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The stronger manuscript construction has properties beyond the formal C/D nonexistence statements.","assessment":"Assess the forced C/D proofs and exact statement correspondence. CMI’s official evaluation is a separate process.","source_entry_ids":["navier-stokes:entry:paper:v1","fluid-regularity:entry:announcement:v1","fluid-regularity:entry:lean:v2","navier-stokes:entry:clay:v1","navier-stokes:entry:clay-announcement:v1","navier-stokes:entry:lean-r3:v2","navier-stokes:entry:lean-periodic:v2"]}},"representations":[{"hash_profile":"rfc8785","schema_version":"visual-representation-v1","id":"navier-stokes-state-ladder","version":1,"subject":{"kind":"problem","id":"problem-navier-stokes","sha256":"554b69f01db7eadb9cbc0ed6fc2ff3f43a7c0add3bf7024d1e5f4fff75a2c1a0"},"purpose":"setup","title":"From what is proved to what is asked","caption":"Read from the bottom: what holds for every solution, what holds under extra conditions, then the two ways the problem can be settled. 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a local build and independent correspondence check are not attached.","related_claim_ids":["claim-navier-stokes-forced-r3","claim-navier-stokes-forced-periodic"]}],"curation":{"checked_on":"2026-10-13","selection":"Original question, distinct results, proof provenance, selected history and an explanatory schematic.","search_scope":"Selected primary papers, exact formal declarations, project provenance and official descriptions checked on 13 October 2026.","formal_status":"Pinned Lean C/D nonexistence theorems are reported complete, with zero sorry and standard logical axioms. They are distinguished from the manuscript’s stronger detailed singular-flow construction; OpenMath has not executed the project.","ai_status":"OpenAI is the printed manuscript author and credits an internal multi-agent research-model workflow for the mathematical argument. The announcement and formal metadata credit GPT-6 Astra/Codex for formalization. Fefferman and CMI own the official statement and evaluation process; Lean, Mathlib and Formal Conjectures/Comparator contributors support the proof infrastructure. 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":3,"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. The manuscript and complete formal theorem are source-reported; mathematical assessment remains pending.","source_entry_ids":["navier-stokes:entry:paper:v1","navier-stokes:entry:lean-r3:v2","fluid-regularity:entry:lean:v2","navier-stokes:entry:clay:v1"],"formal_declarations":["NavierStokes.Comparator.navier_stokes_breakdown_R3"]},{"statement":"For every ν > 0 there exist a smooth force compactly supported in ℝ³ × (0,∞), a fixed compact spatial set K, and smooth incompressible Navier–Stokes velocity and pressure on ℝ³ × [0,1), starting with u₀ = 0 and supported in K, such that kinetic energy stays uniformly bounded while limsup as t ↑ 1 of ‖u(t)‖∞ is infinite. 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The linked formal nonexistence theorem does not by itself formalize every property in this statement.","source_entry_ids":["navier-stokes:entry:paper:v1"],"formal_declarations":[]},{"statement":"For every ν > 0 there are smooth divergence-free periodic initial data and smooth periodic forcing for which no global smooth periodic solution of the incompressible Navier–Stokes equations exists on ℝ³/ℤ³.","id":"claim-navier-stokes-forced-periodic","slug":"navier-stokes-forced-periodic","version":2,"kind":"theorem","title":"Forced breakdown on the three-torus","assumptions":["Positive viscosity and smooth periodic data/force.","The spatial domain is the unit three-torus."],"scope":"Corollary 10.6 and official alternative D; no unforced conclusion is asserted.","source_entry_ids":["navier-stokes:entry:paper:v1","navier-stokes:entry:lean-periodic:v2","fluid-regularity:entry:lean:v2","navier-stokes:entry:clay:v1"],"formal_declarations":["NavierStokes.Comparator.navier_stokes_breakdown_periodic"]},{"statement":"If a smooth velocity field on ℝ³ × [0,T) is supported in one fixed compact spatial set and its spatial supremum norm has infinite limsup as t ↑ T < ∞, then it has no smooth continuation to ℝ³ × [0,∞) that agrees with it before T.","id":"claim-navier-stokes-compact-continuation-obstruction","slug":"navier-stokes-compact-continuation-obstruction","version":1,"kind":"theorem","title":"Unbounded compactly supported velocity prevents smooth continuation","assumptions":["The same compact set contains the velocity support for every time before T.","Continuation must agree with the original velocity before the finite time T."],"scope":"A conditional consequence of continuity on a compact space-time set; it does not certify existence of a singular Navier–Stokes solution.","source_entry_ids":["navier-stokes:entry:paper:v1"],"formal_declarations":[]},{"statement":"In two space dimensions, the analogues of assertions (A) and (B) hold: smooth data lead to global smooth solutions with bounded energy. The same holds for the Euler equations.","id":"claim-navier-stokes-two-dimensional","slug":"navier-stokes-two-dimensional","version":1,"kind":"theorem","title":"Global regularity in two dimensions","assumptions":["Two space dimensions.","Analogues of the unforced alternatives on the whole plane and the periodic torus."],"scope":"A different dimension. The official description notes it gives no hint about three dimensions, where the main difficulties lie.","source_entry_ids":["navier-stokes:clay-two-dimensions:v1"],"formal_declarations":[]},{"statement":"In three space dimensions, assertions (A) and (B) hold provided the initial velocity satisfies a smallness condition.","id":"claim-navier-stokes-small-data","slug":"navier-stokes-small-data","version":1,"kind":"theorem","title":"Global regularity for small initial data","assumptions":["Three space dimensions; no external force, as in (A) and (B).","The initial velocity satisfies a smallness condition."],"scope":"Small data only; arbitrary large data remain open.","source_entry_ids":["navier-stokes:clay-small-data:v1"],"formal_declarations":[]},{"statement":"For initial data not assumed small, assertions (A) and (B) hold, also for ν = 0, on a time interval [0, T) with T depending on the initial data. For ν > 0, if the maximal time of existence is finite, the velocity becomes unbounded near it.","id":"claim-navier-stokes-short-time","slug":"navier-stokes-short-time","version":1,"kind":"theorem","title":"Smooth solutions on a short time interval","assumptions":["Three space dimensions; smooth divergence-free initial velocity as in (A) and (B).","T depends on the initial data."],"scope":"Short time only; whether the maximal time can be finite is the open question.","source_entry_ids":["navier-stokes:clay-short-time:v1"],"formal_declarations":[]},{"statement":"In three space dimensions, the Navier–Stokes equations always have a weak solution (p, u) with suitable growth properties.","id":"claim-navier-stokes-leray-weak-solutions","slug":"navier-stokes-leray-weak-solutions","version":1,"kind":"theorem","title":"Global weak solutions (Leray)","assumptions":["Three space dimensions.","Weak solutions satisfy the equations in the integrated sense (12)–(13) of the official description."],"scope":"Weak solutions need not be smooth, and their uniqueness is not known.","source_entry_ids":["navier-stokes:clay-leray:v1"],"formal_declarations":[]},{"statement":"For Leray–Hopf weak solutions, ∫₀ᵀ ‖u(τ)‖_{L^q}^{4q/(3(q−2))} dτ ≤ C K₀^{2q/(3(q−2))} for 2 ≤ q ≤ 6, where K₀ is the initial kinetic energy.","id":"claim-navier-stokes-leray-hopf-bound","slug":"navier-stokes-leray-hopf-bound","version":1,"kind":"bound","title":"Energy bounds for Leray–Hopf solutions","assumptions":["Leray–Hopf weak solutions on the domain of the cited source.","2 ≤ q ≤ 6; C is a generic constant."],"scope":"These are the norms every weak solution controls. Each satisfies 2/p + 3/q = 3/2, above the regularity threshold 2/p + 3/q ≤ 1.","source_entry_ids":["navier-stokes:kang-protas-energy:v1"],"formal_declarations":[]},{"statement":"Let u be a weak solution of the Navier–Stokes equations satisfying suitable growth conditions, and let E be its singular set. Then the one-dimensional parabolic Hausdorff measure P¹(E) is zero. 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