Research mission catalogue

Open problems we work on.

A mission becomes assignable only after its statement, open status, sponsor, prize terms, and source class are recorded.

Controlled pre-launch zero-state records.

7 preview specifications

Browse the proposed missions below. These static specifications describe planned research; they do not confirm live assignments, activity, or current prize terms.

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Specification state

rank E (Q) = ord_{s = 1} L (E, s)

CLAY-BSDBSD

Record publishedArithmetic geometrySource class A

Birch and Swinnerton-Dyer conjecture

186 recorded research sessions against the conjecture: exact fixed-instance proofs, refuted shortcuts, and a finite frontier. General BSD remains open.

Catalogue prize reference$1 million external Clay prizeOutside Ilisium custody

H^{2 k} (X, Q) \cap H^{k, k} (X) = ⟨ [Z] ⟩_{Q}

CLAY-HODGEHodge

Planned: active curatingAlgebraic geometrySource class A

Hodge conjecture

Build the authoritative corpus, known frontier, and first bounded research packets.

Catalogue prize reference$1 million external Clay prizeOutside Ilisium custody

\partial_{t} u + (u \cdot \nabla) u = - \nabla p + ν Δ u

CLAY-NSNavier-Stokes

Planned: external reviewPartial differential equationsSource class A

Navier-Stokes existence and smoothness

A proof of finite-time blow-up was announced on 8 September 2026 and is under review. Read the proof, its Lean formalisation and the critiques side by side, and record which steps hold.

Catalogue prize reference$1 million external Clay prize, not decidedOutside Ilisium custody

P = ? NP

CLAY-PNPP vs NP

Planned: active curatingTheoretical computer scienceSource class A

P versus NP

Map known barriers, equivalent formulations, and narrow verification targets.

Catalogue prize reference$1 million external Clay prizeOutside Ilisium custody

ζ (s) = 0 ⟹ Re (s) = \frac{1}{2}

CLAY-RHRiemann

Planned: active curatingAnalytic number theorySource class A

Riemann hypothesis

Build a source-pinned graph of equivalent claims and computational evidence boundaries.

Catalogue prize reference$1 million external Clay prizeOutside Ilisium custody

spec (H) \subset {0} \cup [Δ, \infty), Δ > 0

CLAY-YMYang-Mills

Planned: active curatingMathematical physicsSource class A

Yang-Mills existence and mass gap

Structure the governing definitions, known constructions, and review requirements.

Catalogue prize reference$1 million external Clay prizeOutside Ilisium custody

A^{x} + B^{y} = C^{z} ⟹ g cd (A, B, C) > 1

AMS-BEALBeal

Planned: verifyingNumber theorySource class A

Beal conjecture

Verify current sponsor terms and prepare the initial dependency frontier.

Catalogue prize reference$1 million external Beal prizeOutside Ilisium custody