Tagged lean-formalized
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Paper Annular variation and dyadic absolute bounds for the triangular Hilbert transform
Proves maximal and annular r-variation bounds, for every r > 2, from complex L^3(\mathbb R^2)\times L^3(\mathbb R^2) to L^{3/2}(\mathbb R^2). The maximal estimate controls both hard truncation endpoints and…
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Paper Correspondence coloring with a fixed forbidden subgraph
Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph F need OF(\Delta/\log\Delta) colors when their maximum degree Δ is sufficiently large. Also…
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Paper The ℓ¹-Bass conjecture for all discrete groups
Proves the ℓ1-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over \ell^1(G) are supported on finitely many finite-order conjugacy classes. The algebraic companion proves…
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Paper Nagata’s conjecture and maximal Seshadri constants
Proves Nagata's strict inequality \sumi mi\lt d\sqrt r for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities mi. It also proves maximal…
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Paper Quasipolynomial algorithms for mean-payoff, stochastic and parity games
Gives deterministic algorithms using 2^{O((\log(L+2))^2)} bit operations, for complete binary input length L, for ordinary mean-payoff games and two separate extensions. They compute exact values and optimal…
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Paper Directional zero–one laws beyond iid environments and iid ballisticity
On ℤd, d ≥ 3, directional escape has probability zero or one for iid strictly elliptic nearest-neighbor environments, and for stationary ergodic finite-range-dependent environments under uniform ellipticity. In iid…
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Paper Amenability, unitarizability, and strong Ulam stability
Resolves Dixmier's problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation. For countable discrete groups, amenability…
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Paper Sharp finite-matrix Lieb–Thirring inequalities and all equality cases
Proves the sharp one-dimensional Lieb–Thirring inequality for 1/2\lt \gamma\lt 3/2 and arbitrary finite-matrix potentials W ≥ 0 with \int\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty: the optimal…
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Paper Uniform Laughlin gap and stability under bounded scalar disorder
Proves the fermionic Laughlin spectral-gap conjecture for the full V1 interaction at filling 1/3 on the round sphere. The unique ground state remains uniformly gapped under sufficiently weak bounded real scalar one-body…
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Paper Counterexamples to stable-Morse and strong Arnold fixed-point bounds
Disproves stable-Morse lower bounds for nondegenerate Hamiltonian fixed points: on simply connected closed Kähler manifolds of real dimension 22, the deficit below the stable Morse number is unbounded. A separate…
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Paper The quasi-Riemann hypothesis
Proves that every Dirichlet L-function, including \zeta(s), is zero-free in \Re s\gt 7/8, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over…
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Paper A stable-coordinate counterexample in four variables
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…
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Paper Real ultraflat Littlewood polynomials and unbounded binary merit factors
Constructs polynomials with N consecutive coefficients in \{-1,1\} whose modulus is (1+o(1))\sqrt N uniformly on the entire unit circle, for every sufficiently large integer length N. Thus real Littlewood…
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Paper Hyperbolicity cones without semidefinite lifts
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…
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Paper Approximate counting of common integer polymatroid bases
Gives a fully polynomial randomized approximation scheme for counting common integer bases of two integral polymatroids of equal total rank, supplied by exact rank-value oracles. Capacities are binary-encoded, each…
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Paper Talagrand’s expectation thresholds, discrete convexity, and graph decompositions
Proves that integral and fractional expectation thresholds differ by at most a universal factor, and resolves Talagrand's discrete-convexity conjecture. An application proves the Ascoli–He–Park–Talagrand…
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Paper Canonical O(3) continuum limit and exact O(4) mass asymptotics
Constructs the canonical continuum limit of the two-dimensional nearest-neighbor O(3) model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice O(4)…
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Paper Critical SK autocorrelation processes and dynamics across the temperature transition
For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the \log n scale for fixed 0\le\beta\lt 1, mixing time n^{2/3+o(1)} at β = 1, and…
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Paper Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability
Proves equivariant Jiang–Su stability for every countable discrete amenable group action on a simple separable unital infinite-dimensional nuclear stably finite Jiang–Su-stable C∗-algebra, resolving this case of Szabó's…
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Paper Joint metric and connection recovery from one boundary patch
Zero-frequency measurements on any nonempty open boundary patch determine a smooth metric and smooth unitary connection on a trivial Hermitian rank-two bundle over a compact connected manifold of dimension at least…
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Paper A torsion-free group algebra that is not directly finite
Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽2 is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective…
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Paper Uniform black-box noncommutative identity testing across characteristics
For each characteristic, constructs in deterministic polynomial bit time a polynomial-dimensional matrix tuple detecting every nonzero division-free noncommutative formula of bounded size over any field of that…
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Paper Matrix multiplication with exponent at most 9/4
Proves \omega\le9/4 over ℂ, giving O\varepsilon(n^{9/4+\varepsilon}) arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension na with a > 0.465 permits…
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Paper Invariant projections, hyperinvariant subspaces, and transitive algebras
Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator. The construction also…
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Paper Memory–sample lower bounds for noiseless Gaussian regression
For fixed A > 0, a one-pass learner with Ad^2 persistent bits needs \OmegaA(d\log(1/\epsilon)) noiseless Gaussian samples to recover a unit vector to angular error 0\lt \epsilon\le1/10 with probability…
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Paper Counterexamples to Ryser’s covering conjecture
Disproves Ryser's covering conjecture by constructing intersecting (q+1)-partite, (q+1)-uniform hypergraphs with covering number q+1, rather than the predicted bound q, for every sufficiently large prime q.…
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Paper Reflexive midpoint convexity and diamond distortion
Constructs a real reflexive Banach space with an asymptotically midpoint uniformly convex norm but no equivalent asymptotically uniformly convex norm, extending Baudier's separation to reflexive spaces. In the same…
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Paper The exact factor-of-IID threshold for free Ising spins on trees
Determines when the free zero-field ferromagnetic Ising state on the infinite d-regular tree is a factor of independent vertex labels: exactly when \tanh\beta\le(d-1)^{-1/2}, including equality, for d ≥ 3 and β ≥ 0.…
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Paper The complete Crouzeix conjecture
Resolves the complete Crouzeix conjecture: for every bounded operator A on a complex Hilbert space and every finite matrix-valued polynomial P, one has \lVert P[A]\rVert\le2\sup{z\in W(A)}\lVert P(z)\rVert, where…
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Paper Failure of integer-degree harmonic dimension comparison
Disproves Yau's proposed Euclidean dimension bound for harmonic functions of integer growth on manifolds with nonnegative Ricci curvature. For every sufficiently large integer k, a complete smooth metric on ℝ3 has at…
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