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639 source-grounded records. Verification labels remain separate from source authentication and publication status.
Imported from VibeMathed under CC BY 4.0. Each record links to its registry entry and named primary source. Registry verification is preserved verbatim.
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.
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 $n$.
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.
Kim and Roush did not claim uniqueness; the classification of equality cases is new alongside the conjecture itself.
the conjectured cutoff location of 2n log n is wrong
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.
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…
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.
information-theoretic; a polynomial-time implementation remains open
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
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.
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?
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.
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.
a record upper bound; the exact constant remains unknown
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{…
Harris had settled the generalized problem in dimension four; this reaches dimension three
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…
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…
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…
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.
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.
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.
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.
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.
Verified by author of the conjecture
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.
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.
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.
Han and Jiang asked whether being of klt type is an open condition in flat families of varieties. It is not.
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.
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.
the true order is determined, and it is not the conjectured one
New arXiv preprint with an author-provided Lean formalization; not yet peer-reviewed.
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…
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.