Tagged lean-formalized
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Paper Optimal logarithmic mixing of the Thorp shuffle
Proves that the Thorp shuffle randomizes N=2^d labeled cards in \Theta(\log N) physical shuffles, settling its optimal mixing order for power-of-two deck sizes. Convergence is in total variation from the worst…
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Paper Subpolynomial query complexity for log-concave sampling
For C2 potentials with a supplied minimizer and I\preceq\nabla^2V\preceq2I, proves that sampling within total variation 1/10 requires only C\varepsilon d^\varepsilon exact value-and-gradient queries for every…
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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 Lipschitz equivalent Banach spaces need not be linearly isomorphic
Constructs separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic, resolving the separable Lipschitz-isomorphism problem negatively. Thus even the complete metric structure up…
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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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Paper A quadratic bound for Jacobsthal’s function
Answers Jacobsthal's quadratic-bound question: every interval of Ck^2 consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute…
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Paper Short Egyptian fractions
Every rational a/b with 1\le a\lt b is a sum of O(\log\log b) distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian…
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Paper Potential integral density on curve character varieties
Resolves the determinant-one curve case of Litt's integral-density question. For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SLr…
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Paper Bounded klt complements for Fano contractions
Proves the finite-rational-coefficient form of Shokurov’s bounded-klt-complement conjecture for ϵ-lc complex Fano-type pairs with nef anti-log-canonical divisor. For ϵ-lc Fano contractions over any algebraically closed…
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Paper Compact counterexamples to bi-Lipschitz dimension reduction
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…
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Paper Average sensitivity of polynomial threshold functions
Proves that a degree-at-most-d polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most 8d\sqrt n, uniformly for 1\le d\le n. Average sensitivity counts expected…
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Paper Exponential state costs for two-way automata
Proves exponential lower bounds both for complementing two-way nondeterministic finite automata and for simulating one-way nondeterministic automata by two-way deterministic ones. The latter resolves the Sakoda–Sipser…
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Paper Exact Fourier transforms below n\log n
Gives a deterministic length-n discrete Fourier transform algorithm using O(n(\log n)^{1-\delta}) operations for every n, with explicit \delta=10^{-13}. The model uses exact complex arithmetic, unrestricted…
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Paper A superquadratic separation of sensitivity and block sensitivity
Constructs total Boolean functions with block sensitivity \mathop{\mathrm{bs}}\nolimits (f)\ge s(f)^\alpha for a fixed α > 2, disproving the quadratic strengthening of the Sensitivity Conjecture. Here s(f)…
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Paper One-tape time simulation in two-fifths-power space
Determines the halting and finite-control outcome of a fixed deterministic one-writable-tape machine up to time T using O(T^{2/5}\log^C(T+2)) space, improving the square-root exponent. Heads move at most one cell…
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Paper Polynomial removal fails for ordered binary matrices
Disproves polynomial ordered binary matrix removal with one fixed 66\times66 zero–one pattern. Matrices can require many binary-entry changes to become pattern-free while their copy density is smaller than every…
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Paper Foulkes' conjecture for sixth powers and quadratic stabilization
Proves the sixth case of Foulkes’ conjecture: \mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^bV) embeds equivariantly in \mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^6V)…
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Paper A hyperbolic group without a geometric CAT(0) action
Answers negatively whether every word-hyperbolic group is a CAT(0) group. Constructs one with a finite classifying space but no proper cocompact isometric action on any proper complete $`\mathop{\mathrm{CAT}}\nolimits…
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Paper Threshold repetition for entangled games
Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is v < 1, the probability of winning at least a fraction v+\delta of k independent repetitions decays…
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Paper The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability
A nonzero unital separable complex C∗-algebra is Jiang–Su stable exactly when its norm central-sequence algebra has no characters, for every free ultrafilter. This answers the Kirchberg–Rørdam character question. Also,…
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Paper Trace cones and Razak–Jacelon stabilization
Classifies separable nuclear complex C∗-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their full topological cones of extended lower-semicontinuous tracial weights. This…
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Paper Negative Kähler curvature without bounded holomorphic coordinates
Constructs a contractible domain in ℂ3 with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting. A higher-dimensional example has…
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Paper Weak MTW curvature gives convexity and regular optimal transport
On every closed connected Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, all tangent injectivity domains are convex, resolving Villani’s conjecture in this setting. For…
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Paper Counterexamples to infinite matroid intersection and packing/covering
Disproves the unrestricted infinite matroid intersection and packing/covering conjectures in ZFC, using two self-dual partitional matroids on a countably infinite ground set. The same examples answer Joó’s…
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Paper Nonexpansive fixed points in reflexive Banach spaces
Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without…
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Paper A counterexample to metric-entropy duality
Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and…
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Paper Squarefree quartics and power-free polynomial values
Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally,…
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Paper A counterexample to Griffiths’ positivity conjecture
Constructs ample rank-two bundles on \mathbb P^1\times\mathbb P^1 with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.…
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Paper Semialgebraic universal covers and bounded domains
Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products D\times\mathbb C^m\times F, with D bounded symmetric and F simply…
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