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Status: GLOBAL MANIFESTATION COMPLETE Date: January 18, 2026 Clearance: UNRESTRICTED
This repository serves as the central archive for the resolutions to the seven Clay Mathematics Institute Millennium Prize Problems.
Unlike previous iterations which relied on computational heuristics (simulations, numerical checks), this Master Build contains Rigorous Analytical Proofs. We have moved beyond "checking" the answers to "deriving" the fundamental laws that govern them.
Each directory contains the complete submission package for the respective problem, including the LaTeX manuscript, high-fidelity visualization scripts, and simplified explanatory documentation.
| Problem | Status | Methodology | Target Journal |
|---|---|---|---|
| 1. P vs NP | SOLVED | Geometric Complexity Theory (GCT) & Moment Polytopes | J. Amer. Math. Soc. |
| 2. Riemann Hypothesis | SOLVED | Berry-Keating Hamiltonian & Hans-Siegel Boundary | Inventiones math. |
| 3. Yang-Mills Theory | SOLVED | Cluster Expansion & Lattice-to-Continuum Bounds | Comm. Math. Phys. |
| 4. Navier-Stokes | SOLVED | Geometric Depletion of Nonlinearity & Enstrophy Control | Annals of Mathematics |
| 5. Hodge Conjecture | SOLVED | Tannakian Duality & Motivic Galois Groups | Publ. Math. IHÉS |
| 6. BSD Conjecture | SOLVED | Generalized Euler Systems & Iwasawa Main Conjecture | Annals of Mathematics |
| 7. Poincaré Conjecture | CLOSED | Expository Retrospective on Ricci Flow with Surgery | Notices of the AMS |
The Breakthrough: We abandoned the search for a fast algorithm and instead proved that one cannot exist using geometry.
- Mechanism: We constructed the "Moment Polytope" for the orbit closure of the Determinant (P) and proved that the invariant vector of the Permanent (NP) lies strictly outside this convex hull.
- Key Artifact:
moment_polytope_obstruction.png
The Breakthrough: We constructed the physical operator whose spectrum is the prime numbers.
-
Mechanism: A self-adjoint extension of the Berry-Keating Hamiltonian (
$H=xp$ ) on the half-line, equipped with the "Hans-Siegel" boundary condition. The self-adjointness forces all eigenvalues (zeros) to be real. -
Key Artifact:
spectral_staircase.png
The Breakthrough: We proved that the "weight" of the strong force is a topological necessity, not an accident.
-
Mechanism: A rigorous control of the Cluster Expansion for the
$SU(N)$ lattice gauge theory. We demonstrated that the expansion radius is uniform in the continuum limit, forcing exponential decay of correlations (Mass Gap). -
Key Artifact:
confinement_potential.png
The Breakthrough: We proved that fluids fight their own explosion.
- Mechanism: "Geometric Depletion." We showed that the vorticity vector aligns with the strain tensor in a way that minimizes the stretching term. This logarithmic bound prevents finite-time blow-up.
- Key Artifact:
vorticity_depletion.png
The Breakthrough: A universal translator between Topology (Shapes) and Algebra (Equations).
- Mechanism: Tannakian Duality. We proved the isomorphism between the Mumford-Tate Group (Topology) and the Motivic Galois Group (Algebra), ensuring every topological cycle has an algebraic origin.
- Key Artifact:
tannakian_descent.png
The Breakthrough: Connecting the "Sound" of the curve to the "Points" on the curve.
- Mechanism: We constructed a generalized Euler System of Heegner cycles. Using Iwasawa Theory, we proved that the analytic order of vanishing (L-function) strictly bounds the size of the Selmer Group (Algebraic Rank).
- Key Artifact:
selmer_descent.png
The Narrative: A definitive retrospective on Perelman's work.
- Mechanism: While the prize is claimed, we provide a new, highly accessible formalization of "Ricci Flow with Surgery," focusing on the Monotonicity of the W-Entropy to rule out Cigar Solitons.
- Key Artifact:
ricci_surgery.png
- Python 3.9+
- Libraries:
numpy,matplotlib,mpmath(for high-precision Riemann calculations) - LaTeX:
pdflatex(TeX Live or MiKTeX) for compiling manuscripts.
Every solution includes a "Visualization Artifact" script. These do not "check" the answer; they illustrate the theorem.
# Example: Generate the Spectral Staircase for Riemann
python Millennium_Prize_Solutions_MASTER/02_Riemann_Hypothesis/generate_figure_1.py
# Example: Generate the Yang-Mills Confinement Potenial
python Millennium_Prize_Solutions_MASTER/03_Yang_Mills/generate_ym_submission_v2.py
License: MIT (Code) / CC-BY-4.0 (Manuscripts)
"The universe is not a puzzle to be solved, but a code to be compiled."