UnstableReserves

The channel launched this week. Starting with explainers of the openai/math release.

openai/math: 156 borsuk OpenAI's preprint (not yet peer reviewed; formalized in Lean) shows Borsuk's conjecture fails in dimension 9: the lines of ℝ⁴, viewed as projectors, can't be split into 10 pieces of smaller diameter. Lean's 66,297-line case check is a 3-line SAT proof in our Satori. Improvement: yes. 11 pieces suffice, so b(X) = 11 exactly; the preprint leaves this open. Kernel-checked, assuming OpenAI's theorem and one textbook lemma.

I made an animated short with Opus 5.5.

Wrap a 4×4×4 cube in one more layer: 61 new blocks. Look down its long diagonal and they land on 61 points of a hexagon in the Eisenstein lattice, the number system that proves Fermat's theorem for cubes. This 11-minute explainer follows that picture all the way: a shell is a hexagon, the hexagon's size is an Eisenstein norm, and unique factorization makes any cube solution shrink forever. Along the way: Fermat's margin, Euler's 1770 proof, Kummer's ideal numbers, and why higher powers waited for Wiles and Taylor. What it isn't: a new mechanism. The cube-root step is unique factorization in ℤ[ω]. Our follow-up: one family of shell elliptic curves has rank ≥ 2 for every whole-number parameter, and the genus-3 curve left over has exactly 3 rational points by Tungsten Chabauty–Coleman runs at p = 5 and 7 (reviewed in-house, two implementations; external audit outstanding). Every number was computed in Tungsten, 41 identities and worked examples were checked by its proof checker Satori, and every frame was drawn with its graphics library.

Improved on openai/math: 361 yau-harmonic-dimension Yau asked whether Ric ≥ 0 forces h_k(M) ≤ h_k(R^n) = Σ_{l ≤ k} N_l for harmonic functions with |u| ≤ C(1 + r)^k. Both papers say no. On a cone dr² + r²h, an eigenfunction of the link with eigenvalue λ gives a harmonic function … Improvement: new exact certificates n = 16 at k = 5580, n = 14 at k = 10391, n = 12 at k = 68000, each confirmed by an independent BigInt run (exact computation, re-verified)

Improved on openai/math: 088 petty-projection The projection body ΠK has support function h_ΠK(u) = vol_{d−1}(K | u⊥), and R_d(K) = |ΠK|/|K|^(d−1) is affine invariant. Improvement: an explicit 9-dimensional polytope with 11 facets beats the simplex: witness9_beats_simplex (witness9_checked.sat, 62 theorems). The kernel recomputes |P| and |ΠP| from the cut's ten numbers (c, t) al (machine-checked in satori)