Functions, operators, differential equations, dynamics, and random systems, including analytic number theory and mathematical physics.
24 records · Includes focused interests · How we rate progress
Alpöge and Furman report an unconditional lower bound of 0.6725 for the proportion of nontrivial zeta zeros that are simple and lie on the critical line.
Original report: 2026-08-10OpenAI claims disappearance of infinite bond-percolation clusters at criticality on every infinite connected locally finite quasi-transitive graph whose critical threshold is below one.
Read original source Pinned primary manuscript · family 213github.com
Original report: 2026-10-06Parisi and Zamponi give an analytic proof of a+b=1 from the scaling full replica-symmetry-breaking equations for infinite-dimensional hard-sphere jamming. Together with the earlier relation b=(1+c)/2, it yields the scaling identities linking force, gap and overlap exponents.
Read original source Parisi–Zamponi: precise identity, assumptions and AI disclosurearxiv.org
Original report: 2026-06-02The pinned Anthropic development reports θ(p_c)=0 for nearest-neighbour Bernoulli bond percolation on Zᵈ in every integer dimension d≥2. It proves Kozma–Nitzan’s Conjecture 3 through an additive gluing inequality and formalizes the reduction to the lattice statement.
Read original source Pinned Anthropic development: exact lattice statementgithub.com
Original report: 2026-08-28Christopher D. Long reports explicit complex polynomials P and Q in three independent real Gaussian variables with E(P^m)=0 and E(QP^m)=m! for every positive integer m. A five-term degree-four example refutes the conjecture in every dimension at least three; a six-term cubic example works in four variables.
Read original source Original paper and AI-provenance sectionarxiv.org
Original report: 2026-07-20Alpöge and Buckmaster report finite-time blowup for the inviscid Boussinesq equations on R² with smooth spatially compact forcing in both equations. Temperature remains bounded, while its gradient and the vorticity become unbounded.
Read original source Alpöge–Buckmaster manuscript and AI statementcims.nyu.edu
Original report: 2026-09-07The manuscript reports smooth compactly supported divergence-free initial data on R³ whose unforced three-dimensional incompressible Euler solution has finite maximal lifespan, with unbounded velocity gradient and divergent time integral of maximum vorticity.
Read original source Original Euler manuscript, Theorem 1.1cdn.openai.com
Original report: 2026-09-08For every positive viscosity, the manuscript reports a three-dimensional incompressible Navier–Stokes flow starting from rest with smooth compactly supported forcing, bounded kinetic energy, and unbounded velocity in finite time. It claims Clay alternatives C on R³ and D on the periodic torus.
Read original source Original manuscript, Theorem 1.1cdn.openai.com
Original report: 2026-09-08OpenAI reports area-integrability of the s-th power of the derivative of every conformal bijection from a simply connected planar domain to the unit disk whenever 4/3<s<4, together with the stated inverse-square integral-means exponent.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports that every compact subset of Euclidean d-space, d at least two, with Hausdorff dimension strictly greater than d/2 has a distance set of positive Lebesgue measure.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports the three-dimensional Kakeya maximal L³ estimate with tube-radius loss at most C_ε δ^(-ε) for every positive ε, uniformly over unit tubes of radius δ.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports a nondegenerate limiting law for centered and variance-normalized total free energy in the zero-field Gaussian Ising Sherrington–Kirkpatrick model at each fixed β>1, with variance asymptotic to a positive constant times n^(1/3).
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports purely absolutely continuous spectrum of nonzero weight on an open energy interval for the lattice Anderson model with independent uniform site potentials at each fixed dimension d≥3 and sufficiently small positive disorder.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports that the maximum number of mutually unbiased orthonormal bases in complex dimension six is three, using an exhaustive computer-assisted exclusion of four bases under specified arithmetic and compiler conditions.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports a uniform positive spectral gap above the ground energy for the pure antiferromagnetic spin-one Heisenberg chain on even periodic rings.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports that every bounded complex-linear unital homomorphism from a unital complex C*-algebra to operators on a Hilbert space is conjugate by a bounded invertible operator to a *-homomorphism.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports that every nonzero first-positive Neumann eigenfunction on a smooth bounded simply connected planar domain has nonvanishing interior gradient, placing all global extrema on the boundary even for a multiple eigenvalue.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI claims globally smooth unique evolution for arbitrary admissible large initial data in the three-dimensional one-species relativistic Vlasov–Maxwell model, under its support and field assumptions.
Read original source Pinned primary manuscript · family 362github.com
Original report: 2026-10-06OpenAI reports the sharp constant-two numerical-range inequality for matrix-valued polynomials, uniformly over all finite base and coefficient dimensions.
Read original source Pinned primary manuscript · family 325github.com
Original report: 2026-10-06OpenAI claims a normal trace-preserving *-isomorphism between the free group factors L(F₂) and L(F₃), giving the affirmative branch of the classical isomorphism alternative.
Read original source Pinned primary manuscript · family 287github.com
Original report: 2026-10-06OpenAI reports the sharp invariant-mass lower bound in terms of minimum enclosing area for one-ended spacetime initial-data exteriors in every dimension n≥3, under the paper’s energy, trapping and decay assumptions.
Read original source Pinned primary manuscript · family 260github.com
Original report: 2026-10-06OpenAI claims a finite degree-dependent upper bound for the total number of isolated periodic trajectories of any planar polynomial vector field.
Read original source Pinned primary manuscript · family 143github.com
Original report: 2026-10-06OpenAI claims a conformal circle-domain model for an arbitrary planar domain, allowing uncountably many complementary components.
Read original source Pinned primary manuscript · family 071github.com
Original report: 2026-10-06OpenAI’s manuscript reports that all Dirichlet L-functions and finite-order Hecke L-functions over Q(√−3) have no zeros with real part greater than 7/8.
Read original source Original manuscript · family 003github.com
Original report: 2026-10-06Functions, operators, differential equations, dynamics, and random systems, including analytic number theory and mathematical physics.
Related focused topics and problems retain their own statements and baselines. A contribution may belong to more than one field; it is counted once within each topic.
Background review: 2026-10-11. Selected sources:
Publication dates and editorial changes.
Claude: a reported improvement in the proportion of zeta zeros on the critical line
OpenAI reports: Critical percolation on quasi-transitive graphs
Parisi–Zamponi–Claude: an analytic proof of the jamming exponent identity
Claude reports: no infinite critical cluster on integer lattices
AI-assisted small counterexamples to the Gaussian Moments Conjecture
OpenAI reports: Strict hot spots on simply connected planar domains
OpenAI reports: The three-dimensional Kakeya maximal conjecture
OpenAI reports: Brennan’s conformal-integrability conjecture
OpenAI reports: Global classical relativistic Vlasov–Maxwell solutions
OpenAI reports: The enclosing-area spacetime Penrose inequality
OpenAI reports: Critical percolation on quasi-transitive graphs
OpenAI reports: Uniform bounds for planar polynomial limit cycles
Claude: a reported improvement in the proportion of zeta zeros on the critical line
Claude: a reported improvement in the proportion of zeta zeros on the critical line
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