Perplexity
Perplexity develops AI tools for search, browsing, research, content creation, organization, and workflow automation, including perplexity.ai, Comet, and Computer.
Real Littlewood Polynomials Attain the √N Peak Bound at Every Large Length
An OpenAI preprint proves that for every sufficiently large length, a real Littlewood polynomial—one whose coefficients are all +1 or −1—can have a maximum modulus on the unit circle arbitrarily close to the unavoidable lower bound of √N. The result holds across all sufficiently large lengths, not just a subsequence, and concerns the largest peak rather than uniform flatness. The proof constructs high-energy coefficients, separates their leading spectral contributions, then uses coordinated rounding to turn them into signs.
Gaussian Partitions Have a Sharp First-Moment Bound in Every Dimension
An OpenAI preprint proves that, in every positive dimension, the sum of the squared first-moment vectors of any finite partition of Gaussian space is at most 9/(8π). The bound is attained by dividing a plane into three 120-degree sectors and extending them across any remaining dimensions. The paper’s main argument rules out a better partition by reducing a hypothetical improvement to geometric constraints on the moments and probabilities of its cells; it also derives a sharp bound for the expected maximum of finitely many centered Gaussian scores.
Planar Graph Metrics Admit Constant-Distortion Embeddings Into L1
An OpenAI preprint claims that every finite, connected, undirected planar graph with positive real edge lengths can be embedded into \(L_1\) with distortion bounded by a universal constant, independent of graph size and edge-length ratios. Its proof uses weighted cuts and argues that planarity helps coordinate local separation rules and preserve their gains across scales. Perplexity Computer adapted the manuscript into an animated video, writing the script and generating the narration and visuals.
Bochner–Riesz Multipliers Are Bounded on L³ for Every Positive Smoothing Order
An OpenAI preprint argues that in three dimensions, the Bochner–Riesz multiplier is bounded on \(L^3\) for every positive smoothing order \(\delta\), though the bound need not stay uniform as \(\delta\) approaches zero. Its proof uses cancellation in oscillatory wave packets to improve the decay of the operator on distant spatial shells; summing those bounds gives the result. The manuscript also derives a strict range of bounds for other exponents, but does not establish the sharp-cutoff case.
Zero-Sum Vector Sets Can Be Ordered With Prefix Radius O(√d)
An OpenAI preprint on the Euclidean Steinitz–Bergström theorem argues that bounded vectors in \(d\) dimensions can be signed so every partial sum stays within an absolute constant times \(\sqrt d\) of the origin, regardless of sequence length. For vectors that sum to zero, it also proves they can be reordered to achieve the same scale without changing their signs; a regular simplex shows that the square-root dependence on dimension is necessary up to constant factors. Perplexity Computer adapted the preprint into an animated video.
Triangular Hilbert Transform Has Annular Variation Bounds for Every r Greater Than 2
An OpenAI preprint on the triangular Hilbert transform proves an \(L^{3/2}\) bound for its annular \(r\)-variation when \(r>2\), controlled by the product of the inputs’ \(L^3\) norms and uniform over choices of scales. The manuscript reduces variation to a counting estimate for disjoint annuli, whose sublinear growth makes the bound possible; its argument does not establish the endpoint \(r=2\). The estimate also yields almost-everywhere and \(L^{3/2}\) convergence of the joint principal value as the two truncation endpoints move independently.
Irrational Levels Yield a Global Quantum Langlands Equivalence
An OpenAI manuscript claims a global equivalence between the full twisted D-module categories for a simple complex group and its Langlands dual at every irrational complex level, with the two parameters related by a negative reciprocal scaled by the groups’ lacing number. The authors’ argument uses irrationality to eliminate local modes that would otherwise survive, then combines a categorical cutoff with reconstruction and global comparison results to establish the equivalence. The local calculation is only one part of that proof.
A Nonspectrahedral Hyperbolicity Cone Has an Exact 307-Variable Semidefinite Lift
An OpenAI preprint gives an exact semidefinite lift for a hyperbolicity cone that a companion paper shows cannot be represented directly by a positive semidefinite matrix in its original coordinates. The authors add 307 auxiliary variables and impose one 100-by-100 matrix inequality; projecting away those variables recovers the cone, including its singular boundary points. The construction establishes that the obstruction is to a direct representation, not to semidefinite representation by lifting.
Brennan’s Conjecture Holds for Derivative Integrals Between 4/3 and 4
An OpenAI manuscript dated September 24, 2026, argues that Brennan’s conjecture holds for conformal maps from any simply connected plane domain onto the unit disk: the integral of the map’s derivative to the power \(s\) is finite for \(4/3<s<4\), without assumptions that the domain is bounded or has a smooth boundary. Its key estimate makes one the sharp limiting radial growth exponent for the inverse-square derivative mean, with the Koebe map showing why the exponent cannot be lower.
A Hyperbolicity Cone in 23 Variables Has No Finite LMI Representation
An OpenAI preprint constructs a hyperbolicity cone in 23 real variables that cannot be represented directly by any finite-size real symmetric linear matrix inequality. The authors’ obstruction applies to the region itself, not merely to a particular polynomial used to define it: even a different defining polynomial cannot make the cone spectrahedral. The proof turns on a nonnegative form that cannot dominate the square of any nonzero bilinear form; the result does not rule out representations using auxiliary variables.
The Volume Product of General Convex Bodies Is Minimized by Simplices
An OpenAI preprint on the Mahler conjecture claims a sharp lower bound for the volume product of every convex body, without requiring symmetry or a smooth boundary. The bound is \((n+1)^{n+1}/(n!)^2\), attained exactly by simplices. The manuscript reduces the geometric claim to an inequality for weighted cone integrals and outlines Gaussian projection estimates for proving it, while noting that the full estimates and technical arguments are not reproduced.
Every Four-Dimensional Kakeya Set Has Hausdorff Dimension Four
An OpenAI preprint claims that every Kakeya set in four-dimensional space has Hausdorff dimension four, even when the set is nonmeasurable and no measurable rule selects a line segment in each direction. The result rules out sets of dimension below four; it does not establish positive volume or the stronger four-dimensional maximal estimate. The proof turns a hypothetical dimension deficit into a problem about finite, populated charts, then seeks improvements in each of three regimes to contradict an extremal choice.
Fourier Extension on the Sphere Is Bounded for Every p Above Three
An OpenAI preprint establishes an \(L^p\) bound for the Fourier extension of bounded complex data on the three-dimensional unit sphere when \(p>3\). A constant-data example shows that the estimate fails at and below the cubic threshold. For the positive result, the manuscript controls wave-packet concentration through elliptic-capacity estimates and a selection argument that retains more than any fixed negative power of the scale, asymptotically, without assuming velocity and position are independent.
Every Convex Body Has a Lattice Covering of Density O(n log n)
An OpenAI preprint proves that every compact convex body in dimension \(n \ge 2\) can be covered by copies arranged on a single lattice with density at most \(C n \log n\), for an absolute constant \(C\). The bound applies even to bodies that are neither symmetric nor smooth; it limits the worst case rather than describing the density needed for every shape. The proof coordinates weighted choices across many sections, then uses one global shear to keep the construction within a single lattice.
Some Hyperbolicity Cones Admit No Exact Finite Semidefinite Lift
An OpenAI preprint argues that some hyperbolicity cones cannot be represented exactly as projections of finite-dimensional semidefinite constraints, even with auxiliary variables and arbitrary real coefficients. The paper’s counterexample is designed to rule out every finite lift, not just a particular matrix representation; its conclusion concerns exact representation, not approximation.
A Compact Doubling Metric Space Can Defy Every Finite-Dimensional Embedding
An OpenAI manuscript constructs a compact subset of Hilbert space that is doubling—every ball can be covered by a uniformly bounded number of smaller balls—but cannot be embedded into any finite-dimensional Euclidean space with bounded distortion. The authors’ result shows that uniform control of local covering complexity alone does not ensure that distances can be represented in finitely many coordinates.
Optimal L1 Distortion for Edit Distance Grows as exp(Θ(√(log d log log d)))
An OpenAI preprint argues that edit distance on strings of length at most \(d\) requires exponentially growing distortion when represented in real \(L_1\), even if the embedding’s dimension and computation time are unrestricted. For every finite alphabet with at least two symbols, it bounds the best possible distortion above and below by \(\exp(\Theta(\sqrt{\log d\,\log\log d}))\), with the constants in the exponent unspecified; the lower bound already holds for binary strings of a common length.
A Fixed-Slope Trilinear Hilbert Transform Maps Three L³ Inputs to L¹
An OpenAI preprint proves an \(L^3\times L^3\times L^3\to L^1\) bound for the trilinear Hilbert transform at the fixed slopes 1, 2 and 3. The result is specific: it does not settle every exponent or version of the transform. The paper’s proof tackles the multiscale interactions that defeat direct estimates, using quadratic structure to obtain bounds uniform across dyadic depths.
Uniform Sub-Gaussian Projection Tails Yield Dimension-Free Log-Sobolev Bounds
An OpenAI manuscript argues that a centered, log-concave distribution has a dimension-free logarithmic Sobolev inequality when every one-dimensional projection has uniformly controlled sub-Gaussian tails at a common scale. The claimed bound applies to all smooth, compactly supported functions, with an entropy-to-gradient-energy coefficient set by that scale rather than by the ambient dimension; the authors’ proof uses a contradiction argument based on bounds on mutual information.
Some Convex Bodies Require Translative Covering Density Of Order n log n
An OpenAI manuscript claims that, in every sufficiently large dimension, some centrally symmetric convex body requires translative covering density of at least \(c n \log n\), for an absolute constant \(c>0\). The result identifies the worst-case order, matching the known \(O(n \log n)\) upper bound for all convex bodies; it does not say that every body is this difficult. The proof constructs a random body and aims to rule out every sufficiently low-density covering, including arrangements chosen after the body is known.
OpenAI Paper Claims a Fixed Zero-Free Boundary at Re s = 7/8
OpenAI’s September 2026 paper claims that the Riemann zeta function and every Dirichlet L-function have no nontrivial zeros to the right of Re s = 7/8, a fixed boundary inside the critical strip. The result is weaker than the Riemann hypothesis, which places every nontrivial zero on Re s = 1/2, but would improve on zero-free regions whose width shrinks with height. Lean has accepted a formalized version of the zeta statement; the 199-page paper itself had not received public expert review.
Cuspidal Hecke Parameters Share an Orbit-Controlled Growth Pattern
A preprint on cuspidal automorphic functions at full finite level argues that unramified Hecke parameters may be non-tempered, but that any noncompact growth must follow a pattern determined by a nilpotent orbit. It decomposes the cuspidal function space into orbit-indexed pieces, with the same orbit governing every place outside the chosen level within each piece. For split, connected, adjoint, absolutely simple groups, the paper further shows that a unitary cuspidal representation with one local component that is both unramified and generic has tempered unramified components.
Mod-ℓ Milnor Products Recover Function Fields Up to Frobenius
An OpenAI preprint claims that a finitely generated field extension of an algebraically closed field, with transcendence degree at least two and characteristic different from ℓ, can be reconstructed from its nonzero elements modulo ℓ-th powers and the Milnor products among those classes. The authors argue that the product’s Steinberg relation retains enough information about addition to recover the field’s perfect closure and constant field, up to a global scalar and, in positive characteristic, powers of Frobenius. The proof uses valuations and geometry to identify one-variable subfields before recovering the field operations.
Low Selmer Corank Yields the Exact Birch–Swinnerton-Dyer Formula
An OpenAI preprint claims the exact Birch–Swinnerton-Dyer formula for elliptic curves over the rationals whose full \(q\)-power Selmer group has corank zero or one for some prime \(q\). The authors distinguish this hypothesis from a rational-point rank assumption, relying on a companion result that relates Selmer corank to analytic and rational-point rank and establishes finiteness of the Tate–Shafarevich group. The formula matches the leading L-function coefficient to the period, regulator, Sha order and Tamagawa factors, divided by the square of the rational torsion order.