1033 matching entries · Page 11 of 21
For f(x,y)=(x−y)²+Mx with a fixed suitable M, there are arbitrarily large finite real sets A with |f(A,A)|≤|A|^(2−c) for some c>0, although f has neither additive nor multiplicative special form.
For every n≥1, the number of nonnesting permutations of {1,1,…,n,n} avoiding both 1132 and 3312, under the published containment convention, equals 3ⁿ−3·2ⁿ⁻¹+1.
For every transcendental radius r>2, a seven-point set on a circle of radius r, including r=π, has a finite coloring of Euclidean space with no monochromatic congruent copy in any dimension.
For the nearest-neighbor isotropic quantum Heisenberg ferromagnet on ℤᵈ with d≥3 and every positive half-integer spin S, a translation-invariant zero-field KMS equilibrium state exists at sufficiently low positive temperature with magnetization at least S/4.
There are full-dimensional 0/1-polytopes of dimension 4n+1 whose graph edge expansion decreases exponentially with dimension.
For every feasible set of row and column margins on an m×n binary matrix, with m,n≥2, the lazy 2×2 swap chain has spectral gap at least 1/[binom(m,2)binom(n,2)].
For each fixed q>4, critical finite Fortuin–Kasteleyn planar maps with n edges, distances scaled by a q-dependent constant times n⁻¹ᐟ², and normalized degree measure converge in distribution to the Brownian continuum random tree in Gromov–Hausdorff–Prokhorov topology.
An explicit independent set in the 213rd strong power of the 11-cycle gives Shannon capacity C₁₁ ≥ 5.295526013632343.
For iid Bernoulli(p) bits with 0<p≤1/2 and a full-row-rank k×n binary linear encoder H, its optimal bitwise-MAP distortion D(H) obeys k/n≥h(p)(1−D(H)/p). A nonasymptotic entropy bound for every affine subspace drives the converse.
For Z={α>1: some nonzero real ξ makes every floor ⌊ξαⁿ⌋ even for n≥1}, Stephan constructs a witness showing √3∈Z and proves that among integer square roots √m with m≥2, only √2 lies outside Z.
Every finite connected Eulerian multigraph of minimum degree at least four has, for a fixed Euler tour, an edge partition into connected 2-regular circuits compatible with that tour’s transitions.
KEMENY SCORE is NP-complete for exactly three unweighted complete rankings. Peters reduces from SIMPLE MAX CUT with long even edges to a unit-margin majority tournament realizable by three voters.
For an absolutely almost simple simply connected group G over a number field k, every noncentral abstract normal subgroup of G(k) is the inverse image of an open normal subgroup of the product of G(kᵥ) over anisotropic nonarchimedean places.
Liu constructs a smooth polarized complex projective fivefold that is K-polystable for all normal ample algebraic test configurations of every positive exponent, yet has no constant-scalar-curvature Kähler metric.
For any centered Gaussian vector, including singular covariance, and positive real exponents αᵢ, the expected product of |Xᵢ|^αᵢ is at least the product of the corresponding marginal expectations.
For n≥k+1, every simple graph with at least (k−1)n/2+1 edges contains every tree on k+1 vertices. A separate counting lemma also reaches the classical strict threshold when (k−1)n is odd.
For every sufficiently large genus g, Cai constructs a closed hyperbolic surface with systole at least log g−12 log log g. The lower limit of the largest possible systole divided by log g is therefore at least one.
Huang, Liu and Liu give a finite two-receiver discrete memoryless broadcast channel T for which Marton’s two-use sum-rate quantity M(T⊗2) exceeds twice its one-use value M(T), with a certified gap above 1.8821×10⁻⁶ nats. Its one-letter Marton region is therefore strictly smaller than capacity.
For odd n≥3, Lu and Yang obtain a global Lipschitz family of exact hyperplane-section isometries after a positive-degree bundle pullback. Combined with their degree argument, this proves that a real Banach space whose n-dimensional subspaces are all linearly isometric is Hilbert when 2≤n<dim X.
Bostanci and coauthors construct a unitary quantum oracle U for which QMA(2)^U differs from QMA^U. Their oracle problem has a one-query QMA(2) protocol with linear-size unentangled proofs, while QMA needs exponential resources in query count or proof size.
For every fixed positive diameter k, the maximum number nₖ(d) of vertices in a graph of maximum degree at most d and diameter at most k satisfies nₖ(d)/dᵏ→1 as d grows. Halved-flag graphs supply the construction.
For the three-dimensional liquid-drop energy given by De Giorgi perimeter plus half the Coulomb self-interaction, balls uniquely minimize at every fixed volume 0<V≤5(2−2^(2/3))/(2^(2/3)−1)≈3.51, up to translation and null sets. Above that threshold, no minimizer exists.
For every complex polynomial of degree at least two with all zeros in the closed unit disk, each zero lies within distance one of a zero of its derivative, including repeated and boundary zeros.
For every complete first-order theory, SOP₂ implies SOP₃. The reverse implication was already known, so the two dividing lines coincide.
On the Boolean cube {−1,1}^n with uniform measure, for every fixed 0<a<1, n≥1 and u>1, convolution by the biased-coin product measure μ_a satisfies sup_{f≥0, ||f||₁=1} u·λ({T_{μ_a}f≥u})≤C_a/√(log u), with an a-dependent constant. The bound is dimension free.
For n≥r≥2, t≥1 and any n-vertex r-uniform hypergraph H containing no tight r-tree T with t edges, |E(H)|≤(t−1)|∂H|/r≤(t−1)binom(n,r−1)/r, where ∂H is the (r−1)-shadow. The bound is attained for infinitely many n.
Claude computed the nine-loop six-particle MHV amplitude in planar N=4 super Yang–Mills using established bootstrap methods. The released symbol uses quintuple coproducts modulo two 31-bit primes; most, but not all, nonzero coordinates are reconstructed as certified rationals. A separate direct bootstrap agrees on compared coefficients, and a nine-loop function is reported.
ChatGPT 5.6 suggested the two-way trace graph TG_{d,h} of Remark 2.2 on projective points over F_(2^h). Ihringer and Mattheus bound its clique size by binom(d−2,ceil((d−2)/2)) and independence number by (d^h+1)². Their stronger one-way ordered graph and Theorem 1.3 are human-developed refinements.
In ℤ/13ℤ, A={0,1,3,4,5,6,10} and B={0,1,4,6} have |A|+|B|=11≤12. Their restricted sumset is {1,3,4,5,6,7,9,10,11}, of size 9=|A|+|B|−2, but neither A nor B is an arithmetic progression. This contradicts the forward conclusion of Li–Liang Theorem A/3.1 in arXiv:2609.10148v2.
For even N=2m≥2, pairwise distinct real numbers x₁,…,x₂ₘ and odd D=2r+1≥N−1, Ma proves that the matrix [(xⱼ−xᵢ)^D] is nonsingular. After sorting the nodes, the proof gives the stronger sign (−1)^(m choose 2) Pf[((xⱼ−xᵢ)^D)]>0.
A group built as EL₃(LC(X,F₂)⋊ℤ), with X a binary Toeplitz subshift, is infinite and simple, has Kazhdan property (T), and is locally embeddable into finite groups. It is therefore sofic and hyperlinear. The Lean project compares two existential statements using this same witness.
Kegel, Li and Ren give an effective family of finite presentations with two generators and ten relators for which deciding triviality is impossible. The count is an achieved presentation size, not a minimum.
Kegel, Li and Ren give an effective family of finite presentations with four generators and eleven relators for which deciding triviality is impossible. These are fixed presentation slots, not a minimality result.
Kegel, Li and Ren give a finite presentation with three generators and nine relators whose word problem is unsolvable. The count is an achieved presentation size, not a minimum.
Every family of 14-subsets of a 24-point set that covers every 4-subset has at least 20 blocks, so C(24,14,4)≥20. The Schönheim point-derivation bound then gives C(25,15,5)≥34.
For independent nonnegative random variables X₁,…,Xₙ with E[Xᵢ]≤1 and S=ΣXᵢ, Fu, Han, Wang, Yan, Zhang and Zhou prove P(S<E[S]+δ)≥δ(n/(n+δ))ⁿ for 0<δ<1, and ≥(1−1/(n+δ))ⁿ for δ≥1. The second finite-n bound is sharp and implies Feige’s unit-slack e⁻¹ conjecture.
For the three-dimensional space-homogeneous Boltzmann equation with collision kernel B=|v−v*|^γ/(4π), 0≤γ≤1, Silvestre constructs smooth positive radial two-Maxwellian initial densities f_R=(1−p)M₁+pM_R. For 0<p<p_γ and sufficiently large R, the entropy production D initially increases: dD(f_R)/dt>0 at t=0, with a positive leading term proportional to R^γ log R. The range includes constant-angle Maxwell molecules and hard spheres.
For m≥2 labeled supplies on [0,1], independent uniform demands and independent uniform replacement after each online match, the project proves a policy whose one-period root-mean-square cost, from every fixed initial inventory, has limsup at most (72000/log 2)·log(m+1)/m. After a scheduled m-period replacement phase, expected average cost including that phase obeys the same bound for every horizon N≥2m².
For every positive dimension n, the Euclidean kissing number τₙ is an attained maximum and satisfies τₙ≤(√6)ⁿ. A planar hexagon gives τ₂=6, so √6 is the least base in a bound τₙ≤αⁿ valid in every dimension without a prefactor.
For five, six and seven freely rotated, interior-disjoint unit squares in a closed disk, the project reports exact least radii √(5/2), √q* and √13/2, respectively, where q* is its specified quartic root. It classifies all optimal packings up to rotation and relabelling: the five- and six-square optima are unique, while seven squares admit a three-parameter family.
For every real Hilbert space H, including nonseparable spaces, the one-dimensional Besicovitch rectifiability threshold σ₁(H) is at most 0.6934. In particular, a planar Borel set of finite H¹ measure is countably one-rectifiable when its lower one-density exceeds any fixed threshold greater than 0.6934 almost everywhere.
For every connected finite loopless multigraph of genus g≤6 and integers d,r≥0 with ρ=g−(r+1)(g−d+r)≥0, a degree-d divisor of Baker–Norine rank at least r exists. The new case is genus six, proved by descending special divisors from odd subdivisions to the original graph.
Stempin, Llorens and Huber refute the proposed universal bound χ_f(B_ε(ρ)) = O(ε⁻²). Their Theorem A constructs states and Pauli sets from lexicographic powers of the complement of a seven-cycle for which χ_f(B_ε(ρ)) grows faster. Theorem B gives a broader counterexample criterion using anticommutation graphs with α(G) < β(G).
Yoshito Ishiki constructs real Banach spaces X and Y and a bijection U from X to Y that preserves distances between every pair of points in each closed ball of radius 1/4, but is not a global isometry. This answers the unrestricted Scottish Book Problem 155 negatively.
A deterministic exact algorithm counts linear extensions of any n-element poset in O*(1.89^n) time.
Every finite planar point set of at least 10^11055931 points has four collinear points or six pairwise visible points.
Freitag and Mutchnik give a stable theory of an infinite-dimensional vector space over the division ring of fractions of the quantum graph algebra of the random graph, refuting stable forking.
For every natural k≥0, the convergent series Σₙ≥1 σₖ(n)/n! is irrational, where σₖ(n) is the sum of kth powers of positive divisors of n. The k>0 cases answer Erdős Problem 252; the Lean theorem also handles k=0.
Every compact connected planar set that can be moved continuously through both a right-turning and left-turning unit-width hallway from the same starting orientation has area at most M=1+4Y²+arctan(Y), where Y>0 solves 4Y³+3Y=1. Equality holds exactly for rotated and translated copies of Romik’s sofa, and the sofa attains M.
Every set of six distinct points on a circle satisfies the FieldCriterion tensor condition of the OpenAI Euclidean Ramsey classification. More generally, every spherical finite set of at most m+4 points, where m is its affine-span dimension, satisfies that criterion in its own coordinate field; every spherical set with algebraic coordinates also satisfies it.
These entries have recorded screening decisions and are not counted as curated contributions. Importance, evidence, interest, and date filters apply only to curated contributions.
View all matching screening decisionsThese dates show source checks, not the latest research. Failed checks can leave gaps.
arXiv · algebra-geometry — last successful metadata check: 21/10/2026, 07:20:46 UTC
arXiv · algorithms-formal — last successful metadata check: 21/10/2026, 07:19:48 UTC
arXiv · analysis — last successful metadata check: 21/10/2026, 07:21:51 UTC
arXiv · combinatorics — last successful metadata check: 21/10/2026, 07:18:49 UTC
arXiv · geometry-topology — last successful metadata check: 21/10/2026, 07:22:23 UTC
arXiv · logic-foundations — last successful metadata check: 21/10/2026, 07:23:04 UTC
arXiv · optimization-information — last successful metadata check: 21/10/2026, 07:23:10 UTC
arXiv · probability-dynamics-physics — last successful metadata check: 21/10/2026, 07:22:36 UTC
arXiv · theoretical-physics — last successful metadata check: 21/10/2026, 07:22:50 UTC
Allow optional Google Analytics to help improve AIProofWatch? Privacy details.
