1033 matching entries · Page 4 of 21
For p=3 and every prime q≠2, no finite or infinite group G satisfies Glauberman–Norton Hypothesis (B): it would need an embedding of H=(additive F_{3^q})⋊(norm-one units), a finite abelian subgroup Q of order prime to 3 and y∈Q, with the embedded prime-field line normalizing Q and its y-conjugate normalizing the embedded U.
For a quadratic real root α>1 of X²−aX−b whose conjugate has modulus below one, the Route A exponent condition A(α)<1 gives an open interval missed by every orbit (ξαⁿ mod 1) with ξ in the paper’s class C(α), so none is uniformly distributed. The criterion is equivalent to (log₂α−1)(log₂(α/|b|)−1)>1 and yields an explicit gap for α=2+√5.
Among real quadratic Pisot numbers α, the golden ratio is the only one in Dubickas’s set S: every other α admits a nonzero ξ for which all floors ⌊ξαⁿ⌋ are even for n≥1. For α=1+√2, ξ=(2−√2)/4 gives the exact Pell identity ⌊ξ(1+√2)⁽ⁿ⁺¹⁾⌋=Pₙ−[n odd].
For every nonzero real ξ, the fractional parts of ξ(3/2)ⁿ cannot all lie in the closed interval [2/7,5/7]. Equivalently, ||ξ(3/2)ⁿ||<2/7 infinitely often. The certificate has depth 15 and 166 blocks across 32 rank strata.
For positive ξ, the fractional parts of ξ(3/2)ⁿ cannot all lie in [1/6,13/24), a specific single interval of length 3/8. A finite block certificate and a general soundness argument establish this exclusion.
For positive coprime unequal integers a and b, an AlphaProof Nexus Lean proof shows that Sharipov’s degree-eight cuboid polynomial Pₐ,ᵦ(t) is irreducible over the integers. Its route uses quartic factor analysis and an infinite descent for x⁴+34x²y²+y⁴=z².
An eleven-block construction yields α∞≤liminf log D(N)/log N for the largest square-difference-free subset D(N) of {1,…,N}. Every exponent ρ<α∞ holds eventually, and the formal proof certifies α∞>0.7537.
Lean proofs establish Dimitrov’s Conjectures 1 and 3, the stated rational lower bounds involving Euler’s totient φ(n), Dedekind’s ψ(n) and σ(n), for every integer k≥1 and n≥2. Equality in each bound holds exactly when n is prime.
Sauers constructs a separable unital C*-algebra with compact nonempty trace space whose uniform tracial completion has a trace discontinuous in the uniform 2-norm, and a factorial tracially complete pair with the same failure.
Hegarty, Ognissanti and Wedin give convergent heterogeneous Hegselmann–Krause trajectories in the synchronized bounded-confidence and bounded-influence models whose proximity digraphs never stabilize, plus constant-digraph trajectories that fail eventual pseudostability. These counterexamples refute Mirtabatabaei and Bullo Conjectures 2.2 and 2.3 and Theorem 6.4(iv).
Let K be a field of characteristic two with |K|=2ⁿ, gcd(k,n)=1 and dₖ=4ᵏ−2ᵏ+1. If Δ is the image of b↦(b+1)^dₖ+b^dₖ+1, then for distinct nonzero v₁,v₂∈K exactly 2^(2n−3) triples (x,y,z)∈Δ³ satisfy v₁x+v₂y+(v₁+v₂)z=0.
Claude agents devised an integer proof route for the s=1, index-two clauses of Theorem D in Jones’s Hadamard border note, replacing its rank and positive-semidefiniteness route. The formal development checks those Theorem D clauses.
For k≥3 and q≥k−1, let m=2kq+1. The numerical semigroup minimally generated by m,m+1,m+k,m+2k−1,m+2k has toric-ideal generator count 4k+2. The formal development also gives a nonredundant presentation.
For nonconstant g∈Q[t] of degree at most four, no u,v∈Q[t] satisfy u(t)v(t)(u(t)+v(t))=g(t)³+g(t)²+3g(t)−1. This excludes only those low-degree polynomial families.
For every n>0 and every set of n² lattice points, the induced square-grid graph has at most as many spanning trees as the n×n square. Equality holds exactly for translates of that square.
For H≤G of finite index n>1, Palcoux’s manuscript and formal development give a finite rational upper envelope for the number of intermediate subgroups and a strict bound c(p)n^((logₚn)/4), where p is the least prime divisor of n. Selected results also address prime-power and elementary-abelian sharpness.
For diag(1,1,1,−1,1,−1) and the local Hodge operator on coordinate plane 0123, fifteen orthogonal adjoint actions together with that operator generate sl(15,K) over every field of characteristic other than two. The same formal result gives dimension 224 and scalar rigidity for commuting endomorphisms, with determinant c²²⁴.
Birkbeck and Torzewski give uniform, strongly sheafy, nonnoetherian Tate domains that are not stably uniform, answering Kedlaya’s Question 7. One finite-jet example has a nonreduced rational localization; a weighted-parity example has a nonuniform localization that remains a domain.
For nonzero integers r,s with at least one odd, the word xʳyˢ is universal in Sₙ once n is sufficiently large and at least C log m(r,s), where m is the product of the distinct primes dividing rs. A separate construction shows the logarithmic order cannot generally be reduced.
For every nonnegative rational n×n matrix, Anari’s formal development constructs one rational-output algorithm and an absolute constant c<√2 with a two-sided cⁿ approximation guarantee for the permanent. It also proves deterministic polynomial time in the ordinary binary input size.
For each integer r≥3, Dai and coauthors construct an infinite r-differential poset that agrees with Yʳ through rank three but has rank-four size |(Yʳ)₄|−⌊r/3⌋. At r=3 the first sizes are 1, 3, 9, 22, 50, compared with 51 for Y³.
Long gives explicit rational-coefficient polynomials in four variables defining a Poisson endomorphism of the canonical algebra P₂ in characteristic zero that is not an automorphism. An exact three-point fiber proves noninjectivity; extending the map by identity gives examples for every Pₙ with n≥2.
For every non-eventually-periodic sequence over a finite alphabet, the project constructs an infinite labelled population satisfying Alexander’s finite-root, finite-outdegree, finite real-birthdate-sublevel and incoming-label conditions but with no matching infinite path from any start. With Alexander’s earlier positive theorem, biological unavoidability is equivalent to eventual periodicity.
In the paper’s small-ε regime, a randomized adaptive method using one independent fresh sample per query needs Ω(ΔL/ε² + ΔLσ²/ε⁴) queries to reach expected gradient norm at most ε for smooth nonconvex objectives with gap Δ, L-Lipschitz gradient and uniformly bounded, unbiased fixed-law gradient noise.
Zhou constructs two nonisomorphic countable ICC property-(T) groups whose group von Neumann algebras are isomorphic. A quadratic fiber shear identifies the dual actions, while the groups have distinct module structures.
For every order-six complex Hadamard matrix outside Tao’s cubic equivalence class, the normalized character sum vanishes at every permutation of (1,1,1,−1,−1,−1), allowing row and column permutations and arbitrary unit phases.
For x′=y−F(x), y′=−x and every real polynomial F of degree at most five, there are at most two geometrically distinct isolated periodic orbits, including degenerate cycles. Some such system has exactly two.
Assuming the randomized rational-weighted exact-volume Small-Set Expansion Hypothesis, no randomized polynomial-time algorithm can guarantee both additive ε accuracy in rational sparse least squares and support O(k κₛ₊ₖ^(1−γ)) for any fixed γ∈(0,1]. The hard matrices have full column rank and at most two nonzeros per row.
For every real α, the sequence of fractional parts of α times consecutive primes is not well-distributed in the uniform-in-starting-index interval sense. This does not say ordinary equidistribution fails.
An explicit basis A of order two has the property that every partition A=A₁⊔A₂ leaves at least one of A₁+A₁ or A₂+A₂ with arbitrarily long gaps. The construction uses powers-of-five blocks and special points that force missing intervals.
For every sufficiently large dimension n, an OpenAI manuscript constructs a centrally symmetric convex body whose unrestricted translative covering density exceeds c n log n for an absolute c>0. With Rogers’ O(n log n) upper bound, this fixes the worst-case unrestricted order and rules out a universal O(n) bound.
A connected cubic triangle-free graph on 14 vertices has domination number γ=4 and zero-forcing number Z=7. Since it is diamond-free, Z exceeds Davila’s proposed γ+2 bound.
For planar graphs at sufficiently small fixed hard-core activity, including λ≤1/1443, Guo and Zhang give a randomized polynomial-time approximation scheme for the partition function without a maximum-degree bound.
GP(9,2) has a non-real-rooted independence polynomial although k is even. GP(7,3) and GP(3,1) are real-rooted although k is odd. These examples refute both directions of Pandey’s claim for generalized Petersen graphs.
For every smooth integral complex projective surface S with ample L, the manuscript claims that all sufficiently large r have maximal multipoint Seshadri constant ε(S,L;p)=√(L²/r) at very general distinct r-tuples, with a nef boundary divisor on the blowup.
From a property-(T) group Γ with an injective non-surjective endomorphism, Eckhardt forms its ascending HNN extension G and generalized wreath product W=(⊕_(G/Γ) Z/2Z)⋊G. His Theorem 4.3 gives a specified nonidentity element killed by every homomorphism of W to a matrix-ultraproduct quotient, showing W is not MF.
Liu and Wang construct a 19-dimensional smooth complex projective X with B=0, nef Cartier ξ, K_X linearly equivalent to −16ξ and nef part M_X descended to 17ξ. Its generalized log-canonical ring R(X,ξ), and consequently R(X,−K_X), is not finitely generated.
The manuscript claims an absolute integer C such that every finite nonempty simple graph G has list chromatic number at most C times its largest clique-minor order h(G). The constant is not optimized.
For the relative SU(2) BFSS Hamiltonian defined by closing its supercharge quadratic form, the manuscript claims an orthonormal sequence of square-integrable gauge-invariant eigenvectors with positive eigenvalues tending to infinity.
The manuscript claims arbitrarily large finite graphs G with independence number at most two and μ(G)+1<|V(G)|/2≤χ(G), contradicting the proposed inequality χ(G)≤μ(G)+1.
For each dimension d and t>0, the manuscript claims a bound N(d,t) on [k_S:k] for a boundary prime S with coefficient at least t on an ordinary projective log-canonical Q-pair (X,B) over a characteristic-zero field, where X is normal integral, H⁰(X,O_X)=k, B is effective and K_X+B is Q-linearly trivial.
For projective generalized log-canonical Q-pairs over an algebraically closed characteristic-zero field with a fixed nef rational b-divisor, the manuscript claims a birational model that extracts no divisors and preserves discrepancy inequalities, with nef adjoint when pseudo-effective, or a relative-Picard-number-one Mori fibre space otherwise.
For fixed local dimension q, interaction bound J and global spectral gap Δ>0, the manuscript claims a PEPS with bond dimension at most CLᶜ and normalized vector error at most L⁻¹ for the unique ground state of any on-site plus nearest-neighbour Hamiltonian on an open L×L square grid.
For each r≥4, the r-extension of the four-cycle has product exponent X(F)=r but admits Ω(n^(r−1/2)) F-free configurations, exceeding the conjectured exponent r−1. The AI-origin contribution is identifying this counterexample family.
For increasing positive integers aₙ with liminf aₙ^(1/2ⁿ)>1, Aletheia’s accepted proof shows that Σₙ 1/(aₙaₙ₊₁) is irrational. This is the original Erdős #1051 statement.
For the least possible largest distinct factor f(n) in n! = a₁⋯aₖ with n<a₁<⋯<aₖ, Wang’s manuscript claims that (f(n)−2n)log(n)/n tends to 4029639598/25970038185. Its new step is a matching upper-bound construction.
Every oriented graph on n=2δ+2 vertices with minimum outdegree δ has a vertex whose exact second outneighborhood is at least as large as its first. With Fisher’s tournament theorem, this covers n≤2δ+2, without proving Seymour’s conjecture for all oriented graphs.
The proper hat-guessing number of K₈−e is 14. A PGL(2,13)-equivariant fourteen-colour strategy uses 990 normalized twin decisions and a 48,510-edge residual-orbit matching; exhaustive checking covers all 138,378,240 proper colourings, and a general upper bound gives equality.
For sufficiently large N, Sawhney bounds the size of A⊂[N] with ab+1 nonsquarefree for every a,b∈A by the size of the residue class 7 mod 25, and identifies the extremal 7 and possibly 18 mod 25 classes. The model-origin part is a squarefree-density sieve with prime exclusions depending on b and the progression difference.
The OpenAI manuscript claims a domain in some sufficiently large fixed finite complex dimension, diffeomorphic to real Euclidean space, with a complete Kähler metric of real sectional curvature at most −1 and only constant bounded holomorphic functions. The curvature is not bounded below.
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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
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