number-theoryAug 10, 2026Significance 68/100Registry: lean checked
Prior state unknown→proved
An unconditional record, not a resolution: the Riemann hypothesis is untouched, and Anthropic states it does not expect these techniques to lead to a proof of it. The paper is explicit that these are lower bounds only - the remaining third of the zeros are not shown to be off the line, merely not reached by the certificate.
What it does settle is a question that was posed. Goldston and Suriajaya had reduced Montg…
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geometry-topologyAug 24, 2026Significance 65/100Registry: unreviewed
Prior state unknown→proved
Claims an explicit compact complex threefold $X$, fibred over $\mathbb{P}^1$ by complex 2-tori via period functions on the $(3,4,\infty)$ orbifold, degenerating to a del Pezzo-of-degree-six fibre (identified opposite sides of its hexagon) at one point and to bielliptic multiple fibres of multiplicities 3 and 4 at the other two. Argues $X$ is simply connected with $H_*(X;\mathbb{Z})=H_*(S^6;\mathbb{Z})$, hence diff…
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algebraJul 20, 2026Significance 65/100Registry: expert verified
Prior state unknown→disproved
n ≥ 3; plane case open
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geometry-topologyAug 19, 2026Significance 60/100Registry: unreviewed
Prior state unknown→disproved
The paper's appendix draws a distinction worth keeping: a counterexample may reduce to a finite certificate, checkable once the object is written down, or it may itself be a theorem quantified over all degenerations. This is the second kind. The method field records construction, because the resolution exhibits an explicit fivefold, but the difficulty lay elsewhere - candidate manifolds of this shape have been ava…
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algebraAug 1, 2026Significance 60/100Registry: lean verified
Prior state unknown→disproved
answered no: non-sofic groups exist
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theoretical-computer-scienceAug 17, 2026Significance 55/100Registry: unreviewed
Prior state unknown→proved
A record, not a resolution, and a small one by design. The bound moves from $2.371339$ to $2.371177$, about $1.6 \times 10^{-4}$, and the authors describe it as a small step. Whether $\omega = 2$ is untouched, and nothing here suggests the laser method can reach it.
The interesting claim is methodological rather than numerical. The bottleneck in this line of work is a hard optimization problem, and the paper repo…
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combinatoricsJul 10, 2026Significance 55/100Registry: lean checked
Prior state unknown→proved
Conjectures that every bridgeless graph has a collection of cycles covering each edge exactly twice.
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combinatoricsMay 27, 2026Significance 55/100Registry: unreviewed
Prior state unknown→disproved
the model's contribution is one simplifying lemma; the authors state the main ideas are human
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geometry-topologyAug 19, 2026Significance 55/100Registry: lean checked
Prior state unknown→disproved
Only the smooth case falls. Hamburger's real-analytic theorem is untouched, and the counterexample is explicitly a $C^\infty$ object, so the conjecture's classical analytic form remains true. The gap between the two is the whole content of the result.
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analysisAug 5, 2026Significance 53/100Registry: lean verified
Prior state unknown→disproved
Also refutes the 1929 Pompeiu problem: Corollary 1.2 applies Williams' classical 1976 equivalence to the same constructed domains, so this is one construction settling both, not two separate results.
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theoretical-computer-scienceAug 20, 2026Significance 50/100Registry: unreviewed
Prior state unknown→disproved
Marton's inner bound, proposed in 1979, is the best known achievable region for a general discrete memoryless broadcast channel, and whether it always achieves the capacity region had been open ever since. It does not: there is a finite two-receiver discrete memoryless broadcast channel whose two-letter Marton value strictly exceeds twice its one-letter value, so the complete one-letter Marton region is strictly c…
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number-theoryAug 20, 2026Significance 50/100Registry: unreviewed
Prior state unknown→proved
Two tiers, and only the first is the record. Rank $\ge 30$ is unconditional, being thirty explicit independent points. Rank exactly 30 is conditional: applying Bober's bound (arXiv:1112.1503) with $\Delta = 4.25$ gives an analytic rank of at most 31, and the root number is $+1$ so the rank is even, hence 30 - but that argument assumes GRH, and equating analytic rank with rank assumes BSD. The entry is a partial re…
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geometry-topologyAug 1, 2026Significance 50/100Registry: lean verified
Prior state unknown→proved
upper bounds reach the Cohn-Elkies threshold; the true asymptotic density remains open
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number-theoryAug 23, 2026Significance 50/100Registry: unreviewed
Prior state unknown→proved
Two tiers, and only the first is the record. Rank $\ge 31$ is unconditional: 31 explicit points, independence asserted via the leaderboard's stated general practice of exact 2-descent (not reproduced here - see the verification note). Rank exactly 31 is conditional on GRH and BSD, per the submitters' commentary, in the same style as the sibling record's Bober-bound argument; no numeric derivation has been publishe…
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geometry-topologyJul 27, 2026Significance 45/100Registry: unreviewed
Prior state unknown→proved
with constant 2; also improves the global KLS bound to $O(\log^{1/4} n)$
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analysisAug 24, 2026Significance 45/100Registry: unreviewed
Prior state unknown→disproved
The paper constructs a real meromorphic function $F$ on $\mathbb{C}$ satisfying $F^{-1}(\{0,1,\infty\})\subset\mathbb{R}$, with each of the three fibers $F^{-1}(0)$, $F^{-1}(1)$, and $F^{-1}(\infty)$ infinite, such that for every $a\in\widehat{\mathbb{C}}\setminus\{0,1,\infty\}$, the $a$-point divisor in each of the upper and lower half-planes fails the Blaschke condition. Consequently, $F$ is not of bounded type…
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analysisAug 1, 2026Significance 45/100Registry: lean verified
Prior state unknown→disproved
Are ICC property (T) groups remembered by their von Neumann algebras - if $L(\Gamma) \cong L(\Lambda)$ for such groups, must $\Gamma \cong \Lambda$? A counterexample refutes Connes' conjecture that these groups are uniquely determined by their group von Neumann algebras.
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analysisAug 13, 2026Significance 40/100Registry: unreviewed
Prior state unknown→proved
The paper proves the previously unresolved odd-dimensional real cases of Banach's isometric conjecture. Combined with Gromov's earlier theorem for even dimensions and previous results, this completes the conjecture for real Banach spaces.
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combinatoricsJul 23, 2026Significance 40/100Registry: site confirmed
Prior state unknown→disproved
This preprint was not the first disproof. A 68-vertex counterexample was posted to X on 23 July 2026 by @NeuralReformist, credited to GPT-5.6 Sol Ultra, sixteen days earlier. This site decoded that sparse6 string and checked it independently: 68 vertices, 102 edges, simple, cubic, connected, bridgeless, girth five, and no Petersen coloring under the same encoder used for the 112-vertex graph. Whether the two are i…
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geometry-topologyMay 1, 2026Significance 40/100Registry: expert verified
Prior state unknown→disproved
Conjectured upper bound on how many pairs among $n$ points in the plane can be exactly one unit apart.
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analysisAug 5, 2026Significance 40/100Registry: lean verified
Prior state unknown→proved
Sendov's conjecture is resolved for every degree n >= 2, closing a gap that had stood since 1959: degrees up to eight were settled piecemeal between 1969 and 1999, and Tao's 2020 result covered all sufficiently large degrees without ever specifying the threshold, leaving the middle range open. Tao's digestion establishes the stronger interior form of the statement, which resolves the Phelps-Rodriguez conjecture in…
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combinatoricsJul 2, 2026Significance 40/100Registry: unreviewed
Prior state unknown→disproved
White conjectured that the symmetric exchange binomials generate the toric ideal of a matroid. This is now known to be false; a rank $9$ binary matroid constitutes a counterexample.
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theoretical-computer-scienceAug 1, 2026Significance 39/100Registry: lean verified
Prior state unknown→proved
exponential improvement over the 1977 MRRW bounds; the exact rate-distance trade-off remains open
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geometry-topologyAug 13, 2026Significance 38/100Registry: unreviewed
Prior state unknown→proved
Gromov's 1986 question drew three independent proofs within about 24 hours, two with AI in the loop. Ge's non-AI proof (heat-kernel Fisher metric, Nash entropy) came first, 13 August. Antonelli's proof here (14 August, GPT-5.6 Sol) takes a different route, Hodge obstruction and rank improvement, and its headline addition is the general family: for every $0\le m\le n-2$, nonnegative Ricci plus positive $(m{+}1)$-in…
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analysisAug 18, 2026Significance 38/100Registry: unreviewed
Prior state unknown→proved
The theorem is the vector-valued endpoint bound $\|Rf\|_{L^{1,\infty}} \le 2\|f\|_{L^1}$ for $R = (R_1,\ldots,R_n)$, so the constant 2 also serves each component $R_j$ uniformly in the dimension; the best previously known component bound grew like $c\log n$.
The mechanism is a decomposition theorem stated as Theorem 1.2: for every nonnegative $f \in L^1 \cap L^2$ and every $\lambda > 0$, write $f = \mu + (-\Delta…
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combinatoricsAug 4, 2026Significance 38/100Registry: lean verified
Prior state unknown→proved
Settles two conjectures. Theorem 1.1 proves Bollobas's asymptotic degree-diameter conjecture, in the stronger liminf form rather than the conjectured limsup. Corollary 1.2 proves Conjecture 3 of Cambie, Cames van Batenburg, de Joannis de Verclos and Kang on the edge variant, again in the stronger liminf form, and is tight for bipartite graphs.
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theoretical-computer-scienceAug 1, 2026Significance 38/100Registry: lean verified
Prior state unknown→proved
an n^4/log n formula lower bound; VP vs VNP remains wide open
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probability-statisticsAug 16, 2026Significance 37/100Registry: unreviewed
Prior state unknown→proved
The claim is the exact conjectured decay: $\psi_{\mu_a}(u) \le C_a/\sqrt{\log u}$ for every $u > 1$ and every $n$, with $C_a$ dimension-free - concretely $\lesssim \kappa_a^2(\log\frac{\kappa_a}{\kappa_a-1})^{1/2}$ where $\kappa_a = (1+a)/(1-a)$.
What is new is one step in a three-paper chain rather than a proof from scratch, and the paper is explicit about it. Chen's reverse-heat and Boolean-bridge framework and…
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probability-statisticsMay 11, 2026Significance 37/100Registry: unreviewed
Prior state unknown→proved
Talagrand's convexity problem asks whether a universal number of Minkowski sum operations turns any set of large Gaussian measure into one containing a convex body of comparable measure. It is equivalent to a question about subgaussian vectors: is every centered $1$-subgaussian random vector in $\mathbb{R}^n$ the sum of a universal number of standard Gaussian vectors? Both are answered affirmatively, via the sharp…
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combinatoricsJul 24, 2026Significance 36/100Registry: unreviewed
Prior state unknown→disproved
Is the sequence $W_0, W_1, \dots, W_n$ counting the flats of each rank of a matroid always unimodal? Rota conjectured yes in 1970.
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quantum-information-computingJul 23, 2026Significance 35/100Registry: unreviewed
Prior state unknown→disproved
Two-copy distillable if and only if already one-copy distillable.
Four independent papers settled this within five days of each other, and no single one of them is the account of record. Fu, Gao and Park posted first on 23 July (arXiv:2607.21367), followed by Song and Chen on 26 July (arXiv:2607.23416), then on 27 July both Fraser, Huber, Pozsgay and Vona (arXiv:2607.24309) and Bharti, Gajjala and Haug (arXiv:260…
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geometry-topologyMay 13, 2026Significance 35/100Registry: unreviewed
Prior state unknown→disproved
Kinoshita conjectured that every embedded projective plane in $S^4$ is reducible. False: an irreducible embedded projective plane exists in $S^4$. The construction also answers both parts of Problem 4.37 of the Kirby problem list.
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analysisJul 31, 2026Significance 35/100Registry: unreviewed
Prior state unknown→proved
The ball case. For arbitrary domains Pólya's conjecture remains open; this continues the authors' programme after the planar disk, circular sectors, and the Dirichlet case in arbitrary dimensions.
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probability-statisticsJul 13, 2026Significance 35/100Registry: unreviewed
Prior state unknown→proved
the last conditional step in the Ding-Sun and Huang bounds is discharged
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algebraAug 3, 2026Significance 35/100Registry: unreviewed
Prior state unknown→disproved
Settles the ICC property (T) case. The paper records that the result was obtained independently of and concurrently with work by OpenAI.
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number-theoryMay 20, 2026Significance 35/100Registry: unreviewed
Prior state unknown→disproved
false in general; the positive direction holds for k large relative to n
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