# Named conjectures

> 85 AI-made results on named conjectures, each explained in plain English. Results whose title names a conjecture: a statement mathematicians believed but…

Canonical: https://closedform.nishachay.tech/topic/named-conjectures/
Last updated: 2026-10-14

Results whose title names a conjecture: a statement mathematicians believed but could not prove.

85 AI-made results on named conjectures, from 120 papers in the OpenAI math collection, each explained in plain English. 43 have a formal Lean proof listed; 42 are claimed in a paper and not yet machine-checked. They span 16 fields, led by algebraic and complex geometry (12), combinatorics (12) and algebra (11). The oldest question here dates back to 1902: [The Hilbert–Smith conjecture in every dimension](https://closedform.nishachay.tech/e/openai-math-2026-304/). Results rated Landmark include [The rational Hodge conjecture for CM abelian varieties](https://closedform.nishachay.tech/e/openai-math-2026-032/) and [Nagata’s conjecture and maximal Seshadri constants](https://closedform.nishachay.tech/e/openai-math-2026-039/).

## Results (85)

- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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 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 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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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 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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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 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)
- [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 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)
- [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)
- [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)
- [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)
- [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)
- [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 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)
- [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)

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