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405 source-grounded records. Verification labels remain separate from source authentication and publication status.
Imported from VibeMathed under CC BY 4.0. Each record links to its registry entry and named primary source. Registry verification is preserved verbatim.
Tensorisation and parabolic rescaling carry the one-dimensional example to real symmetric isotropic counterexamples on R^d and on every bounded domain, in every dimension.
Does two-terminal reliability, the probability that $s$ still reaches $t$ when edges fail independently, admit a fully polynomial-time randomised approximation scheme? Asked explicitly in Kannan's 1994 survey and left open while the all-terminal cases were settled by Karger and by Guo and Jerrum. Answered positively for general graphs, both directed and undirected. The complementary unreliability question is shown…
Whether the real Kalton-Peck space $Z_2$ is isomorphic to its hyperplanes. It is not: no hyperplane of $Z_2$ is isomorphic to $Z_2$, proved through a rank parity theorem for symplectic spaces applied to the Kalton-Swanson symplectic structure.
Can one construct a plain-model, efficient, information-theoretically secure one-time unclonable-encryption scheme for one classical bit with exponentially small adversarial advantage?
A record lower bound; the kissing number in dimension 19 remains unknown.
a barrier result about one proof strategy, not the conjecture itself
in polynomial edge-density regimes; the general question remains open
At the conjectured density, must every $k$-uniform hypergraph contain a short nontrivial even cover - a set of hyperedges covering each vertex an even number of times - with no superfluous polylogarithmic factors? Known up to polylog factors since 2022; now proved exactly for every $k \ge 3$.
rules out the two-point LP method in this dimension; the optimal packing in dimension 36 remains unknown
The counterexample is an ordinary algebra concentrated in degree zero, with the strongest possible vanishing in positive degrees, so the phenomenon needs no grading or differential. Liu and Shen had already disproved the differential-graded version in December 2025 without any AI involvement; the classical case is the one that fell with a model in the loop.
Strict convexity and real-analytic boundaries are what make this sharp: the classical Gordon-Webb-Wolpert drums are non-convex polygons, so the obvious escape routes are closed off.
first counterexample of the form D^b(X) for X smooth projective
integral unimodular lattices of rank at most 32; the general conjecture is open
Records, not resolutions: the exact values of these Ramsey numbers remain unknown.
Establishes that every PPT channel is eventually entanglement-breaking (finite EB index), in full generality, and bounds the index by 3 uniformly in dimension for a family strictly containing the 2-superpositive maps. The PPT-squared conjecture itself - index at most 2 - remains open; the paper presents its results as strong evidence toward the cubed version.
A conjecture attributed to Kontsevich holds that strata of quadratic differentials are aspherical, that is orbifold $K(\pi,1)$ spaces. False: when there are at least four zeros or poles, no connected component of a genus-one stratum is an orbifold $K(\pi,1)$, giving infinitely many counterexamples, along with counterexamples for associated stability spaces.
The near-quadratic Elekes-Ronyai expander conjecture over $\mathbb{R}$ predicts that a nonspecial polynomial expands any finite set to near-quadratic size. False: a fixed nonspecial quadratic polynomial, together with arbitrarily large finite sets of real algebraic integers, has image with a fixed power saving from quadratic size.
Does every synchronizing one-cluster automaton on $n$ states admit a reset word of length at most $(n-1)^2$? The new bound $(m-1)(n-1) + m\ell \le (n-1)^2$ settles the one-cluster case of the Černý conjecture.
Extending the minimal model program beyond threefolds in positive characteristic is a standing goal of birational geometry. Assuming the log resolution conjecture for all log pairs birational to $X$, the cone theorem holds for projective log canonical, $\mathbb{Q}$-factorial fourfold pairs $(X, \Delta)$ with $K_X + \Delta \equiv M \ge 0$, over bases of positive and mixed characteristic $p > 5$.
Are the Kazhdan-Lusztig polynomials of matroids always unimodal - in particular log-concave, or even real-rooted, as conjectured? No: representable matroids obtained by deleting points from finite projective geometries have non-unimodal Kazhdan-Lusztig polynomials over every finite field, so the log-concavity and real-rootedness conjectures are both false.
Can a noetherian ring have a local cohomology module whose support is not closed - equivalently, one with infinitely many minimal primes? Huneke and Lyubeznik asked; the paper constructs such rings, so the answer is yes.
The Howland-Kato conjecture that every nonzero positive commutator $i[f(P),g(Q)]$ must arise from functions in appropriate Kato classes is false: $i[\arctan(P),\arctan(Q)]$ is nonzero and nonnegative.
Maz'ya and Shaposhnikova introduced a non-classical maximal operator $M^\diamond$, the maximal convolution with the vector-valued signum kernel truncated to centered balls. One of Maz'ya's 75 open problems in analysis asks whether it can be separated from the sharp maximal operator $M^\sharp$. It can: there is a translation-invariant Banach space of locally integrable functions on which $M^\diamond$ is bounded but…
Banks and Martin conjectured in 2013 that for a primitive set $A$ and any set $Q$ of primes, the Erdos sum of the members of $A$ composed only of primes in $Q$ is at most the corresponding sum over $Q$ itself. The unrestricted form turned out to be false once $Q$ is allowed to contain $2$; Lichtman proposed a revised form restricted to odd primes. That revised conjecture, long viewed as a unifying master theorem f…
negative below the 1/3 threshold; the endpoint case is still open
The domain has dihedral symmetry of order 26 and is neither a disc nor centrally symmetric, and its eigenfunction changes sign - which is why an additional sign assumption rescues the statement.
Does every nonlocal game admitting a perfect entangled strategy admit one using a maximally entangled state? Described in the paper as one of the longstanding open problems in quantum nonlocality. Answered negatively by an explicit counterexample game.
Charts the bounded-slope regime of the arithmetic Kakeya program; the conjecture itself remains open.
The theorem improves the best lower bound valid in *every* sufficiently large genus from asymptotic constant $2/9$ to $1$. The every-genus ladder it climbs is Katz-Sabourau's $19/120$ and then Liu-Petri's $2/9$, the latter also by a random construction. Constant $1$ was already reached by Petri-Walker along a subsequence of genera, following Erdos-Sachs, so the new contribution is achieving it uniformly rather tha…
the equality case; the inequality was settled separately and is tracked on its own entry
The paper records how stuck this was: the first author had discussed the question with Caucher Birkar, Osamu Fujino, Christopher D. Hacon, Junpeng Jiao, Vladimir Lazic and Lingyao Xie, and writes that despite a general feeling that a negative answer was likely, no precise counterexample could be found. The authors also note that, given the limitations of generative AI, they may have missed related literature and w…
Fully resolves the posed coordinate-wise question; the main Courtade-Kumar conjecture itself remains open outside the extended high-noise range.
One finite-dimensional construction settles several related questions. Besides the inverse generator problem, it gives a generator whose Cayley transforms satisfy the ordinary Kreiss resolvent condition but are neither strongly Kreiss bounded nor power bounded, and it shows the Crank-Nicolson scheme is unstable in operator norm both over long times at fixed step size and under mesh refinement at fixed final time.…
The counterexamples come with divisibility bounds on the Hodge-theoretic index, at 2-torsion and 5-torsion, on very general hyperkahler fourfolds.
Amdeberhan, Shareshian and Stanley showed a function from the theory of partition Eisenstein series counts alternating permutations with a given record partition, and asked whether a similar theory exists for record compositions, suggesting a role for noncommutative symmetric functions. The paper solves that open problem with a product formula.
For single-source unsplittable flow, every fractional flow can be rounded to an unsplittable flow whose cost is no higher than the fractional cost, while each arc's load is exceeded by at most the maximum demand. (The cost version of Goemans' unsplittable-flow conjecture.)