decaying initial velocity and a smooth force whose space-time derivatives decay faster than every inverse power, for which no global smooth incompressible Navier–Stokes velocity and pressure with uniformly bounded kinetic energy exist on ℝ³.","id":"claim-navier-stokes-forced-r3","slug":"navier-stokes-forced-r3","version":2,"kind":"theorem","title":"Forced whole-space nonexistence","assumptions":["Every positive viscosity and whole-space domain ℝ³.","Smooth divergence-free initial data with all spatial derivatives rapidly decreasing.","Smooth external force with all space-time derivatives rapidly decreasing.","Uniformly bounded kinetic energy is part of the excluded global solution class."],"scope":"Theorem 1.1 and official alternative C. 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