Smale-Derived Structural Challenges
Curated subset of Smale's 18-problem list testing arithmetic addressability, algebraic complexity, dynamical stability, optimisation, and the formal limits of intelligence.
Smale-derived structural challenges
Smale’s list is not imported wholesale. We use it as a curated source for structural mathematical challenges: problems that test arithmetic addressability, algebraic complexity, dynamical stability, global invertibility, optimisation, spectral distribution, and the formal limits of intelligence. Items already covered by Clay are not duplicated. Solved items are treated as recovery checkpoints.
Open challenges
| ID | Challenge | τ Response |
|---|---|---|
| S4 | Sparse Arithmetic Zero Challenge | further investigation |
| S5 | Diophantine Addressability and Height-Bound | further investigation |
| S6 | Central Configuration Finiteness | further investigation |
| S7 | Spherical Capacity and Optimal Distribution | structurally constrained |
| S9 | Strongly Polynomial Optimisation | further investigation |
| S11 | Generic Hyperbolicity | further investigation |
| S13 | Algebraic Cycles and Limit-Cycles | further investigation |
| S16 | Local-to-Global Invertibility | further investigation |
| S18 | Formal Limits of Intelligence | further investigation |
Recovery checkpoints
These Smale items are externally solved; τ recovery is tested without claiming new external solutions.
| ID | Checkpoint | External Status |
|---|---|---|
| S14 | Lorenz Attractor Recovery | externally solved by Tucker (computer-assisted proof) |
| S17 | Average Polynomial Solving Recovery | externally resolved (Beltrán–Pardo and follow-up) |
Smale items already covered elsewhere
- S1 Riemann Hypothesis — see CB-RIEMANN
- S2 Poincaré Conjecture — see CB-POINCARE
- S3 P vs NP — see CB-PVSNP
- S15 Navier–Stokes — see CB-NAVIER-STOKES
S8 (economic dynamics), S10 (Pugh closing lemma), S12 (centralizer of diffeomorphisms), and later additions are not imported as standalone canonical challenges per the briefing.
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