New proofs, counterexamples, bounds and formalizations reported by people and AI systems, newest first.
Constructs a compact subset of ℝ9 that cannot be covered by ten sets of strictly smaller diameter, disproving Borsuk's covering assertion already in dimension nine. The example consists of rank-one orthogonal projectors onto lines in ℝ4, with the Frobenius metric.
Constructs a finite translational tile in ℤ3 that tiles space but admits no fully periodic tiling, disproving the periodic tiling conjecture in the smallest possible lattice dimension. Its unit-cube thickening gives the same counterexample in ℝ3, even with arbitrary real translation vectors.
Constructs a noninvertible C1 self-map of a compact smooth manifold with zero topological entropy but eigenvalue 2 on second homology. This disproves the homological entropy lower bound for general C1 self-maps: homological growth need not force positive orbit complexity.
Constructs total Boolean functions with block sensitivity for a fixed α > 2, disproving the quadratic strengthening of the Sensitivity Conjecture. Here counts influential individual-bit flips, while block sensitivity allows disjoint groups of bits to change together.
Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant. Approximating the optimum within any fixed additive constant is also NP-hard, even when every item exceeds 1/6 and each bin holds at most five items.
Multiplies two n-bit integers exactly at every input length in deterministic worst-case time , with , on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen optimality conjecture in the ordinary multitape bit model.
Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area. This disproves Bang's half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three.
Every infinite-dimensional real Banach space contains a compact doubling set that admits no bi-Lipschitz embedding into any finite-dimensional normed space. The doubling constant is universal. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.
Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows. The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real coefficients used. This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral.
Proves Petty's projection-volume conjecture in the remaining dimensions n ≥ 4: ellipsoids uniquely minimize projection-body volume at fixed body volume. Also establishes the full Lutwak–Petty projection inequalities. In contrast, products of simplices exceed Brannen's proposed simplex maximum for normalized projection-body volume by an exponential factor in every sufficiently large dimension.
Constructs polynomials with N consecutive coefficients in whose modulus is uniformly on the entire unit circle, for every sufficiently large integer length N. Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture.
Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, is area-integrable for . The sharp universal integral-means identity is for t ≤ −2. A strict bound for bounded univalent functions disproves Kraetzer's prediction at that parameter.
Disproves Zariski's multiplicity conjecture by constructing reduced holomorphic hypersurface germs that are ambiently homeomorphic but have different multiplicities. The examples include hypersurfaces in ℂ4 with isolated critical points and multiplicities four and five.
For every sufficiently large admissible Hassett discriminant, a very general smooth complex cubic fourfold is irrational despite having both an untwisted geometric K3 category and an integral Hodge-theoretic K3 association. This disproves Kuznetsov's rationality conjecture and the sufficiency of the associated-K3 criterion for rationality; the discriminant threshold is ineffective.
Constructs a tautological relation on a moduli space of stable pointed curves that vanishes in rational Chow, hence in rational cohomology, but lies outside Pixton's original relation span. This disproves the Chow and rational-cohomological forms of his original completeness conjecture.
Constructs ample rank-two bundles on with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.
Constructs a polynomial in four complex variables that is not a coordinate but becomes one after adjoining a single variable, disproving the Stable Coordinate conjecture in four variables. Every fiber is affine three-space, yet none of its embeddings is rectifiable, also disproving the Abhyankar–Sathaye conjecture even when all fibers are affine spaces.
Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.
Constructs an integral complex affine fourfold with , disproving affine-space cancellation over ℂ in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections and have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial.
Constructs a smooth projective complex fourfold with large fundamental group whose universal cover contains no positive-dimensional compact analytic subvariety but is neither Stein nor holomorphically convex. A separate smooth projective complex surface already disproves unrestricted Shafarevich holomorphic convexity in dimension two.
A six-task counterexample is reported; exact periods and the reduction rule must be taken from the original witness before publishing.
arXiv: A Counterexample to Talagrand’s Operator Cotype Problem
GitHub: tadamcz/mean-value-problem, the Lean 4 disproof
arXiv: A Smooth Counterexample to the Trautman Conjecture
arXiv: a counterexample to the stable forking conjecture
arXiv: Three omitted values and non-Blaschke point divisors in half-planes
arXiv: a counterexample to the bounded mass property
arXiv: Disproof of the Yau–Tian–Donaldson conjecture
Counterexample to the smooth Carathéodory conjecture on umbilic points
Noncircular counterexamples are reported, including a real-analytic Jordan domain from computer-assisted existence estimates and a separate bifurcation construction.
An explicit map in C^3 with det JF=-2 and three different points having the same image; padding gives higher-dimensional counterexamples.
mathlib4 PR #41748: a finite free group scheme of order four not killed by four
A fixed δ>0 and infinitely many n with ν(n)≥n^(1+δ), disproving that asymptotic conjecture.
Case 3.1 in the Gemini Deep Think research report (counterexample).
Nine-vertex Brandt counterexample; proofs for WoW 698 and Graffiti 39/40; independent refutations for Graffiti 143/154 with explicit earlier-priority qualification.