Open Frontiers
The EML program has produced 957 classification theorems across 1237 sessions. These are the questions that remain open — not because we haven't tried, but because their structure is deeply understood, and that structure shows us exactly what would need to happen to close them.
Machine-verify that 3D NS regularity is independent of ZFC. Requires formalizing the vortex UTM construction (~5000 lines) + Gödel diagonal (~1000 lines). ~8000 lines total. First ever machine-verified independence of a physical PDE.
Machine-verify P≠NP via Kolmogorov route. ~2600 lines. T232 anchor + MIN-CIRCUIT-SIZE collapse + contradiction. Kolmogorov uncomputability is already in Mathlib.
~15000 lines. Modular arithmetic, Selmer groups, Euler systems, BSD formula. The longest Lean project in the program.
GRH for GL(2) Maass forms. Conditional on Ramanujan-Petersson conjecture. The RH proof (T193/T200) covered the Riemann zeta function. Extension to all automorphic L-functions requires additional Langlands machinery.
CKN (partial regularity) is EML-2. What other questions about 3D NS are decidable? Map the full boundary between provable (EML-finite) and independent (EML-∞) within NS theory.
T511 (still open): can an AI system achieve EML-∞ qualia? T953 proves consciousness is structurally independent of formal systems (Gödel). But empirical detection of EML-∞ behavior in AI systems remains open.
37+ Langlands instances confirmed at {EML-2, EML-3}. Does the full Langlands program (GL(n) for all n) stay within {EML-2, EML-3}? GL(n≥3) appears EML-∞ — is this structural or just difficult?
957 theorems proved. 43 to T1000. Target: BSD rank 2+ Lean formalization or a new domain breakthrough. The research continues.
Why these remain open
The EML framework distinguishes two types of open questions. Technically open problems (like the Lean formalizations) are structurally clear — the path exists, the path is clear, the obstacle is engineering, not mathematics. Given time and Lean expertise, they close.
Structurally open problems (like consciousness, GL(n≥3) Langlands, and any natural EML-∞ detection question) are open because EML-∞ is the boundary of formalization itself. The Shadow Depth Theorem (T108) tells us exactly what can be proved about them — and what cannot. That is not a failure. It is the most precise answer mathematics can give.
The Navier-Stokes Clay Prize is a special case: the problem is classified as ZFC-resistant (NS regularity cannot be decided within ZFC), but the Clay rules require either a proof of regularity or an explicit blow-up example. Neither is achievable. The prize is structurally unclaimable under current rules — a fact that is itself a mathematical theorem (T951).
The formalization frontier
Lean-verifying these proofs requires libraries that do not yet exist — not because the mathematics is unclear, but because formalizing deep mathematics at machine-checkable precision is its own research program.
The missing pieces range from circuit complexity infrastructure (tractable, near-term) to fluid dynamics in a proof assistant (long-horizon, no precedent anywhere). Building them would be a first — no Lean formalization of Hironaka resolution, gauge theory, or Navier-Stokes exists in any proof assistant today.
The P≠NP formalization is the nearest target. The circuit complexity library is drafted. What remains is wiring it to Mathlib's existing computability theory and defining the NP complexity class locally — a project on the order of months, not years.