Gromov and Weiss's Question on Sofic Groups
Prior state unknown→disproved
answered no: non-sofic groups exist
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Open registry federation
Mathematical findings
136 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.
Schiffer's Conjecture and the Pompeiu Problem
Prior state unknown→disproved
Also refutes the 1929 Pompeiu problem: Corollary 1.2 applies Williams' classical 1976 equivalence to the same constructed domains, so this is one construction settling both, not two separate results.
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Upper Bounds for High-Dimensional Sphere Packing
Prior state unknown→proved
upper bounds reach the Cohn-Elkies threshold; the true asymptotic density remains open
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Connes' Rigidity Conjecture
Prior state unknown→disproved
Are ICC property (T) groups remembered by their von Neumann algebras - if $L(\Gamma) \cong L(\Lambda)$ for such groups, must $\Gamma \cong \Lambda$? A counterexample refutes Connes' conjecture that these groups are uniquely determined by their group von Neumann algebras.
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Sendov's Conjecture
Prior state unknown→proved
Sendov's conjecture is resolved for every degree n >= 2, closing a gap that had stood since 1959: degrees up to eight were settled piecemeal between 1969 and 1999, and Tao's 2020 result covered all sufficiently large degrees without ever specifying the threshold, leaving the middle range open. Tao's digestion establishes the stronger interior form of the statement, which resolves the Phelps-Rodriguez conjecture in…
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Upper Bounds for Binary and Spherical Codes
Prior state unknown→proved
exponential improvement over the 1977 MRRW bounds; the exact rate-distance trade-off remains open
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Asymptotically attaining the Moore bound
Prior state unknown→proved
Settles two conjectures. Theorem 1.1 proves Bollobas's asymptotic degree-diameter conjecture, in the stronger liminf form rather than the conjectured limsup. Corollary 1.2 proves Conjecture 3 of Cambie, Cames van Batenburg, de Joannis de Verclos and Kang on the edge variant, again in the stronger liminf form, and is tight for bipartite graphs.
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Lower Bounds for the Permanent in Arithmetic Circuits
Prior state unknown→proved
an n^4/log n formula lower bound; VP vs VNP remains wide open
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Record Lower Bounds for the Shannon Capacity of Odd Cycles
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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Polynomial-Factor Hardness for the Closest Vector Problem
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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Feige's Conjecture
Prior state unknown→proved
Let $X_1,\ldots,X_n$ be independent nonnegative random variables with $\mathbb{E}X_i \le 1$, and let $S$ be their sum. Is $\mathbb{P}(S < \mathbb{E}S + 1) \ge 1/e$? Feige proved the constant $1/13$ and conjectured the sharp $1/e$. Three independent July 2026 proofs settle it, both building on the Vlassis-Thomas calibration theorem; the sharper one determines the optimal small-deviation bound for every deviation $\…
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Grothendieck's Finite Flat Group Scheme Order Question
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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Kannan–Tetali–Vempala conjecture (bipartite/binary-matrix case)
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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Phelps–Rodriguez Conjecture
Prior state unknown→proved
Phelps-Rodriguez implies Sendov, so this entry records the stronger of the pair; the companion Sendov entry records the weaker statement and Mazur's original formalization, which proved Sendov but never stated the equality classification. The exceptional family is genuinely attained rather than an artefact of the proof: for p = z^n - 1 and a = 1 the only critical point is the origin, at distance exactly 1. Both co…
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The Divisible Rank-Three Case of the Kajitani–Ueno–Miyano Conjecture
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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Quantum Parallel Repetition
Prior state unknown→proved
Does the value of a two-player quantum game decay exponentially under parallel repetition, as Raz's theorem gives for classical games? Yes: an exponential parallel repetition theorem holds for arbitrary finite two-player quantum games.
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Ehrhart's Volume Conjecture
Prior state unknown→proved
What is the maximum volume of a convex body in $\mathbb{R}^n$ whose centroid is its only interior lattice point? Ehrhart conjectured the extremal value in 1964; the sharp maximum is now determined in every dimension.
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Explicit Presentation of the 2-adic Absolute Galois Group
Prior state unknown→proved
Give an explicit profinite presentation of $\operatorname{Gal}(\overline{\mathbb{Q}}_2 / \mathbb{Q}_2)$. The tame local cases were settled by the early 1980s; the dyadic case was the last one missing. The new presentation has four generators, two word relations and a pro-$2$ condition on the wild generators.
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Strichartz's Question on Fourier Frames for the Cantor Measure
Prior state unknown→disproved
Does the middle-third Cantor measure admit a Fourier frame, that is, a countable set of exponentials giving two-sided frame bounds on its $L^2$ space? No. The Cantor measure with base $b$ admits no Fourier frame for any odd integer $b > 1$, which answers Strichartz's question for the middle-third case.
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Erdős Problem #146: Degeneracy Conjecture
Prior state unknown→disproved
If $H$ is bipartite and $r$-degenerate, is $\mathrm{ex}(n;H) \ll n^{2-1/r}$ (a \$500 Erdős-Simonovits prize conjecture)? A counterexample refutes the degeneracy conjecture.
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Erdős Problem #183: Multicolor Triangle Ramsey
Prior state unknown→proved
Let $R(3;k)$ be the least $n$ such that every $k$-colouring of the edges of $K_n$ contains a monochromatic triangle. Determine $\lim_{k\to\infty} R(3;k)^{1/k}$ (a \$250 Erdős prize problem). A superexponential lower bound resolves the problem: the limit is infinite.
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Gaussian product inequality conjecture
Prior state unknown→proved
Let $\boldsymbol{X} = (X_1,\ldots,X_n)$ be a centered Gaussian vector, not necessarily nondegenerate. Then, for every $\alpha_1,\ldots,\alpha_n > 0$, $$\mathsf{E}\left[\prod_{i=1}^n |X_i|^{\alpha_i}\right] \geq \prod_{i=1}^n \mathsf{E}\left[|X_i|^{\alpha_i}\right].$$ Moreover, if $\mathsf{Var}(X_i) > 0$ for every $i$, then equality holds if and only if $X_1,\ldots,X_n$ are independent.
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Erdős Problem #707: Sidon Sets and Perfect Difference Sets
Prior state unknown→disproved
Hall's 1947 counterexample predates the problem itself; this paper's counterexample is independent, smaller, and Lean-certified.
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Optimal Exponent Relating Sumsets and Difference Sets
Prior state unknown→proved
New arXiv preprint with an author-provided Lean formalization; not yet peer-reviewed.
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Erdős Problem #180: Compactness Conjecture
Prior state unknown→disproved
For every finite family $\mathcal{F}$ of graphs, is there a single $G \in \mathcal{F}$ with $\mathrm{ex}(n;G) \ll_{\mathcal{F}} \mathrm{ex}(n;\mathcal{F})$? A counterexample refutes the Erdős-Simonovits compactness conjecture.
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Kourovka Problem 3.46 - Maximal Locally Soluble Normal Subgroups
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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Sabidussi's Compatibility Conjecture
Prior state unknown→proved
Can the edges of a finite connected multigraph, given a closed eulerian trail, be partitioned into circuits so that no circuit contains two edges used consecutively in the trail? The proof in fact four-colours the edges to satisfy the constraints.
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The Ballantine-Beck-Feigon-Maurischat Conjectures on Subsum Polynomials
Prior state unknown→proved
six of the ten conjectures proved; one was found false as printed and its corrected form remains open
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Ben Green's Open Problem 57
Prior state unknown→disproved
intended complex form disproved, with a certified strict support-function gap
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The Erdos-Sos Pairwise-Sums Problem
Prior state unknown→proved
Let $f_3(N)$ be the least size forcing a set $A \subseteq \{1,\ldots,N\}$ to contain distinct $a,b,c$ with $a+b$, $a+c$ and $b+c$ all in $A$. The upper bound $f_3(N) \le 5N/8 + O(1)$ matches the standard construction $[N/8,N/4] \cup [N/2,N]$, so $f_3(N) = 5N/8 + O(1)$.
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Erdős Problem #1196: Primitive Sets
Prior state unknown→proved
Bounds the weighted sum $\sum 1/(a \log a)$ taken over primitive sets of integers (sets where no element divides another).
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Kourovka Problem 19.25 - Totient Sums and Simplicity
Prior state unknown→disproved
Do a finite group's order together with $\sum_{g \in G} \varphi(|g|)$ determine whether the group is simple? A simple and a non-simple group of order $6048$ share the statistic $23984$.
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FGG Conjecture for QAOA on the Ring of Disagrees
Prior state unknown→proved
For an even cycle of size $N$ and depth $p$ with $2p + 2 \le N$, is the optimal QAOA approximation ratio for MaxCut exactly $\frac{2p+1}{2p+2}$, as Farhi, Goldstone and Gutmann conjectured?
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Kourovka Problem 18.50 - Prescribed Permuted-Product Cardinality
Prior state unknown→proved
Given $n$ and $1 \le c \le n!$, can $n$ distinct group elements be chosen so that their $n!$ ordered products take exactly $c$ distinct values? Constructions realize every $c$.
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Kemeny Rank Aggregation for Three Voters
Prior state unknown→proved
Is computing a Kemeny-optimal aggregate ranking NP-hard when the input consists of exactly three complete rankings? Hardness was known for every even $n \ge 4$; three voters was the minimal open case, and $n = 2$ is polynomial-time solvable.
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Kourovka Problem 20.125 - Noninjective Rota-Baxter Operator
Prior state unknown→proved
Can a nonabelian group admit a Rota-Baxter operator that is surjective but not injective? A construction shows yes.
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