probability-statisticsMay 12, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
The classical problem of maximizing the Shannon entropy of a sum of independent random variables supported on a finite alphabet, settled in the ternary case. For independent X1,…,Xn taking values in {0,1,2}, the entropy of Sn=X1+⋯+Xn is maximized when X1,…,Xn−1 are uniform on {0,2} and Xn has an explicitly described three-point distribution. This extends the Shepp-O…
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analysisJul 16, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Two free ergodic measure-preserving flows whose L1 full groups are isomorphic as abstract groups are conjugate up to a scalar time change. This proves the flow analogue of Belinskaya's theorem, answering a question posed by François Le Maître and the author.
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combinatoricsAug 7, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
For a forbidden configuration F, the Anstee-Sali conjecture predicts that forb(m,F) is Θ(mX(F)−1), where X(F) comes from an explicit product construction. Disproved: the 4-uniform family on six vertices formed by a two-vertex core joined to the edges of a 4-cycle has X(F)=4, so the conjecture predicts Θ(m3), while a random-alteration argument gives…
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probability-statisticsJul 20, 2026Significance 20/100Registry: lean verified
Prior state unknown→proved
Let X=(X1,…,Xn) be a centered Gaussian vector, not necessarily nondegenerate. Then, for every α1,…,αn>0,
E[i=1∏n∣Xi∣αi]≥i=1∏nE[∣Xi∣αi].
Moreover, if Var(Xi)>0 for every i, then equality holds if and only if X1,…,Xn are independent.
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analysisJul 29, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Let μ be a probability measure on the unit circle with Verblunsky coefficients α. Lukic conjectured that a weighted entropy condition with finitely many critical points is equivalent to a decomposition of α into components localized at those points. A counterexample with two critical points of multiplicity three refutes it: the sequence satisfies the decomposition conditions while the corresponding…
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combinatoricsJul 16, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
the true order is determined, and it is not the conjectured one
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mathematical-physicsJul 30, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
a record upper bound; the exact constant remains unknown
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analysisJul 29, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
A partial result. Fuglede's conjecture remains open for finite cyclic groups generally; this settles an infinite family and, for the spectral-to-tiling direction, only under rapid growth of the prime factors.
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analysisJul 13, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Harris had settled the generalized problem in dimension four; this reaches dimension three
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probability-statisticsMay 3, 2026Significance 20/100Registry: site confirmed
Prior state unknown→disproved
the paper proposes that the cosine matrix is the true minimizer for all p and n, and proves a stronger stochastic domination statement conditional on a new volumetric extension of Fejes Toth's zone conjecture
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analysisJul 24, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
The dissipative barrier method suppresses spectral pollution when a differential operator is truncated, but can it hide genuine spectral points? Known as the graveyard problem, the question stayed open in dimension two and above for more than a decade. It cannot: for Schrodinger operators in dimensions d≥2 no spectral point becomes invisible, which together with the known one-dimensional theorem settles no-inv…
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analysisJul 23, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
A convex body is in Faber-Krahn position if it minimizes the first Dirichlet eigenvalue within its volume-preserving linear orbit. The paper proves this position is unique up to orthogonal transformations, answering a question of Schmuckenschläger from 2011, via a new log-convexity property of the first eigenvalue under positive definite linear deformations.
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combinatoricsJul 16, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
improves the classical recurrence; the exact Schur numbers beyond S(5) remain unknown
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combinatoricsAug 12, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Disproves the Bowler-Brown-Fenner bound of 2*floor((n-1)/3) on common cards between nonisomorphic graphs: an explicit connected 78-vertex pair shares at least 51 cards against the predicted 50, and for every even r >= 4 there are families with overlap fraction asymptotically at least 1 - 1/r, so the attainable fraction approaches the full deck. The Kelly-Ulam reconstruction conjecture itself is untouched.
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geometry-topologyAug 4, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Chen, Nie and Xu prove a Nakai–Moishezon-type numerical criterion for a broad class of complex Hessian-type equations on compact projective manifolds. In particular, Corollary 1.3 gives a uniform version of Székelyhidi’s conjecture for complex Hessian quotient equations, while Corollary 1.5 proves the uniform version, formulated by Murakami, for complex k-Hessian equations. However, the results assume projectivity a…
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geometry-topologyMay 20, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Kollár and Kovács asked whether the first cohomology of the structure sheaf of the fibers must be constant for a flat projective morphism to a smooth curve whose fibers are Cohen-Macaulay and reduced and whose generic fiber is smooth. It need not be: such a morphism exists with non-constant first cohomology.
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theoretical-computer-scienceMay 29, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Steurer conjectured in 2010 that any family of n unit vectors with polynomially small average correlation Ei,j∣⟨vi,vj⟩∣≤n−ε contains linear-sized constant-separated sets. Refuted in a strong sense, using sparse high-dimensional expanders.
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combinatoricsAug 30, 2025Significance 20/100Registry: unreviewed
Prior state unknown→disproved
An ℓ-Oddtown is a family of subsets of an n-element set whose set sizes are not divisible by ℓ while all pairwise intersection sizes are. Berlekamp and Graver showed the maximum size is n for prime ℓ, Babai and Frankl extended this to prime powers and asked whether n still holds for other moduli, a question open even for ℓ=6. Bukh, Chao and Zheng answer it negatively with an explicit supe…
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theoretical-computer-scienceMay 4, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Odifreddi asked, as Problem 3 in his surveys "Strong Reducibilities" (1981) and "Reducibilities" (1999), whether every computably enumerable tt-degree contains a c.e. irreducible m-degree, meaning an m-degree consisting of a single 1-degree. Answered negatively: there is a c.e. tt-degree containing no c.e. irreducible m-degree. This also shows Jockusch's 1969 theorem, which produces an irreducible m-de…
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combinatoricsJun 23, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
an improved lower bound; the true order of g(r) remains open
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algorithms-optimizationJul 22, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
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.)
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probability-statisticsJul 15, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Among d+1 equiprobable equal-energy signals in Gaussian noise, is the regular simplex optimal for average error probability? Yes. The underlying comparison is that for any m×m correlation matrix R with R−11T/m⪰0 and X∼N(0,R), the maximum of the Xi is stochastically dominated by the maximum of m independent standard Gaussians.
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theoretical-computer-scienceAug 6, 2026Significance 18/100Registry: unreviewed
Prior state unknown→proved
Does planarity help approximate counting? The paper gives an FPRAS for the planar hard-core partition function at small activity, proves that approximately counting q-colourings on planar graphs is NP-hard for every constant q≥4, and completely characterizes when an FPRAS exists for 2-spin systems on planar graphs at small external field.
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logic-foundationsAug 27, 2026Significance 18/100Registry: unreviewed
Prior state unknown→disproved
The paper proves that not every Heyting algebra can occur as the lattice of subterminal objects of an elementary topos. Specifically, the free Heyting algebra F2 on two generators cannot occur.
Using Bellissima’s representation F2↪O↑(K2), the authors construct an upward-closed subset A⊆K2 with A∈/F2. They show that if some elementary topos E satis…
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analysisMar 23, 2026Significance 18/100Registry: unreviewed
Prior state unknown→proved
Localizing Bernstein theory to prove lower bounds for the Lebesgue constants of Lagrange interpolation, with application to a problem of Erdős and Turán and to a conjectured bound from the interpolation literature.
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number-theoryAug 7, 2026Significance 18/100Registry: unreviewed
Prior state unknown→disproved
An n-divisor set contains a multiple of every integer from 1 to n. Umans and Wang proposed, as the arithmetic-progression form of their Strong (α,β)-Divisor Conjecture, that such a progression exists with few terms of bounded magnitude, which would imply faster algorithms for polynomial and integer factorization. Refuted unconditionally, including its exponent-level relaxation.
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combinatoricsAug 3, 2026Significance 18/100Registry: unreviewed
Prior state unknown→disproved
one construction disproves both the product conjecture and its weak form
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theoretical-computer-scienceAug 8, 2026Significance 18/100Registry: unreviewed
Prior state unknown→proved
An average-case claim about the Gaussian model, not a contradiction of the worst-case NP-hardness of integer least squares. If it holds, no computational-statistical gap separates polynomial-time detection from exhaustive maximum likelihood at first order in this model.
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combinatoricsMay 19, 2026Significance 17/100Registry: unreviewed
Prior state unknown→proved
A collection of open problems from the algebraic and enumerative combinatorics literature, resolved in one paper: a conjecture of Defant, Jiang, Marczinzik, Segovia, Speyer, Thomas and Williams on the echelonmotion operator on modular lattices, which also yields a new algebraic bijective proof of Dilworth's theorem; conjectures of Hopkins on parking function statistics studied by Stanley and Yin; and two conjectures…
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quantum-information-computingAug 20, 2026Significance 16/100Registry: unreviewed
Prior state unknown→disproved
Conjecture 13 of King, Gosset, Kothari and Babbush asserts that for the set Bε(ρ) of Pauli observables with expectation value at least ε in magnitude, the fractional chromatic number of the induced anticommutation graph is O(ε−2); it would give a triply efficient Pauli shadow tomography algorithm. False: there are states and observables for which no finite constant bounds…
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combinatoricsAug 11, 2026Significance 16/100Registry: unreviewed
Prior state unknown→proved
Conjectures 2a and 2b of Kauers and Zeilberger, on the asymptotics of a family of restricted lattice walks. Both are obtained from a local limit theorem for excursions of Markov-modulated random walks in cones.
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algebraJul 20, 2026Significance 16/100Registry: lean verified
Prior state unknown→proved
Does there exist a group with more than one but only finitely many maximal locally soluble normal subgroups? An explicit group with exactly two settles it.
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geometry-topologyAug 21, 2026Significance 16/100Registry: unreviewed
Prior state unknown→disproved
Disproved on the Hopf threefold X=(C3∖{0})/⟨z↦e−1z⟩: Xia and Zhang construct a smooth Hermitian form ω and smooth functions φj with ω+ddcφj>0 whose Monge-Ampère masses tend to infinity, so the universal bounded mass property fails already in complex dimension three. The construction uses the Hopf threefold's elliptic fibration, an exact mass…
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combinatoricsJun 28, 2026Significance 15/100Registry: lean verified
Prior state unknown→proved
Let f3(N) be the least size forcing a set A⊆{1,…,N} to contain distinct a,b,c with a+b, a+c and b+c all in A. The upper bound f3(N)≤5N/8+O(1) matches the standard construction [N/8,N/4]∪[N/2,N], so f3(N)=5N/8+O(1).
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geometry-topologyAug 10, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Makeev conjectured it to be true for all dimensions. This result disproves it for dimensions 4 and 5. Dimensions 6 and above remain open.
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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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