geometry-topologyAug 27, 2026Significance 22/100Registry: unreviewed
Prior state unknown→proved
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…
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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 $F_2$ on two generators cannot occur.
Using Bellissima’s representation $F_2\hookrightarrow\mathcal O_\uparrow(K_2)$, the authors construct an upward-closed subset $A\subseteq K_2$ with $A\notin F_2$. They show that if some elementary topos $\mathcal E$ sat…
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quantum-information-computingAug 27, 2026Significance 10/100Registry: site confirmed
Prior state unknown→disproved
For two qubits, the paper considers
$\rho=\frac12|\Phi^+\rangle\langle\Phi^+|+\frac12|01\rangle\langle01|$,
where $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2$. This rank-2 state has no global supporting affine functional for Entanglement of Formation.
Setting $\rho_t=(1-t)\rho+t|10\rangle\langle10|$, Wootters’ formula gives
$C(\rho_t)=\frac12-\sqrt{2t}+O(t)$.
Consequently,
$\lim_{t\to0^+}[E_F(\rho)-E_F(\r…
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combinatoricsAug 26, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
For fixed $\delta\in(0,1)$ and all sufficiently large $\Delta$ depending only on $\delta$, Glauber dynamics for proper $q$-colorings mixes rapidly on every graph of girth at least $5$ whenever $q\ge(1+\delta)\Delta$: spectral gap $\Omega_\delta(1/n)$ and $t_{\mathrm{mix}}(\varepsilon)=O_\delta(n^2\log q+n\log(1/\varepsilon))$. An analogous theorem holds for the anti-ferromagnetic Potts model at $q\ge(1+\delta)(1-\…
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combinatoricsAug 26, 2026Significance 4/100Registry: site confirmed
Prior state unknown→proved
Answered in full: for every $k\ge4$, a target with exactly two nonadjacent zero digits has $F_k(t)=(k+23)3^{k-4}$, independently of the distance between the zeros. Exact at every width, no error term, no hypothesis on $k$ (Theorem 1.1). This is an evaluation, not an extremal result, and the paper is explicit about the difference: the plateau value is not maximal. At $k=12$ it reads $35\cdot3^8=229{,}635$ while $F_…
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analysisAug 25, 2026Significance 27/100Registry: unreviewed
Prior state unknown→proved
If a bilinear form admits an $(r,s)$-sparse bound, its coordinate-wise extension to $\mathbb C^n$-valued functions admits an $(r,s)$-convex body sparse bound, for $1\le r,s<\infty$ with $\tfrac1r+\tfrac1s>1$. It holds both in a fixed dyadic lattice (Theorem 2.6, constants independent of the ambient dimension) and for arbitrary cubes (Theorem 3.4), via a randomization of the good part of the form.
On scope, two th…
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logic-foundationsAug 25, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Let $T$ be a complete first-order theory in discrete or continuous logic, let $M\\prec\\mathcal{U}$, and let $\\mu\\in\\mathfrak{M}_{x}(\\mathcal{U})$ be Borel-definable over $M$. The paper proves that the following three conditions are equivalent:\n\n$(i)$ $\\mu$ is a frequency interpretation measure (fim) over $M$;\n\n$(ii)$ $\\mu$ is definable over $M$ and its canonical random extension $r_{\\mu}$ is genericall…
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algebraAug 25, 2026Significance 14/100Registry: site confirmed
Prior state unknown→proved
Let $S=k[x_1,x_2,x_3,x_4]$ over any characteristic-zero field. For each $d\in\{5,7\}$ and every $r\ge1$, the paper proves that $r$ general degree-$d$ forms satisfy Fröberg’s predicted Hilbert series
$$
\operatorname{HS}_{S/(F_1,\ldots,F_r)}(t)
=
\left[\frac{(1-t^d)^r}{(1-t)^4}\right]_+.
$$
The genuinely new ranges are $6\le r\le11$ for quintics and $6\le r\le21$ for septics. These are reduced to finitely many ex…
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geometry-topologyAug 24, 2026Significance 65/100Registry: unreviewed
Prior state unknown→proved
Claims an explicit compact complex threefold $X$, fibred over $\mathbb{P}^1$ by complex 2-tori via period functions on the $(3,4,\infty)$ orbifold, degenerating to a del Pezzo-of-degree-six fibre (identified opposite sides of its hexagon) at one point and to bielliptic multiple fibres of multiplicities 3 and 4 at the other two. Argues $X$ is simply connected with $H_*(X;\mathbb{Z})=H_*(S^6;\mathbb{Z})$, hence diff…
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analysisAug 24, 2026Significance 45/100Registry: unreviewed
Prior state unknown→disproved
The paper constructs a real meromorphic function $F$ on $\mathbb{C}$ satisfying $F^{-1}(\{0,1,\infty\})\subset\mathbb{R}$, with each of the three fibers $F^{-1}(0)$, $F^{-1}(1)$, and $F^{-1}(\infty)$ infinite, such that for every $a\in\widehat{\mathbb{C}}\setminus\{0,1,\infty\}$, the $a$-point divisor in each of the upper and lower half-planes fails the Blaschke condition. Consequently, $F$ is not of bounded type…
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number-theoryAug 23, 2026Significance 50/100Registry: unreviewed
Prior state unknown→proved
Two tiers, and only the first is the record. Rank $\ge 31$ is unconditional: 31 explicit points, independence asserted via the leaderboard's stated general practice of exact 2-descent (not reproduced here - see the verification note). Rank exactly 31 is conditional on GRH and BSD, per the submitters' commentary, in the same style as the sibling record's Bober-bound argument; no numeric derivation has been publishe…
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number-theoryAug 22, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
The manuscript claims $C_{a,b}$ is transcendental for every $a\ge1$ and $b\ge1-a$, settling the positive integer-valued affine subclass of Erdős Problem 270. Two pieces of context matter. Problem 270 as Erdős and Graham posed it, for every $f(n)\to\infty$, was already answered no by Crmarić and Kovač in 2025: for any $\alpha>0$ some such $f$ makes the series sum to $\alpha$. What survives is the non-decreasing cas…
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number-theoryAug 22, 2026Significance 8/100Registry: site confirmed
Prior state unknown→proved
Proves Haglund's Conjecture 4 for $k=1$: every non-real first-quadrant zero of $\Phi_1+t\Phi_2$ is simple with strictly decreasing imaginary part, no branch escapes forward, and every finite-multiplicity real collision stays real afterwards. The cases $k\ge2$ remain open. Two readings worth separating: Conjecture 4 asserts the monotone descent alone, so the no-escape and stays-real statements are this paper's own…
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geometry-topologyAug 21, 2026Significance 16/100Registry: unreviewed
Prior state unknown→disproved
Disproved on the Hopf threefold $X=(\mathbb{C}^3\setminus\{0\})/\langle z\mapsto e^{-1}z\rangle$: Xia and Zhang construct a smooth Hermitian form $\omega$ and smooth functions $\varphi_j$ with $\omega+dd^c\varphi_j>0$ 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 ma…
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number-theoryAug 21, 2026Significance 20/100Registry: lean checked
Prior state unknown→proved
Answers Problem 3 and generalizes it: the classification $\sqrt{m} \in \mathcal{S} \iff m = 2$ covers every square root, and a further theorem replaces parity by divisibility by any $p \ge 2$. Note the scope of the machine-checking, which is narrower than the paper: the author states that the case $m = 3$ is what is verified in Lean, and the repository flags the thickness computation of section 4.1 and all of sect…
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number-theoryAug 20, 2026Significance 50/100Registry: unreviewed
Prior state unknown→proved
Two tiers, and only the first is the record. Rank $\ge 30$ is unconditional, being thirty explicit independent points. Rank exactly 30 is conditional: applying Bober's bound (arXiv:1112.1503) with $\Delta = 4.25$ gives an analytic rank of at most 31, and the root number is $+1$ so the rank is even, hence 30 - but that argument assumes GRH, and equating analytic rank with rank assumes BSD. The entry is a partial re…
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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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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 $B_\varepsilon(\rho)$ of Pauli observables with expectation value at least $\varepsilon$ in magnitude, the fractional chromatic number of the induced anticommutation graph is $O(\varepsilon^{-2})$; 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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theoretical-computer-scienceAug 20, 2026Significance 50/100Registry: unreviewed
Prior state unknown→disproved
Marton's inner bound, proposed in 1979, is the best known achievable region for a general discrete memoryless broadcast channel, and whether it always achieves the capacity region had been open ever since. It does not: there is a finite two-receiver discrete memoryless broadcast channel whose two-letter Marton value strictly exceeds twice its one-letter value, so the complete one-letter Marton region is strictly c…
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geometry-topologyAug 19, 2026Significance 60/100Registry: unreviewed
Prior state unknown→disproved
The paper's appendix draws a distinction worth keeping: a counterexample may reduce to a finite certificate, checkable once the object is written down, or it may itself be a theorem quantified over all degenerations. This is the second kind. The method field records construction, because the resolution exhibits an explicit fivefold, but the difficulty lay elsewhere - candidate manifolds of this shape have been ava…
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combinatoricsAug 19, 2026Significance 8/100Registry: unreviewed
Prior state unknown→proved
The claim is $R(c) = 40c+41$ for every $c \ge 2$, reduced to three finite facts: the base value $R(2) = 121$, and the unsatisfiability of a 321-position and a 521-position spoke template. The reduction is Lean-checked and holds for every $D \ge 1$; the two unsatisfiability results carry DRAT proofs.
This completes the partial entry for the same conjecture, which proved it for roughly two thirds of integers via a…
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analysisAug 19, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
Question 6.1 of Chalmoukis, Tsikalas and Yakubovich asks how far the Power boundedness constant $P(T)$ of a matrix can exceed its ordinary Kreiss constant $K(T)$. Answered more strongly: for every $K > 1$ there are matrices whose Cayley transforms satisfy $K(C_h(A_{n,h})) \le K$ while the strong Kreiss constant satisfies $K_s(C_h(A_{n,h})) \ge \tfrac{1}{2}Cn^{\alpha_K}$ with $\alpha_K = (K-1)/(C+K-1)$. Since $P(T)…
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geometry-topologyAug 19, 2026Significance 55/100Registry: lean checked
Prior state unknown→disproved
Only the smooth case falls. Hamburger's real-analytic theorem is untouched, and the counterexample is explicitly a $C^\infty$ object, so the conjecture's classical analytic form remains true. The gap between the two is the whole content of the result.
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algorithms-optimizationAug 19, 2026Significance 28/100Registry: unreviewed
Prior state unknown→proved
Koivisto asked at Dagstuhl in 2013 whether the linear extensions of an arbitrary $n$-element poset can be counted exactly in time $O^*(c^n)$ for some $c < 2$. Yes: a deterministic exact algorithm runs in $O^*(1.89^n)$, breaking the $2^n$ barrier for the general problem.
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combinatoricsAug 19, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Every finite simple connected graph $G$ with
$$
|V(G)|=2d+1,\qquad \operatorname{diam}(G)=d\ge 3
$$
satisfies
$$
W(G)\le W(C_{2d+1})
=\frac{(2d+1)d(d+1)}2.
$$
The claimed equality cases are exactly $C_{2d+1}$ for every $d\ge3$, the double star $D_{2,3}$ when $d=3$, and the nine-vertex tree $T_{1,2,2}=S(2,3,3)$ when $d=4$.
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logic-foundationsAug 19, 2026Significance 14/100Registry: lean verified
Prior state unknown→independent
Independent of ZFC, which is why this entry is the first to carry that result rather than proved or disproved. Both directions are formalized: Hechler's 1972 construction gives a model where the answer is no, and adding $\mathfrak{c}^+$ random reals over a model of CH gives one where it is yes.
The credit is shared and mostly human. Newelski, Pawlikowski and Seredynski settled the problem's second question in 198…
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algebraAug 19, 2026Significance 10/100Registry: lean checked
Prior state unknown→proved
For the Kasami APN function $F(x) = x^{4^k - 2^k + 1}$ on $\mathrm{GF}(2^n)$ with $\gcd(k, n) = 1$, the conjecture asserts that for $\Delta = \{F(b) + F(b+1) + 1\}$ and all distinct nonzero $v_1, v_2$, the number of triples in $\Delta^3$ with $v_1 x + v_2 y + (v_1 + v_2) z = 0$ is exactly $2^{2n-3}$. Proved for $k \bmod n \in \{1, 2, n-2, n-1\}$ and verified exhaustively for $n \le 13$; the general case remains open.
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combinatoricsAug 19, 2026Significance 28/100Registry: unreviewed
Prior state unknown→proved
The big-line-big-clique conjecture of Kára, Pór and Wood asserts that for all $k, \ell$ there is an $n$ such that every finite point set of at least $n$ points contains $\ell$ collinear points or $k$ points that pairwise see each other. True for $\ell = 4$, $k = 6$, the first case left open: every finite point set of size at least $10^{11055931}$ has four collinear points or six pairwise visible points.
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analysisAug 18, 2026Significance 38/100Registry: unreviewed
Prior state unknown→proved
The theorem is the vector-valued endpoint bound $\|Rf\|_{L^{1,\infty}} \le 2\|f\|_{L^1}$ for $R = (R_1,\ldots,R_n)$, so the constant 2 also serves each component $R_j$ uniformly in the dimension; the best previously known component bound grew like $c\log n$.
The mechanism is a decomposition theorem stated as Theorem 1.2: for every nonnegative $f \in L^1 \cap L^2$ and every $\lambda > 0$, write $f = \mu + (-\Delta…
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probability-statisticsAug 18, 2026Significance 8/100Registry: unreviewed
Prior state unknown→proved
Four results, and the first is partly a refutation. Jakimiuk conjectured $c_p = \mu_p - 1$ is optimal for every $p \ge 3$; the paper proves that for $p \ge 4$ and gives a counterexample for every $2 < p < 4$, so the conjecture is false as posed and the corrected range is $p \ge 4$. The witness is the two-coordinate vector $S_2 = (\varepsilon_1+\varepsilon_2)/\sqrt2$.
The Baranski-Murawski-Nayar-Oleszkiewicz flat-…
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theoretical-computer-scienceAug 17, 2026Significance 55/100Registry: unreviewed
Prior state unknown→proved
A record, not a resolution, and a small one by design. The bound moves from $2.371339$ to $2.371177$, about $1.6 \times 10^{-4}$, and the authors describe it as a small step. Whether $\omega = 2$ is untouched, and nothing here suggests the laser method can reach it.
The interesting claim is methodological rather than numerical. The bottleneck in this line of work is a hard optimization problem, and the paper repo…
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probability-statisticsAug 16, 2026Significance 37/100Registry: unreviewed
Prior state unknown→proved
The claim is the exact conjectured decay: $\psi_{\mu_a}(u) \le C_a/\sqrt{\log u}$ for every $u > 1$ and every $n$, with $C_a$ dimension-free - concretely $\lesssim \kappa_a^2(\log\frac{\kappa_a}{\kappa_a-1})^{1/2}$ where $\kappa_a = (1+a)/(1-a)$.
What is new is one step in a three-paper chain rather than a proof from scratch, and the paper is explicit about it. Chen's reverse-heat and Boolean-bridge framework and…
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number-theoryAug 15, 2026Significance 10/100Registry: lean checked
Prior state unknown→proved
For every real $\xi>0$ the sequence of integer parts $[\xi 7^{n}]$, $n=0,1,2,\dots$, contains infinitely many composite numbers. Second, there is no infinite right truncatable prime in base~$7$.
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mathematical-physicsAug 14, 2026Significance 3/100Registry: lean checked
Prior state unknown→proved
Finite-jet theorems about compiled truncated EMD equation certificates: the exact shear-orbit fiber classification of the complete first seed channels, an explicit collision family with one metric three-jet realized by an actual cubic metric germ (genuine Frechet Ricci value and first derivative), the compiled impossibility theorem, and the fourth-order recovery with equality fiber $a=\pm b$. Not settled here: pro…
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combinatoricsAug 14, 2026Significance 4/100Registry: unreviewed
Prior state unknown→disproved
Sivaraman asked whether perfect divisibility is characterized by its chromatic consequence: is a graph $G$ perfectly divisible if and only if $\chi(H) \le \binom{\omega(H)+1}{2}$ for every induced subgraph $H$ of $G$? False: the Paley graph $P(17)$ satisfies the chromatic bound hereditarily but is not perfectly divisible.
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algebraAug 14, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
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 $p > 5$.
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algebraAug 14, 2026Significance 8/100Registry: unreviewed
Prior state unknown→disproved
Every purely-maximal ideal of a commutative ring is purely-prime, and the converse holds for several important classes of rings; Tarizadeh conjectured (Conjecture 5.8 of his earlier published paper) that in a commutative ring every purely-prime ideal is purely-maximal. False: there is a commutative ring with a purely-prime ideal that is not purely-maximal.
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theoretical-computer-scienceAug 14, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
Among classes of tournaments for which neither hardness nor polynomial-time solvability of isomorphism was known, bounded VC dimension stood out as an open problem of Neuen and Grohe. Resolved: isomorphism of tournaments of VC dimension $d$ is decidable in time $n^{O(d \log d)}$, so automorphism groups of bounded-VC tournaments are computable in polynomial time; isomorphism of tournaments of bounded chromatic numb…
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combinatoricsAug 14, 2026Significance 3/100Registry: unreviewed
Prior state unknown→proved
A tournament orients every pair in a round-robin (winner → loser). The score sequence is the sorted win-count list. Reversing a directed 3-cycle never changes scores, so score-equivalent tournaments can look structurally different.
Question: Which linear combinations of induced k-subtournament type-counts are score-determined — identical across all tournaments sharing a score sequence, at any host size?
Answer:…
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algebraAug 14, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
The Hessian conjecture $HC_n$ asks whether every polynomial $f$ with $\det \mathrm{Hess}(f) \in \mathbb{C}^\times$ has a polynomial gradient inverse. It is known for $n \le 3$, false for $n \ge 5$, and open exactly in dimension four, where it implies the plane Jacobian conjecture. Proved for every quartic polynomial in dimension four: the quartic case reduces to $f = P(x_1,x_2,x_3) + x_4 Q(x_1,x_2,x_3) + a x_4^2$…
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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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analysisAug 14, 2026Significance 4/100Registry: site confirmed
Prior state unknown→disproved
Nothing in the paper's proof is affected. At the witness, inequality (4) holds with slack +27.0, inequality (5) holds with equality (f is inner), and the theorem itself holds with slack 2 - kappa = +0.80. Only the shortcut Remark 2 floats is refuted.
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combinatoricsAug 14, 2026Significance 8/100Registry: unreviewed
Prior state unknown→proved
Twenty-eight individual exact values, each proved by SAT certificate (coloring at n-1, UNSAT at n). The synthesis theorem covers every c >= 2 whose c+1 is divisible by 3, 4, 5, or 7 (~66% of integers). The prime-reduction corollary shows the full conjecture (R(c)=40c+41 for all c >= 2) is equivalent to checking primes p >= 89; all primes through 83 are settled. What stays open: the conjecture at c=88 (p=89) and ev…
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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 ori…
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algorithms-optimizationAug 14, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
After Chen-He-Ye-Yuan's counterexample to direct three-block ADMM, the subclass in which the third constraint block is the identity matrix remained unresolved: the literature contained neither a convergence proof nor a counterexample. Disproved: an explicit rational counterexample exists in which the first two blocks are strongly convex quadratics and direct three-block ADMM produces a bounded nonconvergent orbit…
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combinatoricsAug 14, 2026Significance 8/100Registry: unreviewed
Prior state unknown→proved
Pavez-Signe (2024) conjectured a Dirac-type condition for spanning $H$-subdivisions and asked whether the subdivision paths can additionally be required to have similar lengths; Lee (2025) resolved the existence conjecture in the stronger digraph setting. Answered affirmatively with epsilon-room: for every $\varepsilon > 0$ there is $C_0$ such that every $n$-vertex digraph $D$ with $n \ge C_0 h$ and minimum semi-d…
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combinatoricsAug 13, 2026Significance 15/100Registry: site confirmed
Prior state unknown→proved
Part III of a series, and the first unconditional positive results in it. Settled exactly: the local envelope ladder $E(2)=1$, $E(3)=9/8$ and $E(4)=(299-41\sqrt{41})/32=1.13974707\ldots$, which recasts the earlier record constants as exact envelopes of the general theory rather than isolated instances, plus exact constants for four classes - out-trees 0, two-layer hubs 1, outerplanar two-exit interval spines 1 (sh…
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analysisAug 13, 2026Significance 40/100Registry: unreviewed
Prior state unknown→proved
The paper proves the previously unresolved odd-dimensional real cases of Banach's isometric conjecture. Combined with Gromov's earlier theorem for even dimensions and previous results, this completes the conjecture for real Banach spaces.
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combinatoricsAug 13, 2026Significance 8/100Registry: site confirmed
Prior state unknown→proved
The dihedral case only, for every $a \ge 4$ and $b \ge 1$; the substance is the upper bound, which the source paper's own computations could not reach. Together with the sibling a = 3 entry this proves Conjecture 4.9's claim $1+(a-1)(b-1)$ for all $a \ge 3$; the conjecture's trivial a = 1, 2 cases are unaddressed by either entry, and the cyclic analogue $R_{cyc}(P_a^{alt}, K_b)$ for $a \ge 4$ remains open. The eng…
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combinatoricsAug 13, 2026Significance 12/100Registry: unreviewed
Prior state unknown→disproved
The Foregger–Sinkhorn tie-point conjecture, Conjecture 41 in Minc's survey, asserts that if a nearly decomposable doubly stochastic matrix minimizes the permanent on a face and the permanental cofactor at a prescribed zero exceeds its permanent, then that zero is a tie point. False: an explicit $8 \times 8$ counterexample exists, built on the unique root $\beta$ of $7t^3 - 13t^2 + 12t - 4$ in $(59/100, 3/5)$.
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logic-foundationsAug 13, 2026Significance 35/100Registry: unreviewed
Prior state unknown→proved
The new content is $SOP_2 \Rightarrow SOP_3$; the converse implication was known from the start. Dzamonja and Shelah asked whether either implication in $SOP_3 \Rightarrow SOP_2 \Rightarrow SOP_1$ reverses: Mutchnik answered the second ($SOP_1 = SOP_2$), and this answers the first, collapsing the bottom of the hierarchy to $SOP_1 = SOP_2 = SOP_3$. The $SOP_n$ hierarchy for $n \ge 3$ remains, as does everything abo…
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geometry-topologyAug 13, 2026Significance 38/100Registry: unreviewed
Prior state unknown→proved
Gromov's 1986 question drew three independent proofs within about 24 hours, two with AI in the loop. Ge's non-AI proof (heat-kernel Fisher metric, Nash entropy) came first, 13 August. Antonelli's proof here (14 August, GPT-5.6 Sol) takes a different route, Hodge obstruction and rank improvement, and its headline addition is the general family: for every $0\le m\le n-2$, nonnegative Ricci plus positive $(m{+}1)$-in…
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quantum-information-computingAug 13, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
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.
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combinatoricsAug 13, 2026Significance 5/100Registry: site confirmed
Prior state unknown→proved
Nineteen individual exact values, each decided by SAT certificate: unsatisfiable at the claimed $n$, witnessed satisfiable at $n-1$. They close cells in DD26's Tables 3-13 but settle no infinite family - the sibling entries do that for the $K$ column. The three overlap cells are $R_{dih}(P_4^{alt},K_6)=16$, $R_{dih}(P_3^{alt},K_9)=17$ and $R_{dih}(P_9^{alt},K_3)=17$, each an instance of a sibling theorem; the rema…
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combinatoricsAug 12, 2026Significance 5/100Registry: site confirmed
Prior state unknown→proved
The a = 3 slice is settled outright. The parent conjecture's dihedral side has since been resolved for every a >= 4 as well (see the related entry), so Conjecture 4.9's claim 1 + (a-1)(b-1) now stands proved for all a >= 3; the trivial a = 1, 2 cases and the cyclic analogue for a >= 4 remain formally unaddressed.
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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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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 the…
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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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analysisAug 12, 2026Significance 30/100Registry: lean verified
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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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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analysisAug 12, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
The first explicit counterexample rather than a first suspicion: the abstract is clear that the failure was widely expected and that what was missing was a witness. It gives an infinite family, one for each d >= 2, all Lamplighter groups over free groups.
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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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theoretical-computer-scienceAug 12, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
The manuscript claims a polynomial-time reduction from 3-SAT proving $(2,1)$-C1P NP-hard; together with membership in NP, this establishes NP-completeness and closes the sole unresolved $(k,\delta)$ case from the earlier classification. It also implies NP-completeness of the equivalent completion problem.
The proof package further shows that, within its specific nested-prefix/internal-local gadget architecture, n…
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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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probability-statisticsAug 11, 2026Significance 8/100Registry: unreviewed
Prior state unknown→proved
The general principle is the interesting part: every semialgebraic property of a bounded fixed-dimensional mean parameter is eventually almost surely predictable. Against merely integrable matrix laws it fails from dimension two.
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combinatoricsAug 11, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
Ekhad and Zeilberger's second computational Chomp challenge asks for a Chomp position with three winning opening moves. Answered by exhibiting a bar with three winning opening moves.
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combinatoricsAug 11, 2026Significance 14/100Registry: unreviewed
Prior state unknown→proved
The first rigorous solid standard Young tableaux challenge asks for a proof of a conjectured second-order recurrence for the number of solid standard Young tableaux. The conjectured recurrence is proved.
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differential-equationsAug 11, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
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.
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algorithms-optimizationAug 11, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Recorded as partial: the bound is Omega(T^-1.9319) against an achievable O(T^-1.2716), so it rules out reaching the optimal rate without pinning down the true one.
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analysisAug 11, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
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.
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combinatoricsAug 11, 2026Significance 14/100Registry: unreviewed
Prior state unknown→proved
The five-dimensional case of the Geode challenge of Amdeberhan, Kauers and Zeilberger, concerning the geode factor attached to a family of multivariate generating functions. Settled in dimension five.
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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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combinatoricsAug 11, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Spahn and Zeilberger's third challenge asks whether the restricted permutation counts $a_{r,s}$ and $b_{r,s}$ are holonomic for all $r, s > 1$. Answered affirmatively.
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algorithms-optimizationAug 10, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
Arjevani et al. asked whether almost-surely bounded oracle error permits a better rate than bounded variance for smooth nonconvex stochastic optimization. It does not: every randomized adaptive algorithm still needs Omega(dL/eps^2 + dL sigma^2/eps^4) queries, matching the standard upper bound.
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geometry-topologyAug 10, 2026Significance 22/100Registry: unreviewed
Prior state unknown→disproved
The counterexamples come with divisibility bounds on the Hodge-theoretic index, at 2-torsion and 5-torsion, on very general hyperkahler fourfolds.
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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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combinatoricsAug 10, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
The key step is a lower bound on the truncated imbalance sum, which yields every Erdos-Gallai inequality for the sorted imbalance list; a parity computation finishes it.
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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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number-theoryAug 10, 2026Significance 68/100Registry: lean checked
Prior state unknown→proved
An unconditional record, not a resolution: the Riemann hypothesis is untouched, and Anthropic states it does not expect these techniques to lead to a proof of it. The paper is explicit that these are lower bounds only - the remaining third of the zeros are not shown to be off the line, merely not reached by the certificate.
What it does settle is a question that was posed. Goldston and Suriajaya had reduced Montg…
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combinatoricsAug 10, 2026Significance 15/100Registry: site confirmed
Prior state unknown→disproved
Teschner's universal bound b(G) <= (3/2)Delta(G) is false: the 18-vertex cubic bipartite graph has b(G) = 5 against a bound of 4.5. What survives is the restricted statement Teschner actually proved, that the bound holds for graphs of domination number at most three, and Gagarin and Zverovich's 2013 result that it holds for almost all graphs. The counterexample does not suggest a replacement bound, and the correct…
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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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mathematical-physicsAug 9, 2026Significance 8/100Registry: lean checked
Prior state unknown→proved
Answers the obstruction rather than the whole question: it says exactly when the Matrix-Tree Theorem can be applied and classifies the chambers, and gives a compact five-graviton formula outside the decay region. Simplifying the general solution, which is what the source paper left to future work, remains open.
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algebraAug 9, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Kim and Roush did not claim uniqueness; the classification of equality cases is new alongside the conjecture itself.
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analysisAug 9, 2026Significance 22/100Registry: unreviewed
Prior state unknown→disproved
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.
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algorithms-optimizationAug 9, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
Improves every prior result for p >= 2 and matches the classical extragradient method at p = 1.
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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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algebraAug 8, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
The footnote is worth reading on its own: an author saying in print that the model earned coauthorship and that policy is what prevents it.
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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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algorithms-optimizationAug 7, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Three immediate consequences follow for undirected unweighted planar graphs: better compression of the Okamura-Seymour metric, less space for constant-time exact distance oracles, and a faster distributed algorithm.
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analysisAug 7, 2026Significance 4/100Registry: unreviewed
Prior state unknown→proved
Agulnick and Busick-Warner exhibited a family of fifth-order ambiguities on the exact unit support $U_{30}$ and conjectured that it was not a complete classification because it did not use the full field $\mathbb Q(\zeta_{30})$. This work proves the complete classification. The larger field enlarges the common amplitude $\alpha$, while every relative ambiguity remains a norm-one parameter in $\mathbb Q(\zeta_6)$.…
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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 $\mathrm{forb}(m, F)$ is $\Theta\left(m^{X(F)-1}\right)$, 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 $\Theta(m^3)$, while a random-alteration argument gives $\Omega(m^{\frac{…
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combinatoricsAug 7, 2026Significance 12/100Registry: unreviewed
Prior state unknown→disproved
One conjecture each way: the second proved, the first disproved. Two further Chen-Lawrencenko conjectures remain open and are flagged as such in the paper.
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analysisAug 7, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
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.
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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 $(\alpha,\beta)$-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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algebraAug 7, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Zhao's Generalized Vanishing Conjecture asks whether, for a differential operator with constant coefficients, $\Lambda^m(P^m) = 0$ for all large $m$ forces $\Lambda^m(P^m Q) = 0$ for all large $m$. Refuted by an explicit five-variable counterexample.
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geometry-topologyAug 7, 2026Significance 12/100Registry: unreviewed
Prior state unknown→disproved
The paper also proves the conjecture in the unimodular case and characterizes equality there, so the boundary between true and false is drawn rather than just crossed.
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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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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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geometry-topologyAug 6, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
Settles subspace dimensions 2 and 3; the generalized problem stays open for larger m.
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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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algorithms-optimizationAug 6, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
The paper records how the collaboration actually went: the authors first aimed at a counterexample showing EF1 and PO incompatible under matroid constraints, and when the model surfaced fundamental difficulties with that plan they redirected toward proving the positive result instead. The paper also extends the technique to category constraints and leaves a pseudopolynomial-time algorithm open.
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analysisAug 6, 2026Significance 22/100Registry: unreviewed
Prior state unknown→disproved
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.…
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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 $q \geq 4$, and completely characterizes when an FPRAS exists for 2-spin systems on planar graphs at small external field.
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quantum-information-computingAug 6, 2026Significance 12/100Registry: site confirmed
Prior state unknown→proved
Five existence statements, all by explicit construction: $\mathrm{AME}(12,5)$, $\mathrm{AME}(17,11)$, $\mathrm{AME}(18,11)$, $\mathrm{AME}(17,13)$ and $\mathrm{AME}(18,13)$. The $[12,6,7]_{25}$ code came from a direct search with no symmetry imposed; its automorphism group turned out to have a regular $\mathbb{Z}_3^2$ coordinate orbit, and imposing that translation symmetry on two nine-coordinate orbits collapses…
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number-theoryAug 6, 2026Significance 12/100Registry: lean checked
Prior state unknown→proved
Proves the a = 3 layer; the conjecture is layered in a and remains open for larger a.
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mathematical-physicsAug 6, 2026Significance 28/100Registry: unreviewed
Prior state unknown→proved
For the Ising pure $p$-spin glass where $p \geq 3$, it was predicted by Gardner that there exists critical inverse temperatures $0<\beta_1^p<\beta_2^p <\infty$ such that: (1) When $0<\beta\leq \beta_1^p$, the Parisi measure is replica symmetric (RS); (2) When $\beta_1^p<\beta \leq \beta_2^p$, the Parisi measure is one-step replica symmetry breaking (1-RSB); (3) When $\beta>\beta_p^2$, the Parisi measure is full r…
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combinatoricsAug 6, 2026Significance 15/100Registry: lean checked
Prior state unknown→proved
Dimension 5 only; public AI-generated candidate with no independent specialist review.
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algorithms-optimizationAug 5, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Removes the constant factor from the previous best guarantee; whether $k/2^k$ is optimal is not settled here.
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analysisAug 5, 2026Significance 53/100Registry: lean verified
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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algorithms-optimizationAug 5, 2026Significance 5/100Registry: unreviewed
Prior state unknown→disproved
Does a single ReLU neuron trained on modular addition align to one Fourier frequency? Open Problem MAIS-O60, itself posed by Claude Fable 5 under the direction of Lionel Levine, is answered negatively: an explicit construction reaches a frozen state whose Fourier energy is spread equally across all nonzero frequency classes, on an open set of initial conditions.
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combinatoricsAug 5, 2026Significance 12/100Registry: lean checked
Prior state unknown→proved
The multivariate independence polynomial is the partition function of the hard-core model with per-vertex fugacities. The paper proves a lower bound extending to the multivariate setting a result Tao proved in the univariate case, and settles a conjectured generalization for a multiaffine version of the semiproper colouring partition function with two proper colours.
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quantum-information-computingAug 5, 2026Significance 22/100Registry: unreviewed
Prior state unknown→disproved
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.
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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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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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analysisAug 5, 2026Significance 40/100Registry: lean verified
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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analysisAug 5, 2026Significance 25/100Registry: lean checked
Prior state unknown→proved
For which lattice parameters does a totally positive window function generate a Gabor frame? Gröchenig and Stöckler initiated the program in 2013; this paper gives the complete characterization, together with a Kadets-type theorem for shift-invariant spaces.
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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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analysisAug 5, 2026Significance 8/100Registry: unreviewed
Prior state unknown→proved
Nazarov conjectured that for $s \in (1, 3/2)$ the quadratic form of the spectral fractional Dirichlet Laplacian strictly increases under $u \mapsto |u|$ when $u$ changes sign. Proved and substantially generalized, with the same conclusion for the restricted form.
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algorithms-optimizationAug 5, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
What is the optimal competitive ratio for online vertex cover when edges arrive one at a time? The paper proves a tight factor-2 lower bound via a reduction in the blueprint framework of Assadi, Jiang and Xiang, closing the gap left by prior work.
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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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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 $q_A$ and $q_B$ 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 $O((q_A + q_B)^5)$ classic…
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logic-foundationsAug 4, 2026Significance 7/100Registry: unreviewed
Prior state unknown→proved
subDL satisfies the signed depth relevance property, answering an open question posed by Øgaard (2026). More precisely, every valid inference in subDL contains a propositional variable that occurs in both the premises and conclusion with matching sign and at matching implicational depth.
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algorithms-optimizationAug 4, 2026Significance 4/100Registry: unreviewed
Prior state unknown→proved
Fixed four-terminal planar DAG; positive route-cost differences for m≥2, with non-attained suprema; signed/zero only for m=1, whose attainment is unstated. No topology-wide/unrestricted-planar claim.
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combinatoricsAug 4, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
The quartet distance counts the four-leaf subsets on which two binary phylogenetic trees display different topologies. Bandelt and Dress conjectured the maximum over trees on $n$ leaves. Proved: it is $(2/3 + o(1))\binom{n}{4}$, by reducing arbitrary pairs of trees to caterpillars through a common-root planarization and an identity on five-leaf trees.
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combinatoricsAug 4, 2026Significance 38/100Registry: lean verified
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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algebraAug 4, 2026Significance 22/100Registry: unreviewed
Prior state unknown→disproved
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…
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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…
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combinatoricsAug 4, 2026Significance 12/100Registry: expert verified
Prior state unknown→proved
Ji, Li and Wang conjectured in 2024 that every parallel chip-firing game on a finite connected graph whose chip count lies strictly between $2|E|-|V|$ and $2|E|$ has period exactly 2, generalizing the middle rung of Levine's devil's staircase from complete graphs to all graphs. Known before only for trees, cycles, complete and complete bipartite graphs.
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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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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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algorithms-optimizationAug 3, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Conditional on the randomized exact-volume Small-Set Expansion Hypothesis, and stated for least-squares objectives rather than sparse convex optimization in general.
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combinatoricsAug 3, 2026Significance 5/100Registry: unreviewed
Prior state unknown→proved
A problem from Fajtlowicz's Graffiti program, studied by Erdős and Staton, on the Havel-Hakimi residue of common-divisor graphs. The paper resolves the problem and extends it, determining the residue's first-order scale and its nontrivial constant from the degree sequence.
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algebraAug 3, 2026Significance 35/100Registry: unreviewed
Prior state unknown→disproved
Settles the ICC property (T) case. The paper records that the result was obtained independently of and concurrently with work by OpenAI.
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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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combinatoricsAug 3, 2026Significance 5/100Registry: lean verified
Prior state unknown→proved
The Formal Conjectures pull request flipping this from open to solved is still open rather than merged, so the canonical repository has not yet accepted it.
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algorithms-optimizationAug 3, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
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…
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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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analysisAug 3, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
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.
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analysisAug 3, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Denjoy's 1932 theorem says a $C^{1+\mathrm{bv}}$ circle diffeomorphism with irrational rotation number has no wandering interval. Whether it is sharp in regularity: for every concave modulus of continuity $\omega$ weaker than Lipschitz, there is a $C^{1+\omega}$ circle diffeomorphism with irrational rotation number and a wandering interval. The case $\omega(t) = t\log(1/t)$ settles an open problem going back to He…
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combinatoricsAug 2, 2026Significance 10/100Registry: unreviewed
Prior state unknown→disproved
A graph $G$ is maximal non-Hamiltonian if it is non-Hamiltonian but $G + e$ is Hamiltonian for every nonedge $e$. In 1994 Vu Dinh Hoa conjectured a property of $G - V(C)$ for a longest cycle $C$ of such a graph. Disproved by an explicit base graph on 56 vertices, extended to larger orders.
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geometry-topologyAug 2, 2026Significance 22/100Registry: unreviewed
Prior state unknown→proved
the equality case; the inequality was settled separately and is tracked on its own entry
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combinatoricsAug 2, 2026Significance 5/100Registry: lean verified
Prior state unknown→proved
The formalization proves the statement under the weaker hypothesis n >= 2; the pull request marking the conjecture solved is open, not merged
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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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combinatoricsAug 1, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Monical, Tokcan and Yong conjectured that Schubitopes, the generalized permutahedra arising as Newton polytopes of Schubert polynomials and of Demazure characters of $\mathrm{GL}_n$, are Ehrhart positive. Disproved by an explicit Schubitope whose Ehrhart polynomial has a negative coefficient in its monomial expansion.
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geometry-topologyAug 1, 2026Significance 50/100Registry: lean verified
Prior state unknown→proved
upper bounds reach the Cohn-Elkies threshold; the true asymptotic density remains open
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algebraAug 1, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Claimed in a self-published research draft; a standalone by-product is the transcendence of the integral of exp(q) between distinct algebraic endpoints for nonconstant algebraic q
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combinatoricsAug 1, 2026Significance 20/100Registry: lean verified
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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quantum-information-computingAug 1, 2026Significance 25/100Registry: lean verified
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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combinatoricsAug 1, 2026Significance 21/100Registry: lean verified
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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number-theoryAug 1, 2026Significance 11/100Registry: lean checked
Prior state unknown→proved
For every integer-valued strongly $b$-additive $g$ with $\gcd(g(1),\dots,g(b-1))=1$ and digit mean $\mu_g\ge0$: $g(p)$ is prime for infinitely many primes $p$. For $\mu_g>0$, $\sum 1/p$ over $p<X$ with $g(p)$ prime is $(d_g/\varphi(d_g))\log_3X + C_{g,1} + O(1/\log\log X)$, likewise for the first $j$ iterates. Also $\#\{p\le x: g(p)\text{ prime}\}\ll\pi(x)/\log\log x$, of that exact order on a large set of $x$, an…
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combinatoricsAug 1, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
Subbarao and Verma asked in 1999 (Problem 5.7, first part) whether the complementary Bell numbers $f(n) = B_n(-1)$ take any given value only finitely many times. Campbell proves they do: for every fixed integer the fiber is finite, a result whose techniques connect to Wilf's conjecture on the vanishing of $f(n)$.
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analysisAug 1, 2026Significance 45/100Registry: lean verified
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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number-theoryAug 1, 2026Significance 12/100Registry: lean checked
Prior state unknown→proved
New theorems, not a formalisation of previously known results. For every base $b\ge2$ the Erdős–Kac law is established for the $\lambda$-digit base-$b$ palindromes and for the base-$b$ reversals of the $\lambda$-digit primes, for $\omega$ and $\Omega$ and for $\omega_S,\Omega_S$ with any regular set $S$ of primes; with normal order $\log\log n$ on both families, and, for $\omega$, all moments of order up to $\tfra…
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geometry-topologyAug 1, 2026Significance 25/100Registry: lean verified
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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algebraAug 1, 2026Significance 60/100Registry: lean verified
Prior state unknown→disproved
answered no: non-sofic groups exist
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theoretical-computer-scienceAug 1, 2026Significance 38/100Registry: lean verified
Prior state unknown→proved
an n^4/log n formula lower bound; VP vs VNP remains wide open
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combinatoricsAug 1, 2026Significance 25/100Registry: lean verified
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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theoretical-computer-scienceAug 1, 2026Significance 39/100Registry: lean verified
Prior state unknown→proved
exponential improvement over the 1977 MRRW bounds; the exact rate-distance trade-off remains open
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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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combinatoricsJul 31, 2026Significance 15/100Registry: site confirmed
Prior state unknown→proved
Record lower bound only. The sub-2 ceiling is the codimension-two case and does not bound the record ladder (k=17 has complement mass 11). 4/3 and 2 are conjectures; the proved gap is [1.28249, 2].
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algebraJul 31, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
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.
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number-theoryJul 31, 2026Significance 14/100Registry: unreviewed
Prior state unknown→proved
The sharp valuation bounds and optimal $\gamma(m)$ are proved for ALL loop quivers $m\ge2$, hence for the extremal BPS invariants of all twist knots (both rows, matching every twist-knot entry of GKS Table 1). Scope limits: the $m=3$/figure-eight divisibility $2n_r/r\in\mathbb Z$ was previously proved by Basor–Conrey–Morrison (arXiv:1703.00990), whose per-$r$ $2$-adic characterization for $m=3$ is finer than the u…
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quantum-information-computingJul 31, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Online Shadow Tomography with $\log m$ dependence, while retaining $\mathrm{poly}(\log(d)/\epsilon)$ dependence. Also, matching the best classical bounds for Adaptive Data Analysis
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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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combinatoricsJul 31, 2026Significance 11/100Registry: unreviewed
Prior state unknown→disproved
Akbari, Alikhani, Oboudi and Peng conjectured in 2010 that 0 and -2 are the only integer roots of the domination polynomial $D(G, x)$, proven for trees and unicyclic graphs and verified exhaustively for small orders. The paper gives a counterexample of order 33 with an integer domination root at $x = -4$, built from an S-unit branch cancellation mechanism.
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theoretical-computer-scienceJul 31, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
Assuming the Unique Games Conjecture, it is NP-hard to approximate MAX-3-CUT better than the Frieze-Jerrum semidefinite program does, and similarly for Quantum MAX-CUT: the sharpness question in the Khot-Kindler-Mossel-O'Donnell line, connected to the Plurality is Stablest problem.
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combinatoricsJul 31, 2026Significance 5/100Registry: lean checked
Prior state unknown→proved
Settles the conjecture for a large family and reduces the rest to unit fractions; the general unit-fraction case remains open.
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combinatoricsJul 30, 2026Significance 5/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Written on the Wall II, Graph Conjecture 217.” The registry entry and named primary source contain the available statement and scope.
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combinatoricsJul 30, 2026Significance 15/100Registry: lean checked
Prior state unknown→proved
With $N = 2^k - 1$ and $\mathrm{wt}(n)$ the binary Hamming weight, Tu and Deng conjectured that for every $1 \leq t \leq N-1$ at most $2^{k-1}$ pairs $(a,b)$ satisfy $a + b \equiv t \pmod N$ and $\mathrm{wt}(a) + \mathrm{wt}(b) < k$. Proved in full.
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combinatoricsJul 30, 2026Significance 6/100Registry: unreviewed
Prior state unknown→disproved
Does every nontrivial finite simple graph have noninteger Sombor energy? If $\rho_1,\ldots,\rho_n$ are the eigenvalues of the Sombor matrix of a graph $G$, its Sombor energy is
$$E_{\mathrm{SO}}(G)=\sum_{i=1}^{n}|\rho_i|.$$
The conjecture asserted that $E_{\mathrm{SO}}(G)\notin\mathbb Z$ for every nontrivial graph. A connected graph on nine vertices is exhibited with $E_{\mathrm{SO}}(G)=64$, disproving the conje…
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mathematical-physicsJul 30, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
the sharp bound for three charges; the general Maxwell bound was separately disproved in July 2026
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number-theoryJul 30, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
leading asymptotic determined up to a bounded q-dependent term
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combinatoricsJul 30, 2026Significance 5/100Registry: unreviewed
Prior state unknown→disproved
Infinite family of counterexamples; mathematical argument internally checked, with external verification and novelty review pending.
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combinatoricsJul 30, 2026Significance 4/100Registry: unreviewed
Prior state unknown→disproved
no constant-bound repair of the conjecture is possible
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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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algebraJul 30, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
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.
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mathematical-physicsJul 29, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
For the Sachdev-Ye-Kitaev Hamiltonian on $n$ Majorana modes with $k$-body interactions, the paper proves $\mathbb{E}\|H\|_{op} = (1-o(1))\sqrt{2n}/k$ for super-constant $k \le o(\sqrt{n})$, confirming predictions of Garcia-Garcia, Jia and Verbaarschot and answering a question of Feng, Tian and Wei.
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geometry-topologyJul 29, 2026Significance 5/100Registry: unreviewed
Prior state unknown→proved
Casalaina-Martin and Zhjeqi proved that the first Chern class of every torsion-free coherent quotient of a tensor power of the logarithmic cotangent sheaf is pseudo-effective, noting in Remark 4.5 that torsion-freeness was imposed only for technical reasons. Can it be dropped? Yes.
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combinatoricsJul 29, 2026Significance 5/100Registry: unreviewed
Prior state unknown→disproved
WOW-284 asserts that the minimum dual degree of every connected graph of order at least three and girth at least five is at most the negative of its least distance eigenvalue. The paper refutes it with exact counterexamples of orders 38, 39, 40, 42 and 50, and develops a structural theory of the failure.
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geometry-topologyJul 29, 2026Significance 7/100Registry: unreviewed
Prior state unknown→proved
Does a general pencil of plane cubics over $\mathbb{C}$ have exactly $12$ common flex lines? Ciliberto, Miranda and Roé asked this in Remark 5.3 of their paper; the answer is yes.
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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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geometry-topologyJul 29, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
The key ingredients were already in the literature, decades old and in distant fields; what was missing was anyone connecting them to this question.
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analysisJul 29, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
This is the MODIFIED conjecture, not the original Lyons-Sidorova one, and it is proved for continuous bounded-variation paths. Prior work had a line-image result under the stronger assumption of infinite radius on every subinterval; this removes that assumption.
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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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analysisJul 29, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Let $\mu$ be a probability measure on the unit circle with Verblunsky coefficients $\alpha$. Lukic conjectured that a weighted entropy condition with finitely many critical points is equivalent to a decomposition of $\alpha$ 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 correspondi…
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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 $f(\varnothing)=1$ 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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algebraJul 29, 2026Significance 20/100Registry: expert verified
Prior state unknown→disproved
Verified by author of the conjecture
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geometry-topologyJul 29, 2026Significance 11/100Registry: lean verified
Prior state unknown→disproved
If $f(n)$ is the maximum total side length of $n$ interior-disjoint squares packed in the unit square, is $f(k^2 + 1) = k$? An exact rational configuration packs $17$ squares with total side length greater than $4$, refuting the identity at $k = 4$.
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number-theoryJul 29, 2026Significance 20/100Registry: lean verified
Prior state unknown→proved
New arXiv preprint with an author-provided Lean formalization; not yet peer-reviewed.
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geometry-topologyJul 29, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Given planks of fixed total width, how should they be placed to cover as much of a convex body's volume as possible? Karoly Bezdek asked whether, for a Euclidean ball, the optimum is a single plank centred at the origin. It is, and the paper also settles every planar convex body.
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number-theoryJul 29, 2026Significance 13/100Registry: unreviewed
Prior state unknown→disproved
For the Erdős–Pomerance functions $F(n)$ and $h_{\mathbb{P}}(n)$ counting how many consecutive integers are needed to contain a distinct multiple of each integer, respectively prime, up to $n$, the paper proves $F(n) \ge h_{\mathbb{P}}(n) \ge n\exp\left(\left(\frac{\log 2}{2} - o(1)\right)\frac{\log n}{\log\log n}\right)$, disproving Kominers' conjecture that $F(n) \ll n\log n$. The paper also significantly impro…
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quantum-information-computingJul 29, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Is the irreversibility of entanglement manipulation robust in the strong-converse sense - a strict separation between the exponential strong-converse distillable entanglement and the entanglement cost, as conjectured by Lami and Regula? Yes: there are states for which any attempt to restore reversibility incurs an error growing exponentially in the number of copies, and the irreversibility persists even at polynom…
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combinatoricsJul 29, 2026Significance 10/100Registry: unreviewed
Prior state unknown→disproved
Around the Kemeny median problem, which stays open for $m=3$ and $m=5$ voters, the paper refutes three conjectures on tournament inducibility: both conjectures of Milosz, Hamel and Pierrot (the 3-cycle extension for odd $m\ge5$, and $\mathrm{FAS}=\mathrm{HS}_3$ at $n=11$), and Shepard's threshold conjecture.
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quantum-information-computingJul 29, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
Holevo and Werner's 1999 lower bound on the quantum capacity of the bosonic thermal attenuator comes from thermal inputs. Is it optimal? The paper proves it is exactly the supremum over single-mode Gaussian states, then exhibits a non-Gaussian state that beats it, giving positive quantum capacity in a region where every single-mode Gaussian input yields none.
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algebraJul 28, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
first counterexample of the form D^b(X) for X smooth projective
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theoretical-computer-scienceJul 28, 2026Significance 7/100Registry: unreviewed
Prior state unknown→proved
Gill introduced the probabilistic automatic complexity $A_P(w)$ of a string: the least number of states of a probabilistic finite automaton for which $w$ is the unique most probably accepted string of its length. He asked whether $A_P$ is unbounded, no string with $A_P>3$ being known. The paper proves $A_P(w)\le 3$ for every string over every finite alphabet, with an explicit three-state witness.
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theoretical-computer-scienceJul 28, 2026Significance 15/100Registry: lean verified
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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algebraJul 28, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Does the analytic Bertini restriction theorem for multiplier ideals hold locally, outside a pluripolar exceptional set of fibers? Proved in full generality.
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combinatoricsJul 28, 2026Significance 5/100Registry: lean verified
Prior state unknown→disproved
Must every connected graph satisfy the proposed upper bound on its independence number in terms of residue and largest induced-bipartite-subgraph order? The family $\overline{K}_{2r+1} \vee (K_r \sqcup K_r)$ violates it for every $r \ge 3$.
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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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combinatoricsJul 28, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
For a differential poset $P$, must the weighted $2$-multichain series $M_{P,2}(q)$ be a rational multiple of $F_P(q)^2$, the square of its rank generating series?
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combinatoricsJul 28, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Must every $r$-differential poset have at least as many elements in each rank as $Y^r$, the $r$-th Cartesian power of Young's lattice? For $r = 3$ the new construction has fourth-rank size $50$ against $51$ for $Y^3$.
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geometry-topologyJul 28, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
improved bounds; the exact Hadwiger-Debrunner numbers remain open
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theoretical-computer-scienceJul 28, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Can the $k$-distinct language - words over $[n]$ of length at most $k$ with no repeated symbol - be recognized by an acyclic NFA of size $c^k n^{O(1)}$ for some $c < 4$? A construction of size $2^{1.96992k} n^{O(1)} < 3.918^k n^{O(1)}$ answers yes.
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quantum-information-computingJul 28, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
Does bipartite bound information exist: classical correlations between two parties and an eavesdropper that cost secret bits to create, yet from which no secret key can ever be distilled?
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combinatoricsJul 28, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
girth >= 5 case; the triangle-free case remains open
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probability-statisticsJul 27, 2026Significance 35/100Registry: lean verified
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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differential-equationsJul 27, 2026Significance 10/100Registry: unreviewed
Prior state unknown→disproved
the named open question is answered negatively; the paper's positive theory goes further
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quantum-information-computingJul 27, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
What is the optimal uniform continuity bound for quantum conditional entropy in trace distance, depending only on the dimension of the conditioned system? The sharp bound $h_2(\delta) + \delta \log(d^2 - 1)$ up to $\delta = 1 - d^{-2}$, conjectured by Wilde, is proved.
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probability-statisticsJul 27, 2026Significance 10/100Registry: unreviewed
Prior state unknown→disproved
Chafai, Dadoun and Youssef asked whether the quadratically penalised logarithmic energy of mean empirical spectral distributions is monotone in the dimension, for Wigner matrices and for matrices with i.i.d. entries. Neither holds: a finite-energy Wigner counterexample and a one-parameter family of Gaussian-regularised Bernoulli entry laws answer both questions negatively.
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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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combinatoricsJul 27, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
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.
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algebraJul 27, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
For a perfect field $k$ and a representation-infinite finite-dimensional $k$-algebra $A$, the Auslander–Reiten quiver of $A$ has infinitely many connected components. This establishes a conjecture of Auslander, Reiten and Smalø, for finite-dimensional algebras over perfect fields.
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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-topologyJul 27, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
How many pairwise non-overlapping infinite circular cylinders of unit radius can simultaneously touch a unit ball? Kuperberg conjectured in 1990 that the maximum is six.
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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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geometry-topologyJul 27, 2026Significance 45/100Registry: unreviewed
Prior state unknown→proved
with constant 2; also improves the global KLS bound to $O(\log^{1/4} n)$
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algebraJul 26, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
the locally noetherian case; whether such a category can fail to admit a projective generator is left open as Problem 1.3
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algebraJul 26, 2026Significance 25/100Registry: lean verified
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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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 $k \ge 4$ there is an explicit lattice point of the Newton polytope of $a_{\delta_k}^k$ with vanishing coefficient, obtained from a Dyson constant-term identity; the odd case follows by alternation, proving the conjecture.
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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 $G = \operatorname{SmallGroup}(128, 859)$ over $\overline{\mathbb{F}}_2$ the ring has depth $2$ while every associated-prime quotient has dimension at least $3$.
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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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combinatoricsJul 26, 2026Significance 5/100Registry: unreviewed
Prior state unknown→disproved
Is the zero forcing number of every connected graph with maximum degree $3$ at most its independence number plus one? A connected 24-vertex subcubic graph with independence number $9$ and zero forcing number $11$ refutes this 2017 TxGraffiti conjecture, and a 36-vertex cubic variant refutes the cubic form: $Z = \alpha + 2$ is attained.
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combinatoricsJul 26, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
closes a one-block gap in the covering tables; the analogous next case is not reachable by this method
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analysisJul 26, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
negative below the 1/3 threshold; the endpoint case is still open
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number-theoryJul 25, 2026Significance 10/100Registry: contested
Prior state unknown→proved
For the least $k$ at which the small-prime part of $\binom{n}{k}$ exceeds $n^2$, how large can $f(n)$ be?
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review! Disputed
number-theoryJul 25, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
If $A(x)$ counts integers satisfying the Sylow divisor condition, determine the constant $c$ in $A(x)/x = \exp(-(c + o(1)) \sqrt{\log x} \log\log x)$. The claimed exact value is $c = 1/(2\sqrt{\log 2})$.
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theoretical-computer-scienceJul 25, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
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.
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analysisJul 24, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Energy measures of any two nonconstant harmonic functions on the standard Sierpinski gasket are mutually absolutely continuous. Strichartz and Tse reported numerical evidence that the Radon-Nikodym densities are $L^p$-integrable for $1 < p < \log 15 / \log 9$. That range is confirmed: the associated quantities are uniformly bounded for arbitrary ordered pairs of nonconstant harmonic functions.
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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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theoretical-computer-scienceJul 24, 2026Significance 5/100Registry: unreviewed
Prior state unknown→proved
conditional on a plausible number-theoretic conjecture; unconditional through s = 15
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combinatoricsJul 24, 2026Significance 36/100Registry: unreviewed
Prior state unknown→disproved
Is the sequence $W_0, W_1, \dots, W_n$ counting the flats of each rank of a matroid always unimodal? Rota conjectured yes in 1970.
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combinatoricsJul 24, 2026Significance 3/100Registry: unreviewed
Prior state unknown→proved
Opus 4.8 constructed a branch of the stated depth, giving a lower bound, and believed it had a matching upper bound; that proof was wrong and the statement stayed a conjecture. FABLE 5 later proved it. In the author's summary of the method: "The proof turns conjugation, near $s$, into base-$p$ arithmetic." A cycle near the fixed point splits into a coarse base permutation and a vector of carries in $\mathbb{Z}/p$,…
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theoretical-computer-scienceJul 24, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
the PIR consequence is conditional on a number-theoretic conjecture implied by either the generalized repunit conjecture or Schinzel's hypothesis H, and is unconditional for s <= 15
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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 $d \ge 2$ no spectral point becomes invisible, which together with the known one-dimensional theorem settles no-i…
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number-theoryJul 24, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
The new content is the upper bound; the matching N^(1/5) construction is prior work of Erdős and Csaba. erdosproblems.com has not accepted the claim
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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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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 23, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
If a chromatic symmetric function is Schur positive, must every finite-variable specialization $X_G(x_1, \dots, x_k)$ have a saturated Newton polytope? A $12$-vertex bipartite graph realizes weights $(6,6,0)$ and $(8,2,2)$ but omits their midpoint $(7,4,1)$.
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algebraJul 23, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Han's conjecture itself remains open; this settles the Liu-Morin extension case
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combinatoricsJul 23, 2026Significance 40/100Registry: site confirmed
Prior state unknown→disproved
This preprint was not the first disproof. A 68-vertex counterexample was posted to X on 23 July 2026 by @NeuralReformist, credited to GPT-5.6 Sol Ultra, sixteen days earlier. This site decoded that sparse6 string and checked it independently: 68 vertices, 102 edges, simple, cubic, connected, bridgeless, girth five, and no Petersen coloring under the same encoder used for the 112-vertex graph. Whether the two are i…
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combinatoricsJul 23, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
corank 3; the general conjecture remains open, corank 2 being McConville's case
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algorithms-optimizationJul 23, 2026Significance 10/100Registry: unreviewed
Prior state unknown→disproved
restricted planar four-terminal case
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combinatoricsJul 23, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
For a finite forbidden triple system $G$, what exact uncountable chromatic cardinalities occur among $G$-free triple systems, and how do those spectra interact? The revised manuscript answers the three exact-cardinal questions and claims a complete spectrum dichotomy.
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combinatoricsJul 23, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
Is the chromatic symmetric function $X_G$ Schur positive for every claw-free graph $G$? Two explicit $12$-vertex line graphs have Schur coefficients $-64$ and $-40$ at $s_{(3,3,3,3)}$.
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combinatoricsJul 23, 2026Significance 10/100Registry: site confirmed
Prior state unknown→disproved
The target-free clique conjecture asserts that the supports of stable fixed points of a nondegenerate combinatorial threshold-linear network are exactly its target-free cliques, the bidirected cliques no outside vertex receives an edge from every member of. An explicit six-neuron counterexample refutes it.
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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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quantum-information-computingJul 23, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
Can one construct a plain-model, efficient, information-theoretically secure one-time unclonable-encryption scheme for one classical bit with exponentially small adversarial advantage?
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combinatoricsJul 23, 2026Significance 5/100Registry: site confirmed
Prior state unknown→proved
The evenly-divided reading of WOWII Conjecture 72 holds:
$ \lceil(A + L)/3\rceil \le t, $
where $ t = $ tree($ G $) (order of a largest induced tree), $ A = $ average eccentricity and $ L = $ maximum neighbourhood independence number.
A stronger reading that divides only $ L $ by three is false. The argument rests on two elementary observations (a diametral path is chordless and therefore induces a t…
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combinatoricsJul 23, 2026Significance 13/100Registry: lean verified
Prior state unknown→proved
Which finite triple systems occur in every triple system of uncountable chromatic number? The claimed characterization: exactly those that, after removing isolated vertices, are linear, have every hyperedge-node of their Levi graph meeting a bridge, and have every Berge cycle even.
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theoretical-computer-scienceJul 23, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Two open problems about extracting order from trees in real-valued functions. A quantitative function analogue of Hodges's tree-to-order extraction yields an at most double-exponential bound on dual sequential fat-shattering dimension, resolving the first. A new proof of Daskalakis-Golowich tight-threshold extraction, avoiding multicolored Ramsey numbers, resolves the second, which concerned repairing the bound in…
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algebraJul 23, 2026Significance 25/100Registry: site confirmed
Prior state unknown→disproved
the counterexample to the original Jacobian conjecture does not specialize to characteristic 2; this is a modification that does
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algebraJul 23, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Baumslag asked whether a one-relator group $G=F/\langle\langle r\rangle\rangle$ with $r$ a commutator is Hopfian, residually finite or automatic. The paper constructs a family $G_m=\langle a,t \mid [t,a[a,t]^{-m}]\rangle$ answering all three negatively.
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combinatoricsJul 22, 2026Significance 10/100Registry: unreviewed
Prior state unknown→disproved
A graph on $n$ vertices is very well-covered if every maximal independent set has size $n/2$. Levit and Mandrescu conjectured that the independence polynomial $i(G,x)$ of every very well-covered graph is unimodal, i.e. its coefficient sequence is nondecreasing and then nonincreasing.
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mathematical-physicsJul 22, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Determine the leading asymptotic of the largest eigenvalue of the $N$-Majorana quartic SYK Hamiltonian as $N \to \infty$. The preprint proves $\lambda_1/\sqrt{N} \to 4\int_0^\infty g_0(t)^4\,dt \approx 0.32504$ almost surely, via the limiting free energy at every fixed positive temperature.
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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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theoretical-computer-scienceJul 22, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
the construction is probabilistic; an explicit uniformly samplable example remains open
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algebraJul 22, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Dogon, Levit and Vigdorovich asked for an explicit upper bound on the stability radius of an infinitely presented group. The lamplighter group provides the first: explicit polynomial bounds on both its Hilbert-Schmidt stability rate and its stability radius, obtained through approximately invariant measures and an effective marker construction.
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theoretical-computer-scienceJul 22, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
information-theoretic; a polynomial-time implementation remains open
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geometry-topologyJul 22, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
also disproves a separate finite-field conjecture of Hunter, Pohoata, Verstraete and Zhang
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combinatoricsJul 22, 2026Significance 15/100Registry: site confirmed
Prior state unknown→disproved
Chromatic quasisymmetric functions of natural unit interval graphs were conjectured to have log-concave coefficients in the elementary basis. A connected $13$-vertex example refutes it: for the Hessenberg function $h=(2,4,4,6,7,10,10,10,10,12,12,13,13)$ and $\lambda=(6,5,1,1)$ the coefficients of $q^5,q^6,q^7$ are $1,6,38$, and $6^2 < 1 \cdot 38$. The coefficient is still positive, palindromic and unimodal, so log…
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combinatoricsJul 22, 2026Significance 5/100Registry: unreviewed
Prior state unknown→disproved
If a finite graph has girth at least five, must its minimum dual degree satisfy $\delta^*(G) \le -\partial_n(G)$, where $\partial_n(G)$ is the smallest eigenvalue of its distance matrix? The Hoffman-Singleton graph violates it: dual degree $7$ against eigenvalue bound $4$.
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combinatoricsJul 22, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Improves the exponent rather than answering the asked question: whether an $N^{o(1)}$ colouring exists remains open.
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combinatoricsJul 22, 2026Significance 5/100Registry: lean verified
Prior state unknown→disproved
For every connected graph $G$, is $\alpha(G) \le \lfloor b(G) - \log(\operatorname{ecc}_{avg}(G)) \rfloor$, where $b(G)$ is the largest induced-bipartite-subgraph order? An $11$-vertex counterexample - a triangle with four leaves on each of two vertices - has $\alpha = 9$ against bound $8$.
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probability-statisticsJul 21, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Is the density of Chernoff's distribution - the law of $\operatorname{argmax}_t \{W(t) - t^2\}$ for two-sided Brownian motion $W$ - strongly log-concave, as conjectured by Balabdaoui and Wellner in 2014?
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combinatoricsJul 21, 2026Significance 5/100Registry: lean verified
Prior state unknown→proved
For every finite connected graph, is $\operatorname{girth}(G) + 1$ at most the product of its largest induced-tree order and its second-smallest degree?
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combinatoricsJul 21, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Norine conjectured that every red-blue edge-colouring of the $n$-dimensional hypercube $Q_n$ in which antipodal edges get opposite colours contains a monochromatic path from some vertex to its antipode. The paper proves it, via a chain-level Borsuk–Ulam obstruction.
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algebraJul 21, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
For a projective variety $X$ with at worst Gorenstein canonical singularities whose stringy $E$-function $E_{\mathrm{st}}(X; u, v)$ is a polynomial, all stringy Hodge numbers $h^{p,q}_{\mathrm{st}}(X)$ are non-negative. (Batyrev 1998, Conjecture 3.10.)
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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 $n \ge 9$. Tait and Tobin proved it for sufficiently large $n$ in 2017; the conjecture now holds for all $n \ge 9$.
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algebraJul 21, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
answered for 3-cocycles, inside a broader partial classification of twisted Deligne products
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algebraJul 21, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Partial: this settles n=16 only. Combined with Pang's n>=17 the conjecture holds for all n>=16, leaving the small cases open.
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algebraJul 21, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
For the integral $\mathcal{I}(h)$ over the unit interval and the torus, the paper gives the three-term Laurent polynomial $f(x,z)=(1-z^{-1})((1-x)+xz)$ with $\mathcal{I}(f^n)=0$ but $\mathcal{I}(z^{-1}f^n)=(-1)^{n-1}/(n+1)\neq0$. This disproves the $xz$-conjecture with one interval and one torus variable, shows $\ker\mathcal{I}$ is not a Mathieu–Zhao subspace, and by padding yields counterexamples for SU(2).
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geometry-topologyJul 21, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
the mod 4 form; the general conjecture is false by Ermotti, Hongler and Weber
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number-theoryJul 21, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
Does the sum of the reciprocals of all primitive pseudoperfect numbers converge?
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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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number-theoryJul 20, 2026Significance 13/100Registry: lean verified
Prior state unknown→proved
Proves positive lower density. The Formal Conjectures encoding asks for Set.HasPosDensity, a density that exists and is positive; erdosproblems.com says Erdos most likely meant lower density.
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probability-statisticsJul 20, 2026Significance 30/100Registry: unreviewed
Prior state unknown→disproved
The Benjamini-Hochberg procedure is known not to control the false discovery rate at its nominal level under arbitrary dependence. A folklore conjecture in the FDR literature held that it must at least control the FDR up to a universal multiplicative constant. It does not: there are finite Gaussian models whose FDR divided by $q$ diverges as $q \downarrow 0$, with an explicit two-sided lower bound $q\sqrt{\log(1/q…
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algebraJul 20, 2026Significance 13/100Registry: lean verified
Prior state unknown→proved
If $\operatorname{CT}_{(k)}$ is generated by all horizontal class transpositions with modulus at most $k$, is $\operatorname{CT}_{(k)} \cong S_{\operatorname{lcm}(2,\dots,k)}$ for every $k \ge 4$?
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algebraJul 20, 2026Significance 13/100Registry: lean verified
Prior state unknown→disproved
Must the right-relatively convex subgroups of a right-orderable nonabelian group form a sublattice of its subgroup lattice? A construction shows they need not.
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algebraJul 20, 2026Significance 13/100Registry: lean verified
Prior state unknown→proved
If the power graph of a finite group contains no induced path on four vertices, must it also contain no induced cycle of length at least four - that is, is every cograph power graph chordal?
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algebraJul 20, 2026Significance 15/100Registry: lean verified
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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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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algebraJul 20, 2026Significance 65/100Registry: expert verified
Prior state unknown→disproved
n ≥ 3; plane case open
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probability-statisticsJul 20, 2026Significance 20/100Registry: lean verified
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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probability-statisticsJul 20, 2026Significance 25/100Registry: site confirmed
Prior state unknown→disproved
Explicit counterexamples in dimensions 3 and 4, so GMC(n) fails for every n >= 3; GMC(1) was already known, and a separate human-authored preprint claims the remaining n = 2 case affirmatively
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algebraJul 20, 2026Significance 14/100Registry: lean verified
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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algebraJul 20, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
If a subgroup of a product of groups of type $F_k$ virtually surjects onto every $k$-tuple of factors, must it be of type $F_k$ itself? Yes, for discrete groups, and likewise for $FP_k$. The homological $n$-$(n+1)$-$(n+2)$ Conjecture follows for discrete groups when the common quotient is finitely presented, and that hypothesis cannot be dropped.
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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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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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algebraJul 20, 2026Significance 13/100Registry: lean verified
Prior state unknown→disproved
For an extension $G = A \rtimes B$ of elementary abelian $p$-groups with $a \in A$ satisfying $C_B(a) = 1$, must $H = \langle a, B\rangle$ satisfy $\operatorname{rank}(Z(H) \cap H') \le \operatorname{rank}(B)$? An explicit extension violates the bound.
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algebraJul 20, 2026Significance 15/100Registry: lean verified
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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theoretical-computer-scienceJul 20, 2026Significance 35/100Registry: unreviewed
Prior state unknown→proved
Can any single-pass semi-streaming algorithm beat the naive greedy $1/2$-approximation for maximum matching? No. No single-pass semi-streaming algorithm, deterministic or randomized, achieves a better-than-half approximation, so greedy is optimal. The same construction settles the optimal competitive ratio of online matching with preemption at $1/2$.
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analysisJul 19, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Pełczyński's duality between strictly singular and strictly cosingular operators fails without weak compactness. Beanland asked, in work with Androulakis and later on MathOverflow, for the separable-range case: the paper answers it affirmatively and shows that for separable $X$, $T$ is strictly cosingular exactly when $T^{*}$ is strictly singular.
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number-theoryJul 19, 2026Significance 13/100Registry: lean verified
Prior state unknown→proved
The headline is the constant, and its two halves have different histories. The lower bound, $\liminf (f(n)-2n)/(n/\log n) \ge 4029639598/25970038185$, is not new here: it is Mausberg's thirteen-layer valuation cut, posted to the erdosproblems.com forum in May 2026 and credited as such in the paper. Its author wrote there that it "does not prove an upper bound, nor does it prove that an asymptotic constant exists."…
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analysisJul 19, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Davis, Figiel, Johnson and Pełczyński showed their interpolation space admits a Schauder basis when the range space has a shrinking one. Can the DFJP space always be chosen with a basis whenever the range space has a basis? The paper proves it can.
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analysisJul 19, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
The Elton–Odell theorem gives, in every infinite-dimensional normed space, a unit-sphere sequence with mutual distances at least $1+\varepsilon$. Over $\mathbb{C}$, identifying vectors differing by a unimodular scalar gives a toroidal distance. Does every infinite-dimensional complex normed space admit such a uniformly separated sequence for that distance? Yes.
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analysisJul 19, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
The realisation problem asks which unital Banach algebras arise as the Calkin algebra $\mathcal{B}(X)/\mathcal{K}(X)$ of some Banach space. Recorded in Tarbard's thesis and studied by Horváth and Kania. The paper exhibits a unital Banach algebra that cannot be one.
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analysisJul 19, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
A Banach space is primary if in every decomposition into two complemented subspaces one summand is isomorphic to the whole. Lechner, Motakis, Müller and Schlumprecht identified the primariness of $L_p(L_1)$ as a prominent remaining open case; the paper proves it is primary for $1<p<\infty$.
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algorithms-optimizationJul 18, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Prim-Dijkstra routing interpolates between a minimum spanning tree and a shortest-path tree, and has been used and improved in VLSI physical design since the early 1990s, but the complexity of the terminal-only Manhattan decision problem was never settled. It is weakly NP-complete. A continuous cost-radius tradeoff with a balanced $(2,2)$ guarantee accompanies the classification.
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combinatoricsJul 18, 2026Significance 9/100Registry: site confirmed
Prior state unknown→proved
Donner proved in 1992 that the list color function $P_\ell(G,k)$ equals the chromatic polynomial $P(G,k)$ once $k$ is large. Kaul and Mudrock asked whether the analogue holds for Hanlon's unlabeled chromatic polynomial, and could not settle even the edgeless graph, which they posed as a conjecture. The conjecture is true, and it implies that a disconnected graph satisfies the unlabeled analogue of Donner's result…
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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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analysisJul 18, 2026Significance 9/100Registry: unreviewed
Prior state unknown→proved
Douglas and Yang attach to each nonzero vector $x$ of a quasinilpotent operator $T$ a local resolvent-growth exponent $k_x$, giving the power set $\Lambda(T) = \{k_x : x \ne 0\}$. Ji and Zhang asked whether $1$ always belongs to $\Lambda(T)$. It does, for every quasinilpotent operator on every Banach space. Moreover $\Lambda(T) = [0,1]$ for every backward unilateral weighted shift on $\ell^p$ with strictly decreas…
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combinatoricsJul 17, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
the companion pattern Fulek proposed alongside L_3 is not covered by this method
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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…
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combinatoricsJul 17, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
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 $k \ge 3$.
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number-theoryJul 16, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Erdos and Graham asked whether a positive-density subset of $\{1,\ldots,N\}$ can avoid having any two distinct elements $a,b$ whose unit fractions average to a unit fraction. It can: there is a constant $c>0$ such that for all large $N$ some $A \subseteq \{1,\ldots,N\}$ of size $> cN$ has that property, which also gives the best known lower bounds for related unit-fraction avoidance problems.
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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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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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combinatoricsJul 16, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Gao, Huo and Ma asked whether for every fixed $k \ge 3$ there is a function $f_k(n) \to \infty$ such that every $n$-vertex $(k+1)$-critical graph contains $f_k(n)$ consecutive cycle lengths. The paper settles this and two related problems on cycle lengths and cycles with chords under chromatic and degree constraints.
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analysisJul 16, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Two free ergodic measure-preserving flows whose $\mathrm{L}^1$ 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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combinatoricsJul 15, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Settles the case p=5. The posed formula for every odd prime remains open.
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quantum-information-computingJul 15, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Does quantum memory give a query-complexity advantage for learning an unknown quantum channel, when protocols without it must measure after each channel use and keep only a classical transcript? It does, and the paper also determines how little coherent memory suffices for the advantage to appear.
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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 $m \times m$ correlation matrix $R$ with $R - \mathbf{1}\mathbf{1}^{\mathsf T}/m \succeq 0$ and $X \sim \mathcal{N}(0,R)$, the maximum of the $X_i$ is stochastically dominated by the maximum of $m$ independent standard Gaussians.
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probability-statisticsJul 15, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
the conjectured cutoff location of 2n log n is wrong
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number-theoryJul 14, 2026Significance 7/100Registry: unreviewed
Prior state unknown→proved
For an irreducible crystallographic root system of rank $r$ with Coxeter number $h$, the paper proves that Au's normalized Witten zeta function has a simple pole at $2/h$ and evaluates its residue in closed form in terms of the Cartan determinant, the Weyl group order and the invariant degrees.
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combinatoricsJul 14, 2026Significance 15/100Registry: lean verified
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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combinatoricsJul 14, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
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.
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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 $\Omega(d)$ and $O(d^2 \log^2 d)$ since 1996. The paper proves a near-quadratic lower bound $\Omega(d^2 / \log(d+1))$, closing the gap: $Q(d, \sim d^{-1/2}) = \Theta(d^2)$, a polynomial separation from full first-order information.
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number-theoryJul 14, 2026Significance 10/100Registry: site confirmed
Prior state unknown→disproved
A monic prime $P$ of $\mathbb{F}_q[T]$ is a $c$-Wieferich prime if $\rho_P(1) \equiv 1 \bmod P^2$ for the Carlitz module $\rho$. On limited data and proofs in degrees $2$ and $3$, Thakur suggested in 2015 that in odd characteristic every $c$-Wieferich prime has degree divisible by $p$. It is false: an explicit irreducible $c$-Wieferich prime has degree not divisible by $p$, and the resulting common factor has a cl…
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combinatoricsJul 14, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
The interchange graph $G(R,S)$ has the $(0,1)$-matrices with row sums $R$ and column sums $S$ as vertices, adjacent when they differ by a single $2\times 2$ interchange. Brualdi asked whether $G(R,S)$ is always Hamiltonian. It satisfies more: it is maximally Hamiltonian, Hamilton-laceable when bipartite and Hamilton-connected when not.
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number-theoryJul 13, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
The paper gives counterexamples in dimensions eight and nine to two problems: the Cartesian-product problem posed by Cassels for critical determinants and formulated by Zong for lattice packings, and a question raised by Sarnak, formulated as a conjecture by Chiu, on whether height among unit-volume flat tori is minimized by a lattice maximizing its shortest nonzero vector.
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geometry-topologyJul 13, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
rules out the two-point LP method in this dimension; the optimal packing in dimension 36 remains unknown
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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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combinatoricsJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
the literal wording is refuted; the intended variant remains open
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number-theoryJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
If $h(r)$ is the maximal finite exact order attainable by an additive basis of order at most $r$, what is $\lim_{r \to \infty} h(r)/r^2$? The candidate proof identifies the sharp limit $1/3$.
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number-theoryJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
How large must $y(\varepsilon, n)$ be so that every interval $(x, x+y)$ contains at most $\varepsilon y$ integers having a divisor in $(n, 2n)$? The candidate proof gives the sharp fixed-$\varepsilon$ order $y = \Theta_\varepsilon(n)$, uniformly in the translate.
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number-theoryJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
exact largest modulus 3·2^{k-3} for k ≥ 5, near-linear least maximum, reciprocal mass Θ(log k), and an infinite divisor family; the counting asymptotic rests on the cited BBMST theorem
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number-theoryJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
If $A \subseteq \mathbb{N}$ has unbounded dyadic-shell counts and $\sum_{n \in A} \|\theta n\| = \infty$ for every $0 < \theta < 1$, must $A$ be complete - is every sufficiently large integer a sum of distinct elements of $A$?
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geometry-topologyJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
natural readings of the ambiguous historical statement are disproved
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number-theoryJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
For the least $t_k(n)$ with $n \mid t_k(n)(t_k(n)+1)\cdots(t_k(n)+k-1)$, do the conjectured logarithmic-saving and adjacent-length estimates hold on average? Both answered affirmatively, with $c = 1/2048$ admissible in the $t_2$ bound.
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combinatoricsJul 13, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
exact k = 3 constant established, settling the Parrilo-Robertson-Saracino conjecture for 3-APs; general k remains open
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geometry-topologyJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
the infinite-chromatic subquestion is proved; the rest of the problem remains open
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number-theoryJul 13, 2026Significance 7/100Registry: unreviewed
Prior state unknown→disproved
Ross introduced $\mathcal{S}$-perfect numbers, integers expressible as $1 + \sum \lambda_j d_j$ over their proper divisors with coefficients in $\mathcal{S}$, and conjectured that they have the same density as the nondeficient numbers, plus a second conjecture relating odd nondeficient numbers to $\mathcal{S}$-perfection. Both are false.
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probability-statisticsJul 13, 2026Significance 35/100Registry: unreviewed
Prior state unknown→proved
the last conditional step in the Ding-Sun and Huang bounds is discharged
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number-theoryJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
If each integer has at most $r$ representations $m = pa$ with $p$ prime and $a \in A \subseteq [1, N]$, what is the best upper bound for $\sum_{a \in A} 1/a$? The candidate proof gives the matching order $\Theta_r(\log N / \log\log N)$.
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number-theoryJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
If $A$ is a forbidden-divisor set with $|A \cap [1,x]| = o(\sqrt{x})$ and $B = \{b_1 < b_2 < \cdots\}$ the sifted set, must $x^{-1} \sum_{b_i < x} (b_{i+1} - b_i)^2$ converge to a finite limit?
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number-theoryJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
If $n_1 < n_2 < \cdots$ with $n_{k+1}/n_k \ge c > 1$, must $\sum_k 1/F_{n_k}$ be irrational? The proposed proof closes the range $1 < c < 2$ left open by earlier criteria.
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number-theoryJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
Estimate the number $F(x)$ of minimal distinct covering systems whose moduli all lie in $[1, x]$. The candidate proof gives $\log\log F(x)/\log x \to 1$, i.e. $F(x) = \exp(x^{1+o(1)})$.
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geometry-topologyJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
superlinear lower bound M(n) ≥ n^{1 + 1/(50000 log log n)}, improving Ω(n log n); the exact order remains open
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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-topologyJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
the conjectured lower bound is disproved; good bounds for c(n) remain open
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probability-statisticsJul 13, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Do the one-species current marginals of type-D ASEP have the predicted Tracy-Widom long-time asymptotics despite the model's two-species interactions?
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probability-statisticsJul 13, 2026Significance 14/100Registry: unreviewed
Prior state unknown→disproved
Does the Benjamini-Hochberg procedure always control the false-discovery rate at its nominal level for correlated two-sided Gaussian p-values? A factor model gives $\mathrm{FDR} > 0.0104$ at nominal level $\alpha = 0.01$.
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number-theoryJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
If $g_3(n)$ is the largest size of $A \subseteq [1,n]$ with fewer than three representations of every product $a_1 a_2$, does its conjectured second-order normalized term converge? The candidate proof gives an explicit limit constant.
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number-theoryJul 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
upper bound improved to f(n) ≤ 14n^{3/7} with an explicit logarithmic lower bound; matching bounds remain open
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combinatoricsJul 12, 2026Significance 5/100Registry: lean verified
Prior state unknown→proved
Is the number of nonnesting permutations of $\{1,1,\dots,n,n\}$ avoiding both $1132$ and $3312$ equal to $3^n - 3 \cdot 2^{n-1} + 1$ for every $n \ge 1$?
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combinatoricsJul 12, 2026Significance 7/100Registry: unreviewed
Prior state unknown→proved
Coble and Barg introduced binary Coxeter codes, the span of indicators of standard cosets of fixed rank in a finite Coxeter system, generalizing Reed-Muller codes, and proposed a conjectural value for the minimum distance of a general Coxeter code. The conjecture is true, and it yields a decoding consequence.
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combinatoricsJul 12, 2026Significance 9/100Registry: unreviewed
Prior state unknown→proved
the odd-cycle half; the even-cycle half is a companion paper by the same authors
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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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combinatoricsJul 10, 2026Significance 55/100Registry: lean checked
Prior state unknown→proved
Conjectures that every bridgeless graph has a collection of cycles covering each edge exactly twice.
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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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combinatoricsJul 10, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
Improved bounds rather than a settled question: the asked-for inequality is not established in general.
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theoretical-computer-scienceJul 10, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
Klopp and Zadik gave an exponential-time node-private algorithm for exact community recovery in stochastic block models and asked whether a polynomial-time algorithm could match it. One can: a Lipschitz surrogate for the penalized likelihood plus an accept-reject sampler gives a high-probability polynomial-time node-private algorithm that nearly matches the exponential-time guarantee.
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combinatoricsJul 10, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
The paper constructs an exact cluster $F\subseteq\mathbb{Z}^2$ of cardinality 8 with full affine span and an $F$-tiling whose orbit closure contains no 1-periodic $F$-tiling, giving a non-degenerate counterexample to Nivat's conjecture for non-convex windows. This answers negatively a question of Kari and Moutot from 2023.
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combinatoricsJul 10, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
asymptotic form only; Kotzig's conjecture itself remains far from solved
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analysisJul 9, 2026Significance 25/100Registry: lean verified
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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theoretical-computer-scienceJul 9, 2026Significance 7/100Registry: unreviewed
Prior state unknown→proved
Can the minimum edge-outerplanarity of a finite loopless planar graph, minimized over all planar embeddings, be computed in polynomial time? Asked by Bentz in 2009.
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geometry-topologyJul 9, 2026Significance 15/100Registry: site confirmed
Prior state unknown→disproved
the refined form for essential arrangements in projective three-space
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combinatoricsJul 8, 2026Significance 5/100Registry: unreviewed
Prior state unknown→proved
For the Fubini numbers $a(n)$, is $a(n) = \sum_{k=0}^{2^{n-1}-1} A284005(k)$ for every $n > 0$, as conjectured on the OEIS in 2018?
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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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mathematical-physicsJul 8, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
A variant rather than the BKMP conjecture itself: the target lies outside the toric setting the conjecture addresses.
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number-theoryJul 8, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
h₄(n) = 4 for every n ≥ 331,777, with improved global bounds; the broader problem remains open
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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 $w:[n]\to[d]$ is isolating when a unique edge attains minimum weight. Faber and Harris conjectured that the number of isolating assignments is at least $n\sum_{j=0}^{d-1} j^{n-1}$, 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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number-theoryJul 7, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
density-one set of n; the conjecture itself remains open
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analysisJul 7, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
After $L^2$ normalization, stable phase retrieval holds over the $L^2$-spans of independent real-valued centered random variables exactly when all but possibly one coordinate satisfies a uniform two-sided $L^1$ bound. This confirms the characterization conjectured by Calderbank, Daubechies, Freeman and Freeman.
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geometry-topologyJul 6, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Moraga conjectured, and Kollár and Zhuang recorded, an odd-dimensional extension of the rank bound for faithful abelian $p$-group actions on smooth Calabi–Yau varieties. The paper disproves it.
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theoretical-computer-scienceJul 6, 2026Significance 9/100Registry: unreviewed
Prior state unknown→proved
Given online vectors $v_t \in \mathbb{R}^d$ with $\|v_t\|_2 \le 1$, can signs $\varepsilon_t \in \{-1, 1\}$ be chosen in $O(dT)$ total time so that every prefix has $\ell_\infty$ discrepancy $O(\sqrt{\log T})$ with high probability? The previous optimal algorithm ran in time exponential in $T$ and $d$.
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geometry-topologyJul 4, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
For every $n\ge2$ the paper exhibits an $n$-dimensional K-polystable toric $\mathbb{Q}$-Fano variety whose alpha invariant is exactly $\tfrac{2}{2n+1}$, answering a question of Liu and Zhuang on whether a K-semistable example exists with alpha invariant between $\tfrac{1}{n+1}$ and $\tfrac1n$.
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probability-statisticsJul 3, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
For a multidimensional reflected diffusion, does the basic adjoint relationship uniquely characterize the stationary distribution? The question had stood unresolved for more than thirty-five years since the BAR approach was introduced. For stable Harrison-Reiman data with a nonsingular $M$-matrix reflection matrix, the finite-signed uniqueness problem is settled, via pathwise differentiability of the reflected pro…
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probability-statisticsJul 3, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
uniqueness proved for stable Harrison-Reiman systems with a nonsingular M-matrix reflection; an infinite-dimensional obstruction is shown in the larger completely-S class
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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 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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combinatoricsJul 2, 2026Significance 40/100Registry: unreviewed
Prior state unknown→disproved
White conjectured that the symmetric exchange binomials generate the toric ideal of a matroid. This is now known to be false; a rank $9$ binary matroid constitutes a counterexample.
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combinatoricsJul 2, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Both bounds move, and the gap stays enormous: the lower bound rises from $\exp(\Omega(\log^2 k))$ to $\exp(\Omega(k^{1/3}))$ and the upper falls from $\exp(O(k^4))$ to $\exp(O(k^2))$, so $A(k)$ is still undetermined between an exponent of $k^{1/3}$ and one of $k^2$. The paper's own closing discussion argues its lower-bound construction is near the limit of the method and that beating it needs additional randomness…
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analysisJul 1, 2026Significance 13/100Registry: site confirmed
Prior state unknown→proved
For unit-modulus complex numbers $z_i$, let $p_n(z)=\prod_{i\le n}(z-z_i)$ and $M_n=\max_{|z|=1}|p_n(z)|$. Erdős's prize question: is there $c>0$ with $\sum_{k\le n} M_k > n^{1+c}$?
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number-theoryJul 1, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
Let $F(n)$ be the largest $A\subseteq\{1,\dots,n\}$ with $a\nmid bc$ for distinct $a,b,c\in A$. Is $F(n)=\pi(n)+(C+o(1))\,n^{2/3}(\log n)^{-2}$ for some constant $C$?
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number-theoryJul 1, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
Let $S(N)$ count the distinct values of $\sum_{n\in A} 1/n$ over $A\subseteq\{1,\dots,N\}$. Estimate $S(N)$.
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number-theoryJul 1, 2026Significance 11/100Registry: lean verified
Prior state unknown→proved
Let $a,b,c>1$ be pairwise coprime integers. Is every large integer a sum of distinct numbers of the form $a^k b^l c^m$ ($k,l,m\ge 0$), none dividing another?
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number-theoryJul 1, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
What is the largest $A\subseteq\{1,\dots,N\}$ such that all subset sums $\sum_{n\in S}1/n$ (over $S\subseteq A$) are distinct?
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analysisJul 1, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Among all nonconstant monic polynomials $f$ whose roots lie in $[-1, 1]$, determine $\inf_f |\{x \in \mathbb{R} : |f(x)| < 1\}|$.
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geometry-topologyJun 30, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Ziegler proved every simplicial $d$-dimensional 0/1-polytope has at most $2d$ vertices, and asked whether attaining $2d$ vertices forces central symmetry (i.e. a 0/1 cross-polytope). Known true for $d \le 6$; open since ~2000.
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probability-statisticsJun 30, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
For an unkilled Levy process $\xi$ drifting to $+\infty$ with all positive exponential moments, let $I_\xi = \int_0^\infty e^{-\xi_t}\,dt$ and $X_\xi = 1/I_\xi$. Bertoin and Yor proved $X_\xi$ is moment-determinate when $\xi$ has no positive jumps and conjectured that this condition is necessary. The conjecture is settled.
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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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combinatoricsJun 29, 2026Significance 32/100Registry: unreviewed
Prior state unknown→proved
minimum out-degree 7; the conjecture is open in general
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combinatoricsJun 29, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Can a finite set of lattice points determine many rectangles but few isosceles triangles? Both parts of the governing question have negative answers, quantified by explicit blowup rates, and the resulting configurations give obstructions in the Mizohata-Takeuchi circle of problems.
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quantum-information-computingJun 29, 2026Significance 15/100Registry: lean verified
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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probability-statisticsJun 28, 2026Significance 10/100Registry: unreviewed
Prior state unknown→disproved
the published multivariable claim fails; the single-variable theorem stands
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combinatoricsJun 28, 2026Significance 15/100Registry: lean verified
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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number-theoryJun 27, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
resolved under an explicit formalization of 'reasonable', not in full generality
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geometry-topologyJun 26, 2026Significance 15/100Registry: site confirmed
Prior state unknown→disproved
Lassak conjectured that a reduced planar convex body of thickness $\Delta$ has area at most $(\pi/4)\Delta^2$, the value for the disc. False: an explicit reduced body of thickness $1$ has area $0.786215\ldots > \pi/4 = 0.785398\ldots$, given by a closed-form support function.
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geometry-topologyJun 26, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Erdos asked whether a finite unit-distance graph in the plane can have independence ratio below $1/4$. One exists, built on the geometric fractional chromatic number framework of Matolcsi, Ruzsa, Varga and Zsamboki plus a carefully chosen two-vertex augmentation.
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combinatoricsJun 26, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
A subset $A$ of the pointwise-ordered cube $[0,1]^n$ is a $k$-antichain when it meets every chain in at most $k$ points. The conjecture concerns the largest possible $(n-1)$-dimensional Hausdorff measure of such a set; it is settled here, following work of Janzer.
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analysisJun 24, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Does every lattice of density above one admit a Gabor frame with a nice window? No. For every dimension $d > 1$ there are explicit criteria on lattices $\Lambda \subset \mathbb{R}^{2d}$ with $D(\Lambda) > 1$ such that no function with continuous Zak transform generates a Gabor frame along $\Lambda$, which answers the existence problem negatively for Schwartz-class windows.
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theoretical-computer-scienceJun 24, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Furthest Pair and its relatives admit $f(d)\,n^{2-\Theta(1/d)}$ algorithms, making them the standard examples of barely subquadratic computation, and whether that is optimal in superconstant dimension was open. Under SETH it is: Furthest Pair requires quadratic time once the dimension is superconstant.
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number-theoryJun 24, 2026Significance 10/100Registry: unreviewed
Prior state unknown→disproved
For $S(x) = \#\{(a,b) : a + b \le x,\ \sigma(a) + \sigma(b) = \sigma(a+b)\}$, is $S(x) \sim cx$? The preprint claims $S(x)$ grows faster than $x (\log x)^R$ for every fixed $R$, ruling out the linear asymptotic.
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number-theoryJun 24, 2026Significance 10/100Registry: unreviewed
Prior state unknown→disproved
Let $S(x)$ count ordered pairs $(a,b)$ with $a+b \le x$ and $\sigma(a)+\sigma(b) = \sigma(a+b)$. Erdos asked whether $S(x) \sim cx$. The opposite extreme holds: for every $R > 0$, $S(x)/(x(\log x)^R) \to \infty$, so the count beats every fixed logarithmic scale.
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combinatoricsJun 23, 2026Significance 13/100Registry: unreviewed
Prior state unknown→proved
settles two numbered Erdos problems at once
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geometry-topologyJun 23, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
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.
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number-theoryJun 23, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Let $A(x)$ count $n \le x$ such that every prime $p \mid n$ has a divisor $d > 1$ of $n$ with $d \equiv 1 \pmod p$. Erdos asked whether $A(x)/x = \exp(-(c+o(1))\sqrt{\log x}\log\log x)$. It does, with $c = 1/(2\sqrt{\log 2})$.
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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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analysisJun 22, 2026Significance 12/100Registry: lean verified
Prior state unknown→proved
Both parts claimed answered affirmatively, with a Lean formalization; two proof claims are filed on erdosproblems.com but the problem is still listed open
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number-theoryJun 22, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #865.” The registry entry and named primary source contain the available statement and scope.
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combinatoricsJun 22, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Claimed proved in a preprint of E. Li; erdosproblems.com still lists the problem open pending human review
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algebraJun 22, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Baker asked, as recorded by Poonen, whether a fixed smooth quasiprojective variety over a finite field must acquire a smooth rational hyperplane section after every sufficiently high-dimensional linearly nondegenerate embedding. Poonen predicted no for every positive-dimensional variety, and that prediction is correct.
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analysisJun 22, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
a record exponent; the conjectured range is not yet reached
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number-theoryJun 21, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
The problem statement is ambiguous: the limit-exists reading is claimed proved (Lean), while the convergence-from-hypotheses reading was disproved by a Lean-checked construction of Price that the community classes as a variant
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analysisJun 21, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
VibeMathed records this result as “Erdős Problem #1197.” The registry entry and named primary source contain the available statement and scope.
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number-theoryJun 21, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Zhi-Wei Sun conjectured a closed evaluation of a truncated Legendre-symbol determinant. For every prime $p \equiv 3 \pmod 4$ it equals $\lfloor (p-2)/3 \rfloor^2 x$, proved by reducing to inverse data for Chapman's full Legendre-symbol matrix and evaluating that with Vsemirnov's factorization and a Schur-Pfaffian resolvent identity.
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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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combinatoricsJun 21, 2026Significance 11/100Registry: lean verified
Prior state unknown→proved
a polynomial bound for N(k,2), stronger than the exponential bound asked for; the two-parameter problem remains open
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number-theoryJun 21, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #948.” The registry entry and named primary source contain the available statement and scope.
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number-theoryJun 19, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
If $a/b \in \mathbb{Q}_{>0}$ and $b$ is squarefree, can $a/b$ always be written as a finite sum of reciprocals of distinct products of two distinct primes?
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number-theoryJun 18, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Let $n_k$ be the least $n > 2k$ such that $(n-k)(n-k+1)\cdots(n-1)$ has no prime factor in $(k, 2k)$. Erdos conjectured a superpolynomial lower bound; for all large $k$, $n_k > e^{\log^2 k / (20 \log\log k)}$.
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number-theoryJun 18, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
the conjectured superpolynomial growth is established; the sharper order remains open
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algorithms-optimizationJun 18, 2026Significance 7/100Registry: lean checked
Prior state unknown→disproved
Ramachandra and Natarajan conjectured a bound on the pairwise independent correlation gap in their 2025 Operations Research Letters paper. An explicit counterexample refutes it.
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analysisJun 18, 2026Significance 8/100Registry: unreviewed
Prior state unknown→disproved
Aldroubi, Cabrelli, Krishtal and Molter conjectured that for a bounded normal operator $T$ and any vector $g$, the normalized orbit $\{T^k g / \|T^k g\| : k \ge 0\}$ is never a frame. It can be: an explicit construction produces a normalized orbit that is a frame.
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geometry-topologyJun 18, 2026Significance 10/100Registry: unreviewed
Prior state unknown→disproved
Sabok asked whether the compact convex set $S'(X)$ attached to a separable metric space of diameter at most one is always a simplex, and whether $S'(\mathbb{U}_1)$ is the Poulsen simplex. Both answers are negative, with obstructions already visible for finite $X$ and for the Urysohn space.
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geometry-topologyJun 16, 2026Significance 25/100Registry: site confirmed
Prior state unknown→disproved
Wegner conjectured in 1965 that every finite family $\mathcal{R}$ of axis-parallel rectangles satisfies $\tau(\mathcal{R}) \le 2\nu(\mathcal{R}) - 1$, where $\tau$ is the minimum number of piercing points and $\nu$ the largest pairwise-disjoint subfamily. False, by an explicit triangle-free counterexample.
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analysisJun 16, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
part (1) of the conjecture
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combinatoricsJun 16, 2026Significance 11/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #986.” The registry entry and named primary source contain the available statement and scope.
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algorithms-optimizationJun 16, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Yun, Sra and Jadbabaie posed as a COLT 2021 open question whether, for well-conditioned symmetric matrices, the operators encoding the expected iterate of single-shuffle SGD, random-reshuffle SGD and gradient descent on a quadratic finite sum satisfy $\|W_{ss}\| \le \|W_{rs}\| \le \|W_{gd}\|$. They do.
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number-theoryJun 16, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
the average order; the uniform bound Erdos asked about is not settled
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combinatoricsJun 16, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
in polynomial edge-density regimes; the general question remains open
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combinatoricsJun 15, 2026Significance 25/100Registry: unreviewed
Prior state unknown→disproved
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.
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probability-statisticsJun 14, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
a structured special case, proved the same week as the independent group version
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number-theoryJun 14, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
lower bound improved to ≫ log n/(log log n · log log log n) infinitely often; the extremal order remains open
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analysisJun 14, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
in the range s in (1/4,1); Brezis's Problem 5.4 outside that range is untouched
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number-theoryJun 14, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
Affirmative answer claimed, contrary to the negative answer Erdős and Graham conjectured; erdosproblems.com still lists the problem open
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number-theoryJun 13, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
the omega = 2 integer case, the one Butler, Erdos and Graham left open
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number-theoryJun 12, 2026Significance 5/100Registry: unreviewed
Prior state unknown→disproved
For an odd prime $p$, do Sun's normalized trigonometric permanents satisfy $s_p < 0 \iff p \equiv 5 \pmod{12}$ and $s'_p < 0 \iff p \equiv 7 \pmod 8$? Exact computation at $p = 29$ refutes both sign laws.
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combinatoricsJun 12, 2026Significance 5/100Registry: unreviewed
Prior state unknown→disproved
For a bipartite graph without isolated vertices and with a unique minimum dominating set, does the proposed function $m(n, \gamma)$ bound the number of edges whenever $\gamma \ge 2$ and $n \ge 3\gamma$? A $13$-vertex bipartite graph with $22$ edges exceeds the conjectured maximum of $21$.
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combinatoricsJun 12, 2026Significance 5/100Registry: unreviewed
Prior state unknown→disproved
For every connected graph, is the variance of its positive adjacency eigenvalues at most its order divided by its average distance? Exact dumbbell-graph certificates refute the bound under both conventions for average distance.
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combinatoricsJun 12, 2026Significance 5/100Registry: unreviewed
Prior state unknown→disproved
If $G$ is connected, cubic and diamond-free, must the zero-forcing number satisfy $Z(G) \le \gamma(G) + 2$? A connected cubic triangle-free $14$-vertex graph has $Z = 7$ and $\gamma = 4$.
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algorithms-optimizationJun 12, 2026Significance 10/100Registry: site confirmed
Prior state unknown→disproved
On the basis of experiments up to 5000 nodes, Papamanthou and Tollis conjectured a relation between the longest paths produced by their MaxSTN and MinSTN algorithms for $st$-orientations of biconnected graphs. A counterexample refutes it.
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combinatoricsJun 12, 2026Significance 5/100Registry: unreviewed
Prior state unknown→disproved
For every $n \ge 2k + 1$, is the independence polynomial of $GP(n, k)$ real-rooted if and only if $k$ is even? Exact Sturm counts refute both directions.
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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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combinatoricsJun 11, 2026Significance 5/100Registry: unreviewed
Prior state unknown→disproved
refuted under the standard-deviation reading; the statement is reading-sensitive and other readings remain open
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geometry-topologyJun 11, 2026Significance 5/100Registry: unreviewed
Prior state unknown→disproved
For a simple $3$-polytope with at least three faces of size at least $7$, must $p_6 \ge \frac{39}{20} + \frac{p_3}{2} - \frac{p_5}{4} - \sum_{k \ge 7} p_k$? Five minimal ten-face counterexamples refute the printed inequality.
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combinatoricsJun 10, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
A cyclic meander induces a cyclic permutation on its $2n$ marked intersection points. Schwartz's conjecture on the quadratic growth of the associated meander number is resolved.
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number-theoryJun 10, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
main exponent determined; sharper subpolynomial factors remain open
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probability-statisticsJun 10, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
the group case; the full Matrix Spencer conjecture remains open
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geometry-topologyJun 9, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
pins the sharp threshold; the conjecture itself was already known false above dimension two
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combinatoricsJun 9, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
VibeMathed records this result as “Erdős Problem #619.” The registry entry and named primary source contain the available statement and scope.
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mathematical-physicsJun 8, 2026Significance 7/100Registry: unreviewed
Prior state unknown→proved
An existence question settled by exhibiting an object, not a general theorem: one model in the family has no exponential degeneracies for generic couplings, and nothing here says which others do.
The route is worth recording because it is not the one anyone was looking down. The author had tried and failed to find such a model directly. It surfaced instead from an unrelated classification of medium-range spin cha…
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geometry-topologyJun 6, 2026Significance 15/100Registry: site confirmed
Prior state unknown→disproved
For a family $F$ of an odd number $n$ of unit disks in the plane, let $\mathrm{OA}(F)$ be the area covered by an odd number of disks. It was conjectured that $\mathrm{OA}(F) \ge \pi$, the area of a single disk. False: configurations exist with smaller odd area.
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number-theoryJun 5, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #696.” The registry entry and named primary source contain the available statement and scope.
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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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combinatoricsJun 4, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Resolved (if correct) by an independence result rather than a proof or disproof in ZFC: consistency of the positive answer is equivalent to a measurable cardinal, of the negative to ZFC alone
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mathematical-physicsJun 2, 2026Significance 20/100Registry: expert verified
Prior state unknown→proved
Can the critical-exponent relation $a + b = 1$ at the jamming transition, observed numerically to high precision in the full replica-symmetry-breaking solution of hard spheres, be derived analytically from the scaling equations?
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number-theoryJun 1, 2026Significance 5/100Registry: unreviewed
Prior state unknown→proved
Define $\varphi_k(n) = \sum_{1 \le a \le n, (a,n)=1} a^k$ and $\mathcal{D}_s = \{k \ge s : \varphi_s(n) \mid \varphi_k(n) \text{ for every } n\}$. Is $\mathcal{D}_1 = \{1, 3, 15\}$, as conjectured by Büyükaşik and collaborators?
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number-theoryJun 1, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Does there exist an integer polynomial $f$ of degree at least two and a set $A \subseteq \mathbb{Z}$ such that every integer has a unique representation $n = a + f(k)$? A manuscript claims the thirteenth powers admit a tiling complement.
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algorithms-optimizationMay 31, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
up to polylogarithmic factors
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number-theoryMay 31, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
integral unimodular lattices of rank at most 32; the general conjecture is open
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algebraMay 29, 2026Significance 9/100Registry: unreviewed
Prior state unknown→disproved
A group is Howson if the intersection of any two finitely generated subgroups is finitely generated, and strongly Howson if the rank of that intersection is bounded in terms of the two ranks. Zhang asked whether the two coincide for finitely generated groups. They do not.
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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 $\mathbb{E}_{i,j}|\langle v_i,v_j\rangle| \le n^{-\varepsilon}$ contains linear-sized constant-separated sets. Refuted in a strong sense, using sparse high-dimensional expanders.
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analysisMay 28, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
How large can a measurable $A \subseteq [0,R]^2$ be while avoiding the vertices of upward-oriented axis-aligned right triangles of area $1/2$? At most $O_c(R^2/(\log R)^c)$, with a matching-shaped lower bound construction.
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algorithms-optimizationMay 28, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Sampling a nearly uniform Eulerian tour of a directed Eulerian multigraph was stuck at $mn$-type running times coming from arborescence sampling. A randomized algorithm achieves $\widetilde{O}(m^{3/2})$ worst case, breaking that barrier on sparse graphs.
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combinatoricsMay 27, 2026Significance 55/100Registry: unreviewed
Prior state unknown→disproved
the model's contribution is one simplifying lemma; the authors state the main ideas are human
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geometry-topologyMay 27, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Aluffi, Chen and Marcolli conjectured that the Poincare polynomial of the Deligne-Mumford moduli space $\overline{\mathcal{M}}_{0,n}$ of stable $n$-pointed rational curves has only real roots. True, with simple roots and strict interlacing between consecutive $n$.
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probability-statisticsMay 26, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
A dimension-independent subgaussian concentration bound for Gaussian vectors under coordinate-wise nonlinear maps, valid for any bounded function under a well-conditioned covariance, which answers a question of Simone Bombari on sign quantization.
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number-theoryMay 26, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
a barrier result about one proof strategy, not the conjecture itself
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combinatoricsMay 26, 2026Significance 30/100Registry: site confirmed
Prior state unknown→disproved
Priority: the result was first obtained by Max Grinsztajn with GPT-5.5 Pro assistance, published 26 May 2026 and recorded as the current best bound on Tao's optimization-problems ledger. The same construction was found again independently in August 2026 by Nicholas Konz working with Claude, with a different derivation and a fuller AI disclosure; the two efforts were evidently unaware of each other, and the submitt…
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combinatoricsMay 25, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
Frankl, Peng, Rodl and Talbot asked in 2007 whether the set of Turan densities of families of $r$-graphs contains intervals. It does: for every $r \ge 3$ the set contains non-degenerate intervals, including one of the form $[1-\delta_r, 1]$.
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combinatoricsMay 25, 2026Significance 9/100Registry: unreviewed
Prior state unknown→proved
Resolved the sharp constant (w.h.p.) for random graphs G(n, d/n)
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analysisMay 25, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Iterates of a firmly nonexpansive operator converge weakly but not strongly, by Genel and Lindenstrauss. Whether their Cesaro means converge strongly was open. They need not: an explicit curve gives a counterexample.
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algebraMay 25, 2026Significance 15/100Registry: site confirmed
Prior state unknown→proved
two of the three remaining cases; one is still unclassified
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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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geometry-topologyMay 24, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
The total Chern class of $\mathrm{Sym}^d(\mathbb{C}^n)$ as a torus representation is a symmetric polynomial whose coefficients were conjectured positive, with a binomial log-concavity refinement. Both are established.
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analysisMay 23, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Two degree inequalities for circle-valued Sobolev maps have constants that degenerate as $p \to 1^+$ or $\delta \to 0^+$. Brezis posed the problem of sharpening them; both are now sharpened, by the same power trick with elementary estimates.
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combinatoricsMay 23, 2026Significance 25/100Registry: site confirmed
Prior state unknown→disproved
Simon conjectured that every skeleton of a simplex is extendably shellable. False: for every $d \ge 3$ there is a pure $d$-dimensional shellable simplicial complex that is not shelling completable.
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number-theoryMay 22, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Does the block $11$ occur infinitely often in the base-$2$ expansion of the Erdős-Borwein constant $E = \sum_{n \ge 1} \frac{1}{2^n - 1}$? Posed by Crandall in 2012.
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probability-statisticsMay 22, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
For every zonotope $Z \subset \mathbb{R}^d$ and vectors $v_1,\ldots,v_n \in Z$, there are signs with $\sum_i x_i v_i \in C\sqrt{d}\,Z$ for a universal constant $C$. This resolves a 2002 conjecture on vector balancing in zonotopes.
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geometry-topologyMay 21, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
Han and Jiang asked whether being of klt type is an open condition in flat families of varieties. It is not.
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number-theoryMay 21, 2026Significance 12/100Registry: lean verified
Prior state unknown→proved
parts (i) and (ii) resolved - a near-linear-density construction exists, refuting the N^{1-c} decay; the reciprocal-sum part remains open
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quantum-information-computingMay 21, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
the diagonal family is ruled out; the broader two-parameter problem remains open
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geometry-topologyMay 21, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
Mauri and Moraga posed a two-part question about log Calabi-Yau pairs whose boundary decomposes into big divisors. Both parts have negative answers.
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combinatoricsMay 21, 2026Significance 14/100Registry: lean verified
Prior state unknown→proved
the stronger question $W(k)^{1/k} \to \infty$ remains open
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quantum-information-computingMay 21, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
the N = 4, D ≥ 4 family is fully ruled out; the general two-parameter problem remains open
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algebraMay 21, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
For a pure O-sequence $h = (h_0, \dots, h_e)$ of codimension three and type two, is $h_i^2 \ge h_{i-1} h_{i+1}$ for every interior index $i$? The stated monomial case is proved; the broader level-Hilbert-function case remains open.
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algorithms-optimizationMay 21, 2026Significance 28/100Registry: unreviewed
Prior state unknown→proved
What is the best deterministic polynomial-time approximation ratio for the permanent of a Hermitian positive semidefinite matrix? Resolved up to lower-order terms in the exponent: an explicit concave maximisation $\widehat P(A)$ satisfies $e^{-\gamma n}\widehat P(A) \le \mathrm{per}(A) \le \widehat P(A)$, giving a deterministic $e^{(\gamma+\varepsilon)n}$-approximation for every $\varepsilon > 0$ and matching the…
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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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combinatoricsMay 21, 2026Significance 5/100Registry: lean verified
Prior state unknown→proved
For a finite connected graph $G$, let $L_s(G)$ be the maximum number of leaves in a spanning tree and $\ell(G)$ the average local independence number. Must $L_s(G) \ge 2(\ell(G) - 1)$?
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analysisMay 21, 2026Significance 15/100Registry: lean verified
Prior state unknown→disproved
intended complex form disproved, with a certified strict support-function gap
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number-theoryMay 20, 2026Significance 35/100Registry: unreviewed
Prior state unknown→disproved
false in general; the positive direction holds for k large relative to n
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number-theoryMay 20, 2026Significance 15/100Registry: lean verified
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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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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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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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 conjectur…
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probability-statisticsMay 18, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
For every smooth positive density $f$ on $\mathbb{R}^d$, must the Fisher information $t \mapsto I(f * \gamma_t)$ be log-convex along the heat flow?
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combinatoricsMay 18, 2026Significance 5/100Registry: unreviewed
Prior state unknown→proved
This settles Steinerberger's third open question and separates the two models: in the disk, full chords cost you a factor growing like $\log L$ over what is achievable without the restriction. It also improves the Steinhaus-type $O(L^{1/3})$ to polylogarithmic within the full-chord class.
It does not settle the Buffon discrepancy problem itself. Steinerberger's first question - whether every convex body admits a…
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algebraMay 17, 2026Significance 5/100Registry: expert verified
Prior state unknown→proved
For a semistable one-parameter family of complex projective varieties with smooth nearby fiber $X_t$ and monodromy $T$, is the map $H^1(X, \mathbb{Z}) \to H^1(X_t, \mathbb{Z})^T$ surjective? True in degree one, although the integral statement fails in higher degree.
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analysisMay 17, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
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…
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analysisMay 17, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
order of magnitude determined; the exact asymptotic constant remains open
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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 $K_n$ isomorphic to $G$. Infinite families were known for $t = 2$, but no example was known for any $t \ge 3$. Resolved in both directions: $3$-edge-balanced graphs exist, and no nontrivial $t$-edge-balanced graphs exist for $t \ge 4$.
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probability-statisticsMay 15, 2026Significance 10/100Registry: expert verified
Prior state unknown→proved
For the switch-walk-switch walk on $\mathbb{Z}_2 \wr \mathbb{Z}$ started at $(0,0)$ and $(0,2)$, prove $\|P_t^x - P_t^y\|_{TV} \asymp t^{-1/2}$.
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probability-statisticsMay 15, 2026Significance 10/100Registry: expert verified
Prior state unknown→proved
For the switch-walk-switch lamplighter walk on $\mathbb{Z}_2 \wr T_d$, prove the sharp asymptotic $p_{2n}(e,e) = \rho_d^{2n} \exp[-(\pi^2 (\log(d-1))^2 + o(1)) \frac{n}{\log^2 n}]$ with $\rho_d = \frac{2\sqrt{d-1}}{d}$.
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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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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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probability-statisticsMay 12, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
also refutes the McKean and Toscani conjectures
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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 $X_1, \ldots, X_n$ taking values in $\{0,1,2\}$, the entropy of $S_n = X_1 + \cdots + X_n$ is maximized when $X_1, \ldots, X_{n-1}$ are uniform on $\{0,2\}$ and $X_n$ has an explicitly described three-point distribution. This extends the Shepp…
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differential-equationsMay 11, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
For the ergodic problem $\tfrac12|D\varphi^\varepsilon|^2 + F(x) - \varepsilon\Delta\varphi^\varepsilon = c(\varepsilon)$ on the torus, normalized by $\varphi^\varepsilon(0) = 0$, Jauslin, Kreiss and Moser asked whether the vanishing-viscosity limit $\lim_{\varepsilon \to 0}\varphi^\varepsilon$ always exists. It need not: there is a one-dimensional example with $F \in C^3$ for which the limit fails to exist.
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probability-statisticsMay 11, 2026Significance 37/100Registry: unreviewed
Prior state unknown→proved
Talagrand's convexity problem asks whether a universal number of Minkowski sum operations turns any set of large Gaussian measure into one containing a convex body of comparable measure. It is equivalent to a question about subgaussian vectors: is every centered $1$-subgaussian random vector in $\mathbb{R}^n$ the sum of a universal number of standard Gaussian vectors? Both are answered affirmatively, via the sharp…
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number-theoryMay 8, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #690.” The registry entry and named primary source contain the available statement and scope.
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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 $S_n$ whose order complex is not contractible even though $w$ has no such form, detected by a nonzero Mobius function value.
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combinatoricsMay 8, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Escobar, Klein and Weigandt proved that gradedness of an ASM weak order interval, constancy of Coxeter length across its fibres, and equidimensionality of the associated ASM varieties are mutually equivalent, and conjectured (Conjecture 3.21) that Cohen-Macaulayness of those varieties belongs on the same list. Proved, via a $0$-Hecke monoid action on the MacNeille completion of Bruhat order and vertex-decomposabil…
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number-theoryMay 7, 2026Significance 12/100Registry: contested
Prior state unknown→proved
Can there be a finite covering system of the integers with distinct moduli, all of which are odd and greater than $1$?
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review! Disputed
combinatoricsMay 7, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
a new density-degree inequality gives δ(G) ≤ (3/10 + o(1))|V(G)|, improving 0.328; existence of a linear construction remains open
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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 $m$-…
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number-theoryMay 3, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #351.” The registry entry and named primary source contain the available statement and scope.
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combinatoricsMay 3, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #750.” The registry entry and named primary source contain the available statement and scope.
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number-theoryMay 3, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #283.” The registry entry and named primary source contain the available statement and scope.
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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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number-theoryMay 2, 2026Significance 10/100Registry: unreviewed
Prior state unknown→disproved
A total refutation is claimed for all k>=3, building on the Larsen-Larsen resolution of problem #868; erdosproblems.com still lists the problem open
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number-theoryMay 1, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #694.” The registry entry and named primary source contain the available statement and scope.
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geometry-topologyMay 1, 2026Significance 40/100Registry: expert verified
Prior state unknown→disproved
Conjectured upper bound on how many pairs among $n$ points in the plane can be exactly one unit apart.
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combinatoricsMay 1, 2026Significance 30/100Registry: unreviewed
Prior state unknown→proved
How dense can a sum-free subset of the lattice cube $\{1,\dots,n\}^d$ be? Aydinian and Cameron asked for the limiting density, which is also Problem 6 in Ben Green's list of 100 open problems. The natural conjecture is that the optimum is a slice $\{x : 1 \le L(x) < 2\}$ for a linear map $L$, previously known only for $d \le 4$. Proved for all $d$. The paper also shows the same phenomenon fails if the cube is repl…
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analysisMay 1, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
exponent 2 fails for every p > 2; the sharp exponent p' form is proved for integer p ≥ 2
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number-theoryMay 1, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
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…
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analysisApr 30, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
An elementary solution via a primitive-row decomposition of the Chebyshev-node measures; the main theorem is formalized in Lean, but erdosproblems.com still lists the problem open
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combinatoricsApr 30, 2026Significance 8/100Registry: unreviewed
Prior state unknown→proved
Every minimally generically globally rigid graph in $\mathbb{R}^d$ containing a subgraph isomorphic to $K_{d+2}$ is itself isomorphic to $K_{d+2}$, confirming Conjecture 6.3 of Garamvölgyi, Jackson and Jordán (2025).
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
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 $k^{-\delta}(1+o(1))$ times a smooth positive function, sharpening a line of estimates running through Łuczak-Pyber and Eberhard-Ford-Green.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 30, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
As Tao notes on the problem page, the claim establishes natural LOWER density at least 1-eta but not that the natural density exists, so the problem as stated remains technically open
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
analysisApr 29, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Must every sufficiently large node set admit bounded labels that force any polynomial fitting almost all labels at degree below $(1+\varepsilon)n$ to have arbitrarily large uniform norm? Claimed via Beurling density for Bernstein spaces.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
combinatoricsApr 29, 2026Significance 5/100Registry: unreviewed
Prior state unknown→proved
The directed five-dimensional torus $D_5(m)$ has a Hamilton decomposition for every odd $m \geq 3$, extending the decomposition program for directed tori beyond the three-dimensional case.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
combinatoricsApr 28, 2026Significance 10/100Registry: site confirmed
Prior state unknown→disproved
VibeMathed records this result as “Erdős Problem #1092.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
geometry-topologyApr 28, 2026Significance 20/100Registry: unreviewed
Prior state unknown→disproved
resolves the noncommutative half of the Kirby-list problem
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 27, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #42.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 27, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
a subexponential good sequence is constructed; the polynomial-growth question remains open
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
geometry-topologyApr 27, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
What is the largest possible measure of a subset of a radius-$R$ disk in $\mathbb{R}^2$ containing no pair of points at a positive integer distance? A Poisson-Bessel kernel argument gives $M(R) \ll R^{1/2}$; with Sárközy's lower construction, $M(R) = R^{1/2 + o(1)}$.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 27, 2026Significance 11/100Registry: site confirmed
Prior state unknown→disproved
both proposed bounds fail
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 26, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #896.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
probability-statisticsApr 26, 2026Significance 20/100Registry: unreviewed
Prior state unknown→proved
Kac's walk on the rotation group, introduced by Hastings in 1970, is a central high-dimensional Markov chain in statistical physics and computational science. The paper proves it mixes in $n^2 \log n$ steps, the conjectured optimal rate, closing the gap left by a long line of successive improvements.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 25, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
VibeMathed records this result as “Erdős Problem #1138.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 25, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #888.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 25, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #38.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
analysisApr 25, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Two independent affirmative claims (Adriano's, posted first, and a GPT-5.5 Pro note); Erdős himself wrote in 1982 that the problem had been solved affirmatively long before, without a locatable reference
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 24, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #330.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
probability-statisticsApr 24, 2026Significance 5/100Registry: lean verified
Prior state unknown→proved
In the all-heads coin game a player starts with $n$ coins, each showing heads with
probability $p$; each round all remaining coins are flipped, the player must set aside at
least one head (losing if none shows), and wins once all coins are set aside. Determine
optimal strategies and the winning probability $w_{n,p}$. Resolved: for $p=\tfrac12$ every
strategy achieves $w_{n,1/2}=\tfrac12$; for $p>\tfrac12$ the sing…
●Source○Replay○Reproduced●Formal proof○Statement audit○External check○Expert review○Peer review
combinatoricsApr 23, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #1014.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 23, 2026Significance 12/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #202.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 23, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #1190.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 22, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #863.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
differential-equationsApr 21, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Sharp global and almost-everywhere convergence rates for periodic homogenization of viscous quadratic Hamilton-Jacobi equations, settling the sharpness question left open by the first-order theory.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
combinatoricsApr 21, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #603.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
analysisApr 21, 2026Significance 10/100Registry: unreviewed
Prior state unknown→disproved
Answered negatively in a preprint that also settles the p=2 case of problem #995; erdosproblems.com still lists the problem open
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
combinatoricsApr 21, 2026Significance 11/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #610.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
probability-statisticsApr 20, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
For $P_n(z) = \sum_{k=0}^n \varepsilon_k z^k$ with independent uniform signs, does the number $R_n$ of roots in $|z| \le 1$ satisfy $R_n/(n/2) \to 1$ almost surely? The manuscript proves the strong law with $R_n = n/2 + O_\omega(n^{149/150})$.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 20, 2026Significance 8/100Registry: lean checked
Prior state unknown→proved
Nathanson asked which subsets of $\mathbb{N}$ can occur as product intersection sets of a family of semigroup subsets, for arbitrary and for decreasing families (his Problems 10 and 11). Both are solved by complete classifications.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
analysisApr 20, 2026Significance 8/100Registry: unreviewed
Prior state unknown→proved
Near-optimal rather than optimal: the bounds match up to logarithmic-type factors.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 19, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #1195.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 16, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #741.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
combinatoricsApr 16, 2026Significance 10/100Registry: lean checked
Prior state unknown→disproved
Erdős asked whether every $n$-point set in Euclidean space whose pairwise distances are mutually at least 1 apart must have diameter at least $(1+o(1))n^2$. Disproved: an explicit high-dimensional construction beats the conjectured constant.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
theoretical-computer-scienceApr 16, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
For fixed $d$, can every $d$-dimensional feasible solution of the triangle-strengthened Max-Cut SDP be rounded in polynomial time with ratio strictly larger than $\alpha_{GW}$? A rounding achieving $\alpha_{GW} + 2^{-O(d)}$ answers yes.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
analysisApr 16, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
For a positive projection $P$ on a Dedekind complete Banach lattice whose largest central operator below $P$ is $\alpha\,\mathrm{id}$, Wickstead conjectured $\alpha$ must be $0$ or $1/n$ for some natural $n$, and proved the finite-dimensional case. The paper proves the conjecture in general and settles the representation problem for Banach lattice algebras as a consequence.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 16, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #1217.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 15, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Identifies the exponent as a variational sunflower-capacity constant, sharpening the Tang-Zhang bounds; the value of that constant itself remains open, as does site acceptance
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 15, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #858.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 14, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #258.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
analysisApr 9, 2026Significance 11/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #987.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
combinatoricsApr 9, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #1091.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 9, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
VibeMathed records this result as “Erdős Problem #1141.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
geometry-topologyApr 9, 2026Significance 10/100Registry: site confirmed
Prior state unknown→disproved
VibeMathed records this result as “Erdős Problem #960.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
analysisApr 9, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
VibeMathed records this result as “Erdős Problem #990.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
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.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
algorithms-optimizationApr 7, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
A polynomial-time algorithm for computing an optimal committee under any Thiele voting rule on the Voter Interval domain, resolving a ten-year-old open problem posed for Proportional Approval Voting by Elkind and Lackner and later extended to every Thiele rule.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 6, 2026Significance 10/100Registry: site confirmed
Prior state unknown→disproved
The question as posed was implicit in Davenport–Erdős (1951); the AI result settles Tenenbaum's open variant negatively
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
probability-statisticsApr 5, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
For the adjacent-transposition chain on $\mathfrak{S}_n$ with a regular parameter vector, Fill's spectral gap conjecture (recently resolved) leaves open the characterization of the equality cases. The paper settles them, constructing the additional eigenfunctions in the exceptional regime.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
algorithms-optimizationApr 4, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
For smooth convex-concave min-max problems, can anchored gradient descent-ascent be scheduled so that its exact last-iterate squared-gradient residual is $O(1/t)$, closing the gap left by the 2019 analysis?
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
algebraApr 4, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
Is every weakly quasi-complete Noetherian local ring quasi-complete? Asked by D. D. Anderson in 2014. The ring $A = k^p[[X, Y]][k]$ with $k = \mathbb{F}_p(u_1, u_2, \dots)$ is weakly quasi-complete but not quasi-complete.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 3, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #152.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
theoretical-computer-scienceApr 2, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
The Exact Matching problem asks whether a bipartite graph with edges colored red and blue admits a perfect matching with exactly $t$ red edges. Introduced by Papadimitriou and Yannakakis in 1982, it has been in randomized polynomial time since Mulmuley-Vazirani-Vazirani (1987) while membership in P stayed open for four decades. The paper claims a deterministic polynomial-time algorithm, replacing probabilistic amp…
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 1, 2026Significance 15/100Registry: lean verified
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).
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
analysisApr 1, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
an escape path dominating every power of |z| with bounded initial length is constructed, and universal positive-power lower bounds are ruled out; the broader variant remains open
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryApr 1, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #1202.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
probability-statisticsMar 31, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
A precise asymptotic formula for the number of $n \times 4t$ partial Hadamard matrices in the regimes $t/n^3 \to \infty$ and $t/n^3 \to \Theta$, reaching the cubic regime that previous approaches (de Launey-Levin and successors) could not.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryMar 31, 2026Significance 15/100Registry: lean checked
Prior state unknown→disproved
For $A \subseteq \mathbb{F}_p$ let $A^* = (A+A) \cup (AA)$. Sárközy conjectured that for all large primes, every set of size at least $c\sqrt{p}$ has $A^* = \mathbb{F}_p$-like covering behaviour. Disproved with an explicit construction from the classical cross-ratio orbit, together with the exact extremal value.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryMar 31, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #380.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryMar 31, 2026Significance 11/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #997.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
theoretical-computer-scienceMar 30, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Record bounds on the ratio; the exact approximability of Multiway Cut remains open.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryMar 30, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
VibeMathed records this result as “Erdős Problem #125.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryMar 26, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #369.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
geometry-topologyMar 25, 2026Significance 15/100Registry: lean checked
Prior state unknown→proved
A density-1 obstruction, not a full resolution: the conjecture that the hard window contains no lattice triangles remains open on a density-0 set.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
combinatoricsMar 25, 2026Significance 5/100Registry: lean verified
Prior state unknown→proved
For $D_3(m) = \vec{C}_m \square \vec{C}_m \square \vec{C}_m$, can the full arc set be partitioned into three directed Hamilton cycles for every integer $m \ge 3$?
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
combinatoricsMar 24, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Let $H(n)$ be the largest number of vertices in a hypergraph with no isolated vertices and no partition of size greater than $n$. With $k_1 = 1$ and $k_n = \lfloor n/2 \rfloor + k_{\lfloor n/2 \rfloor} + k_{\lceil n/2 \rceil}$, prove $H(n) \ge c\,k_n$ for some constant $c > 1$, already for $n = 15$, with a constructive algorithm.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
analysisMar 24, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #1153.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
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.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
algebraMar 21, 2026Significance 15/100Registry: expert verified
Prior state unknown→proved
Does the Hodge bundle $\Omega_g$ over the moduli stack of genus $g \ge 2$ curves contain any nontrivial sub-bundles? Posed by Dawei Chen around 2015; the answer is no.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
algebraMar 19, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Swinnerton-Dyer (1981) proved $R$-equivalence trivial on smooth cubic surfaces over $p$-adic fields with good reduction, except for three special types. The paper resolves two long-standing exceptional cases: triviality for the diagonal cubic over $\mathbb{Q}_3$, answering a question from Manin's Cubic Forms (1972), and the cubic with universal equivalence of exponent 2 (Kanevsky, 1982).
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryMar 16, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #1148.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
differential-equationsMar 16, 2026Significance 4/100Registry: lean verified
Prior state unknown→proved
Under smoothness, positivity, decay and score assumptions, are all steady solutions of the Coulomb Vlasov-Maxwell-Landau system on $\mathbb{T}^3 \times \mathbb{R}^3$ necessarily spatially uniform Maxwellians?
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
combinatoricsMar 12, 2026Significance 15/100Registry: unreviewed
Prior state unknown→proved
Exponent improvements toward Pach's conjecture, which remains open.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
geometry-topologyMar 11, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
A record lower bound; the kissing number in dimension 19 remains unknown.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
combinatoricsMar 10, 2026Significance 25/100Registry: unreviewed
Prior state unknown→proved
Records, not resolutions: the exact values of these Ramsey numbers remain unknown.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
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.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
geometry-topologyMar 8, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
A record, not an endpoint: whether fewer vertices suffice is posed as an open question in the same paper.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryMar 7, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #650.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryMar 2, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #457.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
combinatoricsMar 1, 2026Significance 8/100Registry: unreviewed
Prior state unknown→proved
In the Frankl-Pach-Erdős circle of VC-dimension problems, the first arXiv version of the paper posed the $k=3$ case of a witness construction question. ChatGPT 5.4 Pro answered it; the published construction generalizes the model's response, and the conversation transcript is public.
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number-theoryFeb 25, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
Erdős reported in 1975 that Spencer had shown existence but gave no reference; no proof was on record before the AI solution
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geometry-topologyFeb 25, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
VibeMathed records this result as “Erdős Problem #846.” The registry entry and named primary source contain the available statement and scope.
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algebraFeb 24, 2026Significance 5/100Registry: unreviewed
Prior state unknown→proved
Is the exact nonreal spectral region of the four-cycle family of row-stochastic nonnegative matrices determined by the Karpelevich constraint, as Ran and Teng conjectured in 2024?
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combinatoricsFeb 17, 2026Significance 12/100Registry: unreviewed
Prior state unknown→proved
Wellman and Pettie noted that the true leading constant for large-order Davenport-Schinzel sequences was known only to lie in an interval. The paper improves the Roselle-Stanton lower bound to match the pigeonhole upper bound in the leading term, resolving the constant to exactly 1/2.
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geometry-topologyFeb 13, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Question 8 of the First Proof experiment (Abouzaid et al.) asks whether a polyhedral Lagrangian surface with exactly four faces meeting at every vertex necessarily admits a Lagrangian smoothing. The research report assembles ChatGPT-suggested constructions into an affirmative argument for orientable surfaces in $(\mathbb{R}^4, \omega)$: smooth the edges, verify the vertex links are unknots with rot 0 and tb -1, an…
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mathematical-physicsFeb 12, 2026Significance 15/100Registry: unreviewed
Prior state unknown→disproved
nonzero on half-collinear complex kinematics, with a closed formula
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number-theoryFeb 5, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #851.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryFeb 4, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #347.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryFeb 4, 2026Significance 15/100Registry: lean checked
Prior state unknown→proved
In the circle of Kummer's regular primes and Vandiver's conjecture, the paper proves that almost all primes are partially regular, yielding a partial Vandiver theorem for a density-one set of primes, with consequences for Kubota-Leopoldt p-adic L-functions, Eisenstein congruences and K-theory torsion.
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algebraFeb 3, 2026Significance 10/100Registry: lean checked
Prior state unknown→proved
Does the conjectured universal formula for normalized alternating syzygy power sums of numerical semigroup rings hold for every index?
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geometry-topologyFeb 3, 2026Significance 10/100Registry: lean checked
Prior state unknown→proved
For odd $k$ with $\gcd(n,k) = \gcd(n+1,k) = 1$, is $N_k(n) \equiv \lfloor (k+1)/4 \rfloor \pmod 2$, where $N_k(n)$ counts pairs $1 \le b_i \le (k-1)/2$ with $b_1 + b_2 \ge (k+1)/2$ and $b_2 \equiv n b_1 \pmod k$? Conjectured by Chen and Gendron; its proof removes a conditional step in the genus-zero and genus-one spin-parity classification.
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geometry-topologyFeb 1, 2026Significance 10/100Registry: site confirmed
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #1089.” The registry entry and named primary source contain the available statement and scope.
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analysisFeb 1, 2026Significance 10/100Registry: expert verified
Prior state unknown→disproved
capacity alone does not determine the invariant; the zero-measure clause is a separate open question
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geometry-topologyFeb 1, 2026Significance 10/100Registry: expert verified
Prior state unknown→disproved
the strongest form is disproved via configurations where every point sees at most about 3n/4 distinct distances; the weaker improvement remains open
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number-theoryJan 29, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #1051.” The registry entry and named primary source contain the available statement and scope.
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combinatoricsJan 27, 2026Significance 8/100Registry: lean checked
Prior state unknown→proved
A generalization of Boppana's entropy inequality, of the kind used in union-closed-sets arguments, proved and formalized: the sharp form with the extremal constant characterized via the unique positive solution of an explicit equation.
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number-theoryJan 21, 2026Significance 10/100Registry: site confirmed
Prior state unknown→disproved
VibeMathed records this result as “Erdős Problem #543.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryJan 17, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #281.” The registry entry and named primary source contain the available statement and scope.
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probability-statisticsJan 14, 2026Significance 22/100Registry: unreviewed
Prior state unknown→proved
Fully resolves the posed coordinate-wise question; the main Courtade-Kumar conjecture itself remains open outside the extended high-noise range.
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geometry-topologyJan 13, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #659.” The registry entry and named primary source contain the available statement and scope.
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number-theoryJan 11, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #401.” The registry entry and named primary source contain the available statement and scope.
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number-theoryJan 10, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
VibeMathed records this result as “Erdős Problem #205.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryJan 10, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
VibeMathed records this result as “Erdős Problem #397.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryJan 10, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
VibeMathed records this result as “Erdős Problem #729.” The registry entry and named primary source contain the available statement and scope.
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number-theoryJan 6, 2026Significance 10/100Registry: lean verified
Prior state unknown→proved
Whether there are infinitely many integers $a, b, n$ with $a, b \ge \varepsilon n$ such that $a!\cdot b!$ divides $n!\cdot(a+b-n)!$ while $a+b$ exceeds $n$ by more than $C\cdot\log n$.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryJan 5, 2026Significance 10/100Registry: lean verified
Prior state unknown→disproved
VibeMathed records this result as “Erdős Problem #871.” The registry entry and named primary source contain the available statement and scope.
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combinatoricsJan 3, 2026Significance 10/100Registry: unreviewed
Prior state unknown→proved
Asymptotically optimal for powers of 2; the exact extremal answer for general n stays open.
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number-theoryDec 26, 2025Significance 10/100Registry: lean verified
Prior state unknown→disproved
VibeMathed records this result as “Erdős Problem #897.” The registry entry and named primary source contain the available statement and scope.
●Source○Replay○Reproduced○Formal proof○Statement audit○External check○Expert review○Peer review
number-theoryDec 25, 2025Significance 10/100Registry: lean verified
Prior state unknown→disproved
VibeMathed records this result as “Erdős Problem #333.” The registry entry and named primary source contain the available statement and scope.
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combinatoricsDec 16, 2025Significance 12/100Registry: unreviewed
Prior state unknown→proved
An improved lower bound on the maximal product; the sharp maximizer for the 1958 question remains unknown.
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combinatoricsDec 8, 2025Significance 10/100Registry: lean verified
Prior state unknown→proved
For a sequence of $n$ distinct reals, determine the largest constant $c$ such that some monotonic subsequence always has sum exceeding $(c-o(1))\cdot(1/\sqrt{n})$ times the total sum. Resolved as $c = 1$.
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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-statisticsNov 24, 2025Significance 10/100Registry: unreviewed
Prior state unknown→proved
What is the minimax optimal error rate for density estimation when observations are perturbed by Wasserstein-bounded contaminations? Chao and Dobriban's 2023 preprint left a gap between upper and lower bounds; the sharp rate is now derived, closing the problem.
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number-theoryNov 20, 2025Significance 10/100Registry: site confirmed
Prior state unknown→proved
Resolved for all sufficiently large N via a stability theorem; small N remain a finite computation (erdosproblems.com marks the problem DECIDABLE)
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analysisNov 19, 2025Significance 22/100Registry: unreviewed
Prior state unknown→proved
Charts the bounded-slope regime of the arithmetic Kakeya program; the conjecture itself remains open.
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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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differential-equationsOct 26, 2025Significance 8/100Registry: unreviewed
Prior state unknown→disproved
Curto et al. (Advances in Applied Mathematics, 2024) conjectured that every stable fixed point of a threshold-linear network is minimal. Disproved: an explicit competitive 3-neuron TLN has a stable fixed point whose support strictly contains another's, and 3 neurons is proven smallest possible.
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number-theoryOct 22, 2025Significance 20/100Registry: lean verified
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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number-theorySep 30, 2025Significance 6/100Registry: unreviewed
Prior state unknown→proved
22 conjectures of Cohen about cyclic numbers (integers with $\gcd(n, \varphi(n)) = 1$) settled at once - 16 proved, 6 disproved - together with a complete resolution of a related OEIS problem on sequences whose running averages are Fibonacci numbers (Fried's Conjecture 2).
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combinatoricsAug 30, 2025Significance 20/100Registry: unreviewed
Prior state unknown→disproved
An $\ell$-Oddtown is a family of subsets of an $n$-element set whose set sizes are not divisible by $\ell$ while all pairwise intersection sizes are. Berlekamp and Graver showed the maximum size is $n$ for prime $\ell$, Babai and Frankl extended this to prime powers and asked whether $n$ still holds for other moduli, a question open even for $\ell = 6$. Bukh, Chao and Zheng answer it negatively with an explicit su…
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geometry-topologyJul 20, 2025Significance 10/100Registry: unreviewed
Prior state unknown→proved
A paper torus is an embedded polyhedral torus isometric to a flat torus. Schwartz proves no paper torus with 7 vertices exists and constructs one with 8, settling the minimum-vertex question in the flat-torus embedding tradition of Császár-torus combinatorics and the Lazarus-Tallerie universal triangulation.
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