# Solid results

> 50 AI-made math results rated Solid, each explained in plain English. Results rated Solid: real progress on narrower or more technical questions.

Canonical: https://closedform.nishachay.tech/significance/solid/
Last updated: 2026-10-14

Results rated Solid: real progress on narrower or more technical questions.

50 AI-made results rated Solid, from 98 papers in the OpenAI math collection, each explained in plain English. 37 have a formal Lean proof listed; 13 are claimed in a paper and not yet machine-checked. They span 14 fields, led by probability and statistical mechanics (9), combinatorics (8) and mathematical physics (8). The oldest question here dates back to 1943: [The joint critical Ashkin–Teller current limit](https://closedform.nishachay.tech/e/openai-math-2026-233/).

## Results (50)

- [Positive lower density of large prime gaps](https://closedform.nishachay.tech/e/openai-math-2026-026/) - Entry 026 · For any size bar, a fixed share of prime gaps clears it, all the way out. (Formal proof listed)
- [Log-canonical threefold abundance in numerical dimension one](https://closedform.nishachay.tech/e/openai-math-2026-035/) - Entry 035 · In characteristic above 3, a nef log canonical threefold divisor of numerical dimension one is claimed to give a map. (Claimed, not checked)
- [P = W for fixed-determinant SL n moduli spaces](https://closedform.nishachay.tech/e/openai-math-2026-043/) - Entry 043 · Two ways of layering the cohomology of SL_n Higgs moduli are claimed to agree in every composite coprime rank. (Claimed, not checked)
- [The equivariant cohomological Hikita conjecture](https://closedform.nishachay.tech/e/openai-math-2026-044/) - Entry 044 · For every finite quiver, the cohomology of a Nakajima variety is claimed to equal functions on a Coulomb fixed scheme. (Claimed, not checked)
- [Gepner symmetry and large-volume stability on threefolds](https://closedform.nishachay.tech/e/openai-math-2026-055/) - Entry 055 · A stability condition on every smooth quintic threefold with Toda’s exact 2/5 Gepner symmetry. (Claimed, not checked)
- [The Campana–Peternell conjecture in dimension six](https://closedform.nishachay.tech/e/openai-math-2026-067/) - Entry 067 · A claimed proof that every six-dimensional Fano manifold with nef tangent bundle is a symmetric, homogeneous space. (Claimed, not checked)
- [Cylinder coverings below the half-area bound](https://closedform.nishachay.tech/e/openai-math-2026-100/) - Entry 100 · Cylinders can cover a regular tetrahedron for less than half its smallest shadow. (Formal proof listed)
- [One-sample matroid prophet inequalities against an almighty adversary](https://closedform.nishachay.tech/e/openai-math-2026-111/) - Entry 111 · One sample per item lets an online rule keep a fixed share of the best, even against an all-seeing opponent. (Formal proof listed)
- [Homogeneous depth-five lower bounds for iterated matrix multiplication](https://closedform.nishachay.tech/e/openai-math-2026-135/) - Entry 135 · Five-layer recipes for multiplying n matrices need n to a power that grows like √n. (Formal proof listed)
- [Memory–sample lower bounds for noiseless Gaussian regression](https://closedform.nishachay.tech/e/openai-math-2026-140/) - Entry 140 · Exact clues still cost more reading for each extra digit of accuracy when memory is limited. (Formal proof listed)
- [Bounded-degree coboundary expanders](https://closedform.nishachay.tech/e/openai-math-2026-177/) - Entry 177 · Large higher-dimensional expanders with a bounded number of faces at each point exist in every dimension. (Formal proof listed)
- [Deterministic nonbipartite Ramanujan graphs in every fixed degree](https://closedform.nishachay.tech/e/openai-math-2026-178/) - Entry 178 · A fixed recipe builds networks with best-possible mixing, in every degree and every large even size. (Claimed, not checked)
- [The circulant Hadamard and Barker-sequence conjectures](https://closedform.nishachay.tech/e/openai-math-2026-179/) - Entry 179 · Circulant Hadamard matrices exist only in sizes 1 and 4. (Formal proof listed)
- [Barnette’s Hamiltonian-cycle conjecture](https://closedform.nishachay.tech/e/openai-math-2026-180/) - Entry 180 · Every Barnette graph has a round trip through all its points. (Formal proof listed)
- [Counterexamples to infinite matroid intersection and packing/covering](https://closedform.nishachay.tech/e/openai-math-2026-185/) - Entry 185 · Two self-dual infinite matroids break the infinite matroid packing/covering and intersection conjectures. (Formal proof listed)
- [Snaky in 21 Maker moves](https://closedform.nishachay.tech/e/openai-math-2026-187/) - Entry 187 · On an endless grid, the first player can always build the six-square Snaky shape within 21 moves. (Formal proof listed)
- [Polynomial removal fails for ordered binary matrices](https://closedform.nishachay.tech/e/openai-math-2026-190/) - Entry 190 · One fixed 66×66 pattern of zeros and ones has no polynomial removal bound in ordered binary matrices. (Formal proof listed)
- [Boolean functions violate the square-root degree bound by arbitrary factors](https://closedform.nishachay.tech/e/openai-math-2026-192/) - Entry 192 · Boolean functions can beat the proposed square-root-of-degree bound on their linear correlations by any factor. (Formal proof listed)
- [Tensor saturation for even spin groups](https://closedform.nishachay.tech/e/openai-math-2026-204/) - Entry 204 · For even spin groups, an invariant after stretching the weights implies one before, if the weights pass a parity test. (Claimed, not checked)
- [Foulkes' conjecture for sixth powers and quadratic stabilization](https://closedform.nishachay.tech/e/openai-math-2026-210/) - Entry 210 · Foulkes' conjecture is claimed for sixth powers, with a Lean-proved bound past which a natural map is onto. (Formal proof listed)
- [Continuum phase transitions for radial pair potentials](https://closedform.nishachay.tech/e/openai-math-2026-228/) - Entry 228 · Two built pair potentials give particles in space a first-order phase transition across a range of densities. (Formal proof listed)
- [Exact three- and four-state reconstruction thresholds and four-state tree capacity](https://closedform.nishachay.tech/e/openai-math-2026-229/) - Entry 229 · For three and four colors, guessing a tree's root from far leaves works exactly above the Kesten–Stigum bound. (Formal proof listed)
- [The free uniform spanning forest is a factor of IID](https://closedform.nishachay.tech/e/openai-math-2026-231/) - Entry 231 · One fixed rule builds the free uniform spanning forest from independent random labels on every infinite graph. (Formal proof listed)
- [Gaussian fields and interfaces for triangular-lattice Lipschitz heights](https://closedform.nishachay.tech/e/openai-math-2026-232/) - Entry 232 · Random Lipschitz height surfaces on the triangular lattice converge to the Gaussian free field. (Claimed, not checked)
- [The joint critical Ashkin–Teller current limit](https://closedform.nishachay.tech/e/openai-math-2026-233/) - Entry 233 · Heights and current clusters of the critical Ashkin–Teller model converge together to a free field and its local sets. (Claimed, not checked)
- [All-temperature pressure of orthogonally invariant Ising spin glasses](https://closedform.nishachay.tech/e/openai-math-2026-234/) - Entry 234 · A formula for the pressure of orthogonally invariant Ising spin glasses holds at every temperature. (Formal proof listed)
- [The exact factor-of-IID threshold for free Ising spins on trees](https://closedform.nishachay.tech/e/openai-math-2026-236/) - Entry 236 · The free Ising state on a regular tree can be built from independent labels exactly up to tanh β = 1/√(d−1). (Formal proof listed)
- [Optimal logarithmic mixing of the Thorp shuffle](https://closedform.nishachay.tech/e/openai-math-2026-238/) - Entry 238 · Claims the Thorp shuffle on 2^d cards randomizes the whole deck in order d shuffles, the best possible order. (Formal proof listed)
- [Sharp singularity rates for symmetric random sign matrices](https://closedform.nishachay.tech/e/openai-math-2026-239/) - Entry 239 · Claims a random symmetric ±1 matrix is singular with probability (1/2 + o(1))ⁿ, with a matching biased-sign rate. (Claimed, not checked)
- [Weak normalization implies strong normalization in pure type systems](https://closedform.nishachay.tech/e/openai-math-2026-245/) - Entry 245 · Claims that in every pure type system, if every legal term can finish computing, then every way of computing finishes. (Formal proof listed)
- [Sharp finite-matrix Lieb–Thirring inequalities and all equality cases](https://closedform.nishachay.tech/e/openai-math-2026-262/) - Entry 262 · Proves the sharp 1D Lieb–Thirring constant for 1/2 < γ < 3/2; claims the matrix case and all equality cases. (Formal proof listed)
- [Exactly three mutually unbiased bases in dimension six](https://closedform.nishachay.tech/e/openai-math-2026-266/) - Entry 266 · Dimension six fits only three mutually unbiased bases, by a computer-assisted search. (Formal proof listed)
- [QMA-hardness of continuum Coulomb energy](https://closedform.nishachay.tech/e/openai-math-2026-275/) - Entry 275 · Proves that estimating a molecule's exact electronic ground energy, from nuclear positions alone, is QMA-hard. (Formal proof listed)
- [Classical capacity of generalized amplitude damping](https://closedform.nishachay.tech/e/openai-math-2026-276/) - Entry 276 · Proves an exact formula for the classical capacity of every generalized amplitude-damping qubit channel. (Formal proof listed)
- [Threshold repetition for entangled games](https://closedform.nishachay.tech/e/openai-math-2026-277/) - Entry 277 · Proves that winning well above the entangled value in many repeated two-player games is exponentially unlikely. (Formal proof listed)
- [Failure of Kohn–Sham ensemble representation](https://closedform.nishachay.tech/e/openai-math-2026-278/) - Entry 278 · Claims a three-electron molecule whose ground-state density no Kohn–Sham ensemble can reproduce. (Claimed, not checked)
- [Exact quantum factoring over a fixed finite gate set](https://closedform.nishachay.tech/e/openai-math-2026-279/) - Entry 279 · Proves polynomial-size quantum circuits over a fixed gate set factor every integer with probability exactly one. (Formal proof listed)
- [QAOA attains the SK optimum in the thermodynamic-first limit](https://closedform.nishachay.tech/e/openai-math-2026-281/) - Entry 281 · Claims QAOA reaches the Sherrington–Kirkpatrick ground-state energy as size grows first and depth then increases. (Formal proof listed)
- [A ZFC counterexample to Naimark's problem](https://closedform.nishachay.tech/e/openai-math-2026-297/) - Entry 297 · An algebra with one irreducible representation that is not the compact operators, built without extra axioms. (Formal proof listed)
- [Thomason model structures in all strict higher dimensions](https://closedform.nishachay.tech/e/openai-math-2026-317/) - Entry 317 · Strict n-categories carry a Thomason model structure in every dimension, settling an Ara–Maltsiniotis conjecture. (Formal proof listed)
- [Counterexamples to finite generation at chromatic height two](https://closedform.nishachay.tech/e/openai-math-2026-319/) - Entry 319 · A paper claims to disprove the Hahn–Wilson conjecture at height two, for every large enough prime. (Claimed, not checked)
- [The cotype–cotype conjecture under the approximation property](https://closedform.nishachay.tech/e/openai-math-2026-326/) - Entry 326 · For Banach spaces with the approximation property, K-convexity equals finite cotype of the space and its dual. (Formal proof listed)
- [Reflexive midpoint convexity and diamond distortion](https://closedform.nishachay.tech/e/openai-math-2026-331/) - Entry 331 · Proves a reflexive Banach space can be midpoint convex far out yet admit no asymptotically uniformly convex norm. (Formal proof listed)
- [The isoperimetric profile of the cubic three-torus](https://closedform.nishachay.tech/e/openai-math-2026-354/) - Entry 354 · Proves the isoperimetric conjecture for the cubic flat three-torus: balls, tubes, slabs and their complements. (Formal proof listed)
- [Gigli’s characterization of Alexandrov curvature](https://closedform.nishachay.tech/e/openai-math-2026-356/) - Entry 356 · Claims Gigli's conjecture: Alexandrov curvature bounds match RCD spaces with a distributional sectional curvature bound. (Formal proof listed)
- [Bi-Lipschitz coordinates at every regular RCD point](https://closedform.nishachay.tech/e/openai-math-2026-357/) - Entry 357 · Claims bi-Lipschitz coordinate charts around every regular point of a noncollapsed RCD space. (Claimed, not checked)
- [A three-manifold without conjugate points or nonpositive curvature](https://closedform.nishachay.tech/e/openai-math-2026-358/) - Entry 358 · Proves that a closed three-manifold can carry a metric without conjugate points yet no metric of nonpositive curvature. (Formal proof listed)
- [Nonattainment of the three-marginal Coulomb Monge problem](https://closedform.nishachay.tech/e/openai-math-2026-373/) - Entry 373 · Proves that the best three-electron Coulomb arrangement need not follow fixed maps, even for a smooth density. (Formal proof listed)
- [Sharp one-third stability of Brenier maps](https://closedform.nishachay.tech/e/openai-math-2026-374/) - Entry 374 · Proves optimal transport maps from a uniform convex source change at most like the cube root of the target change. (Formal proof listed)
- [Universal computation in forced Navier–Stokes flows](https://closedform.nishachay.tech/e/openai-math-2026-376/) - Entry 376 · Proves a pushed fluid can run any program: a marked particle enters a fixed box exactly when the program halts. (Formal proof listed)

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