The Seven Millennium Problems — Closed Cells and Empty Cells, Certified Computations around the Riemann Hypothesis and Formal Verification in the Lean Kernel

For each of the seven Millennium Prize Problems of the Clay Mathematics Institute we record, as of October 2026, the established results, the unrefereed claims and the open parts, and we certify finite statements, centred on the Riemann hypothesis, by exact arithmetic, ball arithmetic and the Lean 4 kernel. The positivity of the Weil quadratic form is established by ball-arithmetic certificates on twelve multiplicative windows (λ = √2 … 4), in both sectors, with an even and simple ground state in every window. On the window √13, the window of Connes’s 2026 letter, this verifies the letter’s assumption that the lowest eigenvalue is simple and even. For the Davenport–Heilbronn function, which has no Euler product, the Weil form is shown to be negative, and 68 of its first 1,024 zeros are located off the critical line. The final steps of two unrefereed certificates (Liu; Mori) are re-verified with independent code. The seven problems are recounted in twenty-eight cells with the Lean 4 kernel, without Mathlib (KIMBANJI: 1,701 theorems in 60 modules, released under CC0). The infinite-field cell of every one of the seven problems remains empty. Apart from Theorem A, all results are finite statements, and Theorem A gives no new information on the Riemann hypothesis. No Millennium problem is claimed to be solved; of the seven, only the Poincaré conjecture is solved. © 2026 THROMBO PROTOCOL — CC BY 4.0 (Lean sources: CC0 1.0) SHA-256 (PDF): 6aa605c776fb807c7bffc82cc765dde0137210108e7c5223a77a9e3b3b74c496