combinatoricsAug 12, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
The honest reading, which the paper gives itself: the reduction is implicit in earlier work of Aboulker, Oijid, Petit, Rocton and Simon, and the model itself surfaced that reference when asked about originality. So this establishes the conjecture and supplies a polynomial-time algorithm, while the underlying idea is a rediscovery rather than a first. It is a striking record of a model producing an argument and then…
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algorithms-optimizationJul 14, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
For deterministically minimizing a convex 1-Lipschitz function on the d-dimensional ball using only exact function values, the query complexity sat between Ω(d) and O(d2log2d) since 1996. The paper proves a near-quadratic lower bound Ω(d2/log(d+1)), closing the gap: Q(d,∼d−1/2)=Θ(d2), a polynomial separation from full first-order information.
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combinatoricsAug 14, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Reading computed the order dimension of the poset of regions for most finite Coxeter arrangements, observed that an exceptional type whose dimension exceeds its rank would be the first known simplicial arrangement with that property, and recorded the general guess that every simplicial region poset has dimension equal to its rank (Problem 9.3 of his 2016 chapter); Segovia later asked the analogous question for orien…
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combinatoricsMay 8, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Hamaker and Reiner conjectured that the order complex of an open interval (u,w) in the ASM weak order is contractible unless w is the long element of a standard parabolic subgroup, in which case it is homotopy equivalent to a sphere. False: there is an interval in the ASM weak order on Sn whose order complex is not contractible even though w has no such form, detected by a nonzero Mobius function value.
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geometry-topologyJul 4, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
For every n≥2 the paper exhibits an n-dimensional K-polystable toric Q-Fano variety whose alpha invariant is exactly 2n+12, answering a question of Liu and Zhuang on whether a K-semistable example exists with alpha invariant between n+11 and n1.
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differential-equationsJul 17, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Does every globally asymptotically stable homogeneous polynomial vector field admit a homogeneous polynomial Lyapunov function? No. A planar homogeneous cubic vector field with integer coefficients is globally asymptotically stable yet admits no positive definite homogeneous polynomial with nonpositive Lie derivative, and no real-analytic Lyapunov function even locally, though it does have exponential and rational s…
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combinatoricsApr 30, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
An asymptotic formula for p(k), the limiting probability that a random permutation has an invariant set of size k: it is asymptotically k−δ(1+o(1)) times a smooth positive function, sharpening a line of estimates running through Łuczak-Pyber and Eberhard-Ford-Green.
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algebraJul 26, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Is the depth of the mod-p cohomology ring of every finite group realized as the dimension of one of its associated primes? For G=SmallGroup(128,859) over F2 the ring has depth 2 while every associated-prime quotient has dimension at least 3.
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differential-equationsJul 28, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
within the curvature G-equation model, in three dimensions
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differential-equationsJul 26, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Is the O'Shea-Zames-Falb multiplier test necessary for robust stability of Lur'e systems with slope-restricted nonlinearities, as conjectured by Carrasco? No: there is a stable Lur'e interconnection, certified by a full-block multiplier, that admits no OZF multiplier.
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algorithms-optimizationJul 10, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
a lower bound for Tokuda's sequence; the general Shellsort complexity question stays open
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theoretical-computer-scienceJul 18, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Can online vector balancing in the Spencer setting achieve the optimal order of prefix discrepancy with an efficient algorithm?
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combinatoricsJun 5, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
asymptotic in k; the paper also shows the analogous statement is false for d = 2
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probability-statisticsJun 30, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
For an unkilled Levy process ξ drifting to +∞ with all positive exponential moments, let Iξ=∫0∞e−ξtdt and Xξ=1/Iξ. Bertoin and Yor proved Xξ is moment-determinate when ξ has no positive jumps and conjectured that this condition is necessary. The conjecture is settled.
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quantum-information-computingAug 4, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Whether perfectly complete quantum key agreement can be built from quantumly secure one-way functions in a black-box way. It cannot: for any protocol where Alice and Bob exchange only classical messages, make at most qA and qB quantum queries to a Boolean random oracle and agree on a key with certainty, an eavesdropper given the classical messages recovers the key with certainty in O((qA+qB)5) classical…
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algorithms-optimizationMar 10, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
In the list update problem, is the simple transposition rule optimal under IID requests? The question traces to Rivest's 1976 study of self-organizing lists. The paper proves transposition is within a small constant factor of the optimal online algorithm under any IID distribution.
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combinatoricsJul 3, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
disproved at d = 4; the conjecture for smaller d is untouched
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combinatoricsJul 21, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Boots and Royle, and independently Cao and Vince, conjectured that the join of an edge with a path on n−2 vertices is the unique planar graph of maximum adjacency spectral radius for every n≥9. Tait and Tobin proved it for sufficiently large n in 2017; the conjecture now holds for all n≥9.
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combinatoricsAug 4, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
A question of Averkov, Hofscheier and Nill on whether the Ehrhart h∗-polynomial of a lattice polytope of large lattice width is real-rooted. Proved in fixed dimension for sufficiently large lattice width, giving strict log-concavity and unimodality of the h∗-vector, with the analogous statement for the local h∗-polynomial of a lattice simplex.
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algebraMay 24, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
a batch of problems from published lists, resolved in one paper
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combinatoricsJul 26, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Monical, Tokcan and Yong conjectured that every fixed positive power of the Vandermonde determinant fails to have saturated Newton polytope in sufficiently many variables. For every even power k≥4 there is an explicit lattice point of the Newton polytope of aδkk with vanishing coefficient, obtained from a Dyson constant-term identity; the odd case follows by alternation, proving the conjecture.
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combinatoricsMay 16, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
A graph G on n vertices with k edges is t-edge-balanced if every graph on n vertices with t edges is contained in exactly the same number of subgraphs of Kn isomorphic to G. Infinite families were known for t=2, but no example was known for any t≥3. Resolved in both directions: 3-edge-balanced graphs exist, and no nontrivial t-edge-balanced graphs exist for t≥4.
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quantum-information-computingJul 8, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Friedland and coauthors proposed a quantum analogue of the p-Wasserstein distance and conjectured that, though only a semidistance in general, it is a true distance for a particular quantum cost matrix and for cost matrices near it. The paper disproves both conjectures with an explicit family of triples of states violating the triangle inequality.
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combinatoricsAug 11, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Spahn and Zeilberger's third challenge asks whether the restricted permutation counts ar,s and br,s are holonomic for all r,s>1. Answered affirmatively.
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theoretical-computer-scienceApr 8, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Does exhaustive AdaBoost always converge to a finite cycle of weak classifiers and weight vectors on every finite training set? A finite instance whose orbit never becomes periodic answers no.
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geometry-topologyJun 29, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
an independent human proof of the same conjecture appeared the same week
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probability-statisticsJul 27, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Proves general cases of the conjecture rather than every case.
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combinatoricsJul 23, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
the question is answered negatively and the correct threshold is determined
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combinatoricsJul 21, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Pak and Slonim conjectured that stretched Schubert structure constants are eventually polynomial. They are. Monomial coefficients in affine families of key and Schubert polynomials are eventually polynomial, and the Schubert duality of Watanabe carries this to the structure constants. The same result settles the polynomiality half of a conjecture of Alexandersson and Alhajjar for key polynomials.
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theoretical-computer-scienceJul 20, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
A finite closure system can be given by implications or by a list of subsets closed under intersection. Deciding whether one specification of each kind defines the same family had remained open in several settings; the paper proves the problem coNP-complete.
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combinatoricsJul 28, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
For a differential poset P, must the weighted 2-multichain series MP,2(q) be a rational multiple of FP(q)2, the square of its rank generating series?
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combinatoricsJul 29, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Regts and Sevenster conjectured that a complex-valued graph parameter f with f(∅)=1 has exponentially bounded edge-connection rank precisely when it is a mixed partition function. The paper proves it, with the numbers of even and odd colours bounded in terms of the rank bound.
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quantum-information-computingJun 12, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Englert and Rehacek conjectured which measurement is globally information-optimal for an ensemble of equiangular equiprobable pure states. Their conjecture holds, via the remaining entropy inequalities of Holevo and Utkin.
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algorithms-optimizationJul 13, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
The authors describe the tradeoff as nearly settled rather than settled.
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geometry-topologyMay 20, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Ciliberto, Knutsen, Lesieutre, Lozovanu, Miranda, Mustopa and Testa asked a question about effective divisors of positive self-intersection on smooth projective surfaces. The answer is negative, witnessed by a very non-movable effective divisor.
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combinatoricsJul 7, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
For an inclusion-free hypergraph on n vertices, a weight assignment w:[n]→[d] is isolating when a unique edge attains minimum weight. Faber and Harris conjectured that the number of isolating assignments is at least n∑j=0d−1jn−1, attained by the hypergraph of n singleton edges. The bound holds, and extends to a more general class of objective functions.
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