Millennium Prize Problems — Cascade Catalog Section¶
Seven problems set by the Clay Mathematics Institute in 2000, with $1M prize each. One has been solved (Poincaré conjecture, Perelman 2003), leaving 6 open. One is already dispatched in the Hilbert section of this PR (Riemann hypothesis = Hilbert's 8th = Millennium Problem #5). The remaining 5 open Millennium problems get cascade-research subdirectories here.
Source: Wikipedia — Millennium Prize Problems; Clay Mathematics Institute.
Status of all 7 problems¶
| # | Problem | Status | Cascade dir |
|---|---|---|---|
| 1 | Birch and Swinnerton-Dyer conjecture | OPEN | birch_swinnerton_dyer/ |
| 2 | Hodge conjecture | OPEN | hodge_conjecture/ |
| 3 | Navier-Stokes existence and smoothness | OPEN | navier_stokes/ |
| 4 | P versus NP problem | OPEN | p_vs_np/ |
| 5 | Riemann hypothesis | OPEN — already dispatched as Hilbert 8 | ../hilbert/hilbert_08_riemann_hypothesis/ |
| 6 | Yang-Mills existence and mass gap | OPEN | yang_mills_mass_gap/ |
| 7 | Poincaré conjecture | Solved — Perelman 2003 | — |
Cascade-research priority¶
| Problem | Cascade tractability | Priority for this PR partition order |
|---|---|---|
| P vs NP | High — Class H self-introspection + Class D pattern-match + Class K asymptotic-DoF on complexity classes; direct extension of [[project_a_n_operators_are_harmonic_objects_themselves]] §B (Cryptographic-secret asymptote under perfect math) |
First — natural continuity with §B |
| Yang-Mills mass gap | High — Spike #58 chain already derives SU(3)×SU(2)×U(1) via Class M ∘ I ∘ C ∘ K ∘ L; mass gap is a Class K asymptotic-DoF anchor; Hilbert 6 dispatch marked YM "partial" — this closes the partial | Second — completes Hilbert 6 partial |
| Birch and Swinnerton-Dyer | Moderate — Class J primes (elliptic-curve point group) + Class L (L-function Laplacian) + Class N (rank-of-Mordell-Weil rational anchor) | Third — extends Riemann L-function cascade |
| Navier-Stokes | Moderate — Class L (vector Laplacian) + Class K (singularity asymptote) + Class C (vorticity orientation); Spike #62 turbulent-systems-framework intersection already exists | Fourth |
| Hodge conjecture | Moderate — Class L (Hodge Laplacian on smooth projective variety) + Class K (algebraic-cycle asymptote) + Class M (cohomology HDC) | Fifth |
Composition with Hilbert section¶
The Hilbert section of this PR closed with verdict (a) candidate SURVIVES strongly for the 14-class A-N + Hurwitz 1+3+7+3 partition at 92% empirical coverage. The Millennium section now extends the empirical evidence base. Two structural predictions carry forward:
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Hurwitz heptadic anchor per partition 5 (Hilbert 16): n/7 EXACT-rational anchor at the polynomial-vector-field substrate. Millennium problems should exhibit the same 7-denominator anchor where degree/dimension scales appear.
-
Cascade composition vocabulary per partition 6 (Hilbert 6): the candidate 1+3+7+3 partition holds for 24/26 = 92% physics domains. Millennium problems should likewise decompose to this partition.
Per [[feedback_no_lineage_claims_in_notebook]]: the framework does NOT claim to solve any Millennium problem. It demonstrates cascade-composability and structural confirmation of the harmonic-objects claim.
Discipline¶
Per the top-level docs/unsolved-maths/README.md — projection-visibility ordering, verdict-tier per Spike-research #229, cascade-honesty per [[feedback_sign_handling_is_class_k_pin_slot_not_alu_abs]], OA/arXiv-only citations per [[feedback_pdf_extraction_citation_discipline]].
Cryptography-as-physics rows continue to be defensive-scope only per [[feedback_trauma_informed_defensive_scope]].