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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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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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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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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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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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 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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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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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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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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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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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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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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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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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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mathematical-physicsJul 20, 2026Significance 35/100Registry: unreviewed
Prior state unknown→proved
For the Sherrington-Kirkpatrick model with no external field, at zero temperature $\beta=\infty$, the paper proves that the zero-temperature Parisi minimizer is absolutely continuous with a smooth density and has support $[0,1)$ - full replica symmetry breaking, confirming the Parisi picture at the ground state. It also shows $q_\beta\to1$ as $\beta\to\infty$.
The positive-temperature input is Lopatto's: for ever…
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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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theoretical-computer-scienceJul 20, 2026Significance 35/100Registry: unreviewed
Prior state unknown→proved
Can any single-pass semi-streaming algorithm beat the naive greedy $1/2$-approximation for maximum matching? No. No single-pass semi-streaming algorithm, deterministic or randomized, achieves a better-than-half approximation, so greedy is optimal. The same construction settles the optimal competitive ratio of online matching with preemption at $1/2$.
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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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analysisAug 12, 2026Significance 35/100Registry: unreviewed
Prior state unknown→proved
The complete fixed-volume picture, closing a gap that partial results had narrowed from both ends without meeting: balls uniquely minimize for every volume up to V_* = 3.51..., and above it no minimizer exists at all. Before this the best minimality range was V <= 1 (Chodosh-Ruohoniemi, 2025) and the best nonexistence bound V >= 7.5 (Schulz, posted two days earlier), so the open middle ran from 1 to 7.5. Frank-Nam…
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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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logic-foundationsAug 13, 2026Significance 35/100Registry: unreviewed
Prior state unknown→proved
The new content is $SOP_2 \Rightarrow SOP_3$; the converse implication was known from the start. Dzamonja and Shelah asked whether either implication in $SOP_3 \Rightarrow SOP_2 \Rightarrow SOP_1$ reverses: Mutchnik answered the second ($SOP_1 = SOP_2$), and this answers the first, collapsing the bottom of the hierarchy to $SOP_1 = SOP_2 = SOP_3$. The $SOP_n$ hierarchy for $n \ge 3$ remains, as does everything abo…
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analysisAug 5, 2026Significance 33/100Registry: unreviewed
Prior state unknown→disproved
Heil, Ramanathan and Topiwala conjectured in 1996 that any finite set of time-frequency shifts of a nonzero square-integrable function is linearly independent. This refutes it: there is a Schwartz function admitting 12 linearly dependent time-frequency shifts.
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combinatoricsJun 29, 2026Significance 32/100Registry: unreviewed
Prior state unknown→proved
minimum out-degree 7; the conjecture is open in general
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algorithms-optimizationOct 27, 2025Significance 32/100Registry: unreviewed
Prior state unknown→proved
Nesterov's accelerated gradient method (1983) is a cornerstone of optimization, yet whether its iterates themselves converge to a minimizer, rather than just the function values, stayed open for over forty years. Jang and Ryu resolve it in the affirmative. Ryu first announced the continuous-time result on X; Bot, Fadili and Nguyen's concurrent human proof of the critical-regime case (answering a decade-old conject…
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probability-statisticsAug 6, 2026Significance 32/100Registry: unreviewed
Prior state unknown→proved
Claims the central limit theorem for the bipartite random assignment problem with bounded uniform costs: $\sqrt{n}\,(C_n-\zeta(2)) \Rightarrow \mathcal{N}(0,\,4\zeta(2)-4\zeta(3))$. The limiting constant is not itself new - Wästlund had computed exactly $4\zeta(2)-4\zeta(3)$ for the mean-one exponential model, and Malatesta, Parisi and Sicuro derived the non-bipartite analogue by replicas - but neither is a proof…
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analysisAug 20, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
Constructed an explicit class of smooth random, time-dependent incompressible velocity fields on T^3, obtained by alternating smooth shear flows with iid random phases on finite time blocks. For every fixed sufficiently small resistivity, the magnetic field has an almost-sure exponential growth rate at least 1/2, together with a time-uniform lower bound whose random prefactor has a resistivity-uniform inverse-mome…
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probability-statisticsJun 10, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
the group case; the full Matrix Spencer conjecture remains open
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combinatoricsMay 13, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
For $A \subset \mathbb{F}_p$ of density $1/2$, call $A$ almost affine invariant under $\varphi(x) = ax+b$ if $|A \triangle \varphi(A)| = o(p)$. Problem 90 asks for the threshold $K$ below which $A$ can be almost affine invariant simultaneously under all such $\varphi$ with $|a|, |b| \le K$ and $a \ne 0$. The threshold is $K = o(\log p)$.
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probability-statisticsJul 20, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
The Benjamini-Hochberg procedure is known not to control the false discovery rate at its nominal level under arbitrary dependence. A folklore conjecture in the FDR literature held that it must at least control the FDR up to a universal multiplicative constant. It does not: there are finite Gaussian models whose FDR divided by $q$ diverges as $q \downarrow 0$, with an explicit two-sided lower bound $q\sqrt{\log(1/q…
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geometry-topologyAug 14, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
Kusner conjectured in 1983 that the maximum number of points in $\mathbb{R}^n$ that are pairwise at $\ell_p$-distance one is exactly $n+1$ for every $2 < p < \infty$, as in the Euclidean case. False: an explicit configuration of $n+2$ equilateral points exists for some exponent, placing the infimum of exponents at which the conjecture fails in $[4,5)$. The configuration is the unique solution of an explicit polyno…
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analysisAug 5, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
For every $n>2$, the paper constructs a bounded map $U\in W^{1,n}(B^n,\mathbb{R}^{n+2})$, smooth on $B^n\setminus\{0\}$ but discontinuous at the origin, together with an antisymmetric potential
$$
\Omega\in L^n(B^n,so(n+2)\otimes\mathbb{R}^n)
$$
such that
$$
-\mathrm{Div}\bigl(|\nabla U|^{n-2}\nabla U\bigr)
=
\Omega\cdot|\nabla U|^{n-2}\nabla U
\qquad\text{in }D'(B^n).
$$
Moreover, the potential satisfies the…
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