# OpenAI Math 2026

> 372 results, 719 papers, honest trust labels.

Canonical: https://closedform.nishachay.tech/collection/openai-math-2026/
Last updated: 2026-10-14

372 results in 719 papers across 17 fields. Released by OpenAI in the [openai/math](https://github.com/openai/math) repository.

- **Papers:** 719
- **Results:** 372
- **Formal proofs:** 172
- **Fields:** 17

## Fields

- [Number theory](https://closedform.nishachay.tech/science/mathematics/number-theory/) - Field · 31 results
- [Algebraic and complex geometry](https://closedform.nishachay.tech/science/mathematics/algebraic-and-complex-geometry/) - Field · 36 results
- [Real and complex analysis](https://closedform.nishachay.tech/science/mathematics/real-and-complex-analysis/) - Field · 16 results
- [Convex and metric geometry](https://closedform.nishachay.tech/science/mathematics/convex-and-metric-geometry/) - Field · 15 results
- [Theoretical computer science](https://closedform.nishachay.tech/science/mathematics/theoretical-computer-science/) - Field · 40 results
- [Dynamical systems and ergodic theory](https://closedform.nishachay.tech/science/mathematics/dynamical-systems-and-ergodic-theory/) - Field · 12 results
- [Combinatorics](https://closedform.nishachay.tech/science/mathematics/combinatorics/) - Field · 37 results
- [Algebra](https://closedform.nishachay.tech/science/mathematics/algebra/) - Field · 18 results
- [Probability and statistical mechanics](https://closedform.nishachay.tech/science/mathematics/probability-and-statistical-mechanics/) - Field · 29 results
- [Mathematical logic](https://closedform.nishachay.tech/science/mathematics/mathematical-logic/) - Field · 6 results
- [Group theory](https://closedform.nishachay.tech/science/mathematics/group-theory/) - Field · 14 results
- [Mathematical physics](https://closedform.nishachay.tech/science/mathematics/mathematical-physics/) - Field · 25 results
- [Operator algebras](https://closedform.nishachay.tech/science/mathematics/operator-algebras/) - Field · 19 results
- [Topology](https://closedform.nishachay.tech/science/mathematics/topology/) - Field · 18 results
- [Functional analysis](https://closedform.nishachay.tech/science/mathematics/functional-analysis/) - Field · 11 results
- [Differential geometry](https://closedform.nishachay.tech/science/mathematics/differential-geometry/) - Field · 29 results
- [Partial differential equations](https://closedform.nishachay.tech/science/mathematics/partial-differential-equations/) - Field · 16 results

## All results in this collection

### Number theory (31)

- [Milne’s rationality conjecture and algebraic specialization](https://closedform.nishachay.tech/e/openai-math-2026-001/) - Entry 001 · After reduction mod p, Hodge classes on abelian varieties are claimed to give one rational number in every cohomology. (Claimed, not checked)
- [The full BSD formula from low Selmer corank](https://closedform.nishachay.tech/e/openai-math-2026-002/) - Entry 002 · The full Birch–Swinnerton-Dyer formula is claimed for every elliptic curve with a small Selmer group at one prime. (Claimed, not checked)
- [The quasi-Riemann hypothesis](https://closedform.nishachay.tech/e/openai-math-2026-003/) - Entry 003 · No zero of Riemann's zeta function, or of any Dirichlet L-function, lies to the right of real part 7/8. (Formal proof listed)
- [Hilbert’s tenth problem over ℚ](https://closedform.nishachay.tech/e/openai-math-2026-004/) - Entry 004 · No algorithm can decide whether a whole-number polynomial equation has a solution in fractions, the paper claims. (Claimed, not checked)
- [Irrationality of Catalan’s constant](https://closedform.nishachay.tech/e/openai-math-2026-005/) - Entry 005 · Catalan's constant, 1 − 1/9 + 1/25 − 1/49 + …, is not a fraction. (Formal proof listed)
- [Goldfeld’s conjecture: densities and mean analytic rank](https://closedform.nishachay.tech/e/openai-math-2026-006/) - Entry 006 · Among the quadratic twists of any elliptic curve, half are claimed to have analytic rank 0 and half rank 1. (Claimed, not checked)
- [Ordinary two-point correlations and the corrected Elliott conjecture](https://closedform.nishachay.tech/e/openai-math-2026-007/) - Entry 007 · Neighboring numbers stop agreeing on prime-factor parity. (Formal proof listed)
- [The Deligne–Drinfeld conjecture](https://closedform.nishachay.tech/e/openai-math-2026-008/) - Entry 008 · A short list of braid equations has exactly the solutions Deligne and Drinfeld predicted, in every size. (Formal proof listed)
- [Function-field reconstruction from Milnor K-theory and Galois data](https://closedform.nishachay.tech/e/openai-math-2026-009/) - Entry 009 · A field of functions can be rebuilt from its multiplication, blurred by powers, plus one trace of addition. (Formal proof listed)
- [Unrestricted pro-modularity at the prime two](https://closedform.nishachay.tech/e/openai-math-2026-010/) - Entry 010 · Odd two-dimensional 2-adic Galois representations are claimed to come from modular forms, with no condition mod 2. (Claimed, not checked)
- [Prime-factor statistics of p-1](https://closedform.nishachay.tech/e/openai-math-2026-011/) - Entry 011 · The prime factors of p − 1, for a random prime p, are claimed to split like a randomly broken stick. (Claimed, not checked)
- [Independent largest prime factors of consecutive integers](https://closedform.nishachay.tech/e/openai-math-2026-012/) - Entry 012 · The largest prime factors of n and n + 1 act like independent dice, so each wins half the time. (Formal proof listed)
- [Ostmann’s inverse Goldbach conjecture](https://closedform.nishachay.tech/e/openai-math-2026-013/) - Entry 013 · The primes cannot be built by adding two sets of whole numbers. (Formal proof listed)
- [Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan](https://closedform.nishachay.tech/e/openai-math-2026-014/) - Entry 014 · Linked papers claim the restricted geometric Langlands equivalence in characteristic p, plus Ramanujan bounds. (Claimed, not checked)
- [Torus-packet equidistribution in prime, quartic, and sextic degrees](https://closedform.nishachay.tech/e/openai-math-2026-015/) - Entry 015 · Number-field grids spread evenly over all grid shapes in prime degrees from five, and in degrees four and six. (Formal proof listed)
- [Zilber–Pink in abelian varieties and the Siegel threefold](https://closedform.nishachay.tech/e/openai-math-2026-016/) - Entry 016 · Unlikely intersections in abelian varieties are claimed to be finite in number, in every dimension. (Claimed, not checked)
- [The irrationality exponent of π is 2](https://closedform.nishachay.tech/e/openai-math-2026-017/) - Entry 017 · Fractions can never get unusually close to π. (Formal proof listed)
- [The Margulis–Platonov conjecture over global fields](https://closedform.nishachay.tech/e/openai-math-2026-018/) - Entry 018 · Normal subgroups of these simple groups over any global field are claimed to come from finitely many local pieces. (Claimed, not checked)
- [The local p -adic section conjecture and global consequences](https://closedform.nishachay.tech/e/openai-math-2026-019/) - Entry 019 · Points on p-adic curves of genus two or more are claimed to match exactly the sections of their fundamental group. (Claimed, not checked)
- [Squarefree quartics and power-free polynomial values](https://closedform.nishachay.tech/e/openai-math-2026-020/) - Entry 020 · Whole-number quartics like n⁴ + 2 take squarefree values at exactly the share the local guess predicts. (Formal proof listed)
- [A quadratic bound for Jacobsthal’s function](https://closedform.nishachay.tech/e/openai-math-2026-021/) - Entry 021 · Every run of about k² whole numbers holds one sharing no factor with a number that has k prime factors. (Formal proof listed)
- [The weak inhomogeneous Duffin–Schaeffer conjecture](https://closedform.nishachay.tech/e/openai-math-2026-022/) - Entry 022 · For any fixed shift, a weighted sum decides whether almost every number is well approximated. (Claimed, not checked)
- [Patterson's first moment for cubic Gauss sums](https://closedform.nishachay.tech/e/openai-math-2026-023/) - Entry 023 · Cubic Gauss sums over primes lean positive at exactly the rate Patterson predicted in 1978. (Formal proof listed)
- [An asymptotic formula for the number of totients](https://closedform.nishachay.tech/e/openai-math-2026-024/) - Entry 024 · Counting totient values up to x gets an exact leading formula, and doubling x doubles the count in the limit. (Formal proof listed)
- [Short Egyptian fractions](https://closedform.nishachay.tech/e/openai-math-2026-025/) - Entry 025 · Every fraction below 1 is a sum of about log log b distinct unit fractions, as Erdős conjectured. (Formal proof listed)
- [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)
- [Potential integral density on curve character varieties](https://closedform.nishachay.tech/e/openai-math-2026-027/) - Entry 027 · On every curve, whole-number matrix rules spread across the character variety after one enlargement of the integers. (Formal proof listed)
- [Uniformly bounded components of Gaussian-prime graphs](https://closedform.nishachay.tech/e/openai-math-2026-028/) - Entry 028 · No walk through Gaussian primes with steps of bounded length can go on forever. (Formal proof listed)
- [Primitive roots for every admissible integer base](https://closedform.nishachay.tech/e/openai-math-2026-029/) - Entry 029 · Every whole number except −1 and perfect squares is claimed to be a primitive root for infinitely many primes. (Claimed, not checked)
- [Modularity of elliptic curves over imaginary quadratic fields](https://closedform.nishachay.tech/e/openai-math-2026-030/) - Entry 030 · Every elliptic curve over every imaginary quadratic field is claimed to match an automorphic form, place by place. (Claimed, not checked)
- [Uchida’s conjecture for open homomorphisms of Galois groups](https://closedform.nishachay.tech/e/openai-math-2026-031/) - Entry 031 · Every open map between these Galois groups is claimed to come from a unique map between the fields. (Claimed, not checked)

### Algebraic and complex geometry (36)

- [The rational Hodge conjecture for CM abelian varieties](https://closedform.nishachay.tech/e/openai-math-2026-032/) - Entry 032 · The rational Hodge conjecture is claimed for every abelian variety with complex multiplication, in every dimension. (Claimed, not checked)
- [Iitaka subadditivity, variation, and logarithmic additivity](https://closedform.nishachay.tech/e/openai-math-2026-033/) - Entry 033 · Twisting a family of shapes never destroys the forms its fibers and base contribute. (Formal proof listed)
- [Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity](https://closedform.nishachay.tech/e/openai-math-2026-034/) - Entry 034 · A family of 14 papers claims log abundance in characteristic zero and the effective Iitaka fibration conjecture. (Claimed, not checked)
- [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)
- [Numerical semiampleness and generalized minimal models](https://closedform.nishachay.tech/e/openai-math-2026-036/) - Entry 036 · Nef adjoint divisors with an extra nef part are claimed to be numerically semiample; minimal models are claimed too. (Claimed, not checked)
- [The ordinary-double-point volume gap](https://closedform.nishachay.tech/e/openai-math-2026-037/) - Entry 037 · Every singular point of a mild kind is at most as roomy as a double cone, in any dimension. (Claimed, not checked)
- [Fujita’s freeness conjecture](https://closedform.nishachay.tech/e/openai-math-2026-038/) - Entry 038 · On every smooth shape, n + 1 copies of a positive bundle are always enough for Fujita's freeness. (Claimed, not checked)
- [Nagata’s conjecture and maximal Seshadri constants](https://closedform.nishachay.tech/e/openai-math-2026-039/) - Entry 039 · Nagata's 1959 rule for curves through random points holds for every count from 10 points up. (Formal proof listed)
- [Bloch’s conjecture for complex surfaces](https://closedform.nishachay.tech/e/openai-math-2026-040/) - Entry 040 · On a complex surface with no holomorphic two-forms, zero-cycles carry only their degree and Albanese data. (Claimed, not checked)
- [Hyperkähler SYZ and projective-space bases](https://closedform.nishachay.tech/e/openai-math-2026-041/) - Entry 041 · Every nonzero nef isotropic line bundle on a compact hyperkähler manifold is claimed to give a torus fibration. (Claimed, not checked)
- [Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII](https://closedform.nishachay.tech/e/openai-math-2026-042/) - Entry 042 · A claimed proof that every complex K3 surface is Oka: local holomorphic maps into it can be copied by entire maps. (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)
- [Shafarevich counterexamples in dimension two and with large fundamental group](https://closedform.nishachay.tech/e/openai-math-2026-046/) - Entry 046 · Two claimed counterexamples to Shafarevich's conjecture: a surface and a fourfold whose universal covers fail convexity. (Claimed, not checked)
- [Zariski cancellation and affine fibrations over the complex numbers](https://closedform.nishachay.tech/e/openai-math-2026-047/) - Entry 047 · A four-dimensional shape that is not ordinary space, until you add one line. (Formal proof listed)
- [A characteristic-zero counterexample to Lipman–Zariski](https://closedform.nishachay.tech/e/openai-math-2026-048/) - Entry 048 · A singular complex surface whose vector fields still form a free basis, against the Lipman–Zariski conjecture. (Claimed, not checked)
- [A stable-coordinate counterexample in four variables](https://closedform.nishachay.tech/e/openai-math-2026-049/) - Entry 049 · A polynomial that becomes a coordinate with one extra variable, but not without it, disproves two conjectures. (Formal proof listed)
- [A counterexample to Griffiths’ positivity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-050/) - Entry 050 · Algebraic positivity need not come with positive curvature, even on the simple surface built from two spheres. (Formal proof listed)
- [Kobayashi’s canonical-ampleness conjecture](https://closedform.nishachay.tech/e/openai-math-2026-051/) - Entry 051 · Compact Kähler manifolds with no entire curves have ample canonical bundle, as Kobayashi conjectured. (Claimed, not checked)
- [Tangent splittings and product decompositions](https://closedform.nishachay.tech/e/openai-math-2026-052/) - Entry 052 · Two integrable families of directions on a compact Kähler manifold make its universal cover a product. (Formal proof listed)
- [A counterexample to Pixton completeness in Chow](https://closedform.nishachay.tech/e/openai-math-2026-053/) - Entry 053 · A tautological relation in genus 10^60 that Pixton’s original relations miss, in Chow and in cohomology. (Claimed, not checked)
- [Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations](https://closedform.nishachay.tech/e/openai-math-2026-054/) - Entry 054 · Very general cubic fourfolds of large admissible discriminant are irrational despite a K3 derived category. (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)
- [Termination of projective and Kähler fourfold minimal model programs](https://closedform.nishachay.tech/e/openai-math-2026-056/) - Entry 056 · Every permitted minimal model program on a projective log canonical fourfold claims to stop after finitely many steps. (Claimed, not checked)
- [Fundamental groups of special complex varieties and root orbifolds](https://closedform.nishachay.tech/e/openai-math-2026-057/) - Entry 057 · Special compact Kähler manifolds claim virtually abelian fundamental groups in every dimension, as Campana predicted. (Claimed, not checked)
- [Semialgebraic universal covers and bounded domains](https://closedform.nishachay.tech/e/openai-math-2026-058/) - Entry 058 · Unrolled shapes with a polynomial description split into just three kinds of pieces. (Formal proof listed)
- [Counterexamples to Zariski’s multiplicity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-059/) - Entry 059 · Two hypersurface germs with the same embedded topology but different multiplicities, against Zariski’s 1971 question. (Claimed, not checked)
- [The Global Spherical Shell conjecture](https://closedform.nishachay.tech/e/openai-math-2026-060/) - Entry 060 · Every minimal class VII surface with positive second Betti number contains a global spherical shell. (Claimed, not checked)
- [Projective contact classification and the LeBrun–Salamon conjecture](https://closedform.nishachay.tech/e/openai-math-2026-062/) - Entry 062 · Contact Fano manifolds are homogeneous, which settles the LeBrun–Salamon conjecture on quaternionic-Kähler spaces. (Claimed, not checked)
- [The generalized Mukai conjecture](https://closedform.nishachay.tech/e/openai-math-2026-063/) - Entry 063 · A claimed proof that rational curves on a Fano manifold cap its Picard number, with equality only for product spaces. (Claimed, not checked)
- [Topological triviality of μ -constant surface singularities](https://closedform.nishachay.tech/e/openai-math-2026-064/) - Entry 064 · A claimed proof that surface singularities with a constant Milnor number keep the same topology along a family. (Claimed, not checked)
- [Virasoro constraints for complete intersections and projective-bundle towers](https://closedform.nishachay.tech/e/openai-math-2026-065/) - Entry 065 · Claimed proofs that curve counts on complete intersections and on projective bundles obey the Virasoro equations. (Claimed, not checked)
- [Bounded klt complements for Fano contractions](https://closedform.nishachay.tech/e/openai-math-2026-066/) - Entry 066 · One complement index, fixed by dimension and singularity bound, works for every mildly singular Fano contraction. (Formal proof listed)
- [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)
- [Anticanonical nonvanishing in every dimension](https://closedform.nishachay.tech/e/openai-math-2026-068/) - Entry 068 · A claimed proof that a smooth nonnegatively curved anticanonical bundle always has a section of some power. (Claimed, not checked)
- [Global quantum geometric Langlands at irrational level](https://closedform.nishachay.tech/e/openai-math-2026-069/) - Entry 069 · A claimed proof of quantum geometric Langlands duality on any smooth complex curve at every irrational level. (Claimed, not checked)

### Real and complex analysis (16)

- [Koebe’s circle-domain conjecture](https://closedform.nishachay.tech/e/openai-math-2026-071/) - Entry 071 · Every region of the sphere, however many holes it has, can be reshaped so that each hole is round. (Formal proof listed)
- [Brennan's conjecture and the integral-means spectrum](https://closedform.nishachay.tech/e/openai-math-2026-072/) - Entry 072 · Reshaping a region into a disk can never squeeze too hard: Brennan's 1978 conjecture is proved. (Formal proof listed)
- [The Falconer distance conjecture](https://closedform.nishachay.tech/e/openai-math-2026-073/) - Entry 073 · Any closed, bounded set whose dimension is more than half that of its space has distances with positive total length. (Formal proof listed)
- [Kakeya in three and four dimensions](https://closedform.nishachay.tech/e/openai-math-2026-074/) - Entry 074 · Claimed proofs: the Kakeya maximal conjecture in 3D, and full dimension for every Kakeya set in 4D. (Claimed, not checked)
- [The L log L Fourier-convergence conjecture](https://closedform.nishachay.tech/e/openai-math-2026-075/) - Entry 075 · Fourier sums of every L log L function converge to it at almost every point of the circle. (Formal proof listed)
- [Real ultraflat Littlewood polynomials and unbounded binary merit factors](https://closedform.nishachay.tech/e/openai-math-2026-076/) - Entry 076 · Polynomials with only +1 and −1 coefficients can be almost perfectly flat on the unit circle. (Formal proof listed)
- [Fourier restriction for positively curved surfaces](https://closedform.nishachay.tech/e/openai-math-2026-077/) - Entry 077 · Claimed proofs of the full Fourier restriction range p > 3 for the sphere and for positively curved surfaces. (Claimed, not checked)
- [The three-dimensional Bochner–Riesz conjecture](https://closedform.nishachay.tech/e/openai-math-2026-078/) - Entry 078 · A claimed proof of the Bochner–Riesz conjecture in three dimensions, in its strict-order form. (Claimed, not checked)
- [Local smoothing in three dimensions](https://closedform.nishachay.tech/e/openai-math-2026-079/) - Entry 079 · A claimed proof of Sogge's local smoothing conjecture for waves in 3D space, including the critical case p = 3. (Claimed, not checked)
- [The exact Sobolev endpoint for Schrödinger convergence](https://closedform.nishachay.tech/e/openai-math-2026-080/) - Entry 080 · Claimed proofs that Schrödinger waves return to rough data at the exact smoothness threshold, in dimensions n ≥ 2. (Claimed, not checked)
- [Riesz transforms and rectifiability in higher codimension](https://closedform.nishachay.tech/e/openai-math-2026-081/) - Entry 081 · A bounded Riesz transform forces surfaces in higher codimension to contain big flat-like pieces. (Formal proof listed)
- [Annular variation and dyadic absolute bounds for the triangular Hilbert transform](https://closedform.nishachay.tech/e/openai-math-2026-082/) - Entry 082 · A triangle-shaped singular integral stays bounded at the symmetric L³ point, whatever cutoffs each point picks. (Formal proof listed)
- [Hilbert transforms along Lipschitz directions](https://closedform.nishachay.tech/e/openai-math-2026-083/) - Entry 083 · Hilbert transforms along any Lipschitz direction field in the plane obey one energy bound at short lengths. (Formal proof listed)
- [The geometric case of the Erdős similarity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-084/) - Entry 084 · A set covering almost all of [0, 1] can dodge every copy of a geometric list like 1/2, 1/4, 1/8, … (Formal proof listed)
- [Endpoint Sobolev regularity of centered disk averages](https://closedform.nishachay.tech/e/openai-math-2026-085/) - Entry 085 · Taking the best disk average at each point never multiplies a surface's total slope by more than a fixed factor. (Formal proof listed)
- [An L 3 bound for the trilinear Hilbert transform](https://closedform.nishachay.tech/e/openai-math-2026-086/) - Entry 086 · A singular average of three shifted functions obeys an L³ × L³ × L³ to L¹ size bound, the paper claims. (Claimed, not checked)

### Convex and metric geometry (15)

- [The Mahler conjectures, functional inequalities and polar-product symplectic width](https://closedform.nishachay.tech/e/openai-math-2026-087/) - Entry 087 · Symmetric convex bodies cannot beat Mahler's volume bound. (Formal proof listed)
- [Sharp projection-body inequalities and a counterexample to simplex maximization](https://closedform.nishachay.tech/e/openai-math-2026-088/) - Entry 088 · In four or more dimensions, ellipsoids have the smallest shadow body for their volume. (Formal proof listed)
- [Bounded-distortion L 1 embeddings of planar and bounded-treewidth graphs](https://closedform.nishachay.tech/e/openai-math-2026-089/) - Entry 089 · Every planar road network can be redrawn in taxicab distance with stretch below one fixed number. (Formal proof listed)
- [Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy](https://closedform.nishachay.tech/e/openai-math-2026-090/) - Entry 090 · No arrangement of points in the plane beats the triangular lattice for any steadily fading repulsion. (Formal proof listed)
- [Logarithmic and L p Brunn–Minkowski inequalities and the B-conjecture](https://closedform.nishachay.tech/e/openai-math-2026-091/) - Entry 091 · Blending two symmetric shapes the thrifty way never loses volume below their geometric mean. (Formal proof listed)
- [The optimal order of convex-body covering density](https://closedform.nishachay.tech/e/openai-math-2026-092/) - Entry 092 · The worst convex shape needs covering density about n log n in dimension n, lattice or not. (Formal proof listed)
- [Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://closedform.nishachay.tech/e/openai-math-2026-093/) - Entry 093 · Log-concave laws with bell-curve tails in every direction obey a dimension-free log-Sobolev bound, the paper claims. (Claimed, not checked)
- [Subpolynomial dimension reduction in L p](https://closedform.nishachay.tech/e/openai-math-2026-094/) - Entry 094 · Points in Lp space can be squeezed into far fewer coordinates if distances may stretch slightly. (Formal proof listed)
- [Hyperbolicity cones without semidefinite lifts](https://closedform.nishachay.tech/e/openai-math-2026-095/) - Entry 095 · A claimed proof that some hyperbolicity cones have no semidefinite description, even with hidden variables. (Formal proof listed)
- [The Gaussian propeller conjecture in every dimension](https://closedform.nishachay.tech/e/openai-math-2026-096/) - Entry 096 · Three 120-degree wedges are the best way to cut a bell-curve cloud, in every dimension. (Formal proof listed)
- [The Euclidean Steinitz–Bergström bound](https://closedform.nishachay.tech/e/openai-math-2026-097/) - Entry 097 · One choice of signs keeps every running total of unit vectors within a constant times √d, at any length. (Formal proof listed)
- [Compact counterexamples to bi-Lipschitz dimension reduction](https://closedform.nishachay.tech/e/openai-math-2026-098/) - Entry 098 · Some uncrowded point sets in infinite dimensions cannot be copied faithfully into any finite dimension. (Formal proof listed)
- [The sharp exponential scale of edit-distance distortion](https://closedform.nishachay.tech/e/openai-math-2026-099/) - Entry 099 · Edit distance between words can be drawn as ℓ1 distance only with distortion of size exp(√(log d·log log d)). (Formal proof listed)
- [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)
- [The sharp simplex conjecture for isotropic constants](https://closedform.nishachay.tech/e/openai-math-2026-101/) - Entry 101 · In every dimension, simplices alone have the largest isotropic constant among convex bodies, the paper claims. (Claimed, not checked)

### Theoretical computer science (40)

- [The Unique Games Conjecture and optimal approximation thresholds](https://closedform.nishachay.tech/e/openai-math-2026-102/) - Entry 102 · Telling almost-solvable puzzles from hopeless ones can be impossibly hard. (Formal proof listed)
- [Exact derandomization of logarithmic space: L=RL=BPL](https://closedform.nishachay.tech/e/openai-math-2026-103/) - Entry 103 · Coin flips give no extra power to computers with tiny memory: L = RL = BPL. (Formal proof listed)
- [Quasipolynomial algorithms for mean-payoff, stochastic and parity games](https://closedform.nishachay.tech/e/openai-math-2026-104/) - Entry 104 · Who wins a mean-payoff game can be found in quasipolynomial time, however large the scores. (Formal proof listed)
- [Perfect completeness for 2-to-1 games](https://closedform.nishachay.tech/e/openai-math-2026-105/) - Entry 105 · Telling fully solvable 2-to-1 games from nearly worthless ones is NP-hard. (Formal proof listed)
- [Hardness of coloring three-colorable graphs](https://closedform.nishachay.tech/e/openai-math-2026-106/) - Entry 106 · Even a three-colorable graph can be NP-hard to tell from one with only tiny independent sets. (Formal proof listed)
- [Matrix multiplication with exponent at most 9/4](https://closedform.nishachay.tech/e/openai-math-2026-107/) - Entry 107 · Huge n-by-n complex matrices can be multiplied in about n^(9/4) arithmetic steps, up to any tiny slack. (Formal proof listed)
- [A cubic permanent–determinant lower bound](https://closedform.nishachay.tech/e/openai-math-2026-108/) - Entry 108 · Disguising the permanent as a determinant needs a matrix that grows at least like the cube of its size. (Formal proof listed)
- [Integer multiplication below n log n](https://closedform.nishachay.tech/e/openai-math-2026-109/) - Entry 109 · A tape-based computer can multiply huge whole numbers slightly faster than the long-conjectured n log n limit. (Claimed, not checked)
- [Optimal-order randomized k -server on arbitrary metrics](https://closedform.nishachay.tech/e/openai-math-2026-110/) - Entry 110 · Randomized k-server costs at most a squared-logarithm factor on every space, matching the known lower bound. (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)
- [Beyond the square-root exponent for depth-three circuits](https://closedform.nishachay.tech/e/openai-math-2026-112/) - Entry 112 · One fast-to-decide problem needs three-layer circuits bigger than any 2^(A√n) for large inputs. (Formal proof listed)
- [Approximate counting and entropy of perfect matchings](https://closedform.nishachay.tech/e/openai-math-2026-113/) - Entry 113 · A fast coin-flipping method estimates the number of perfect matchings in any network, to any chosen accuracy. (Formal proof listed)
- [Approximate counting of common integer polymatroid bases](https://closedform.nishachay.tech/e/openai-math-2026-114/) - Entry 114 · Shared solutions of two capacity rules can be counted approximately, at a cost set by digits, not size. (Formal proof listed)
- [Sampling and counting contingency tables with arbitrary margins](https://closedform.nishachay.tech/e/openai-math-2026-115/) - Entry 115 · Number grids with fixed row and column sums can be sampled fairly and counted fast, whatever their size. (Formal proof listed)
- [Uniform black-box noncommutative identity testing across characteristics](https://closedform.nishachay.tech/e/openai-math-2026-116/) - Entry 116 · One fixed set of matrices exposes every nonzero small formula whose variables do not commute. (Formal proof listed)
- [Uniform sparsest cut: hardness and semidefinite gaps](https://closedform.nishachay.tech/e/openai-math-2026-117/) - Entry 117 · Finding a network's weakest split even roughly is claimed NP-hard, and a key relaxation can miss by almost √log n. (Formal proof listed)
- [Bin packing and unbounded configuration-LP gaps](https://closedform.nishachay.tech/e/openai-math-2026-118/) - Entry 118 · Rounding up the fractional bin count can be wrong by any fixed number of bins. (Formal proof listed)
- [The Courtade–Kumar and Hellinger conjectures](https://closedform.nishachay.tech/e/openai-math-2026-119/) - Entry 119 · No yes-or-no summary of random bits survives noise better than simply keeping one bit. (Formal proof listed)
- [Almost-linear-time exact matching and prescribed-degree factors in general graphs](https://closedform.nishachay.tech/e/openai-math-2026-120/) - Entry 120 · A randomized algorithm finds a largest matching in any graph in almost-linear time, the paper claims. (Claimed, not checked)
- [Almost-linear approximation of edit distance](https://closedform.nishachay.tech/e/openai-math-2026-121/) - Entry 121 · Edit distance can be estimated to within any fixed accuracy in barely more than linear time. (Formal proof listed)
- [Quantitative trace-reconstruction bounds with a uniform decoder](https://closedform.nishachay.tech/e/openai-math-2026-122/) - Entry 122 · Rebuilding a string from randomly thinned copies needs more than polynomially many copies. (Formal proof listed)
- [Polynomial-time scheduling on three identical machines](https://closedform.nishachay.tech/e/openai-math-2026-124/) - Entry 124 · The best schedule for one-hour jobs with order rules on three machines can be found in polynomial time. (Formal proof listed)
- [The metric k -median approximation threshold and recovery](https://closedform.nishachay.tech/e/openai-math-2026-125/) - Entry 125 · The best possible guarantee for placing k facilities is exactly 1 + 2/e, assuming P ≠ NP. (Formal proof listed)
- [Exponential semidefinite complexity of perfect matching](https://closedform.nishachay.tech/e/openai-math-2026-126/) - Entry 126 · Exact semidefinite descriptions of perfect matching need exponential size, a new paper claims. (Formal proof listed)
- [Average sensitivity of polynomial threshold functions](https://closedform.nishachay.tech/e/openai-math-2026-127/) - Entry 127 · Yes-or-no rules built from degree-d polynomials on n inputs have average sensitivity at most 8d√n. (Formal proof listed)
- [A factor-two approximation for shortest common superstring](https://closedform.nishachay.tech/e/openai-math-2026-128/) - Entry 128 · A fast method builds a string containing all given pieces, never more than twice the shortest possible. (Formal proof listed)
- [Exponential state costs for two-way automata](https://closedform.nishachay.tech/e/openai-math-2026-129/) - Entry 129 · Removing guesses from simple two-way reading machines can cost exponentially many states. (Formal proof listed)
- [Exact Fourier transforms below n log n](https://closedform.nishachay.tech/e/openai-math-2026-130/) - Entry 130 · In exact arithmetic, the discrete Fourier transform can be computed in slightly fewer than n log n steps. (Formal proof listed)
- [Rapid mixing of graph switches for every degree sequence](https://closedform.nishachay.tech/e/openai-math-2026-131/) - Entry 131 · Random rewiring reaches a fair random network quickly, for every possible list of degrees. (Formal proof listed)
- [A superquadratic separation of sensitivity and block sensitivity](https://closedform.nishachay.tech/e/openai-math-2026-132/) - Entry 132 · Block sensitivity can outgrow any constant times sensitivity squared. (Formal proof listed)
- [The computational complexity of Weisfeiler–Leman refinement](https://closedform.nishachay.tech/e/openai-math-2026-133/) - Entry 133 · Deciding Weisfeiler–Leman equivalence at dimension k needs time n to a power proportional to k. (Formal proof listed)
- [Generalized star height at most three](https://closedform.nishachay.tech/e/openai-math-2026-134/) - Entry 134 · Three nested stars are enough to describe any regular language once complement is allowed. (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)
- [A quasilinear PCP theorem for PPAD](https://closedform.nishachay.tech/e/openai-math-2026-136/) - Entry 136 · End-of-Line puzzles turn into number circuits only slightly bigger that tolerate many broken gates, the paper claims. (Claimed, not checked)
- [One-tape time simulation in two-fifths-power space](https://closedform.nishachay.tech/e/openai-math-2026-137/) - Entry 137 · A one-tape machine's run can be replayed with far less memory than the old square-root rule allowed. (Formal proof listed)
- [Subset Sum in O(2^(0.49n)) time](https://closedform.nishachay.tech/e/openai-math-2026-138/) - Entry 138 · A randomized algorithm claims to decide every Subset Sum instance in O(2^(0.49n)) time, below the 2^(n/2) scale. (Claimed, not checked)
- [Subpolynomial query complexity for log-concave sampling](https://closedform.nishachay.tech/e/openai-math-2026-139/) - Entry 139 · Sampling from a gently curved bowl needs no fixed power of the dimension in queries. (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)
- [Existential–universal real sentences in the counting hierarchy](https://closedform.nishachay.tech/e/openai-math-2026-141/) - Entry 141 · Deciding “there is x so that for every y” questions about real numbers claimed at one fixed counting level. (Claimed, not checked)
- [Deterministic polynomial factorization over prime fields](https://closedform.nishachay.tech/e/openai-math-2026-142/) - Entry 142 · A claimed coin-free algorithm that factors any polynomial modulo a prime in time polynomial in the input size. (Claimed, not checked)

### Dynamical systems and ergodic theory (12)

- [Hilbert's sixteenth problem: uniform bounds for limit cycles](https://closedform.nishachay.tech/e/openai-math-2026-143/) - Entry 143 · Claims a finite cap, set by degree alone, on limit cycles of planar polynomial systems. (Formal proof listed)
- [Banach’s simple Lebesgue-spectrum problem](https://closedform.nishachay.tech/e/openai-math-2026-144/) - Entry 144 · A smooth map of a 3D torus where one measurement’s past and future build every average-zero measurement. (Formal proof listed)
- [Rokhlin’s multiple-mixing problem](https://closedform.nishachay.tech/e/openai-math-2026-145/) - Entry 145 · If long waits make any two moments independent, they make every finite group of moments independent. (Formal proof listed)
- [Positive metric entropy for the standard map](https://closedform.nishachay.tech/e/openai-math-2026-146/) - Entry 146 · The standard map is chaotic on a region of positive area for every large enough parameter. (Formal proof listed)
- [The near-boundary Birkhoff conjecture](https://closedform.nishachay.tech/e/openai-math-2026-147/) - Entry 147 · Two papers claim that a smooth billiard table with an unbroken band of caustics along its edge must be an ellipse. (Claimed, not checked)
- [The entropy-rate dimension formula for self-similar measures](https://closedform.nishachay.tech/e/openai-math-2026-148/) - Entry 148 · A single formula gives the dimension of every self-similar measure on the line, even when its pieces collide. (Formal proof listed)
- [Classwise permanence for weakly reversible mass-action systems](https://closedform.nishachay.tech/e/openai-math-2026-149/) - Entry 149 · With fixed rates, every weakly reversible chemical network herds each class of states into one common window. (Formal proof listed)
- [Weak mixing of triangular billiards with an irrational angle](https://closedform.nishachay.tech/e/openai-math-2026-150/) - Entry 150 · A paper claims billiards in every triangle with an irrational angle forget their start: they are weakly mixing. (Formal proof listed)
- [A C 1 counterexample to the entropy conjecture](https://closedform.nishachay.tech/e/openai-math-2026-151/) - Entry 151 · A smooth-enough map can double a shape's homology while its orbits stay orderly. (Formal proof listed)
- [Zero entropy does not guarantee a smooth positive-volume model](https://closedform.nishachay.tech/e/openai-math-2026-152/) - Entry 152 · An abstract zero-entropy system with no smooth, volume-keeping copy in any dimension. (Formal proof listed)
- [Arithmetic classification and non-Pisot singularity for Bernoulli convolutions](https://closedform.nishachay.tech/e/openai-math-2026-153/) - Entry 153 · Claimed singular Bernoulli convolutions beyond Pisot reciprocals, plus an arithmetic test for every parameter. (Claimed, not checked)
- [Pointwise multiple ergodic averages for mixing transformations](https://closedform.nishachay.tech/e/openai-math-2026-154/) - Entry 154 · For mixing systems, averages along k, 2k, …, nk are claimed to settle at almost every starting point, for every n. (Claimed, not checked)

### Combinatorics (37)

- [A counterexample to periodic tiling in dimension three](https://closedform.nishachay.tech/e/openai-math-2026-155/) - Entry 155 · Proves a finite tile in three dimensions can tile space, yet never in a fully repeating pattern. (Formal proof listed)
- [Borsuk's conjecture fails in dimension nine](https://closedform.nishachay.tech/e/openai-math-2026-156/) - Entry 156 · A set in nine-dimensional space cannot be cut into ten smaller pieces, so Borsuk's conjecture fails there. (Formal proof listed)
- [Graph coloring, clique minors, and Colin de Verdière invariants](https://closedform.nishachay.tech/e/openai-math-2026-157/) - Entry 157 · Proves Hadwiger's 1943 coloring conjecture is false, and that a fixed multiple of the minor size still bounds coloring. (Formal proof listed)
- [The Euclidean plane cannot be colored with five colors](https://closedform.nishachay.tech/e/openai-math-2026-158/) - Entry 158 · No way of coloring the plane with five colors keeps every unit-distance pair apart. (Formal proof listed)
- [Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds](https://closedform.nishachay.tech/e/openai-math-2026-159/) - Entry 159 · Any set of whole numbers whose reciprocals add to infinity holds evenly spaced runs of every length. (Formal proof listed)
- [Superexponential van der Waerden numbers](https://closedform.nishachay.tech/e/openai-math-2026-160/) - Entry 160 · Proves van der Waerden numbers grow faster than any exponential, answering a question of Erdős for two colors. (Formal proof listed)
- [Counterexamples to Sidorenko’s conjecture and the forcing conjecture](https://closedform.nishachay.tech/e/openai-math-2026-161/) - Entry 161 · A 35-vertex bipartite pattern appears less often in some graph than random would predict. (Formal proof listed)
- [Counterexamples to Ryser’s covering conjecture](https://closedform.nishachay.tech/e/openai-math-2026-162/) - Entry 162 · Ryser's covering conjecture is false: some intersecting hypergraphs with r parts need r vertices to cover. (Formal proof listed)
- [Hindman’s finite sums and products conjecture](https://closedform.nishachay.tech/e/openai-math-2026-164/) - Entry 164 · However whole numbers are colored, some set of each size has its sums and products in one color, the paper claims. (Claimed, not checked)
- [The Harary–Hill and Zarankiewicz crossing-number formulas](https://closedform.nishachay.tech/e/openai-math-2026-165/) - Entry 165 · The classic crossing formulas for drawing complete networks are exact. (Formal proof listed)
- [The higher-dimensional Erdős distinct-distances conjecture](https://closedform.nishachay.tech/e/openai-math-2026-166/) - Entry 166 · Any n points in d ≥ 3 dimensions span at least a constant times n^(2/d) distinct distances, the paper claims. (Claimed, not checked)
- [Planar distinct distances and unit-distance bounds](https://closedform.nishachay.tech/e/openai-math-2026-167/) - Entry 167 · Proves almost every point in a large planar set sees nearly n distances, and lowers the unit-distance exponent 4/3. (Formal proof listed)
- [Combinatorial invariance of Kazhdan–Lusztig polynomials](https://closedform.nishachay.tech/e/openai-math-2026-168/) - Entry 168 · Proves a Kazhdan–Lusztig polynomial depends only on the shape of its Bruhat interval, settling combinatorial invariance. (Formal proof listed)
- [Shareshian–Wachs elementary positivity](https://closedform.nishachay.tech/e/openai-math-2026-169/) - Entry 169 · A coloring count with an extra q-weight always splits into elementary pieces with nonnegative coefficients. (Formal proof listed)
- [Sharp logarithmic exponents for off-diagonal Ramsey numbers](https://closedform.nishachay.tech/e/openai-math-2026-170/) - Entry 170 · For every fixed s ≥ 5, the Ramsey number r(s,t) now has its exact power of log t, up to a vanishing error. (Formal proof listed)
- [The hypercube Ramsey conjecture](https://closedform.nishachay.tech/e/openai-math-2026-171/) - Entry 171 · Two-coloring the links among a constant times 2ⁿ points always gives a one-color n-dimensional cube, the paper claims. (Claimed, not checked)
- [Classification of finite Euclidean Ramsey configurations](https://closedform.nishachay.tech/e/openai-math-2026-172/) - Entry 172 · An exact algebraic test tells which finite point sets must appear in one color in any coloring of space. (Formal proof listed)
- [Seymour’s second-neighborhood conjecture](https://closedform.nishachay.tech/e/openai-math-2026-173/) - Entry 173 · In every network of one-way arrows, some point reaches at least as many points in two steps as in one. (Formal proof listed)
- [Deterministic construction of strong thin spanning trees](https://closedform.nishachay.tech/e/openai-math-2026-174/) - Entry 174 · Every k-connected network has a spanning tree that uses at most a fixed multiple of 1/k of every cut. (Formal proof listed)
- [Talagrand’s expectation thresholds, discrete convexity, and graph decompositions](https://closedform.nishachay.tech/e/openai-math-2026-175/) - Entry 175 · Spreading credit fractionally beats whole-set covers by at most a fixed factor in Talagrand's threshold theory. (Formal proof listed)
- [The second Kahn–Kalai conjecture with an edge-count bound](https://closedform.nishachay.tech/e/openai-math-2026-176/) - Entry 176 · A random graph contains a target graph soon after simple expected counts say it could, up to a log factor. (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)
- [The Erdős–Gallai cycle-decomposition conjecture](https://closedform.nishachay.tech/e/openai-math-2026-181/) - Entry 181 · Every network on n points splits into at most a fixed multiple of n loops and single links. (Formal proof listed)
- [Power savings for intersective polynomial differences and prime arguments](https://closedform.nishachay.tech/e/openai-math-2026-182/) - Entry 182 · Sets of whole numbers with no square difference are smaller than N by a fixed power of N. (Formal proof listed)
- [Power savings for planar halving lines and k -sets](https://closedform.nishachay.tech/e/openai-math-2026-183/) - Entry 183 · The maximum number of halving lines for n points in the plane is below n to the power 4/3 by a fixed margin. (Formal proof listed)
- [Correspondence coloring with a fixed forbidden subgraph](https://closedform.nishachay.tech/e/openai-math-2026-184/) - Entry 184 · Networks with no fixed-size clique have larger independent sets and need fewer colors, by a log factor. (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)
- [Uniform influence and sharp thresholds for graph and hypergraph properties](https://closedform.nishachay.tech/e/openai-math-2026-186/) - Entry 186 · Every symmetric increasing graph property switches on within a window of width about 1/(log n)². (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)
- [The sharp terminal leave in random triangle removal](https://closedform.nishachay.tech/e/openai-math-2026-188/) - Entry 188 · Randomly erasing triangles from a complete graph leaves about n^(3/2)/(2√2) edges at the end. (Formal proof listed)
- [Cycle–clique Ramsey numbers](https://closedform.nishachay.tech/e/openai-math-2026-189/) - Entry 189 · Exact Ramsey numbers for a cycle versus a clique: R(C_m, K_n) = (m − 1)(n − 1) + 1 when m ≥ n ≥ 3. (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)
- [A power improvement in the Heilbronn triangle lower bound](https://closedform.nishachay.tech/e/openai-math-2026-191/) - Entry 191 · Some n points in a unit square keep every triangle above n^(−2+η) in area, beating 1/n² by a fixed power. (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)

### Algebra (18)

- [Serre’s intersection-multiplicity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-193/) - Entry 193 · Serre's intersection multiplicity is claimed to be strictly positive over every regular local ring. (Claimed, not checked)
- [Lech’s multiplicity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-194/) - Entry 194 · Hilbert–Samuel multiplicity is claimed never to drop along a flat local map, settling Lech's 1960 conjecture. (Formal proof listed)
- [A counterexample to the small Cohen–Macaulay module conjecture](https://closedform.nishachay.tech/e/openai-math-2026-195/) - Entry 195 · A three-dimensional complete local domain over the complex numbers has no small Cohen–Macaulay module. (Formal proof listed)
- [A counterexample to Kaplansky’s zero-divisor conjecture](https://closedform.nishachay.tech/e/openai-math-2026-196/) - Entry 196 · A group with no looping moves still lets two nonzero sums multiply to zero. (Formal proof listed)
- [A torsion-free group algebra that is not directly finite](https://closedform.nishachay.tech/e/openai-math-2026-197/) - Entry 197 · In some group algebras, a left inverse is not a right inverse, which disproves Kaplansky's direct-finiteness conjecture. (Formal proof listed)
- [A counterexample to finitistic-dimension finiteness](https://closedform.nishachay.tech/e/openai-math-2026-198/) - Entry 198 · An algebra with finite resolutions of every length refutes the little finitistic-dimension conjecture. (Formal proof listed)
- [Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures](https://closedform.nishachay.tech/e/openai-math-2026-199/) - Entry 199 · A module with no self-extensions in any degree can still fail to be projective, against Auslander–Reiten and Tachikawa. (Formal proof listed)
- [Eisenbud–Green–Harris and lex-plus-powers](https://closedform.nishachay.tech/e/openai-math-2026-200/) - Entry 200 · Two papers claim the Eisenbud–Green–Harris and lex-plus-powers conjectures over every field of characteristic zero. (Claimed, not checked)
- [A counterexample to Kurosh’s division-ring problem](https://closedform.nishachay.tech/e/openai-math-2026-201/) - Entry 201 · A division ring made from two elements, each element algebraic, can still be infinite-dimensional over its center. (Claimed, not checked)
- [The blockwise Alperin weight conjecture](https://closedform.nishachay.tech/e/openai-math-2026-202/) - Entry 202 · A paper claims to prove Alperin's weight conjecture, block by block, for every finite group and every prime. (Claimed, not checked)
- [Donovan's conjecture over fields and complete mixed-characteristic DVRs](https://closedform.nishachay.tech/e/openai-math-2026-203/) - Entry 203 · Two papers claim Donovan's conjecture: blocks with bounded defect fall into finitely many types, for every prime. (Claimed, not checked)
- [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)
- [Saxl’s conjecture and universal tensor squares](https://closedform.nishachay.tech/e/openai-math-2026-205/) - Entry 205 · The staircase representation's tensor square contains every irreducible representation, proving Saxl's conjecture. (Formal proof listed)
- [Finite lattice representation and undecidability](https://closedform.nishachay.tech/e/openai-math-2026-206/) - Entry 206 · Some finite lattices are not the congruence lattice of any finite algebra, and no program can tell which. (Formal proof listed)
- [The ℓ¹-Bass conjecture for all discrete groups](https://closedform.nishachay.tech/e/openai-math-2026-207/) - Entry 207 · Idempotents in group rings carry no trace at infinite-order elements, for every group; Kaplansky's problem follows. (Formal proof listed)
- [Finite symmetric tensor categories and the Verlinde tower](https://closedform.nishachay.tech/e/openai-math-2026-208/) - Entry 208 · Every finite symmetric tensor category in characteristic p is claimed to map into one higher Verlinde category. (Claimed, not checked)
- [Integral counterexamples to Gersten’s conjecture](https://closedform.nishachay.tech/e/openai-math-2026-209/) - Entry 209 · Two ramified rings are claimed to break Gersten's conjecture: nonzero K-theory classes vanish over their fractions. (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)

### Probability and statistical mechanics (29)

- [The geometric phase diagram, diffusion, and spectra of random planar maps](https://closedform.nishachay.tech/e/openai-math-2026-211/) - Entry 211 · Random FK planar maps are claimed to look like random surfaces for q ≤ 4 and like a random tree above 4. (Formal proof listed)
- [Planar first-passage geometry and the absence of bigeodesics](https://closedform.nishachay.tech/e/openai-math-2026-212/) - Entry 212 · Planar random-metric grids are claimed to have no route that is fastest forever both ways, and a smooth limit shape. (Formal proof listed)
- [Critical percolation on every quasi-transitive graph](https://closedform.nishachay.tech/e/openai-math-2026-213/) - Entry 213 · At the exact tipping point, random networks of a broad, symmetric kind form no endless cluster. (Formal proof listed)
- [The Benjamini–Schramm nonuniqueness conjecture](https://closedform.nishachay.tech/e/openai-math-2026-214/) - Entry 214 · Percolation on every fast-spreading symmetric graph has a range with infinitely many endless clusters. (Formal proof listed)
- [Canonical O(3) continuum limit and exact O(4) mass asymptotics](https://closedform.nishachay.tech/e/openai-math-2026-215/) - Entry 215 · Spin models with three or more components are proved to lose alignment exponentially at every temperature. (Formal proof listed)
- [Critical and near-critical XY scaling and BKT universality](https://closedform.nishachay.tech/e/openai-math-2026-216/) - Entry 216 · Six papers claim exact critical laws for the planar XY model, as the Berezinskii–Kosterlitz–Thouless picture predicts. (Claimed, not checked)
- [The low-temperature Sherrington–Kirkpatrick fluctuation law](https://closedform.nishachay.tech/e/openai-math-2026-217/) - Entry 217 · The typical spread of the low-temperature Sherrington–Kirkpatrick free energy grows like n^(1/6), with a limiting law. (Claimed, not checked)
- [Conformal universality for weakly interacting and random-bond Ising models](https://closedform.nishachay.tech/e/openai-math-2026-218/) - Entry 218 · Weak extra interactions or weak random bonds leave the critical Ising interface's conformal limit, SLE₃, unchanged. (Formal proof listed)
- [GOE bulk universality for regular graphs with weak Anderson disorder](https://closedform.nishachay.tech/e/openai-math-2026-219/) - Entry 219 · Eigenvalue spacings of random regular graphs, even with weak random potentials, match those of Gaussian random matrices. (Claimed, not checked)
- [Directional zero–one laws beyond iid environments and iid ballisticity](https://closedform.nishachay.tech/e/openai-math-2026-220/) - Entry 220 · A walk in a random environment escapes in a given direction with probability 0 or 1, and escape forces positive speed. (Formal proof listed)
- [The Mézard–Parisi formula for diluted spin glasses](https://closedform.nishachay.tech/e/openai-math-2026-221/) - Entry 221 · The Mézard–Parisi cavity formula gives the exact free energy of sparse spin glasses in the Panchenko–Talagrand class. (Formal proof listed)
- [Perceptron free energies and microscopic jamming exponents](https://closedform.nishachay.tech/e/openai-math-2026-222/) - Entry 222 · Exact free energies for Ising and spherical perceptrons at every density, and the gap and force exponents at jamming. (Formal proof listed)
- [Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling](https://closedform.nishachay.tech/e/openai-math-2026-223/) - Entry 223 · Critical random-cluster interfaces on the square grid converge to SLE curves for every cluster weight up to 4. (Claimed, not checked)
- [Critical and quenched near-critical universality for Poisson–Voronoi percolation](https://closedform.nishachay.tech/e/openai-math-2026-224/) - Entry 224 · Cardy's formula holds for Poisson–Voronoi percolation, and near-critical thresholds match the triangular lattice. (Claimed, not checked)
- [Gaussian free field limits throughout the balanced six-vertex regime](https://closedform.nishachay.tech/e/openai-math-2026-225/) - Entry 225 · The balanced six-vertex height surface tends to a Gaussian free field for all c up to 2, with variance 1/arcsin(c/2). (Claimed, not checked)
- [The double-dimer loop ensemble converges to CLE 4](https://closedform.nishachay.tech/e/openai-math-2026-226/) - Entry 226 · Loops from two overlaid random domino tilings of the half-plane converge, as curves, to nested CLE₄. (Claimed, not checked)
- [Critical SK autocorrelation processes and dynamics across the temperature transition](https://closedform.nishachay.tech/e/openai-math-2026-227/) - Entry 227 · Heat-bath mixing in the SK spin glass is fast below β = 1, takes n^(2/3) at β = 1, and slows sharply above it. (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)
- [Exact Hausdorff gauges for SLE](https://closedform.nishachay.tech/e/openai-math-2026-230/) - Entry 230 · An explicit size rule gives every piece of an SLE random curve a positive, finite Hausdorff measure. (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)
- [Limiting random SAT thresholds, sharp variance and computability](https://closedform.nishachay.tech/e/openai-math-2026-235/) - Entry 235 · Random k-SAT has a single limiting satisfiability threshold for every k ≥ 3, and the 3-SAT one is computable. (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)
- [The three-quarter exponent for honeycomb self-avoiding walk](https://closedform.nishachay.tech/e/openai-math-2026-237/) - Entry 237 · Claims random self-avoiding walks of length n on the honeycomb lattice span about n^(3/4), the long-predicted exponent. (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)

### Mathematical logic (6)

- [Shelah's eventual categoricity and the prescribed-threshold obstruction](https://closedform.nishachay.tech/e/openai-math-2026-240/) - Entry 240 · Claims Shelah's eventual categoricity conjecture in ZFC, and shows under CH that its proposed exact threshold fails. (Formal proof listed)
- [Rigidity of the Turing degrees](https://closedform.nishachay.tech/e/openai-math-2026-241/) - Entry 241 · Claims every symmetry of the Turing degrees is trivial, settling the rigidity question. (Formal proof listed)
- [Single-fold Diophantine representations and undecidability under an at-most-one-solution promise](https://closedform.nishachay.tech/e/openai-math-2026-242/) - Entry 242 · Every computably listable set gets a whole-number polynomial test with exactly one solution per member. (Formal proof listed)
- [Separating choiceless counting from polynomial time and witnessed choice](https://closedform.nishachay.tech/e/openai-math-2026-243/) - Entry 243 · Claims a polynomial-time query that choiceless polynomial time with counting cannot express. (Formal proof listed)
- [The Partition Principle does not imply Choice](https://closedform.nishachay.tech/e/openai-math-2026-244/) - Entry 244 · Fitting sets back inside their images does not give you the Axiom of Choice. (Formal proof listed)
- [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)

### Group theory (14)

- [Cannon's conjecture](https://closedform.nishachay.tech/e/openai-math-2026-246/) - Entry 246 · A hyperbolic group whose boundary is a sphere acts geometrically on hyperbolic three-space. (Formal proof listed)
- [An infinite finitely presented residually finite 2-group and a finitely presented nil algebra](https://closedform.nishachay.tech/e/openai-math-2026-247/) - Entry 247 · Claims an infinite finitely presented group where every element has finite order, answering a Burnside question. (Formal proof listed)
- [Thompson's group F is nonamenable](https://closedform.nishachay.tech/e/openai-math-2026-248/) - Entry 248 · Thompson's group F, a famous group of interval stretchings, is not amenable. (Formal proof listed)
- [A finitely generated Eilenberg–Ganea counterexample](https://closedform.nishachay.tech/e/openai-math-2026-249/) - Entry 249 · Claims a group of cohomological dimension two with no two-dimensional classifying space, refuting Eilenberg–Ganea. (Formal proof listed)
- [Boone–Higman embeddings with higher finiteness](https://closedform.nishachay.tech/e/openai-math-2026-250/) - Entry 250 · A group has a word-solving algorithm exactly when it sits inside a finitely presented simple group. (Formal proof listed)
- [Amenability, unitarizability, and strong Ulam stability](https://closedform.nishachay.tech/e/openai-math-2026-251/) - Entry 251 · A discrete group is amenable exactly when every uniformly bounded representation can be made unitary. (Formal proof listed)
- [A torsion-free hyperbolic group that is neither residually finite nor linear over any field](https://closedform.nishachay.tech/e/openai-math-2026-252/) - Entry 252 · A torsion-free hyperbolic group in which some element vanishes in every finite quotient. (Formal proof listed)
- [An infinite finitely presented simple amenable group](https://closedform.nishachay.tech/e/openai-math-2026-253/) - Entry 253 · One infinite group that is simple, amenable and described by finitely many rules at once. (Formal proof listed)
- [Classifying spaces and geometric obstructions for Artin groups](https://closedform.nishachay.tech/e/openai-math-2026-254/) - Entry 254 · Proves the Artin K(π,1) conjecture in finite rank, and finds an Artin group with no geometric CAT(0) action. (Formal proof listed)
- [Quasi-isometric recognition of virtually polycyclic groups](https://closedform.nishachay.tech/e/openai-math-2026-255/) - Entry 255 · Any finitely generated group that looks like a virtually polycyclic group from far away is virtually polycyclic. (Formal proof listed)
- [Nonsingular systems of equations over arbitrary groups](https://closedform.nishachay.tech/e/openai-math-2026-256/) - Entry 256 · Claims every nonsingular system of equations over any group is solvable in a larger group, settling Howie's conjecture. (Formal proof listed)
- [A hyperbolic group without a geometric CAT(0) action](https://closedform.nishachay.tech/e/openai-math-2026-257/) - Entry 257 · A hyperbolic group that cannot act geometrically on any CAT(0) space, in any dimension. (Formal proof listed)
- [Gersten’s conjecture and virtual compact specialness of one-relator groups](https://closedform.nishachay.tech/e/openai-math-2026-258/) - Entry 258 · Claims one-relator groups without Baumslag–Solitar subgroups are hyperbolic and virtually compact special. (Claimed, not checked)
- [A group without fixed price](https://closedform.nishachay.tech/e/openai-math-2026-259/) - Entry 259 · Claims a finitely generated group with two free measure-preserving actions of different costs, so fixed price fails. (Claimed, not checked)

### Mathematical physics (25)

- [Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions](https://closedform.nishachay.tech/e/openai-math-2026-260/) - Entry 260 · Claims the spacetime Penrose inequality: a black hole's enclosing area caps how small the total mass can be. (Formal proof listed)
- [Localization and delocalization in the Anderson model](https://closedform.nishachay.tech/e/openai-math-2026-261/) - Entry 261 · Claims electrons stay trapped on a random square grid at any disorder, and can spread at weak disorder in 3D. (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)
- [The ionization and generalized ionization conjectures](https://closedform.nishachay.tech/e/openai-math-2026-263/) - Entry 263 · A nucleus can hold only a fixed number of extra electrons, however large its charge. (Formal proof listed)
- [Strong cosmic censorship near two-ended Kerr data](https://closedform.nishachay.tech/e/openai-math-2026-264/) - Entry 264 · Claims typical vacuum perturbations of a rotating Kerr black hole cannot be smoothly continued past the inner horizon. (Claimed, not checked)
- [Area laws and tensor networks for two-dimensional gapped systems](https://closedform.nishachay.tech/e/openai-math-2026-265/) - Entry 265 · Claims the 2D area law: gapped ground states on a square grid hold entanglement proportional to boundary, not volume. (Claimed, not checked)
- [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)
- [Positive-temperature Bose–Einstein condensation and exact quantum depletion](https://closedform.nishachay.tech/e/openai-math-2026-267/) - Entry 267 · Proves condensation in every ground state of the dilute hard-sphere Bose gas; claims thermal and depletion laws. (Formal proof listed)
- [The spin-one Haldane gap](https://closedform.nishachay.tech/e/openai-math-2026-268/) - Entry 268 · Claims the spin-one Haldane gap: the Heisenberg chain on even rings has an energy gap that stays above log(20)/784. (Claimed, not checked)
- [Uniform Laughlin gap and stability under bounded scalar disorder](https://closedform.nishachay.tech/e/openai-math-2026-269/) - Entry 269 · Proves a uniform energy gap of at least 1/25 above the Laughlin state on the sphere; claims it survives weak disorder. (Formal proof listed)
- [Threshold and positive-energy bound states of the BFSS matrix model](https://closedform.nishachay.tech/e/openai-math-2026-270/) - Entry 270 · Claims the BFSS matrix model has one zero-energy bound state for every N, and many positive-energy states at N = 2. (Claimed, not checked)
- [Bloch's law, its lattice correction, and the spherical magnetization law](https://closedform.nishachay.tech/e/openai-math-2026-271/) - Entry 271 · Proves 3D quantum Heisenberg ferromagnets magnetize at low temperature; claims Bloch's T^(3/2) law and its correction. (Formal proof listed)
- [Entanglement without distillable secret key](https://closedform.nishachay.tech/e/openai-math-2026-272/) - Entry 272 · Claims an entangled state in local dimension ten that yields no secret key, answering a 2005 open problem. (Formal proof listed)
- [The entropy photon-number inequality](https://closedform.nishachay.tech/e/openai-math-2026-273/) - Entry 273 · Proves the entropy photon-number inequality for two independent finite-energy light inputs mixed on a beam splitter. (Formal proof listed)
- [Parity is not in QAC 0](https://closedform.nishachay.tech/e/openai-math-2026-274/) - Entry 274 · Proves constant-depth quantum circuits with polynomially many qubits cannot compute parity, settling Moore's conjecture. (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)
- [Unitary vertex operator algebras and conformal nets](https://closedform.nishachay.tech/e/openai-math-2026-280/) - Entry 280 · Proves strongly rational unitary vertex operator algebras give conformal nets; claims their representations match. (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)
- [From scale symmetry to local conformal symmetry in four-dimensional QFT](https://closedform.nishachay.tech/e/openai-math-2026-282/) - Entry 282 · Claims that scale symmetry forces local conformal symmetry for a broad class of four-dimensional quantum field theories. (Claimed, not checked)
- [Polynomial-time unitary synthesis from a Boolean oracle](https://closedform.nishachay.tech/e/openai-math-2026-283/) - Entry 283 · Claims a polynomial-size quantum circuit can carry out any n-qubit operation, given a suitable Boolean oracle. (Claimed, not checked)
- [The optimal quartic separation between randomized and quantum queries](https://closedform.nishachay.tech/e/openai-math-2026-284/) - Entry 284 · Claims functions where randomized queries need nearly the fourth power of quantum queries, refuting a cubic conjecture. (Claimed, not checked)

### Operator algebras (19)

- [Counterexamples to Baum–Connes and Kadison–Kaplansky](https://closedform.nishachay.tech/e/openai-math-2026-285/) - Entry 285 · Claims torsion-free groups that break the Kadison–Kaplansky projection conjecture and reduced Baum–Connes. (Claimed, not checked)
- [Rigidity and arithmetic of lattice von Neumann algebras](https://closedform.nishachay.tech/e/openai-math-2026-286/) - Entry 286 · Claims the von Neumann algebra of a property (T) lattice remembers its group, its twist and its scale. (Claimed, not checked)
- [Isomorphism of the free group factors](https://closedform.nishachay.tech/e/openai-math-2026-287/) - Entry 287 · Proves, with a Lean check, that the free group factors L(F₂) and L(F₃) are isomorphic, settling a decades-old problem. (Formal proof listed)
- [Kadison's similarity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-288/) - Entry 288 · Every bounded map from a C*-algebra to operators can be straightened to respect the mirror operation. (Formal proof listed)
- [Strong Kadison–Kastler stability and its spatial boundaries](https://closedform.nishachay.tech/e/openai-math-2026-289/) - Entry 289 · Proves, with a Lean check, that close von Neumann algebras are conjugate by a unitary near the identity, uniformly. (Formal proof listed)
- [Relative bicentralizers and modular spectral recovery](https://closedform.nishachay.tech/e/openai-math-2026-290/) - Entry 290 · Claims every expected inclusion of von Neumann algebras holds an amenable piece with the same commutant in the core. (Formal proof listed)
- [Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability](https://closedform.nishachay.tech/e/openai-math-2026-291/) - Entry 291 · Claims amenable actions on Jiang–Su-stable algebras absorb the Jiang–Su algebra, plus three regularity theorems. (Formal proof listed)
- [Kirchberg's O₂ norm-ultrapower embedding problem](https://closedform.nishachay.tech/e/openai-math-2026-292/) - Entry 292 · Proves, with a Lean check, an explicit C*-algebra that fits in no nuclear norm ultrapower, answering Kirchberg. (Formal proof listed)
- [Invariant projections, hyperinvariant subspaces, and transitive algebras](https://closedform.nishachay.tech/e/openai-math-2026-293/) - Entry 293 · Proves that some operators on Hilbert space have no hyperinvariant subspace, a negative answer to a long-open problem. (Formal proof listed)
- [Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness](https://closedform.nishachay.tech/e/openai-math-2026-294/) - Entry 294 · Proves Kaplansky's quasitrace conjecture false, and that stable finiteness can fail for tensor products. (Formal proof listed)
- [The Kadison–Ringrose cohomology conjecture](https://closedform.nishachay.tech/e/openai-math-2026-295/) - Entry 295 · Proves bounded Hochschild cohomology of von Neumann algebras vanishes in degrees two and up, settling Kadison–Ringrose. (Formal proof listed)
- [The generator problem for finite factors](https://closedform.nishachay.tech/e/openai-math-2026-296/) - Entry 296 · Every type II₁ factor with separable predual is generated by a single operator. (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)
- [Two notions of free entropy differ even when both are finite](https://closedform.nishachay.tech/e/openai-math-2026-298/) - Entry 298 · Proves two standard notions of free entropy can differ by at least 1/2 even when both are finite. (Formal proof listed)
- [The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability](https://closedform.nishachay.tech/e/openai-math-2026-299/) - Entry 299 · Proves a separable C*-algebra absorbs the Jiang–Su algebra exactly when its central-sequence algebra has no characters. (Formal proof listed)
- [Approximation and quadratic strong-operator paving](https://closedform.nishachay.tech/e/openai-math-2026-300/) - Entry 300 · Claims to settle both paving conjectures of Popa and Vaes for every maximal abelian subalgebra of a von Neumann algebra. (Claimed, not checked)
- [Trace cones and Razak–Jacelon stabilization](https://closedform.nishachay.tech/e/openai-math-2026-301/) - Entry 301 · Claims the trace cone determines every separable nuclear C*-algebra once stabilized by the Razak–Jacelon algebra. (Formal proof listed)
- [Radius of comparison equals half the mean dimension](https://closedform.nishachay.tech/e/openai-math-2026-302/) - Entry 302 · Claims the radius of comparison of a minimal crossed product is half the mean dimension, the Phillips–Toms conjecture. (Claimed, not checked)
- [Weak pure infiniteness and Cuntz-algebra absorption](https://closedform.nishachay.tech/e/openai-math-2026-303/) - Entry 303 · Proves pure infiniteness implies strong pure infiniteness in all C*-algebras, part of a Kirchberg–Rørdam question. (Formal proof listed)

### Topology (18)

- [The Hilbert–Smith conjecture in every dimension](https://closedform.nishachay.tech/e/openai-math-2026-304/) - Entry 304 · Claims a proof of Hilbert–Smith: a locally compact group acting faithfully on a connected manifold is a Lie group. (Claimed, not checked)
- [Four-dimensional disk embedding and Wall's conjecture](https://closedform.nishachay.tech/e/openai-math-2026-305/) - Entry 305 · Claims the two-generator free group is not good for 4D surgery, plus a counterexample to Wall's question. (Claimed, not checked)
- [The purely cosmetic surgery conjecture](https://closedform.nishachay.tech/e/openai-math-2026-306/) - Entry 306 · Claims that two different surgeries on a nontrivial knot never give the same oriented three-dimensional space. (Claimed, not checked)
- [Failure of rational injectivity for maximal coarse assembly](https://closedform.nishachay.tech/e/openai-math-2026-307/) - Entry 307 · Graph families where coarse assembly forgets a class of infinite order; Lean checks the reduced version. (Formal proof listed)
- [Finite Smith–Toda complexes at every height](https://closedform.nishachay.tech/e/openai-math-2026-308/) - Entry 308 · Claims Smith–Toda complexes exist at every height if the prime may grow, with an explicit V(4) at the prime 1009. (Claimed, not checked)
- [The Kervaire invariant problem at the prime three](https://closedform.nishachay.tech/e/openai-math-2026-309/) - Entry 309 · Claims to settle which odd-primary Kervaire classes survive at the prime three: stems 10, 106 and 322. (Claimed, not checked)
- [Quillen's conjecture in rational homology](https://closedform.nishachay.tech/e/openai-math-2026-310/) - Entry 310 · Claims Quillen's 1978 conjecture on subgroup posets for every finite group and every prime, in a stronger homology form. (Claimed, not checked)
- [The Hovey–Strickland and Chai conjectures](https://closedform.nishachay.tech/e/openai-math-2026-311/) - Entry 311 · Claims Chai's invariant-ideal conjecture at every prime and height, and with it the Hovey–Strickland classification. (Claimed, not checked)
- [The Grothendieck homotopy hypothesis](https://closedform.nishachay.tech/e/openai-math-2026-312/) - Entry 312 · Grothendieck's algebraic ∞-groupoids capture all shapes of spaces, a paper claims, with its key step checked in Lean. (Formal proof listed)
- [Finite generation for the K(n)-local sphere](https://closedform.nishachay.tech/e/openai-math-2026-313/) - Entry 313 · Claims every homotopy group of the K(n)-local sphere is finitely generated over the p-adic integers. (Claimed, not checked)
- [Cyclic length and chromatic fixed-point loss](https://closedform.nishachay.tech/e/openai-math-2026-314/) - Entry 314 · Claims the exact chromatic loss between fixed points of p-group actions is a count of cyclic steps. (Claimed, not checked)
- [The four-dimensional Singer conjecture](https://closedform.nishachay.tech/e/openai-math-2026-315/) - Entry 315 · Claims the Singer conjecture in dimension four: L²-Betti numbers of aspherical four-manifolds vanish off degree two. (Claimed, not checked)
- [Curtis’s conjecture](https://closedform.nishachay.tech/e/openai-math-2026-316/) - Entry 316 · A paper claims to prove Curtis's 1975 conjecture: only η, ν, σ and Kervaire classes show up in mod-2 sphere homology. (Claimed, not checked)
- [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)
- [Chromatic splitting: filtrations and counterexamples](https://closedform.nishachay.tech/e/openai-math-2026-318/) - Entry 318 · Five papers claim Hopkins' chromatic splitting conjecture fails in its strong and weak forms, and offer a filtered fix. (Claimed, not checked)
- [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)
- [Nonhomeomorphic closed aspherical four-manifolds](https://closedform.nishachay.tech/e/openai-math-2026-320/) - Entry 320 · A paper claims two closed aspherical 4-manifolds with the same homotopy type but no homeomorphism, against Borel. (Claimed, not checked)
- [A counterexample to Wall's finite D(2) problem](https://closedform.nishachay.tech/e/openai-math-2026-321/) - Entry 321 · A paper claims a negative answer to Wall's D(2) problem: a finite 3-complex that passes the test but has no 2-d model. (Claimed, not checked)

### Functional analysis (11)

- [Tingley’s sphere-isometry problem](https://closedform.nishachay.tech/e/openai-math-2026-322/) - Entry 322 · Every distance-keeping map between unit spheres of two real Banach spaces extends to a linear map of the spaces. (Formal proof listed)
- [Independence of the separable quotient problem](https://closedform.nishachay.tech/e/openai-math-2026-323/) - Entry 323 · The separable quotient problem cannot be settled from the usual axioms, a paper claims; Lean checks the CH half. (Formal proof listed)
- [Lipschitz equivalent Banach spaces need not be linearly isomorphic](https://closedform.nishachay.tech/e/openai-math-2026-324/) - Entry 324 · Two separable Banach spaces match distances up to fixed factors but are not linearly the same, answering Kalton. (Formal proof listed)
- [The complete Crouzeix conjecture](https://closedform.nishachay.tech/e/openai-math-2026-325/) - Entry 325 · The complete Crouzeix conjecture holds: matrix-coefficient polynomials of an operator obey the sharp factor 2. (Formal proof listed)
- [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)
- [Markov type characterizes superreflexivity](https://closedform.nishachay.tech/e/openai-math-2026-327/) - Entry 327 · Proves that a Banach space with nontrivial Markov type is superreflexive, answering a question of Naor. (Formal proof listed)
- [Nonexpansive fixed points in reflexive Banach spaces](https://closedform.nishachay.tech/e/openai-math-2026-328/) - Entry 328 · Proves that every nonexpansive map of a closed bounded convex set in a reflexive Banach space has a fixed point. (Formal proof listed)
- [A counterexample to metric-entropy duality](https://closedform.nishachay.tech/e/openai-math-2026-329/) - Entry 329 · Proves that the duality conjecture for metric entropy is false, already when the covering body is a cube. (Formal proof listed)
- [A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces](https://closedform.nishachay.tech/e/openai-math-2026-330/) - Entry 330 · Proves a uniformly discrete metric space can have a Lipschitz-free space lacking the bounded approximation property. (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)
- [Metric Markov cotype of ℓ 1 and Hilbert-space Lipschitz extension](https://closedform.nishachay.tech/e/openai-math-2026-332/) - Entry 332 · Claims to prove that ℓ1 has metric Markov cotype two, settling Ball's extension problem for maps into ℓ1. (Claimed, not checked)

### Differential geometry (29)

- [Smooth isometric immersions of surfaces into ℝ 4](https://closedform.nishachay.tech/e/openai-math-2026-333/) - Entry 333 · Proves that every closed smooth curved surface can be placed in four-dimensional space with all its distances kept. (Formal proof listed)
- [A smooth surface metric with no local isometric immersion in ℝ 3](https://closedform.nishachay.tech/e/openai-math-2026-334/) - Entry 334 · Some smooth rules for measuring length cannot be built as a surface in space, even near one point. (Formal proof listed)
- [Gromov’s integral scalar-curvature bound for simplicial volume](https://closedform.nishachay.tech/e/openai-math-2026-335/) - Entry 335 · Claims Gromov's integral scalar-curvature bound for simplicial volume and the Gromov–Lawson aspherical conjecture. (Claimed, not checked)
- [Spectral scalar curvature, Urysohn width, and macroscopic dimension](https://closedform.nishachay.tech/e/openai-math-2026-336/) - Entry 336 · Claims Gromov's codimension-two width bound for positive scalar curvature in every dimension, and a spectral version. (Claimed, not checked)
- [Sharp Cartan–Hadamard isoperimetry and rigidity](https://closedform.nishachay.tech/e/openai-math-2026-337/) - Entry 337 · Proves sharp Euclidean fillings in CAT(0) spaces and claims the generalized Cartan–Hadamard isoperimetric inequality. (Formal proof listed)
- [Yau's uniformization conjecture](https://closedform.nishachay.tech/e/openai-math-2026-338/) - Entry 338 · Claims to prove Yau's uniformization conjecture: positive bisectional curvature makes a complete Kähler manifold ℂⁿ. (Claimed, not checked)
- [Katok's entropy rigidity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-339/) - Entry 339 · Claims to prove Katok's entropy rigidity conjecture for negatively curved manifolds of dimension three and higher. (Claimed, not checked)
- [A counterexample to the nearby Lagrangian conjecture](https://closedform.nishachay.tech/e/openai-math-2026-340/) - Entry 340 · Claims a counterexample to Arnold's nearby Lagrangian conjecture, in the cotangent bundle of a product of two spheres. (Claimed, not checked)
- [Donaldson's hypersymplectic deformation conjecture](https://closedform.nishachay.tech/e/openai-math-2026-341/) - Entry 341 · Claims every hypersymplectic four-manifold deforms, with its cohomology classes fixed, to a hyperkähler triple. (Claimed, not checked)
- [Donaldson's tamed-to-compatible conjecture](https://closedform.nishachay.tech/e/openai-math-2026-342/) - Entry 342 · Proves Donaldson's tamed-to-compatible conjecture for tamed almost-complex structures on closed four-manifolds. (Formal proof listed)
- [Symplectic ball packing in higher dimensions](https://closedform.nishachay.tech/e/openai-math-2026-343/) - Entry 343 · Proves Siegel and Yao's conjecture: from dimension six, ball packings are limited only by volume and the two-ball test. (Formal proof listed)
- [The metric Blaschke conjecture](https://closedform.nishachay.tech/e/openai-math-2026-344/) - Entry 344 · Claims the metric Blaschke conjecture: injectivity radius equal to diameter forces a round sphere or projective space. (Claimed, not checked)
- [Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds](https://closedform.nishachay.tech/e/openai-math-2026-345/) - Entry 345 · Claims every Riemannian metric on a sphere of any dimension has infinitely many geometrically distinct closed geodesics. (Claimed, not checked)
- [Sharp singular-set bounds for stationary integral varifolds](https://closedform.nishachay.tech/e/openai-math-2026-346/) - Entry 346 · Claims that the singular points of a stationary integral varifold have dimension at most one less than the varifold. (Claimed, not checked)
- [Counterexamples to stable-Morse and strong Arnold fixed-point bounds](https://closedform.nishachay.tech/e/openai-math-2026-347/) - Entry 347 · Claims counterexamples to strong forms of Arnold's fixed-point conjecture; the three-point quadric example is in Lean. (Formal proof listed)
- [Nonnegative-curvature Einstein classification and an L 2 topological gap](https://closedform.nishachay.tech/e/openai-math-2026-348/) - Entry 348 · Classifies positive Einstein four-manifolds with nonnegative curvature, and claims a small-energy topological gap. (Formal proof listed)
- [The Solomon–Yau least-volume conjecture](https://closedform.nishachay.tech/e/openai-math-2026-349/) - Entry 349 · Claims the Solomon–Yau conjecture: after the equator, the next minimal hypersurface of a sphere is a Clifford product. (Claimed, not checked)
- [Yau’s nodal bounds: surfaces and higher dimensions](https://closedform.nishachay.tech/e/openai-math-2026-350/) - Entry 350 · Proves Yau's sharp nodal-length bound on smooth surfaces, and shows it fails for smooth metrics in dimensions 3 to 5. (Formal proof listed)
- [Scalar curvature and finite-time Ricci-flow singularities](https://closedform.nishachay.tech/e/openai-math-2026-351/) - Entry 351 · Claims four-dimensional Ricci flow continues while scalar curvature is bounded, but not in some high dimension. (Claimed, not checked)
- [A finite-time singularity of Calabi flow](https://closedform.nishachay.tech/e/openai-math-2026-352/) - Entry 352 · Claims a smooth starting metric on complex projective 10-space whose Calabi flow breaks down in finite time. (Claimed, not checked)
- [Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample](https://closedform.nishachay.tech/e/openai-math-2026-353/) - Entry 353 · Proves affine Bernstein rigidity for complete graphs in dimensions 3 to 9; claims a smooth dimension-ten counterexample. (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)
- [Unique tangent flows at the first surface singularity](https://closedform.nishachay.tech/e/openai-math-2026-355/) - Entry 355 · Claims that blow-up limits of surface mean curvature flow are unique at the first singular time, in fixed coordinates. (Claimed, not checked)
- [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)
- [Negative Kähler curvature without bounded holomorphic coordinates](https://closedform.nishachay.tech/e/openai-math-2026-359/) - Entry 359 · Proves a complete Kähler threefold with pinched negative curvature can lack bounded holomorphic coordinates. (Formal proof listed)
- [Weak MTW curvature gives convexity and regular optimal transport](https://closedform.nishachay.tech/e/openai-math-2026-360/) - Entry 360 · Proves weak MTW curvature makes every injectivity domain convex, as Villani conjectured, and transport maps Hölder. (Formal proof listed)
- [Failure of integer-degree harmonic dimension comparison](https://closedform.nishachay.tech/e/openai-math-2026-361/) - Entry 361 · Proves Yau's Euclidean bound on polynomial-growth harmonic functions fails under nonnegative Ricci curvature. (Formal proof listed)

### Partial differential equations (16)

- [Global smoothness for relativistic Vlasov–Maxwell](https://closedform.nishachay.tech/e/openai-math-2026-362/) - Entry 362 · Proves the 3D relativistic Vlasov–Maxwell plasma equations have smooth solutions for all time, from any admissible data. (Formal proof listed)
- [Nonuniqueness with local conservation for the hard-sphere Boltzmann equation](https://closedform.nishachay.tech/e/openai-math-2026-363/) - Entry 363 · Proves the periodic hard-sphere Boltzmann equation can have two different solutions from the same initial gas. (Formal proof listed)
- [Kinetic limits and fluctuations over the Boltzmann lifespan](https://closedform.nishachay.tech/e/openai-math-2026-364/) - Entry 364 · Claims the Boltzmann equation follows from Newton's laws for short-range forces for as long as it stays regular. (Claimed, not checked)
- [Joint metric and connection recovery from one boundary patch](https://closedform.nishachay.tech/e/openai-math-2026-365/) - Entry 365 · Claims one boundary patch determines a smooth metric; proves rough 3D conductivities can share all boundary data. (Formal proof listed)
- [The planar Mumford–Shah regularity conjecture and local weak- L 4 gradient bounds](https://closedform.nishachay.tech/e/openai-math-2026-366/) - Entry 366 · Claims interior Mumford–Shah regularity in the plane: edges are smooth arcs, crack tips or 120-degree triple junctions. (Claimed, not checked)
- [The critical dimension for the one-phase Bernoulli problem](https://closedform.nishachay.tech/e/openai-math-2026-367/) - Entry 367 · Claims seven is the critical dimension for one-phase Bernoulli minimizers, so free boundaries are smooth through six. (Formal proof listed)
- [The three-dimensional Ball–Evans approximation problem](https://closedform.nishachay.tech/e/openai-math-2026-368/) - Entry 368 · Claims to solve the Ball–Evans problem in 3D: every Sobolev homeomorphism is a strong limit of smooth diffeomorphisms. (Claimed, not checked)
- [The hot spots conjecture for simply connected planar domains](https://closedform.nishachay.tech/e/openai-math-2026-369/) - Entry 369 · On smooth flat shapes without holes, the lasting hot and cold spots lie on the edge. (Formal proof listed)
- [The Lane–Emden and Hénon–Lane–Emden conjectures](https://closedform.nishachay.tech/e/openai-math-2026-370/) - Entry 370 · Proves the subcritical Hénon–Lane–Emden conjecture: for small enough powers, these equations have no positive solution. (Formal proof listed)
- [Stable blowup for the defocusing Schrödinger equation](https://closedform.nishachay.tech/e/openai-math-2026-371/) - Entry 371 · Proves that blowup is stable for a defocusing Schrödinger equation on a 12-dimensional torus, for some large odd power. (Formal proof listed)
- [Global uniqueness in smooth isotropic elasticity](https://closedform.nishachay.tech/e/openai-math-2026-372/) - Entry 372 · Proves that boundary push-and-measure data fix both stiffness values everywhere inside a smooth 3D elastic body. (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)
- [De Giorgi's conjecture in dimension eight](https://closedform.nishachay.tech/e/openai-math-2026-375/) - Entry 375 · Claims De Giorgi's conjecture in dimension eight: monotone Allen–Cahn solutions there are flat one-dimensional profiles. (Claimed, not checked)
- [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)
- [Interior C^(1,α) regularity for infinity-harmonic functions](https://closedform.nishachay.tech/e/openai-math-2026-377/) - Entry 377 · Claims the gradient of every infinity-harmonic function is Hölder continuous inside its domain, in dimension 3 and up. (Claimed, not checked)

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