# Landmark results

> 49 AI-made math results rated Landmark, each explained in plain English. Results rated Landmark: answers to well-known, long-standing questions in their…

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

Results rated Landmark: answers to well-known, long-standing questions in their field.

49 AI-made results rated Landmark, from 109 papers in the OpenAI math collection, each explained in plain English. 27 have a formal Lean proof listed; 22 are claimed in a paper and not yet machine-checked. They span 14 fields, led by operator algebras (7), real and complex analysis (7) and algebra (5). The oldest question here dates back to 1896: [The quasi-Riemann hypothesis](https://closedform.nishachay.tech/e/openai-math-2026-003/). Results rated Landmark include [The full BSD formula from low Selmer corank](https://closedform.nishachay.tech/e/openai-math-2026-002/) and [Hilbert’s tenth problem over ℚ](https://closedform.nishachay.tech/e/openai-math-2026-004/).

## Results (49)

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