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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theoretical-computer-scienceAug 1, 2026Significance 35/100Registry: lean verified
Prior state unknown→proved
Is the closest vector problem NP-hard to approximate within polynomial factors $n^c$? Yes for some $c > 0$: hardness of approximation reaches polynomial factors, with consequences for decoding and related lattice problems - a foundational question underpinning post-quantum cryptography where hardness had stalled at almost-polynomial factors since the late 1990s.
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combinatoricsJul 31, 2026Significance 35/100Registry: lean verified
Prior state unknown→proved
record lower bounds for seven odd cycles; the exact capacities remain open for every odd cycle beyond C5
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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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analysisJul 27, 2026Significance 35/100Registry: expert verified
Prior state unknown→proved
Two independent proofs within eight days, both with AI in the loop. Jin's (posted 27 July, preprints.org, submitted to Annals) is the first: its decisive theorem came out of an autonomous GPT-5.6 Sol run, and it is the proof Townsend, Greenbaum and Crouzeix have checked. Lorist and Schwenninger's five-page argument (arXiv, 4 August) is a genuinely different route - double-layer potentials plus a perturbation lemma…
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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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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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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 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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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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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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combinatoricsJun 29, 2026Significance 32/100Registry: unreviewed
Prior state unknown→proved
minimum out-degree 7; the conjecture is open in general
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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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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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algebraJul 24, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
The paper exhibits an explicit integer polynomial in five variables, of total degree 14 with constant Hessian determinant 128, whose gradient is not injective. Its formal Legendre transform is therefore not a polynomial, so the Hessian conjecture $\mathrm{HC}_5$ is false. The counterexample comes from a one-variable Schur descent applied to the six-variable doubling of Alpöge's 2026 Jacobian counterexample.
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probability-statisticsAug 10, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
Closes both gaps left open by Bandeira and Maillard: exact fitting, and removal of the operator-norm constraint. The threshold turns out to be governed by the statistical dimension d(d+1)/4 of the PSD cone.
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geometry-topologyMay 21, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
proved in dimension at most three
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combinatoricsAug 6, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
Settles a class, not the conjecture: Tuza's conjecture remains open in general.
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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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mathematical-physicsAug 9, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
Talagrand's Conjecture 11.7.5 is resolved affirmatively for the critical Ising Sherrington-Kirkpatrick model: $N^{2/3}\mathbb{E}\langle R_{1,2}^2\rangle$ converges to a positive finite constant. The paper proves substantially more, showing that the entire quenched distribution of $N^{1/3}R_{1,2}$ converges to an explicit random probability measure defined from the reflected $\mathrm{Airy}_1$ point process. The sam…
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quantum-information-computingAug 9, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
Partial deliberately: the NPT bound entanglement problem itself is untouched. What falls is the conjecture about the canonical family, and the paper is explicit that a substantial neighbouring region remains unresolved while another is known two-copy undistillable.
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combinatoricsAug 6, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
A Cayley graph is minimal when no proper subset of its connection set generates the group. Babai asked whether minimal Cayley graphs have bounded chromatic number. Resolved negatively: finite minimal Cayley graphs exist with arbitrarily large chromatic number.
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number-theoryNov 27, 2025Significance 30/100Registry: unreviewed
Prior state unknown→proved
Settles 9 and 10 runners only; the general conjecture remains open.
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probability-statisticsJun 21, 2026Significance 30/100Registry: lean verified
Prior state unknown→proved
The swap chain flips checkerboard 2×2 blocks to sample 0/1 matrices with fixed row and column sums. Kannan, Tetali and Vempala conjectured in 1997 that it mixes in polynomial time for all feasible margins; the lazy chain is shown to have spectral gap at least $\binom{m}{2}^{-1}\binom{n}{2}^{-1}$ on $m \times n$ matrices, which is worst-case tight and settles the bipartite case.
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combinatoricsAug 5, 2026Significance 30/100Registry: lean verified
Prior state unknown→proved
Proves the divisible rank-three case: rank exactly three and ground-set size a multiple of three, with no restriction to simple, paving, representable or graphic matroids. The part not previously in the literature is the non-simple sub-case, since McGuinness had settled all paving matroids and a rank-three matroid is paving exactly when it has no parallel pairs. Combined with the coprime-case theorem of van den He…
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mathematical-physicsAug 3, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
One variant case of Arnold's 1994 fast-dynamo problem, not the problem itself. Arnold asks for a single velocity field on T^3 that is smooth, divergence-free, autonomous and deterministic, fixed independently of the magnetic diffusivity, and that grows the magnetic field exponentially at every small enough diffusivity. The field constructed here is all of that except smooth: it is Lipschitz, not C^1. The sibling e…
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mathematical-physicsJul 29, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
Do $n$ point charges whose electrostatic potential has only non-degenerate critical points always have at most $(n-1)^2$ of them? A configuration of five charges - three at the vertices of an equilateral triangle plus two small central charges pulled apart into a shallow bipyramid - has at least $24 > 16$ non-degenerate critical points, so the conjecture is false.
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geometry-topologyJul 27, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
Bellman's problem for general regions remains open
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combinatoricsAug 11, 2026Significance 30/100Registry: site confirmed
Prior state unknown→disproved
The ratio 15/31 is not claimed to be optimal, and the paper makes no claim that 31 vertices is the smallest possible counterexample. The construction produces separating triangles by design, so it says nothing about the 4-connected case.
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algebraAug 4, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
For a Brauer class on a variety, the period-index conjecture bounds the index in terms of the period and the dimension. Disproved: for any uncountable algebraically closed field $k$ of characteristic $0$ and any $d \geq 3$ there is a $d$-dimensional variety over $k$ carrying a Brauer class that violates it, for Hodge-theoretic reasons. For $d = 3$ the construction needs no uncountability, so the conjecture fails a…
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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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combinatoricsAug 12, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
A dense case, not the conjecture: it remains open in general. The concrete gain is on the size of any counterexample - combined with the known minimum-outdegree results, this raises the best known lower bound on the order of a counterexample from 16 to 17, and to 19 conditional on the 2026 preprint of Sadhukhan, Sandeep and Sen. The novelty against the earlier dense-case work is that no structure is prescribed on…
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algebraJul 11, 2026Significance 30/100Registry: lean verified
Prior state unknown→disproved
Grothendieck asked whether every finite locally free group scheme of order $n$ is killed by $n$ (its $n$-th convolution power map equals the unit). The counterexample is an order-4 group scheme not killed by 4 (killed only by 8); since Deligne settled the commutative case, it is necessarily non-commutative over a non-reduced base.
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combinatoricsAug 12, 2026Significance 30/100Registry: site confirmed
Prior state unknown→proved
Explicit construction of a Hadamard matrix of order 668, the smallest previously unresolved order, verified exactly by this site from the announcement plus its decoder reply. The same post encodes matrices for all twelve previously-open admissible orders below 2000 (668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964), and this site verified every one of them. The entry records the order-668 existe…
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combinatoricsAug 3, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
Mihail and Vazirani conjectured that the graph of every $0/1$-polytope has edge expansion at least one. Disproved by a family of $0/1$-polytopes whose edge expansion decreases exponentially in the dimension.
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combinatoricsJul 2, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
Refuting log-concavity of the flat counts is weaker than refuting their unimodality, since log-concavity is the stronger property. A counterexample to unimodality followed three weeks later and is tracked separately as Rota's Unimodality Conjecture for Matroid Flats; this paper came first.
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