Mathematics & Science Breakthrough Tracker

An AI-assisted preprint claims a counterexample to Simon's multidimensional L2 conjecture

Becker and Yang construct, for every dimension at least two and every weight exponent below one third, smooth bounded nonnegative potentials with arbitrarily small weighted L2 size whose Schrödinger operators have pure point spectrum throughout the positive axis. If correct, the construction disproves Barry Simon's Problem 8.

Problem or result

Barry Simon's multidimensional weighted-L2 conjecture for preservation of absolutely continuous spectrum

Authors

Simon Becker and Mengxuan Yang

Institutions

Bocconi University; Texas A&M University

Result date

First-version preprint submitted October 22, 2026

Why it matters: The conjecture sought a broad integrability condition guaranteeing that multidimensional Schrödinger operators retain continuous spectrum. A valid counterexample would rule out that program even under a stronger positive spatial weight and with arbitrarily small bounded potentials.

Limits: This is a new unrefereed preprint with no independent specialist correctness assessment located in this cycle. The construction is highly parameter-sensitive, and the tracker has not reproduced its spectral-averaging argument. The record therefore says claimed counterexample, not proved theorem.

Parallel preprints claim the corrected Cheng–Murray deep-hole classification

Fang claims to settle the remaining low-dimensional cases and, with earlier high-dimensional results, prove the corrected Cheng–Murray conjecture for full-length Reed–Solomon codes in every nonexceptional dimension. Chen and Huang independently cover every dimension for fields larger than 19,683, while Fan supplies exact classifications for several interpolation-degree ranges and reproducible finite certificates.

Problem or result

Corrected Cheng–Murray conjecture on deep holes of full-length Reed–Solomon codes

Authors

Weijun Fang; Bocong Chen and Jing Huang; Xiang Fan

Institutions

Shandong University; South China University of Technology; Guangzhou University; Sun Yat-sen University

Result date

Parallel first-version preprints submitted October 21 and 22, 2026

Why it matters: Deep holes describe received words at maximum distance from a code and connect decoding theory to arcs and maximum-distance-separable extensions in finite geometry. A correct full classification would close a problem posed in 2007 while isolating the known characteristic-two exceptions.

Limits: All three manuscripts are recent and unrefereed. Their scopes overlap without being identical: Fang's full claim relies on earlier high-dimensional classification, Chen and Huang impose a field-size threshold, and Fan proves degree-specific and bounded classifications rather than the whole conjecture. The tracker found no external specialist verdict and does not merge those scopes into stronger evidence than the papers supply.

An AI-assisted preprint constructs a critical two-dimensional KPZ process

Berger, Caravenna, Clark and collaborators construct a non-Gaussian distribution-valued process as the critical limit of centred log partition functions for two-dimensional Gaussian directed polymers and as a renormalized Cole–Hopf functional of the critical stochastic heat flow.

Problem or result

Construction and characterization of a critical two-dimensional KPZ process

Authors

Quentin Berger, Francesco Caravenna, Jeremy Clark, Alexander Dunlap, Yu Gu, Ziyang Liu, Shuta Nakajima, Rongfeng Sun, Li-Cheng Tsai, Zuodi Xie and Nikos Zygouras

Institutions

Université Sorbonne Paris Nord; University of Milano-Bicocca; University of Mississippi; Duke University; University of Maryland; Hong Kong University of Science and Technology; Keio University; National University of Singapore; University of Utah; University of Warwick

Result date

First-version preprint submitted October 22, 2026

Why it matters: The two-dimensional KPZ equation is ill-defined at criticality without renormalization. The proposed process supplies a nontrivial critical object, connects it to the previously constructed stochastic heat flow and identifies non-Gaussian local behavior at the marginal dimension.

Limits: This is an 84-page unrefereed preprint and no independent specialist correctness assessment was located. The result is a mathematically constructed scaling limit, not an experimental demonstration or a claim that every two-dimensional KPZ model has been solved. Its extensive AI contribution makes author checking important but is not independent validation.

Independent rewrites support OpenAI's quasi-Riemann and Landau–Siegel claims

OpenAI manuscripts claim a zero-free half-plane Re(s) greater than 11/12 for every Dirichlet L-function and a uniform exclusion of Landau–Siegel zeros. Independent preprints by William D. Banks and Bryce Kerr reconstruct the Landau–Siegel argument, with Banks supplying explicit constants.

Problem or result

Quasi-Riemann hypothesis for Dirichlet L-functions and exclusion of Landau–Siegel zeros

Authors

OpenAI; independent proofs by William D. Banks and Bryce Kerr

Institutions

OpenAI; University of Missouri (William D. Banks); School of Science, UNSW Canberra (Bryce Kerr). Banks also acknowledges support from the Max Planck Institute for Mathematics in Bonn.

Result date

OpenAI manuscripts dated October 1 and 5; independent preprints submitted October 11 and 15; specialist assessment published October 19, 2026

Why it matters: A fixed zero-free half-plane would materially strengthen control of primes in arithmetic progressions and settle the Landau–Siegel exceptional-zero problem. Independently authored, AI-assisted expositions make the argument more inspectable, while their authors retain responsibility for checking it.

Limits: None of the papers is peer reviewed. Specialists quoted by Quanta largely judged the 11/12 argument correct, but Peter Sarnak declined to call it a theorem until every detail is checked, and experts set aside a separate stronger 7/8 manuscript because its exposition was too difficult to audit. The tracker therefore does not mark either claim as an unqualified proved theorem.

Three proof routes and a Lean artifact strengthen the KLS resolution claim

Song and Zhang, Bizeul, Klartag and Lehec, and Balasubramanian and Kasiviswanathan give three proof routes to a dimension-free KLS bound. A separate public Lean project checks a Poincaré constant at most 500 for an encoded statement.

Problem or result

Kannan–Lovász–Simonovits conjecture for isotropic log-concave measures

Authors

Zhao Song and Xinzhi Zhang; Pierre Bizeul, Boaz Klartag and Joseph Lehec; Krishnakumar Balasubramanian and Shiva Kasiviswanathan; Lean artifact by Hongru Zhao

Institutions

Song: independent researcher; Zhang: Microsoft Research. The inspected Bizeul–Klartag–Lehec manuscript did not list affiliations in its author block.

Result date

Three proof preprints posted October 4; public Lean artifact and synthesis documented by October 21, 2026

Why it matters: KLS has guided high-dimensional convex geometry since 1995 and controls concentration, random-walk mixing, sampling and volume computation. Three mathematical routes plus a kernel-checked quantitative statement provide materially stronger evidence than a single manuscript.

Limits: The papers and Lean repository remain recent and unrefereed. The formalization's own evidence says Comparator and Nanoda were not run, and kernel checking validates the encoded theorem rather than every prose-to-formal correspondence or historical claim. Microsoft is an author-affiliated source, so its synthesis is context rather than independent peer review.

A Lean-checked construction disproves Vaught's conjecture for infinitary logic

Ackerman, Freer and Knight construct an L-omega-one-omega sentence with exactly aleph-one countable models up to isomorphism and no perfect set of non-isomorphic models, giving a counterexample to the infinitary version of Vaught's conjecture.

Problem or result

Vaught's conjecture for sentences of infinitary logic L-omega-one-omega

Authors

Nathanael L. Ackerman, Cameron E. Freer and Robin W. Knight

Institutions

Harvard University; Massachusetts Institute of Technology; University of Oxford

Result date

Preprint submitted October 10; Lean correspondence work extended through October 25, 2026

Why it matters: The infinitary statement is one of the oldest open problems connecting model theory and descriptive set theory. The public Lean theorem now proves the manuscript's Theorem 4.36 spectrum statement for the repository's template sentence on models coded by the natural numbers, adding unusually strong artifact evidence to the unrefereed paper.

Limits: The result concerns infinitary logic, not the still-open first-order Vaught conjecture. The repository proves exactly aleph-one isomorphism classes and no perfect set for its template sentence on models coded by the natural numbers, but explicitly does not identify that sentence with its density sentence or transfer the repository's main theorem. The prose paper remains unrefereed and the definitions still require semantic specialist review.

AlphaProof Nexus receives peer-reviewed publication for broad formal proof search

The Science paper reports that AlphaProof Nexus produced Lean-checked resolutions for 9 of 353 selected open Erdős problems and proofs for 44 of 492 selected OEIS conjectures, with public formal and human-readable proof materials.

Problem or result

Large-scale Lean-verified proof search across selected Erdős and OEIS conjectures

Authors

George Tsoukalas, Anton Kovsharov, Sergey Shirobokov and collaborators

Institutions

Google DeepMind, Aarhus University and Google

Result date

Science publication October 8, 2026; underlying preprint first posted May 21, 2026

Why it matters: Peer review and public formal artifacts turn an earlier system announcement into a reproducible, broad demonstration of AI-guided formal proof search across a curated open-problem set.

Limits: The October event is peer review of the system and reported result set, not the first discovery date for every included proof. Problem selection, statement encoding, novelty and mathematical significance vary, and formal checking does not independently validate every natural-language interpretation.

Specialists give a concise preprint proof of Brennan's conjecture

Binder, Ivrii, Prause and Smirnov prove the universal integral-means bound B_b(t) at most t squared over 4 for t between minus 2 and 2, which yields Brennan's integrability range 4/3 less than p less than 4.

Problem or result

Brennan's conjecture via the universal integral means spectrum

Authors

Ilia Binder, Oleg Ivrii, István Prause and Stanislav Smirnov

Institutions

University of Toronto; Tel Aviv University; Åbo Akademi University; University of Geneva

Result date

Preprint submitted October 16, 2026

Why it matters: Brennan's conjecture is a long-standing problem in geometric function theory. Four specialists independently reconstructed a short mathematical route instead of merely repeating an institutional claim.

Limits: The note is not peer reviewed. It announces a stricter interior inequality but defers those details to a later paper, so the tracker does not treat that stronger statement as established here or generalize this validation to the rest of the OpenAI catalogue.

A preprint proves the Erdős–Gallai cycle-decomposition conjecture

Kim proves that the edges of every n-vertex graph can be decomposed into O(n) cycles and single edges, equivalently that every Eulerian n-vertex graph decomposes into O(n) cycles.

Problem or result

Erdős–Gallai cycle decomposition of graph edges

Authors

Jaehoon Kim

Institutions

Korea Advanced Institute of Science and Technology

Result date

First submitted October 6 and revised October 9, 2026

Why it matters: The result settles the asymptotic cycle-decomposition conjecture and supplies a second proof route separate from the formalized OpenAI announcement.

Limits: This does not prove the sharp Hajós bound; it establishes the required linear-order decomposition. Both proof routes are recent preprints, and formal verification of one encoding is not peer review of either mathematical exposition.

The Erdős–Ulam artifact sharpens bounds and narrows its claimed scope

Bhattacharjee, Mandal and Bhattacharya claim answers to the two questions recorded as Erdős Problem 1183. The maintained artifact now reports 11n/20 as a stronger finite lower bound and preserves a subexponential upper construction, while explicitly leaving the exact orders of the associated functions open.

Problem or result

Erdős–Ulam monochromatic union-closed-family problem, Erdős Problem 1183

Authors

Deep Bhattacharjee, Priyabrata Mandal and Ushashi Bhattacharya

Institutions

Formerly EGSPL, Bhubaneswar; Maulana Azad National Institute of Technology Bhopal; formerly National Taiwan University

Result date

Preprint and first artifact published October 2; corrected and sharpened artifact releases October 6–10, 2026

Why it matters: The maintained artifact materially improves a quantitative bound, clarifies which two Erdős questions are answered and exposes a layered verification boundary instead of overstating complete formalization.

Limits: This remains unrefereed. Chojecki had independently proved the subexponential side in March 2026, so that part is not novel to this artifact. Four conventional checkers cover the finite list and Lean checks the colourings-by-size component, but the mixed-level reduction is not formalized; the exact asymptotics remain open. A footnote error was corrected without changing the stated conclusion.

A formalization-backed preprint claims Lehmer’s permutation conjecture

Verhoeff claims that every multiset-permutation neighbor-swap graph admits Lehmer’s imperfect Hamiltonian traversal, proving the equivalent non-stutter Hamiltonicity formulation and identifying the exact path-only exceptions.

Problem or result

Lehmer’s 1965 permutation conjecture for neighbor-swap graphs

Authors

Tom Verhoeff

Institutions

Department of Mathematics and Computer Science, Eindhoven University of Technology

Result date

Manuscript dated September 30; first-version preprint submitted October 1, 2026

Why it matters: The claim resolves a 1965 permutation-generation conjecture recorded as an unsolved Knuth problem and provides executable and kernel-checked artifacts.

Limits: This reconciliation addition predates the prior cutoff but was absent from the dataset. The paper is not peer reviewed, the large formal package lacks an outside semantic audit, and its port from Lean and Mathlib 4.28.0 to a stable toolchain was still in progress.

A preprint claims Simpson’s closedness conjecture in arbitrary rank

Hu claims that the attracting fiber associated with every stable degree-zero graded Higgs bundle is Zariski closed in the de Rham moduli space, extending the known rank-two case to arbitrary rank through determinant-of-cohomology and Quillen geometry.

Problem or result

Simpson’s closedness conjecture for attracting fibers of stable graded Higgs bundles

Authors

Tianzhi Hu

Institutions

School of Mathematics and Statistics, Wuhan University

Result date

Preprint submitted October 2; version 2 posted October 5, 2026

Why it matters: The claimed theorem would establish a strong algebraic separatedness property for Simpson filtrations throughout arbitrary-rank de Rham moduli, closing the gap beyond the established rank-two case.

Limits: The proof is an unrefereed first version. No independent specialist assessment, formal proof or journal acceptance was located, and the disclosed AI role includes intermediate mathematical work.

A preprint claims Mond’s conjecture for corank-one maps from C³ to C⁴

Rimányi claims that every Aₑ-finite corank-one map germ from C³ to C⁴ satisfies the Mond inequality between Aₑ-codimension and image Milnor number, with equality for quasihomogeneous germs.

Problem or result

The corank-one C³-to-C⁴ case of Mond’s conjecture

Authors

Richárd Rimányi

Institutions

No institution was stated in the inspected manuscript author block

Result date

First-version preprint submitted October 2, 2026

Why it matters: The paper claims the first broad three-dimensional corank-one case after more than thirty years of partial results.

Limits: This is a primary-only, unrefereed first version without a formal artifact or independent proof assessment. It covers the corank-one C³-to-C⁴ case, not the full conjecture in higher dimensions, and reduces further cases to another inequality.

A human exposition and Lean artifact support a claimed full proof of the Erdős–Sós conjecture

Riordan and Scott claim that every graph of average degree greater than k−2 contains every k-vertex tree, including a classification of equality cases and an antidirected-tree analogue. Their mathematical exposition builds on a GPT-6 Astra proof, and a separate Lean 4 artifact checks the formal statement.

Problem or result

The full Erdős–Sós conjecture for embedding every k-vertex tree

Authors

Oliver Riordan and Alex Scott; underlying proof credited to GPT-6 Astra; Lean artifact maintained by Tamás C. Adamczewski

Institutions

University of Oxford

Result date

Preprint first submitted September 14; v3 posted October 8, 2026

Why it matters: This is the full 1960s conjecture rather than the dense-regime theorem already tracked. The short human exposition and separate formal artifact materially improve checkability.

Limits: The v3 manuscript remains unrefereed. Its authors describe minor corrections in Section 5 and add an Eulerian-graph result; these changes do not independently validate the claimed full proof or alter the formal artifact's semantic-review caveat.

A preprint claims the remaining case of Sylvester’s rational-cubes conjecture

Burungale and Tian claim that every prime congruent to 4, 7 or 8 modulo 9 is a sum of two rational cubes, closing the previously open residue-8 case through an analytic-rank-one result for an associated elliptic curve.

Problem or result

Sylvester’s rational-cubes conjecture for primes in the 4, 7 and 8 residue classes modulo 9

Authors

Ashay A. Burungale and Ye Tian

Institutions

University of Texas at Austin; Morningside Center of Mathematics, Chinese Academy of Sciences

Result date

Submitted September 14 and revised September 15, 2026

Why it matters: The result would close the last residue class in a rational-cube problem rooted in Sylvester’s 1879 work and connects it with Heegner points and Gross–Zagier methods.

Limits: The long technical preprint is not peer reviewed. No formalization or independent proof-level assessment was located, and its historical formulation is modern rather than a verbatim 1879 statement.

Two internal proof routes claim the Cassels–Swinnerton-Dyer criterion for cubic surfaces

Alexeev and Schreieder claim that a cubic surface over any field has a rational point exactly when it has a point of degree coprime to three, equivalently a zero-cycle of degree one. Their paper gives two proofs of the missing degree-four case and smooth and singular extensions.

Problem or result

The rational-point criterion for cubic surfaces

Authors

Valery Alexeev and Stefan Schreieder

Institutions

University of Georgia; Leibniz University Hannover

Result date

Submitted September 14; current v3 September 25, 2026

Why it matters: It would close the cubic-surface case of a longstanding rational-point criterion across characteristics and singularity cases.

Limits: The two routes are within one coauthored paper, not independent external verification. Expert comments are acknowledged, but the preprint is not peer reviewed.

A Lean development claims to formalize the Hamilton–Perelman proof of 3D Poincaré

Qin and collaborators released a Lean 4 development claiming to formalize the smooth three-dimensional Poincaré theorem, Moise smoothing and the topological endpoint through Ricci flow with surgery and finite-time extinction.

Problem or result

Lean formalization of the Hamilton–Perelman proof of the three-dimensional Poincaré theorem

Authors

Ziyang Qin, Yuan Liao, Bennett Chow and Ayush Khaitan

Institutions

Cornell University; University of California, San Diego; Princeton University and Princeton Language and Intelligence

Result date

Paper submitted September 27; reviewed stable release v0.1.4 on October 5; development preview v0.1.5 on October 9, 2026

Why it matters: If its definitions and interfaces withstand expert audit, this is an unusually large formal-verification milestone for modern geometric analysis.

Limits: Poincaré was not newly proved. Stable v0.1.4 remains the last tracker-reviewed source. The newer v0.1.5 development preview adds claimed geometrization and Mostow–Prasad work but explicitly awaits later integration and build caches. A successful kernel build does not establish that every formal definition matches the intended geometry; independent semantic review remains pending.

An explicit Heisenberg-group element is claimed to refute determinant approximation

Kammeyer gives the element 1−2a+2b in the integral group ring of the discrete Heisenberg group, with infinite Fuglede–Kadison determinant 2 while finite-quotient determinants have limsup at most the cube root of 5.

Problem or result

Determinant approximation for finite quotients of residually finite groups

Authors

Holger Kammeyer

Institutions

Heinrich Heine University Düsseldorf

Result date

First-version preprint submitted September 14, 2026

Why it matters: The example would refute a standing approximation conjecture in a simple class-two nilpotent setting and block a proposed L²-torsion approximation route.

Limits: This is a primary-only, non-peer-reviewed preprint. No independent correctness assessment was located.

OpenAI releases a formalized finite-time Navier–Stokes blowup construction

OpenAI claims an analytical proof of finite-time blowup for the three-dimensional incompressible Navier–Stokes equations, starting from rest with smooth compactly supported forcing and uniformly bounded kinetic energy. The paper says this establishes alternatives C and D in the Clay formulation. Independent PDE papers now explain part of the construction and use it downstream, without completing an outside proof check.

Problem or result

Navier–Stokes existence and smoothness, Clay alternatives C and D

Authors

OpenAI

Institutions

OpenAI

Result date

OpenAI release September 8; independent technical papers through September 2026

Why it matters: Navier–Stokes regularity is one of the seven Millennium Prize Problems. If the analytical statement, formal specification and proof all withstand expert review, the work settles the problem by exhibiting smooth forced data whose velocity becomes unbounded in finite time.

Limits: This remains an institutional preprint rather than a peer-reviewed or Clay-accepted solution. Bastounis, Circelli and Hansen specifically dispute that the Lean theorem corresponds to the natural-language blow-up argument. Their critique is not a counterproof to every analytical step, and the promised Lei–Ren Part II correction was not located, so the record is disputed rather than disproved or retracted.

OpenAI claims a formalized finite-time blowup for the unforced 3D Euler equations

A companion OpenAI paper constructs smooth, compactly supported, divergence-free initial velocity on three-dimensional space whose unforced incompressible Euler evolution develops a finite-time singularity, with unbounded C1 norm and divergent time-integrated maximum vorticity.

Problem or result

Finite-time singularity for the unforced three-dimensional incompressible Euler equations

Authors

OpenAI

Institutions

OpenAI

Result date

Publicly released September 8, 2026

Why it matters: The unforced Euler regularity problem is a central open question in mathematical fluid dynamics. The claimed construction is distinct from concurrent forced-Euler work because no external forcing drives it.

Limits: The 57-page paper and Lean artifact have not yet received located independent semantic review or peer review. Machine checking does not by itself establish that the formalized definitions capture every analytic condition intended by the paper.

Pierce–Birkhoff now has a hand-checkable R30 example and a claimed formal R4 counterexample

Lai, Lim and Ren’s paper gives a continuous piecewise-quadratic function on R30 that cannot be expressed by finite polynomial min/max combinations. Their repository now also claims a four-variable construction with degree-232 partition tests, degree-40 pieces and a Lean endpoint excluding every finite min/max representation.

Problem or result

Pierce–Birkhoff representation of continuous piecewise-polynomial functions

Authors

Zehua Lai, Lek-Heng Lim and Junyu Ren

Institutions

Chinese Academy of Sciences; University of Chicago

Result date

Paper submitted September 9; four-variable artifact added October 4, 2026

Why it matters: The paper’s short R30 construction refutes the 1956 conjecture, while the new R4 artifact would greatly reduce the dimension and supplies machine-checkable evidence for its encoded endpoint.

Limits: The paper is unrefereed and still presents the hand-checkable R30 construction. The repository reports a clean build and kernel replay for R4, but this maintenance pass did not reproduce the large build; human expert review of the four-dimensional argument remains pending.

A preprint claims a counterexample to Köthe's 90-year-old ring-theory conjecture

Tom Adamczewski, Bernhard Böhmler and René Marczinzik construct, over every countable field, a nil algebra whose 2-by-2 matrix ring is not nil. This yields two nil left ideals with a nonnil sum and therefore a counterexample to Köthe's conjecture, while also answering a question of Rowen negatively.

Problem or result

Köthe's conjecture that the sum of two nil left ideals is nil

Authors

Tom Adamczewski, Bernhard Böhmler and René Marczinzik

Institutions

Epoch AI; Leibniz University Hannover; University of Bonn

Result date

Preprint submitted September 7, 2026

Why it matters: Köthe's conjecture has resisted resolution since 1930 and is a foundational question about nil ideals and matrix rings in noncommutative algebra.

Limits: The construction is an unreviewed first-version preprint and applies over countable fields. No independent specialist verification or published version was located in this cycle.

An AI-assisted explicit ideal is claimed to disprove the general Huneke–Wiegand conjecture

Son Pham constructs a one-dimensional Gorenstein local domain and a nonprincipal two-generated ideal whose tensor product with its dual is torsion-free, giving a counterexample to the conjecture's general Gorenstein formulation over every field.

Problem or result

The general one-dimensional Gorenstein Huneke–Wiegand tensor-product conjecture

Authors

Son Pham

Institutions

Independent researcher

Result date

Preprint submitted September 7, 2026

Why it matters: The conjecture has shaped rigidity questions in commutative algebra since the 1990s. The explicit numerical-semigroup example supports both algebraic proofs and small exact certificates.

Limits: The manuscript has not been peer reviewed. Craig Huneke is reported to have independently recomputed the semigroup and colon-ideal data and verified rigidity, but the tracker has not independently run the two supplied Python verifiers. The counterexample does not cover complete-intersection restrictions.

Two independent preprints claim uniform Turán density one half for the tetrahedron

Kielak and collaborators and, independently, Matija Bucić claim that the uniform Turán density of the complete 3-uniform tetrahedron equals one half. The papers use distinct combinatorial and Fourier-analytic routes.

Problem or result

Uniform Turán density of the complete 3-uniform tetrahedron

Authors

Bartłomiej Kielak, Daniel Král’, Ander Lamaison, Hong Liu, Xichao Shu and Zhuo Wu; independently, Matija Bucić

Institutions

Leipzig University; MPI MiS; Public University of Navarre; Institute for Basic Science; UPC

Result date

Independent preprints revised through September 24, 2026

Why it matters: The uniform problem is a long-open extremal-hypergraph question and fixes its exact density. It must not be confused with the original unrestricted 1941 Turán tetrahedron problem, which remains open.

Limits: Both proof routes use supplied exact or computer-checkable certificates and neither is peer reviewed. Independent convergence is strong support but not journal acceptance or a complete outside audit; the unrestricted 1941 Turán problem remains open.

A ChatGPT-generated construction is claimed to refute Yau's scalar-curvature integral bound

Tianze Hao and Jintian Zhu construct a complete one-ended three-manifold, diffeomorphic to R3 and with strictly positive Ricci curvature, for which the scalar-curvature integral over radius-R balls divided by R has infinite limsup. This would refute Yau's unrestricted asymptotic scalar-curvature bound.

Problem or result

Yau's asymptotic scalar-curvature integral bound under nonnegative Ricci curvature

Authors

Tianze Hao and Jintian Zhu

Institutions

Peking University; Westlake University

Result date

Submitted September 6 and announced September 9, 2026

Why it matters: The construction targets a remaining unrestricted scalar-curvature case after positive results under stronger geometric assumptions and would also defeat a related one-ended conjecture.

Limits: This is a long first-version preprint with no external review or formal proof artifact. The authors say they checked the mathematics and take responsibility, but that is not independent validation.

Independent preprints claim a dimension-free Komlós discrepancy bound

Guo, Fang and Lu claim a universal 3√(2π) discrepancy bound for unit-norm columns; Karingula and Lovett give a separate elementary proof with constant 36; and an algorithmic follow-up claims polynomial-time constant discrepancy.

Problem or result

The dimension-free Komlós discrepancy conjecture and an algorithmic construction

Authors

Shengtao Guo, Ethan X. Fang and Junwei Lu; Sankeerth Rao Karingula and Shachar Lovett

Institutions

Harvard T.H. Chan School of Public Health; Agentin AI; University of California, San Diego

Result date

Preprints submitted September 10–20 and revised through September 22, 2026

Why it matters: A universal constant is the conjectured threshold. The separate elementary proof provides meaningful technical corroboration, and the spectral follow-up claims a polynomial-time construction.

Limits: All three papers are unrefereed preprints. The reproof simplifies the original rather than peer-reviewing it, and the algorithm’s stated polynomial is very large rather than a practical benchmark. Ercan’s earlier log-quarter result remains valid historical progress.

A revised preprint claims all three de Bruijn–Erdős consecutive-gap conjectures

Samuel Korsky proves quantitative lower bounds for the largest and smallest r-spans formed by inserting points on a circle and a matching-order bound for their ratio, resolving the three asymptotic conjectures posed by de Bruijn and Erdős in 1949.

Problem or result

The three 1949 de Bruijn–Erdős asymptotic consecutive-gap conjectures

Authors

Samuel Korsky

Institutions

No affiliation stated in the manuscript

Result date

Version 2 submitted September 9, 2026

Why it matters: The revision expands a one-part result into a claimed resolution of the full 77-year-old problem and determines the ratio's order up to an absolute constant.

Limits: Version 1 resolved only one of the three limits; the broader version 2 is a fresh, non-peer-reviewed claim with no located external specialist assessment.

An Astra-assisted preprint claims the remaining case of Erdős Problem 1040

Ioannis Tzachristas proves that every compact plane set of logarithmic capacity one admits monic polynomials with zeros in the set whose unit lemniscates have arbitrarily small area. Combined with prior work and an unbounded-set argument, this answers the vanishing question for every closed infinite set of transfinite diameter at least one.

Problem or result

Vanishing unit-lemniscate area in Erdős Problem 1040

Authors

Ioannis Tzachristas

Institutions

No affiliation stated in the manuscript

Result date

Submitted September 5 and announced September 9, 2026

Why it matters: The capacity-one case was the boundary left open by earlier results; the preprint would complete the affirmative vanishing classification at and above the sharp capacity threshold.

Limits: The author explicitly labels this an AI-assisted proposed result. It has no proof-assistant artifact, independent human verification or peer review, and it does not give degree bounds or explicit root sets.

An Astra-attributed preprint lowers the bounded prime-gap record to 186

An OpenAI preprint proves DHL[40,2] and concludes that infinitely many pairs of consecutive primes differ by at most 186, improving the long-standing published bound of 246 and a concurrent 240 result.

Problem or result

The best unconditional upper bound for the limit inferior of consecutive-prime gaps

Authors

OpenAI; proof attributed to GPT-6 Astra

Institutions

OpenAI

Result date

Paper dated August 30, 2026; publicly announced September 3, 2026

Why it matters: Bounded prime gaps are a central post-Zhang problem in analytic number theory. A correct 186 bound would be the first improvement below 246 in more than a decade and a striking AI-assisted advance.

Limits: The result is an unreviewed preprint. Its Lean derivation remains conditional on three explicit input axioms, including cited finite-field estimates and numerical integral/cap bounds; the numerical certificate does not discharge those axioms. The twin-prime gap of two remains unproved.

An Astra-attributed proof improves the classic lower bound for large prime gaps

An OpenAI preprint proves that the largest prime gap below X is bounded below, up to a constant, by log X times (log₂ X)² times log₄ X divided by (log₃ X)², improving Rankin's classical expression by a factor of log₂ X.

Problem or result

Asymptotic lower bounds for unusually large gaps between consecutive primes

Authors

OpenAI; proof attributed to GPT-6 Astra

Institutions

OpenAI

Result date

Publicly announced September 3, 2026

Why it matters: The improved iterated-log factor changes a term in the classical large-gap scale that OpenAI says had resisted improvement for more than 80 years, while supplying a machine-checkable proof artifact.

Limits: This is an unreviewed institutional preprint, and the tracker did not independently rebuild the Lean repository. The theorem is asymptotic and hides an unspecified positive constant; it is not an explicit new largest numerical prime gap.

A 146-page preprint claims a complete classification of Forsythe's conjecture

Matthew Colbrook, George Stepaniants and Alex Townsend prove that Forsythe's restarted-conjugate-gradient conclusion holds exactly for restart lengths one, two and three, while diagonal positive-definite counterexamples exist for every restart length at least four.

Problem or result

Forsythe's 1968 asymptotic-direction conjecture for restarted conjugate gradients

Authors

Matthew J. Colbrook, George Stepaniants and Alex Townsend

Institutions

University of Cambridge; California Institute of Technology; Cornell University

Result date

Preprint submitted September 4, 2026

Why it matters: The work gives a sharp restart-length boundary for a nearly six-decade-old numerical-linear-algebra problem and combines analytic dynamics with exact rational interval and Sturm-sequence certification.

Limits: The 146-page first-version preprint has not been peer reviewed. Exact programs certify the finite restart-four premises, while the general higher-length construction is analytic; no independent specialist correctness assessment was located.

Dense-graph Erdős–Sós progress also resolves an Erdős–Graham Ramsey problem

Bruce Reed and Maya Stein prove the Erdős–Sós conjecture for sufficiently large host graphs when the tree size is at least a fixed positive fraction of the host size. As a corollary, they report a solution to a 51-year-old Erdős–Graham problem on multicolor Ramsey numbers of trees.

Problem or result

Dense-regime Erdős–Sós theorem and a 51-year-old Erdős–Graham tree-Ramsey problem

Authors

Bruce Reed and Maya Stein

Institutions

Academia Sinica; Universidad de Chile and Centro de Modelamiento Matemático

Result date

Companion preprints submitted September 4, 2026

Why it matters: Erdős–Sós is a central tree-embedding conjecture from the 1960s. The exact dense-regime theorem removes maximum-degree restrictions and yields a separate long-open Ramsey-theory consequence.

Limits: This does not prove the full Erdős–Sós conjecture: it assumes sufficiently large n and k at least a fixed positive fraction of n. Both papers are first-version preprints without located peer review.

A Lean-certified counterexample answers Pommerenke's convexity question negatively

Yuankai Guo and Xiaozhe Hu construct explicit convex functions in the exterior-disk class Σ whose convex linear combination remains univalent but is not convex, answering Pommerenke's 1962 question in the negative.

Problem or result

Pommerenke's Problem 6.10 on convex combinations in the exterior-disk class Σ

Authors

Yuankai Guo and Xiaozhe Hu

Institutions

No current institutional affiliations stated in the manuscript

Result date

Preprint submitted September 3, 2026

Why it matters: The explicit algebraic construction closes a question carried in Hayman's problem collection and pairs a short human-readable argument with a substantial formal proof artifact.

Limits: This is a six-page, non-peer-reviewed first version. The authors report 32 Lean files with no custom axioms or unfinished placeholders, but this maintenance cycle did not independently rebuild the repository.

An AI-generated graph is claimed to disprove Teschner's bondage-number conjecture

Yousof Yavari gives a connected 18-vertex cubic bipartite graph with domination number six and bondage number five, contradicting Teschner's proposed bound b(G) ≤ 3Δ(G)/2.

Problem or result

Teschner's upper-bound conjecture for graph bondage numbers

Authors

Yousof Yavari

Institutions

University of British Columbia

Result date

Preprint submitted September 1 and announced September 7, 2026

Why it matters: A small explicit graph would settle a three-decade-old universal bound negatively, and the complete standard-library verifier makes every finite enumeration independently reproducible.

Limits: The nine-page manuscript is not peer reviewed. The exact verifier checks the stated graph and all four-edge deletion sets, but the tracker did not independently rerun it in this cycle.

A counterexample breaks the Chung–Graham–Spiro gap-set conjecture at step four

Mohsen Aliabadi proves that the four-step gap sets of the up- and down-integers from slow Fibonacci walks differ, specifically with 9 in U₄ but not D₄, disproving equality for every step length.

Problem or result

The Chung–Graham–Spiro conjecture equating all up/down slow-Fibonacci gap sets

Authors

Mohsen Aliabadi

Institutions

Clayton State University

Result date

Preprint submitted September 3, 2026; journal acceptance reported

Why it matters: The result pinpoints the first stated failure of a structural conjecture about the local spacing of the integer partition generated by slow Fibonacci walks.

Limits: The arXiv record says the paper is to appear in the International Journal of Number Theory, but a final journal DOI or version of record was not located. The theorem is narrow and concerns step four.

RSA-260 has been factored, setting a public general-purpose record

Eric Lu published a 130-digit divisor of RSA-260. Dividing the 260-digit, 862-bit challenge number produces a second 130-digit factor, and multiplying the pair exactly reconstructs RSA-260.

Problem or result

Factorization of the 260-decimal-digit, 862-bit RSA challenge number

Authors

Eric Lu

Institutions

Cognition (employer reported by Scientific American)

Result date

Factor announced September 3, 2026

Why it matters: RSA-260 is larger than the previously factored RSA-250 and is readily checkable once a factor is known, making it a clear computational-number-theory milestone.

Limits: The primary post supplies a factor but no algorithm, runtime, hardware, cost or reproducibility record. The result does not threaten modern 2048-bit RSA, and exact multiplication verifies the factorization without explaining how it was found.

A short preprint claims the Daykin–Frankl width conjecture

Kada Kálmán Williams presents a proof that every convex subset of the Boolean cube contains an antichain of at least its cardinality times the central-binomial density, confirming a conjecture of Daykin and Frankl from 1983.

Problem or result

Daykin and Frankl’s 1983 lower bound on the width of convex subsets of the Boolean cube

Authors

Kada Kálmán Williams

Institutions

Independent author in Szeged, Hungary

Result date

First-version preprint submitted September 2, 2026

Why it matters: The inequality gives the conjectured sharp-scale antichain guarantee for every convex family in the Boolean lattice, resolving a four-decade-old extremal-combinatorics question if the proof holds.

Limits: The argument is a four-page first-version preprint generated by an LLM and communicated after author verification. No peer review, independent proof, formal artifact or specialist correctness assessment was located.

A preprint gives a counterexample to McKean’s conjecture for Landau–Coulomb dynamics

Jonathan Junné, Raphael Winter and Havva Yoldaş construct smooth radial perturbations arbitrarily close to Maxwellian equilibrium for which entropy dissipation initially increases in the spatially homogeneous Landau–Coulomb equation. Their argument also transfers to a grazing-collision Boltzmann regime.

Problem or result

McKean’s proposed monotonic entropy-dissipation hierarchy for kinetic equations

Authors

Jonathan Junné, Raphael Winter and Havva Yoldaş

Institutions

University of Rennes–Inria–CNRS; Cardiff University; Delft University of Technology

Result date

First-version preprint submitted September 3, 2026

Why it matters: McKean’s conjecture is a proposed strengthening of Boltzmann’s H-theorem. A counterexample near equilibrium delineates exactly which higher-order monotonicity principles fail in physically important models.

Limits: This is a first-version, non-peer-reviewed preprint. The result concerns the homogeneous Landau–Coulomb equation and a grazing-collision Boltzmann regime; it does not invalidate the ordinary H-theorem, which only asserts monotonic entropy.

A preprint claims a full proof of Fraenkel’s Beatty-sequence conjecture

Hu Tan and Ying Zhang claim that every exact partition of the integers into at least three Beatty sequences with distinct moduli has Fraenkel’s binary density pattern. Their proof reduces the problem to a dimension-free one-third density theorem, exact finite checks and a deletion induction.

Problem or result

Fraenkel’s 1973 classification of exact partitions by distinct Beatty sequences

Authors

Hu Tan and Ying Zhang

Institutions

Academy of Mathematics and Systems Science, Chinese Academy of Sciences; Soochow University

Result date

Preprint v2 submitted October 2, 2026; Lean artifact uploaded the same day

Why it matters: The conjecture links exact integer coverings to a rigid binary pattern and had previously been proved only through seven components. A correct general proof would close the remaining cases uniformly.

Limits: The 38-page v2 remains unrefereed. Its Lean package contains no sorry or admit tokens but declares 13 explicit axioms, including three imported certificate assertions, so it is useful partial formalization evidence rather than an end-to-end derivation from Lean’s foundational axioms.

A preprint claims the truncated octahedron is the unique least-area parallelohedron

Thomas Hales and Lark Song claim that, among three-dimensional parallelohedra of fixed volume, the Archimedean truncated octahedron uniquely minimizes surface area. The final scalar inequality is supported by an exact computer-assisted certificate.

Problem or result

Károly Bezdek’s 2006 fixed-volume surface-area conjecture for three-dimensional parallelohedra

Authors

Thomas Hales and Lark Song

Institutions

University of Pittsburgh; University of Wisconsin–Madison

Result date

Current v2 revised September 11, 2026

Why it matters: The result would give a sharp global isoperimetric classification across all five three-dimensional parallelohedron types and settle a geometric problem tied to space-filling cells.

Limits: The simplified 23-page v2 remains unrefereed. Its exact Bernstein certificate was introduced earlier in the argument, but this maintenance cycle did not independently replay the full computation; the theorem concerns translative polyhedral tilings, not unrestricted Kelvin foams.

A preprint gives a counterexample to the stable forking conjecture

James Freitag and Scott Mutchnik construct a simple theory whose forking is not always witnessed by a stable formula, contradicting the stable forking conjecture posed by Hart, Kim and Pillay in 1996.

Problem or result

The 1996 stable forking conjecture in simple theories

Authors

James Freitag and Scott Mutchnik

Institutions

University of Illinois Chicago

Result date

First submitted August 31; substantially revised October 19, 2026

Why it matters: Stable forking is a prominent proposed bridge between stable and simple model theory. A valid counterexample would settle the general conjecture negatively and sharpen the boundary between the two.

Limits: This remains a non-peer-reviewed preprint without an independent specialist correctness assessment, formal proof or separate construction. Version 2 corrects an algebraic sign in the Ore-condition argument, expands the manuscript from 21 to 26 pages and adds a claimed negative answer to Ziegler's one-variable reduction question without changing the headline counterexample.

A preprint claims a complete, Lean-formalized solution to Colombo’s determinant problem

Qianli Ma proves a necessary-and-sufficient condition for the determinant of the matrix with entries (x_j−x_i)^D to be nonzero for distinct real nodes, completing Colombo’s 1928 problem. The new odd-exponent branch is accompanied by a Lean 4 formalization.

Problem or result

Colombo’s 1928 determinant nonvanishing classification for powers of pairwise differences

Authors

Qianli Ma

Institutions

Zhejiang University; WuJie AI

Result date

First-version preprint submitted August 31, 2026

Why it matters: The result supplies the missing odd-exponent classification for a problem originating in hyperbolic partial differential equations. The independent odd-branch preprint and Lean artifact provide stronger corroboration than an isolated informal claim.

Limits: The complete classification remains a non-peer-reviewed preprint. The repository reports a pinned Lean 4.30/mathlib build with 2,817 jobs and no sorry/admit declarations, but this cycle reviewed the artifact metadata and source rather than independently rebuilding the full formalization.

A preprint claims a full proof of the Arnold–Givental conjecture

Shaoyun Bai, Egor Shelukhin, Yi Wang and Guangbo Xu claim the full Arnold–Givental intersection bound over F₂ for fixed loci of anti-symplectic involutions in closed symplectic manifolds. Their argument combines integral Floer theory, an equivariant PSS map and global Kuranishi-chart techniques.

Problem or result

The Arnold–Givental lower bound for intersections of a real Lagrangian fixed locus with its Hamiltonian image

Authors

Shaoyun Bai, Egor Shelukhin, Yi Wang and Guangbo Xu

Institutions

MIT; Université de Montréal; Hetao Institute of Mathematics and Interdisciplinary Sciences; Rutgers University

Result date

First-version preprint submitted August 27, 2026

Why it matters: The 1989 conjecture is a central symplectic-rigidity statement. The claimed proof gives the full mod-two intersection bound without the restrictions that limited earlier cases.

Limits: This is a first-version, non-peer-reviewed preprint. Amanda Hirschi separately announced joint work in progress with Noah Porcelli on August 26, but no manuscript was available, so that announcement is context rather than independent verification.

Matrix Spencer now has explicit constants, algorithms and a separate exposition

Akbas and Sra’s O(√n) proof claim now has a separate exposition by Bansal and Kook, while Song and Zhang claim existence below 8√n, a randomized algorithm below 12√n in n^(3+o(1)) real-arithmetic operations for dense input, and a (2−o(1))√n lower bound.

Problem or result

The Matrix Spencer conjecture for operator-norm discrepancy of symmetric matrices

Authors

Emrullah Akbas and Suvrit Sra; Zhao Song and Lichen Zhang; Nikhil Bansal and Yunbum Kook

Institutions

Technical University of Munich; MIT; Song lists no institution in the inspected preprint

Result date

Original August 28; technical follow-ups submitted September 14 and revised through September 21, 2026

Why it matters: Matrix Spencer extends classical vector discrepancy to noncommuting matrices. The O(√n) scale matches the conjectured order and would close the general form left open by structured and moderate-rank results.

Limits: All sources remain preprints. The exposition improves checkability but is not peer review. Song–Zhang runtime is stated in a real-arithmetic model, not as a practical benchmark.

A preprint gives a counterexample to Nevanlinna’s century-old half-plane problem

Yixin He and Teng Zhang construct a nonconstant meromorphic function on the complex plane whose zeros, one-points and poles all lie on the real axis but whose restriction to the upper half-plane is not of bounded type. This would answer Nevanlinna’s unrestricted 1925 question negatively.

Problem or result

Nevanlinna’s 1925 half-plane bounded-type problem for entire-plane meromorphic functions

Authors

Yixin He and Teng Zhang

Institutions

Fudan University; Xi’an Jiaotong University

Result date

First-version preprint submitted August 25, 2026

Why it matters: The problem had survived a century of positive results under finite-order and related growth conditions. The construction would show that those hypotheses cannot be removed.

Limits: The manuscript is not peer reviewed and no independent proof or specialist assessment was located. The counterexample concerns the unrestricted problem and does not overturn Nevanlinna’s finite-order theorem or later positive results under explicit growth conditions.

A preprint claims the planar Khavinson–Shapiro conjecture

Feng Shao and Weicheng Zhan claim that a bounded planar domain whose boundary is a finite union of disjoint Jordan curves has polynomial Dirichlet solutions for every polynomial boundary datum only when its boundary is an ellipse and the domain is the ellipse’s interior.

Problem or result

The planar Khavinson–Shapiro classification of domains with polynomial solutions to polynomial Dirichlet data

Authors

Feng Shao and Weicheng Zhan

Institutions

Xiamen University

Result date

First-version preprint submitted August 30, 2026

Why it matters: The conjecture asks whether exact polynomial solvability forces a domain’s geometry to be quadratic. The claimed theorem settles the planar case without assuming a smooth or connected boundary.

Limits: This is a first-version, non-peer-reviewed preprint with no located independent correctness assessment. Its scope is two-dimensional bounded domains with finitely many Jordan boundary components; it does not settle the higher-dimensional conjecture.

A formally verified bound puts more than two thirds of zeta zeros on the critical line and simple

The manuscript proves unconditionally that at least two thirds of the nontrivial Riemann-zeta zeros, counted with multiplicity, are simple and on the critical line and that at least five sixths are distinct. An optimized window raises the constants to 0.6725 and 0.8362, and the results extend to primitive Dirichlet L-functions.

Problem or result

Unconditional lower bounds for simple and distinct nontrivial zeros of the Riemann zeta function

Authors

Levent Alpöge and Ralph Furman; argument credited to Claude

Institutions

Anthropic hosted the project; the manuscript does not print formal author affiliations

Result date

Submitted August 13; manuscript dated August 17, 2026

Why it matters: The result raises the previous unconditional simple-on-line bound from 5/12 to above two thirds and supplies a formal artifact for a major analytic-number-theory advance without assuming the Riemann hypothesis.

Limits: This does not prove the Riemann hypothesis, simplicity of every zero, or any statement about the remaining fraction. It is a very recent preprint rather than a refereed article; formal verification establishes the encoded theorems, while specialists must still assess encoding, novelty and mathematical significance.

Two preprints claim to settle Banach’s isometric conjecture over real, complex and quaternionic spaces

Xinbao Lu and Kaiwen Yang claim the remaining real odd-dimensional cases, while Antonio Acuaviva and Tomasz Kania claim the remaining complex cases and a quaternionic counterpart. Together with earlier results, the two first-version preprints would settle the classical finite-dimensional real and complex isometric-subspace conjectures.

Problem or result

Banach’s 1932 isometric-subspace conjecture over real and complex scalars, plus a quaternionic analogue

Authors

Xinbao Lu and Kaiwen Yang; Antonio Acuaviva and Tomasz Kania

Institutions

Tongji University; Lancaster University; Czech Academy of Sciences and Jagiellonian University

Result date

Real-case preprint submitted August 13 and revised August 31; complex/quaternionic preprint submitted August 18, 2026

Why it matters: If both arguments hold, they close the remaining finite-dimensional cases of a 94-year-old problem connecting normed spaces, ellipsoids, bundle topology and Brouwer degree theory, and extend the mechanism to quaternionic spaces.

Limits: The real-case manuscript is now version 2, with a substantially expanded and restructured proof but an unchanged theorem claim. Both manuscripts remain non-peer-reviewed preprints. No independent specialist correctness assessment or formal proof was located, and the complex/quaternionic paper adapts the real-case mechanism rather than independently confirming it.

Six-author analysis tightens the Petersen Coloring counterexample to 40–52 vertices

Putman’s 112-vertex disproof remains independently supported. Goedgebeur, Jooken, Máčajová, Mattiolo, Mazzuoccolo and Ulyanov now give two 52-vertex counterexamples, a purely theoretical proof, an infinite family for every even order at least 60 and a computational lower bound of 40 vertices.

Problem or result

Jaeger’s Petersen Coloring Conjecture for bridgeless cubic graphs

Authors

Bryce Putman; Jan Goedgebeur, Jorik Jooken, Edita Máčajová, Davide Mattiolo, Giuseppe Mazzuoccolo and Nikolay Ulyanov

Institutions

Leiden University; KU Leuven; Ghent University; Comenius University; University of Modena and Reggio Emilia

Result date

112-vertex manuscripts August 8–14; six-author v4 revised September 30, 2026

Why it matters: The original construction refuted the conjecture. The later theoretical analysis narrows the possible minimum to 40–52 vertices and supplies counterexamples at every even order from 60 onward.

Limits: The current six-author analysis is not peer reviewed. Its theoretical proof and computational lower bound materially strengthen the record, but the exact minimum between 40 and 52 vertices remains unknown.

A preprint claims a fivefold counterexample to the cscK Yau–Tian–Donaldson conjecture

Jihao Liu constructs a smooth polarized projective fivefold claimed to be K-polystable but to admit no constant-scalar-curvature Kähler metric. If correct, it disproves the broad cscK form of the Yau–Tian–Donaldson conjecture in dimension five.

Problem or result

The cscK Yau–Tian–Donaldson conjecture for smooth polarized projective varieties

Authors

Jihao Liu

Institutions

Beijing International Center for Mathematical Research, Peking University

Result date

Preprint submitted August 19; manuscript dated August 21, 2026

Why it matters: The conjecture is a central bridge between algebro-geometric stability and canonical Kähler metrics. A valid counterexample would show that K-polystability alone is insufficient for the general cscK problem.

Limits: This is a 79-page first-version preprint with no located peer review, formal verification or independent correctness assessment. Its scope is the broad cscK formulation; it does not undo established Yau–Tian–Donaldson theorems in more restricted Fano or Kähler–Einstein settings.

Independent preprints claim Pólya’s Neumann bound for every Euclidean ball

Two independently developed preprints claim Pólya’s lower bound for the Neumann eigenvalue-counting function of every Euclidean ball. Together with the published Dirichlet result and Neumann-disk case, the claims would settle both Pólya inequalities for balls in every dimension at least two.

Problem or result

Pólya’s conjectured lower bound for the Neumann Laplacian on Euclidean balls

Authors

Yutian Li; independently, Nikolay Filonov, Michael Levitin, Iosif Polterovich and David A. Sher

Institutions

Nanfang College, Guangzhou; Steklov Institute of Mathematics; University of Reading; Université de Montréal; DePaul University

Result date

Li v3; independent branch v2 revised September 23, 2026

Why it matters: Pólya’s 1954 spectral bound is known for tiling domains and selected non-tiling domains, but higher-dimensional Neumann balls resisted the methods that settled the disk. Two groups now report different dimension-uniform arguments, including exact or computer-assisted finite checks.

Limits: Both manuscripts remain unrefereed. Li is at v3; the independent Filonov–Levitin–Polterovich–Sher branch is at v2 with minor revisions and 42 pages. Concurrent proofs improve confidence but do not replace specialist assessment.

Two preprints claim a proof of Crouzeix’s conjecture

Two preprints give independent arguments for the sharp constant two in Crouzeix’s inequality: for a matrix or bounded operator A and an appropriate function f, the operator norm of f(A) is at most twice the maximum of |f| on A’s numerical range.

Problem or result

Crouzeix’s conjectured optimal constant two for functions of matrices and operators

Authors

Emiel Lorist and Felix Schwenninger; independently, Shanmu Jin

Institutions

Delft University of Technology and University of Twente; Jin’s affiliation was not verified from the inspected source

Result date

Lorist–Schwenninger preprint submitted August 4; Jin manuscript revised through August 7, 2026

Why it matters: The conjecture has been a central sharp-bound problem connecting non-normal matrices, spectral sets and numerical analysis. The factor two is known to be optimal, so a correct proof would close the remaining gap from the published 1 + square-root-of-two bound.

Limits: Both manuscripts are unreviewed. Lorist and Schwenninger’s arXiv posting is a five-page first version, and Jin’s posting is a Preprints.org manuscript rather than an independent specialist assessment. Concurrent arguments strengthen the signal but do not substitute for peer review.

Independent preprints give planar counterexamples to the Schiffer and Pompeiu conjectures

Two independent preprints construct noncircular bounded simply connected planar domains with a Neumann eigenfunction constant on the boundary, contradicting Schiffer’s conjecture and yielding counterexamples to the associated planar Pompeiu conjecture.

Problem or result

The planar Schiffer conjecture and the equivalent rigidity direction of the Pompeiu problem

Authors

Matthew J. Colbrook and George Stepaniants; independently, Gonzalo Cao-Labora and Jaume de Dios Pont

Institutions

University of Cambridge; California Institute of Technology; École Polytechnique Fédérale de Lausanne; New York University

Result date

Preprints submitted August 3 and August 5, 2026

Why it matters: The Pompeiu problem dates to 1929, and Schiffer’s spectral formulation became a prominent rigidity conjecture. The two groups reach the same negative conclusion using different constructions: an exact computer-assisted certificate and an analytic bifurcation family with Lean verification.

Limits: Neither manuscript has been peer reviewed. One argument depends on a rigorous computational certificate and the other on a new bifurcation analysis plus a claimed Lean check; both require specialist scrutiny. The stated counterexamples concern planar simply connected domains.

A preprint claims the period–index conjecture is false in dimension three and higher

Alexander Perry constructs period-two Brauer classes of index 2^d on smooth projective d-folds, violating the proposed period–index bound in every dimension at least three. Version 2 extends the construction to positive characteristic and adds a corrected small-prime formulation with a proved Hodge-theoretic counterpart.

Problem or result

The period–index conjecture for Brauer classes on algebraic varieties

Authors

Alexander Perry

Institutions

University of Michigan

Result date

Preprint submitted August 4 and revised September 3, 2026

Why it matters: Period and index measure different obstructions carried by a Brauer class. A valid Hodge-theoretic counterexample would overturn the proposed general bound in every dimension at least three while leaving the established surface case intact.

Limits: The expanded 41-page version 2 remains a non-peer-reviewed preprint with no located independent correctness assessment. Its positive-characteristic scope excludes characteristic two and depends on specified transcendence-degree hypotheses; the established surface theorem remains intact.

A preprint gives real-valued counterexamples to Landis’s conjecture in dimensions three and higher

Frank and Ivanisvili construct, in every dimension at least three, a nonzero solution of the stationary Schrödinger equation with bounded real-valued potential that decays like exp(−c|x|^(4/3)), giving the claimed negative answer to Landis’s conjecture in those dimensions.

Problem or result

Landis’s unique-continuation conjecture for real-valued Schrödinger potentials

Authors

Rupert L. Frank and Paata Ivanisvili

Institutions

LMU Munich and Munich Center for Quantum Science and Technology; University of California, Irvine

Result date

Preprint submitted August 1, 2026

Why it matters: The exponent four-thirds is the sharp complex-valued decay scale from Meshkov’s construction. Producing the same scale with a real potential resolves the higher-dimensional form of a long-standing unique-continuation question.

Limits: This is a 34-page first-version preprint without located independent specialist assessment or peer review. The construction applies in dimensions three and higher; the real-valued two-dimensional problem remains open.

Claimed counterexamples to matroid Kazhdan–Lusztig real-rootedness and log-concavity

The authors construct representable matroids over every finite field whose Kazhdan–Lusztig polynomials are not even unimodal. The examples therefore refute the stronger conjectures that all such polynomials are log-concave or real-rooted.

Problem or result

Real-rootedness and log-concavity conjectures for Kazhdan–Lusztig polynomials of matroids

Authors

Ronnie Cheng and Shurui Liu

Institutions

Stanford University; the disclosed Rethlas research team

Result date

Preprint submitted July 27, 2026

Why it matters: The conjectures had survived extensive computation and proofs for many important matroid families. The projective-deletion construction gives counterexamples over every finite field and makes the failure stronger than non-real roots alone.

Limits: The manuscript is a first-version preprint and has not been conventionally peer reviewed. A specialist MathOverflow assessment reports that an independent counterexample was announced at ICM 2026 and verified, which supports the negative conclusion but does not review every argument in this paper.

A claimed smooth-projective counterexample to the Bondal–Polishchuk conjecture

A preprint gives a weak Fano threefold whose bounded derived category has two full exceptional collections in different braid-group orbits, claimed as the first smooth-projective counterexample to the Bondal–Polishchuk conjecture. Its expanded second version also reports infinitely many connected components in the threefold’s Bridgeland stability space.

Problem or result

Bondal and Polishchuk’s 1993 transitivity conjecture for full exceptional collections

Authors

Anya Nordskova

Institutions

Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo

Result date

Preprint submitted July 28; substantially expanded second version posted August 3, 2026

Why it matters: Earlier published counterexamples lived in partially wrapped Fukaya categories. The construction targets the smooth-projective setting that motivated the original conjecture, and the added stability-space result would give the first known disconnected Bridgeland stability space for a smooth projective variety.

Limits: This is an 18-page second-version preprint without located independent correctness assessment or peer review. The expanded manuscript retains the same core counterexample and adds refinements and stability-condition results; the broader conjecture had already failed in other triangulated categories.

A claimed complete proof of the Erdős–Gyárfás monochromatic path-cover conjecture

The authors claim that, for every positive integer n, every red–blue edge-colouring of the complete graph K n has a same-colour collection of at most √n paths covering every vertex. This removes the finite-size gap left by the published large-n theorem.

Problem or result

The 1995 Erdős–Gyárfás monochromatic path-cover conjecture

Authors

Hangdi Chen and Yaojun Chen

Institutions

Putian University; Nanjing University

Result date

Preprint submitted July 24, 2026

Why it matters: The conjecture asks for the sharp √n cover bound in every finite red–blue complete graph, a natural extremal question open since 1995. A 2026 Journal of Combinatorial Theory, Series B paper had proved it only for n greater than 20 40 .

Limits: This is a first-version preprint without located independent correctness assessment or peer review. The covering paths may intersect; the statement is not a vertex-disjoint path partition.

A claimed five-variable counterexample to the Hessian conjecture

The preprint gives an explicit degree-14 integer polynomial in five variables with constant Hessian determinant 128 and two distinct rational points having the same gradient. If the identities and bridge arguments stand, the Hessian conjecture is false in every dimension at least five.

Problem or result

The Hessian conjecture in dimension five and its stabilization to higher dimensions

Authors

Guowu Meng and Liang Yang

Institutions

Hong Kong University of Science and Technology; Sichuan University

Result date

Version one dated July 23 and submitted July 24, 2026

Why it matters: Together with known results in dimensions one through three, the construction would leave only the four-variable Hessian conjecture open; that case is linked to the still-open two-variable Jacobian conjecture.

Limits: The draft appeared only days after the Jacobian counterexample on which it builds. It is not peer reviewed, and the authors explicitly request independent computer-algebra checking and expert review. The four-variable Hessian and two-variable Jacobian cases remain open.

Claimed counterexamples to Rota’s matroid-flat unimodality conjecture

The authors construct matroids whose numbers of flats by rank fall and then rise, contradicting the unimodality conjecture attributed to Rota in 1970. Their mechanism applies q-lifts to matroids with sufficiently strong failures of flat-count log-concavity.

Problem or result

Rota’s conjecture that the Whitney numbers of the second kind are unimodal for every matroid

Authors

Alexander Divoux, Chayim Lowen and Shouda Wang

Institutions

Princeton University

Result date

Preprint submitted July 24, 2026

Why it matters: The conjecture is a long-standing structural prediction about matroids. The construction also clarifies how a failure of the stronger Mason log-concavity conjecture can be amplified into a failure of unimodality itself.

Limits: This remains a first-version preprint without conventional peer review. A specialist MathOverflow assessment now describes the result as a disproof of Rota’s conjecture; that is meaningful independent reception, but it is not a line-by-line review of the manuscript’s infinite construction.

The Jacobian conjecture is false in dimension 3

An explicit polynomial map from C³ to C³ has constant Jacobian determinant −2 but is not injective. The example disproves the unrestricted Jacobian conjecture in dimension 3.

Problem or result

Jacobian conjecture

Authors

Levent Alpöge; example credited to Claude Fable after a question from Akhil Mathew

Institutions

Affiliation not stated in the inspected announcement

Result date

Announced July 19, 2026

Why it matters: The conjecture had been open since 1939 and is a central problem in affine algebraic geometry.

Limits: The two-dimensional conjecture remains open. The exact determinant and collision have been checked independently, but no conventional paper or refereed publication by the announcer was located.

A formally verified proof package for the Cycle Double Cover Conjecture

The proof package establishes that every finite loopless bridgeless undirected multigraph has a multiset of cycles covering every edge exactly twice. A Lean formalization and two independent specialist expositions are public.

Problem or result

Cycle Double Cover Conjecture

Authors

OpenAI proof system; independent expositions by Jim Geelen and Sang-il Oum

Institutions

OpenAI; University of Waterloo; Institute for Basic Science

Result date

Announced July 9–10, 2026; expert expositions July 16–17, 2026

Why it matters: The Szekeres–Seymour conjecture is a foundational, decades-old problem in graph theory and a prominent test of AI-generated research mathematics.

Limits: No peer-reviewed journal publication was located. Formal verification checks the encoded theorem and proof, while specialists must still assess exact correspondence with the accepted conjecture and the surrounding mathematical exposition.

Correlated two-sided Gaussian tests can defeat the Benjamini–Hochberg guarantee

An explicit correlated Gaussian factor model, backed by an interval-arithmetic certificate, has asymptotic false discovery rate above 0.0104 when Benjamini–Hochberg is run at nominal level 0.01. A follow-on paper extends the negative result.

Problem or result

False-discovery-rate control for Benjamini–Hochberg under unrestricted Gaussian correlation

Authors

Edgar Dobriban

Institutions

University of Pennsylvania

Result date

Preprint submitted July 13, 2026

Why it matters: The primary counterexamples delimit a standard multiple-testing guarantee under unrestricted Gaussian dependence, while an independent v4 analysis now gives a sharper two-sided upper bound with a matching leading constant for a class of correlations.

Limits: The result does not invalidate Benjamini–Hochberg under independence or its established positive-dependence conditions. Both the primary and follow-on analyses remain unrefereed preprints, and the matching leading constant is proved only for specified correlation classes rather than every dependence structure.

A near-quadratic lower bound in derivative-free convex optimization

The paper proves an Ω(d²/log(d+1)) lower bound for a specified deterministic exact-value-oracle problem at accuracy on the order of d⁻¹ᐟ². This nearly matches the known upper bound up to polylogarithmic factors.

Problem or result

Oracle-complexity gap for deterministic derivative-free convex optimization

Authors

Phillip Kerger

Institutions

Affiliation not stated on the inspected arXiv record

Result date

Preprint submitted July 14, 2026

Why it matters: It sharply narrows a complexity gap dating to the 1990s and limits what derivative-free methods can achieve in the stated oracle model.

Limits: The theorem concerns a specific deterministic oracle model and accuracy regime. Lean verification supports deductive correctness, but independent specialists still need to assess statement correspondence, novelty and the surrounding literature.

Erdős Problem #90: the planar unit-distance conjecture is false

Arbitrarily large planar point sets can determine at least n^(1+c) unit-distance pairs for an absolute c greater than zero, contradicting the proposed n^(1+o(1)) growth. A companion paper gives an explicit exponent greater than 1.014.

Problem or result

Erdős Problem #90, the planar unit-distance conjecture

Authors

Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang and Melanie Matchett Wood

Institutions

Multi-institution collaboration

Result date

Announced and submitted May 20, 2026

Why it matters: The disproved near-linear-growth conjecture had guided the unit-distance problem for decades.

Limits: The extremal unit-distance problem remains open. A large gap persists between the new lower bound and the classical O(n^(4/3)) upper bound.

Erdős Problems #1196 and #1217: primitive sets and divisibility chains

The authors resolve 1966 Erdős–Sárközy–Szemerédi conjectures concerning dense primitive subsets of sufficiently large integers and long divisibility chains. They develop a Markov-chain and von Mangoldt framework for the proofs.

Problem or result

Erdős Problems #1196 and #1217

Authors

Boris Alexeev, Kevin Barreto, Yanyang Li, Jared Duker Lichtman, Liam Price, Jibran Iqbal Shah, Quanyu Tang and Terence Tao

Institutions

Multi-institution collaboration

Result date

Preprint submitted May 1, 2026

Why it matters: The work settles two named Erdős problems and introduces machinery that may apply beyond them.

Limits: The paper also gives a short proof related to the already resolved Erdős primitive-set conjecture #164. That result must not be presented as a third newly solved problem.

Erdős Problem #728: a Lean-formalized resolution of the reconstructed intended statement

The proof establishes an infinite family of middle-range factorial-divisibility triples whose logarithmic gap lies between any prescribed positive constant multiples of log n. A Lean proof is translated into conventional mathematical prose.

Problem or result

Erdős Problem #728

Authors

Nat Sothanaphan; GPT-5.2 Pro and Aristotle operated by Kevin Barreto

Institutions

Harmonic for Aristotle; see the paper for author affiliation

Result date

Submitted January 12, 2026; revised through January 26, 2026

Why it matters: The authors describe it as the first problem in the Erdős Problems database regarded as fully resolved autonomously by an AI system.

Limits: The historical wording of #728 was ambiguous, so the project formalized a reconstructed intended statement. Related older literature also complicates an unqualified novelty claim.

Near-diagonal Ramsey lower bounds improve exponentially

For every fixed C greater than 1, the authors improve the exponential base in the classical Erdős lower bound for r(ℓ,Cℓ). This is the first exponential improvement over the 1947 bound in that near-diagonal regime.

Problem or result

Classical lower bounds for near-diagonal Ramsey numbers

Authors

Jie Ma, Wujie Shen and Shengjie Xie

Institutions

University of Science and Technology of China; Tsinghua University

Result date

Submitted July 17, 2025; revised April 26, 2026

Why it matters: The work breaks decades of stasis and introduces a geometric random-coloring method that has already produced follow-on results.

Limits: The numerical improvement can be extremely small and does not determine Ramsey numbers. It does not improve the equal-size diagonal case C=1.

The Kakeya set conjecture in three dimensions

Every Kakeya set in R³, meaning a set containing a unit line segment in every direction, has both Hausdorff and Minkowski dimension 3.

Problem or result

Three-dimensional Kakeya set conjecture

Authors

Hong Wang and Joshua Zahl

Institutions

NYU Courant and University of British Columbia at announcement

Result date

Preprint submitted February 24, 2025

Why it matters: The theorem resolves the three-dimensional case of a central problem linking harmonic analysis, geometric measure theory, partial differential equations and number theory.

Limits: Higher-dimensional cases remain open. No peer-reviewed publication was located for the long and technically demanding preprint.

Hilbert's tenth problem for rings of integers of number fields

Two independent teams prove that no algorithm can always decide whether an arbitrary polynomial equation has a solution in the ring of integers of a given number field.

Problem or result

Hilbert's tenth problem over rings of integers of number fields

Authors

Peter Koymans and Carlo Pagano; independently Levent Alpöge, Manjul Bhargava, Wei Ho and Ari Shnidman

Institutions

Utrecht University and Concordia University for the first team; see the second manuscript for its exact affiliations

Result date

One proof published online December 1, 2025, in a March 2026 volume; independent proof forthcoming in JAMS

Why it matters: The work extends the negative solution of Hilbert's tenth problem from ordinary integers to the rings of integers of all number fields.

Limits: The quantified domain is rings of integers of number fields. It should not be broadened casually to every ring described as involving algebraic integers.

Erdős's large sum-free subset conjecture

Every n-element set of integers contains a sum-free subset of size at least n/3 + c log log n for an absolute positive c. The guaranteed excess over n/3 therefore tends to infinity.

Problem or result

Erdős's 1965 large sum-free subset conjecture

Authors

Benjamin Bedert

Institutions

University of Oxford at the time of the work

Result date

Preprint submitted February 12, 2025

Why it matters: The result settles a simple-to-state, 60-year-old problem connecting additive combinatorics and harmonic analysis.

Limits: The theorem proves an unbounded improvement of the stated order. It does not determine the optimal excess term.

The bunkbed conjecture is false

The authors construct an explicit finite planar graph with 7,222 vertices that violates the bunkbed percolation inequality.

Problem or result

Bunkbed conjecture in percolation

Authors

Nikita Gladkov, Igor Pak and Aleksandr Zimin

Institutions

University of California, Los Angeles and Massachusetts Institute of Technology at publication

Result date

Preprint October 3, 2024; published June 13, 2025

Why it matters: The peer-reviewed counterexample overturns an intuitively compelling conjecture studied for roughly four decades.

Limits: The disproof does not show that every bunkbed-type inequality fails. Restricted graph classes and related variants may remain true.

A counterexample to the Mizohata–Takeuchi conjecture

The paper constructs counterexamples for hypersurfaces not contained in a hyperplane, disproving the Mizohata–Takeuchi conjecture in its stated generality and the related formulation of Stein's conjecture discussed in the paper.

Problem or result

Mizohata–Takeuchi conjecture

Authors

Hannah Mira Cairo

Institutions

Work developed while studying mathematics at UC Berkeley

Result date

Preprint submitted February 10, 2025

Why it matters: The result changes which weighted estimates researchers can plausibly pursue in Fourier restriction theory.

Limits: Stronger or more restricted replacement statements may survive. The counterexample should not be described as ending the wider restriction-theory research program.

The Friedlander–Iwaniec Gaussian-primes conjecture

For each n congruent to 0 or 4 modulo 6, the authors obtain an asymptotic count of primes of the form p² + nq² with p and q prime. The case n=4 settles the Friedlander–Iwaniec Gaussian-primes conjecture.

Problem or result

Friedlander–Iwaniec Gaussian-primes conjecture

Authors

Ben Green and Mehtaab Sawhney

Institutions

University of Oxford and Columbia University at announcement

Result date

Preprint submitted October 5, 2024; accepted July 22, 2025

Why it matters: The proof overcomes a long-standing parity barrier for simultaneous prime-variable constraints in sieve theory.

Limits: The theorem applies to specified congruence classes of n. It is not a complete classification of all quadratic forms that take prime values. Version three corrected a partial-summation error in Section 8.3 and a minor inaccuracy in Section 8.7. Acta Mathematica lists the paper as accepted and awaiting publication; no final issue or DOI was located.

Modularity theorems for a positive proportion of abelian surfaces

Under specified local, image and ordinary or distinguished hypotheses, the paper proves modularity for a positive proportion of abelian surfaces over the rational numbers.

Problem or result

Modularity of abelian surfaces over the rational numbers

Authors

George Boxer, Frank Calegari, Toby Gee and Vincent Pilloni

Institutions

Imperial College London, University of Chicago and CNRS/Université Paris-Saclay affiliations are represented

Result date

Preprint submitted February 28, 2025

Why it matters: The work extends modularity machinery central to the proof of Fermat's Last Theorem from elliptic curves toward two-dimensional abelian varieties.

Limits: This is not a theorem that every abelian surface over the rational numbers is modular. The positive-proportion and hypothesis restrictions are essential.

The moving sofa problem: a claimed optimality proof

The paper claims that Gerver's 18-section shape has the largest possible area, approximately 2.2195, among planar sofas that can move around a right-angled unit-width hallway corner.

Problem or result

Moving sofa problem

Authors

Jineon Baek

Institutions

Yonsei University at announcement

Result date

Preprint submitted November 29, 2024

Why it matters: If accepted, the proof resolves a famous geometric optimization problem posed in 1966.

Limits: No journal publication was located, and arXiv still showed the original version at the last check. The result should remain labeled as a claimed proof while the long, technical argument is reviewed.

A claimed proof of Godbersen's mixed-volume conjecture

The authors claim the sharp Godbersen bound for the mixed volume of a convex body and its reflection, confirming the conjectured simplex value. They also prove an Lp Rogers–Shephard inequality and classify the equality cases among convex polytopes.

Problem or result

Godbersen's 1938 conjecture on mixed volumes of a convex body and its reflection

Authors

Jan Kotrbatý and Mohamed A. Mouamine

Institutions

Mathematical Institute of Charles University

Result date

Current v2 revised September 14, 2026

Why it matters: The conjecture has stood since 1938 and sits inside the Brunn–Minkowski theory of convex bodies and mixed volumes.

Limits: The current v2 remains unrefereed. It simplifies the main proof and adds a symmetric-body proposition, but no independent specialist assessment, journal acceptance or formal verification was located.

Claimed counterexamples to the Duval–Reiner majorization conjecture

Huang reports strict counterexamples to the majorization assertion for every index r at least 5 in some uniformity and for every uniformity q at least 3 at some index. The paper separately proves the universal second partial-sum inequality and classifies its equality cases.

Problem or result

Duval–Reiner majorization assertion for simplicial up-Laplacian spectra

Authors

Jing Huang

Institutions

Guangzhou University

Result date

Preprint submitted July 22, 2026

Why it matters: The result would delimit how far graph Laplacian majorization extends to higher-dimensional uniform simplicial families.

Limits: The counterexamples appear in a same-day version-one preprint. No independent recomputation, specialist assessment or journal review was located by the maintenance cutoff.

A counterexample to Thakur's conjecture on Carlitz–Wieferich primes

The paper gives an explicit degree-five Carlitz–Wieferich prime over the field with 19 cubed elements. Because 19 does not divide five, the example contradicts the conjectured degree-divisibility condition.

Problem or result

Thakur's conjectural degree-divisibility condition for Carlitz–Wieferich primes in odd characteristic

Authors

David Niedbala Giraudin

Institutions

Independent researcher, Meaux, France

Result date

Version 2 submitted July 22, 2026

Why it matters: It overturns a conjectural pattern in function-field arithmetic that had survived earlier proofs in small degrees and extensive computations.

Limits: Version 2 corrected an unrelated exactness theorem to a conjecture. The counterexample remains in the manuscript, and the author reports an independent Magma check by Dinesh Thakur, but no separate verification artifact or journal review was located.

Both directions of Fuglede’s conjecture fail in dimension two

The paper gives two explicit 60-point constructions in a finite abelian group and transfers them to bounded subsets of the plane. One planar set tiles by translations but is not spectral; the other is spectral but does not tile.

Problem or result

Fuglede’s conjecture in the two-dimensional unrestricted measurable setting

Authors

Tao Zhang

Institutions

Xidian University

Result date

Preprint submitted July 17, 2026

Why it matters: Counterexamples were already known in dimensions three and higher, while the unrestricted two-dimensional case had remained unresolved.

Limits: This is a first-version preprint without located independent specialist assessment or peer review. The result concerns unrestricted measurable sets in the plane; narrower versions of Fuglede’s problem can remain open.

The remaining strict off-diagonal cases of Weissler’s conjecture

The authors give a self-contained proof of Weissler’s two-point inequality for 2 < p ≤ q < infinity and, by duality, 1 < p ≤ q < 2, including the previously known diagonal cases and resolving the missing strict off-diagonal ranges.

Problem or result

The two-point inequality underlying Weissler’s characterization of complex hypercontractivity

Authors

Yi C. Huang and Paata Ivanisvili

Institutions

Nanjing Normal University and University of California, Irvine

Result date

Current v2 revised September 18, 2026

Why it matters: The result claims to complete the open off-diagonal part of a long-studied characterization in complex hypercontractivity.

Limits: The current v2 remains an unrefereed preprint without located independent specialist assessment. Its new self-contained framing includes prior diagonal cases; novelty is the previously missing strict off-diagonal range.

A claimed proof of Fajtlowicz’s graph-energy conjecture

The paper claims a proof that the sum of the absolute adjacency eigenvalues of every graph is at least twice the difference between its order and independence number.

Problem or result

Whether every graph of order n satisfies energy at least 2(n minus its independence number)

Authors

Hitesh Kumar and Shivaramakrishna Pragada

Institutions

Simon Fraser University

Result date

Preprint submitted July 22, 2026

Why it matters: Fajtlowicz’s inequality has been an open spectral graph theory conjecture since the 1980s.

Limits: This is a first-version preprint without located independent specialist assessment or peer review. No formal proof artifact or independently rerun computation accompanies the claim.

A characteristic-two counterexample to Wehlau’s Noether-number conjecture

For the dihedral group D8 in characteristic two, the paper constructs a five-dimensional submodule U of a six-dimensional module V with Noether number 6 for U but 5 for V. Inflation extends the construction to groups admitting D8 as a quotient.

Problem or result

Monotonicity of Noether numbers when a group submodule is included in a larger module

Authors

Muhammad Fazeel Anwar

Institutions

Sukkur IBA University

Result date

Preprint submitted July 20, 2026

Why it matters: The explicit example reverses the inequality proposed by Wehlau in a concrete modular invariant-theory setting.

Limits: This is a first-version preprint without located independent specialist assessment or peer review. The result is characteristic-specific, and the accompanying Magma calculations were not independently rerun in this cycle.

A claimed universal residue formula for Witten zeta functions

The paper claims a universal formula for the leading residue of a normalized Witten zeta function and says this establishes Au’s gamma-product-shape conjecture, including the A4 prediction.

Problem or result

Au’s gamma-product-shape conjecture for irreducible crystallographic root systems

Authors

Jonas Matuzas

Institutions

No institutional affiliation stated in the manuscript

Result date

Preprint submitted July 14; third version posted August 3, 2026

Why it matters: A uniform formula across all irreducible crystallographic root systems would connect representation growth with Coxeter and special-function identities.

Limits: This is a third-version but unreviewed preprint whose mathematical content was generated in an AI workflow. The manuscript now reports independent numerical corroboration of selected identities, but describes those checks as partial, non-rigorous evidence rather than proof. No independent specialist assessment or formal verification was located.

A counterexample to Batyrev’s non-negativity conjecture for stringy Hodge numbers

Satriano and Usatine give a projective variety with a negative stringy Hodge number, refuting Batyrev’s general non-negativity conjecture. A separate Huang–Satriano preprint now claims the exact dimensional threshold: non-negativity through dimension four and counterexamples in every dimension from five onward.

Problem or result

Batyrev’s conjecture that polynomial stringy E-functions of projective varieties with Gorenstein canonical singularities have non-negative stringy Hodge numbers

Authors

Matthew Satriano and Jeremy Usatine

Institutions

University of Waterloo and Florida State University

Result date

Classification source v2 revised October 1, 2026

Why it matters: The counterexample rules out non-negativity in full generality; the newer result, if confirmed, draws the boundary sharply by preserving the conjecture through dimension four and disproving it from dimension five onward.

Limits: Both controlling results remain unrefereed. The classification source is now v2 and still claims non-negativity through dimension four and failure from dimension five, but it shares an author with the original counterexample and is not independent validation.

The minimal-distance problem has sharp exponent two-thirds

Pohoata constructs arbitrarily large point–line configurations whose off-diagonal distances are all at least n^(−2/3−ε). Combined with the published Cohen–Pohoata–Zakharov upper bound, this gives Δ_PL(n) = n^(−2/3+o(1)). The second version also derives finite-field matchings that disprove a related conjecture of Hunter, Pohoata, Verstraëte and Zhang.

Problem or result

Determine the asymptotic exponent of the greatest possible off-diagonal distance in an n-pair point–line incidence configuration in the unit square

Authors

Cosmin Pohoata

Institutions

Emory University

Result date

Preprint submitted July 22; second version posted July 30, 2026

Why it matters: The result determines the exponent of a discrete-geometry problem introduced in connection with the Heilbronn triangle problem. Its added finite-field consequence also overturns a conjectured upper bound for induced point–line matchings.

Limits: The matching lower bound is a 14-page second-version preprint without located independent specialist assessment or peer review. It determines the exponent, not a leading constant, and it does not settle the Heilbronn triangle problem.

Erdős Problem #793: the second-order constant for strongly 2-primitive sets

The preprint claims that the maximum size F(n) is π(n) + (27/2 + o(1))n^(2/3)/(log n)^2. It gives matching upper and lower bounds, and a public Lean package encodes the same constant and allows the two factors in the divisibility condition to coincide.

Problem or result

Erdős Problem #793, asking whether the extremal size of a strongly 2-primitive subset of {1,…,n} has a constant second-order term

Authors

Przemek Chojecki (sole listed author); GPT-5.6 Sol credited in the project record

Institutions

Ulam AI

Result date

Preprint and Lean artifact posted July 14, 2026

Why it matters: The formula identifies the exact second-order constant in an Erdős extremal-number-theory problem whose order of growth was known but whose asymptotic constant was not.

Limits: This is a first-version, non-peer-reviewed proof without located independent specialist assessment. Erdős Problems marks it PROVED (LEAN) while its external metadata still says the statement is not formalized. The self-published Star Fleet Math bundle contains the proof sources and build logs, but omits the pinned external dependency directories referenced by its verifier, so the exact build could not be replayed locally from the ZIP alone. The archive operator reports a clean rebuild; this tracker therefore records a Lean-checked claim, not field acceptance.

Erdős Problem #421: a claimed density-one distinct-products construction

Chojecki gives a prime-gap greedy construction that claims to retain a density-one set while preventing equal products on distinct consecutive blocks. Sneiderman’s separate nine-page reconstruction fills in the cancellation, curve-counting and forest-summation steps and sharpens one rejected-gap estimate.

Problem or result

Erdős Problem #421, asking for a density-one increasing sequence in which all products over distinct consecutive index intervals are different

Authors

Przemek Chojecki; expanded audit and reconstruction by Rob Sneiderman

Institutions

Ulam AI; independent public GitHub repository

Result date

Proof preprint dated July 13; independent audit posted July 20, 2026

Why it matters: A correct construction would answer an Erdős–Graham question by improving the previously recorded density lower bound to natural density one.

Limits: Erdős Problems still labels the problem OPEN and explicitly says its proof-claim listing is not a correctness review. The proof and audit are self-published, not peer reviewed or on arXiv, and no specialist assessment was located. The audit repository is complete and its TeX rebuilt successfully in this maintenance cycle, but that establishes reproducibility of the document, not correctness of the mathematical argument.

A finite counterexample claim for the Dinitz–Garg–Goemans cost conjecture

The claim gives a seven-vertex acyclic network with three demands. Its fractional flow has cost 58, while exhaustive enumeration of all eight unsplittable routings leaves every load-feasible routing with cost at least 60.

Problem or result

Goemans’ conjecture that a fractional single-source flow can always be rounded to an unsplittable flow of no greater cost while increasing every arc load by at most the maximum demand

Authors

GPT-5.6 Pro output shared by Dmitry Rybin

Institutions

Public ChatGPT shared conversation

Result date

Public claim shared July 22, 2026

Why it matters: If confirmed, the example disproves a nearly three-decade-old conjecture in single-source unsplittable flow using a small certificate that can be checked entirely with integer arithmetic.

Limits: No paper, arXiv record, DOI, versioned author repository or independent specialist assessment was located. The public shared conversation exposes the full numerical certificate, but its linked sandbox attachments are not independently downloadable from the public page. Kingy independently reconstructed the graph, enumerated all six source–terminal paths and all eight routings, and reproduced the 58-versus-60 separation; the tracker therefore records a provisional exact counterexample claim, not field acceptance.

A deterministic NP-hardness claim for the Euclidean shortest-vector problem

Wan claims a deterministic polynomial-time reduction proving that γ-GapSVP in the ℓp norm is NP-hard for every finite p and every approximation factor γ below 2^(1/p). At p = 2, this includes exact Euclidean SVP and approximation factors below √2, removing the randomness used in earlier hardness reductions.

Problem or result

Van Emde Boas’s 1981 conjecture on deterministic NP-hardness of the shortest-vector problem in Euclidean space

Authors

Daqing Wan

Institutions

Center for Discrete Mathematics and College of Mathematics and Statistics, Chongqing University

Result date

First submitted March 28, 2026; expanded version 3 submitted July 22, 2026

Why it matters: The Euclidean shortest-vector problem is a foundational lattice problem. Deterministic NP-hardness had remained open since van Emde Boas posed it in 1981, despite Ajtai’s 1998 randomized reduction and later randomized approximation-hardness results.

Limits: This is an expanded version-three preprint, not a peer-reviewed or independently validated theorem. No specialist correctness assessment or journal record was located. The result concerns worst-case computational hardness under the paper’s reductions; it is not a polynomial-time algorithm, a proof that P differs from NP, or evidence that deployed lattice cryptosystems are broken.

Two explicit 64-rectangle counterexamples to Wegner’s packing–piercing conjecture

Two current manuscripts give explicit triangle-free families of 64 axis-parallel rectangles with packing number ν = 16 and piercing number τ at least 32, contradicting Wegner’s bound τ ≤ 2ν − 1. Both also construct recursive families whose packing–piercing ratios and standard clique-LP gaps approach 5/2.

Problem or result

Wegner’s 1965 conjecture that every finite family of axis-parallel rectangles satisfies τ ≤ 2ν − 1

Authors

Deepak Ajwani, Rishikesh Gajjala, Rajiv Raman and Saurabh Ray; independently, Aranya Kumar Bal

Institutions

University College Dublin, NYU Abu Dhabi, IIIT Delhi and Indian Statistical Institute Kolkata

Result date

First preprint submitted June 16, 2026; current companion preprints revised July 9 and July 14, 2026

Why it matters: The conjecture had guided geometric transversal theory since 1965. Besides the compact finite counterexample, the recursive constructions raise the known lower bound for the standard LP relaxation used in maximum-independent-set-of-rectangles algorithms.

Limits: Both sources are revised preprints, not peer-reviewed publications, and no independent specialist correctness assessment was located. Their bibliographic narratives are not yet synchronized: Bal describes an earlier, much larger Ajwani–Gajjala–Raman–Ray construction, while the current Ajwani et al. version now states the same 64-rectangle scale and a 5/2 − ε family. The mathematical claims and priority history should therefore be checked again at formal publication.

Polynomial improvements refute the standard 3SUM and APSP hypotheses

Alman and Vassilevska Williams give deterministic O(n^1.9992) time for polynomial-size integer 3SUM and O(n^2.9995) time for directed APSP with polynomially bounded integer weights, yielding the first fixed-polynomial exponent improvements over the textbook bounds in those models.

Problem or result

Truly subquadratic 3SUM and truly subcubic all-pairs shortest paths

Authors

Josh Alman and Virginia Vassilevska Williams

Institutions

Columbia University; Massachusetts Institute of Technology

Result date

First-version preprint submitted October 5, 2026; independent follow-ups submitted later in October

Why it matters: The result overturns two central fine-grained hardness hypotheses in their standard polynomially bounded integer models and transfers through established reductions to a broad family of graph and geometric problems.

Limits: The exponent gains are small and asymptotic, with restrictions on input size and weight models; they are not evidence of practical speedups at ordinary input sizes. The preprint and supplied Lean artifact are not peer review, although independent follow-up manuscripts reproduce and simplify the route.

An AI-generated preprint claims matrix multiplication exponent at most 9/4

An OpenAI preprint proves an asymptotic arithmetic upper bound omega at most 9/4 over the complex numbers. A later independent manuscript improves a shared-leg step and extends the 9/4 conclusion over algebraic closures in every characteristic.

Problem or result

Upper bound for the square matrix-multiplication exponent

Authors

OpenAI; independent extension by Houxi Lü

Institutions

OpenAI; Lü's affiliation was not stated in the inspected arXiv author block

Result date

Preprint dated October 2; announced October 6; independent extension submitted October 12, 2026

Why it matters: The claimed 9/4 bound would be a major simplification and improvement in algebraic-complexity theory, with a separate manuscript already probing and strengthening one of its central constructions.

Limits: This is an asymptotic arithmetic-operation result, not a practical implementation, bit-complexity bound or claim that ordinary matrix multiplication now runs near n^2. The manuscripts are unrefereed; formal checking certifies encoded statements and does not replace semantic or historical-priority review.

An explicit construction disproves the undirected multiple-unicast conjecture

Braverman and He construct an undirected network with 157 independent sessions whose linear network-coding rate is at least 1 while fractional multicommodity flow is at most 147/157, and use amplification to obtain a polylogarithmic coding gap.

Problem or result

Undirected multiple-unicast network-coding conjecture

Authors

Mark Braverman and Zhongtian He

Institutions

Princeton University

Result date

Preprint submitted October 7, 2026

Why it matters: The paper supplies a finite explicit witness against a long-standing network-coding conjecture rather than only an existential separation.

Limits: The executable checker is author-supplied rather than an independent replication, the amplified family depends on the cited theorem, and the large construction has not been peer reviewed or independently replayed in the sources checked.

A formalized lower bound rules out 21-multiplication recursive integer 3×3 schemes

Rudich and Rousseau prove that recursive 3×3 block-matrix multiplication algorithms using integer constants require at least 22 multiplications. Within that restricted model, no 3×3 scheme can beat Strassen’s recursive exponent.

Problem or result

Minimum multiplication count for recursive 3×3 block algorithms with integer constants

Authors

Isaac Rudich and Louis-Martin Rousseau

Institutions

Polytechnique Montréal; Carnegie Mellon University

Result date

Deposits September 30; first-version preprint submitted October 1, 2026

Why it matters: The result closes the possibility that an integer-constant recursive 3×3 scheme with 21 multiplications can improve on Strassen and materially raises a long-stagnant restricted lower bound.

Limits: This reconciliation addition predates the prior cutoff but was absent from the dataset. It does not determine unrestricted 3×3 tensor rank, rule out non-integer constants or improve the general matrix-multiplication exponent; the author-supplied formalization has no independent semantic audit.

A preprint claims the optimal deterministic k-server bound

Coester, Koutsoupias and Zbysiński claim that the work-function algorithm is k-competitive on every metric space, matching the general lower bound and resolving the deterministic k-server conjecture.

Problem or result

The deterministic k-server conjecture for online algorithms

Authors

Christian Coester, Elias Koutsoupias and Marek Zbysiński

Institutions

University of Oxford

Result date

First-version preprint submitted September 14, 2026

Why it matters: The conjecture has been central to competitive analysis since the late 1980s. The claimed k ratio improves the longstanding general 2k−1 work-function bound to the conjectured optimum.

Limits: This is a first-version preprint. No peer review, formal proof or independent semantic correctness assessment was located; author-page repetition and machine-generated reviews are not independent validation.

A preprint claims the optimal strong matroid secretary guarantee

Huang claims an expected-polynomial-time ordinal algorithm that is 1/e-probability-competitive for every matroid using only comparison and independence oracles on arrived elements.

Problem or result

The strong matroid secretary conjecture in the ordinal oracle model

Authors

Zhiyi Huang; contextual results by Sahil Singla, Abdi and collaborators, and Bérczi and collaborators

Institutions

University of Hong Kong and collaborating institutions

Result date

Primary preprint submitted September 24, 2026

Why it matters: The claim upgrades a nearly two-decade-old constant-factor conjecture to the optimal classical secretary constant in the strongest common information model.

Limits: The surrounding papers establish important weaker or linear-matroid results but do not verify the all-matroid 1/e theorem. No peer review or formal artifact was located.

A scaffolded DNA computer runs programmable calculations at equilibrium

Stérin and collaborators demonstrated a programmable DNA computer whose intended products are favoured at thermodynamic equilibrium. Ten programs and more than 700 experimental computations included 25-bit addition, a 100-bit molecular computation.

Problem or result

Programmable molecular computation favoured by thermodynamic equilibrium

Authors

Tristan Stérin, Abeer Eshra, Constantine Glen Evans, Janet Adio and Damien Woods

Institutions

Maynooth University; Foundation for Molecular Medicine; PRGM DEV

Result date

Published September 16, 2026

Why it matters: The architecture encodes computation into an energy landscape rather than relying only on staged kinetic reactions, showing that useful molecular calculations can reliably settle toward their answer and be cycled again.

Limits: This is a laboratory proof of principle, not an electronic-computer replacement. Synthesis, thermal cycling and runtimes up to about 14 hours are real costs; no end-to-end energy advantage was established, and patent-related interests are disclosed.

An AI-assisted proof yields a new iterative-rounding route for Feedback Vertex Set

Karthekeyan Chandrasekaran, Chandra Chekuri and Shubhang Kulkarni prove a 2021 extreme-point conjecture for strong-density polyhedra and derive polynomial-time iterative-rounding 2-approximation algorithms for Feedback Vertex Set.

Problem or result

Fiorini's extreme-point conjecture and iterative rounding for Feedback Vertex Set

Authors

Karthekeyan Chandrasekaran, Chandra Chekuri and Shubhang Kulkarni

Institutions

University of Illinois Urbana-Champaign

Result date

Preprint submitted September 3, 2026

Why it matters: The factor-two approximation ratio is already optimal under standard assumptions, but this proves a previously open polyhedral property and supplies the first stated iterative-rounding route to that ratio.

Limits: This is a first-version preprint without located peer review. The approximation factor itself is not new, and the paper does not yet identify the AI tools or provide a detailed interaction record.

A preprint disproves the greedy superstring 2-approximation conjecture

Hiroki Shibata constructs instances for which the standard greedy shortest-common-superstring algorithm has approximation ratio approaching 9/4. This refutes the nearly four-decade conjecture that the ratio is at most 2.

Problem or result

The conjectured factor-2 guarantee for the greedy shortest-common-superstring algorithm

Authors

Hiroki Shibata

Institutions

Joint Graduate School of Mathematics for Innovation, Kyushu University

Result date

First-version preprint submitted September 1, 2026

Why it matters: Shortest common superstring is a foundational approximation problem with applications in sequence assembly and compression. The construction raises the greedy algorithm’s known worst-case lower bound from 2 to 9/4.

Limits: The result is a first-version preprint with no located peer review or independent verification. It establishes a lower bound approaching 9/4; it does not determine the exact worst-case ratio, whose known upper bound remains 3.

Separating QMA from QCMA with a classical oracle

The authors constructed a classical-oracle setting in which QMA, the class with quantum witnesses, is strictly larger than QCMA, whose witnesses are classical. The proof uses a decision problem called spectral Forrelation.

Problem or result

Whether quantum proofs can be strictly more powerful than classical proofs checked by a quantum verifier

Authors

John Bostanci, Jonas Haferkamp, Chinmay Nirkhe and Mark Zhandry

Institutions

Columbia, Saarland/Harvard, University of Washington, Stanford and NTT Research

Result date

November 12, 2025; published at STOC 2026 on June 9, 2026

Why it matters: It supplies the oracle separation quantum-complexity researchers had sought for roughly two decades and introduces techniques that may be useful beyond this problem.

Limits: This is a relativized oracle separation, not an unconditional proof that QMA differs from QCMA in the ordinary unrelativized setting. It is not a demonstrated hardware advantage.

Simulating Time With Square-Root Space

Williams proved that every multitape Turing-machine computation running in time t can be simulated in O(sqrt(t log t)) space. This substantially improves the previous O(t/log t) universal bound.

Problem or result

The general relationship between computation time and memory, including the 50-year-old Hopcroft–Paul–Valiant simulation bound

Authors

R. Ryan Williams

Institutions

Massachusetts Institute of Technology

Result date

February 25, 2025; published at STOC 2025

Why it matters: The theorem is the first major improvement in half a century on a foundational time-versus-space simulation.

Limits: It does not prove P is different from PSPACE. Its formal scope is multitape Turing machines and related models, not immediate memory savings for production software.

Breaking the Sorting Barrier for Directed Single-Source Shortest Paths

The authors gave a deterministic O(m log^(2/3) n)-time algorithm for directed single-source shortest paths in the comparison-addition model. It is the first to beat Dijkstra's O(m + n log n) bound on sparse directed graphs in that model.

Problem or result

Single-source shortest paths in directed graphs with non-negative real edge weights

Authors

Ran Duan, Jiayi Mao, Xiao Mao, Xinkai Shu and Longhui Yin

Institutions

Tsinghua University, Stanford University and Max Planck Institute for Informatics

Result date

April 23, 2025; published at STOC 2025

Why it matters: The result shows that the sorting-like overhead in Dijkstra's method is not an unavoidable barrier for this foundational graph problem.

Limits: The improvement is asymptotic and model-specific to non-negative real weights in the comparison-addition model. It does not establish faster performance on current routing workloads or small-integer-weight graphs.

Optimal Bounds for Open Addressing Without Reordering

The authors constructed open-addressed hash tables with better-than-expected search bounds, disproving the central conjecture in Andrew Yao's “Uniform Hashing Is Optimal.” They also proved matching lower bounds for the models studied.

Problem or result

Expected search and insertion costs in open-addressed hash tables without reordering

Authors

Martín Farach-Colton, Andrew Krapivin and William Kuszmaul

Institutions

New York University, University of Cambridge and Carnegie Mellon University

Result date

FOCS 2024; arXiv version January 4, 2025

Why it matters: The paper replaces a presumed 40-year-old barrier for one of computer science's most basic data structures with optimal bounds.

Limits: The result concerns a defined open-addressing model without reordering. It does not show that every real-world hash-table implementation should be replaced, and immediate engineering performance was not demonstrated.

A claimed proof of the complex Permanent Anticoncentration Conjecture

The paper bounds the density and small-ball probability of normalized permanents in real, complex and quaternionic Ginibre ensembles. In the complex case, the authors state that this resolves the Permanent Anticoncentration Conjecture.

Problem or result

Aaronson–Arkhipov Permanent Anticoncentration Conjecture for complex Gaussian matrices

Authors

Frederic Koehler and Pui Kuen Leung

Institutions

University of Chicago for Koehler; Leung’s affiliation was not stated in the manuscript

Result date

Preprint submitted July 22, 2026

Why it matters: Permanent anticoncentration is a central assumption in arguments about the classical hardness of approximate boson sampling.

Limits: This remains a first-version preprint without located independent specialist assessment or peer review. A separate QCE 2026-accepted manuscript submitted July 24 proves only a weaker super-exponential bound and says that an inverse-polynomial Permanent Anticoncentration Conjecture remains open, but it does not cite or evaluate the Koehler–Leung manuscript posted two days earlier. The chronology and formulations require reconciliation; the parallel paper is not evidence that the claimed proof has been accepted or refuted.

A counterexample to the polynomial-time low-degree conjecture

The paper constructs permutation-invariant graph distributions whose low-degree advantage vanishes through polylogarithmic degree, yet a deterministic rank test distinguishes the noisy planted model from the uniform null model in polynomial time.

Problem or result

The standard binary polynomial-time low-degree conjecture after independent noise

Authors

Songtao Mao

Institutions

Johns Hopkins University

Result date

Preprint submitted July 22, 2026

Why it matters: The low-degree method is widely used as evidence of average-case computational hardness. A valid counterexample would identify a missing condition in the standard conjecture.

Limits: This is a first-version preprint without located independent specialist assessment or peer review. The construction refutes the specific formulation stated in the paper; it does not invalidate every low-degree heuristic or lower bound.

A nonlinear fluorescence measurement confirms attosecond hard-X-ray pulses

Inoue, Robles, Halavanau and collaborators report the first direct experimental confirmation of hard-X-ray attosecond pulse generation at a free-electron laser, inferring shot-to-shot durations of about 100 to 400 attoseconds from amplified spontaneous emission in transition-metal foils.

Problem or result

Direct confirmation and duration measurement of hard-X-ray attosecond pulses

Authors

Ichiro Inoue, River Robles, Aliaksei Halavanau, Veronica Guo, Takahiro Sato, Agostino Marinelli and collaborators

Institutions

University of Tokyo; RIKEN SPring-8 Center; SLAC National Accelerator Laboratory; Stanford University; DESY and University of Hamburg; Osaka University; University of Wisconsin–Madison

Result date

Nature Photonics version of record published October 20, 2026; underlying preprint submitted June 9, 2025

Why it matters: The measurement extends directly confirmed attosecond operation to roughly 9-keV hard X-rays, combining electron-timescale resolution with atomic-scale wavelengths and establishing a diagnostic route for future ultrafast scattering experiments.

Limits: Pulse duration was inferred from collective fluorescence yield with simulations and statistical modelling rather than read from a general-purpose pulse meter. The reported 100-attosecond case was nearly transform limited, but routine user operation, damage-free structural imaging and broad experimental access remain future goals.

Two independent teams operated thorium-229 nuclear-clock prototypes

Vienna- and Tsinghua-led teams independently stabilized continuous-wave lasers to the thorium-229 nuclear transition in calcium-fluoride crystals, making the nuclear resonance the feedback reference in operational prototype clocks.

Problem or result

Closed-loop operation of solid-state thorium-229 nuclear clocks

Authors

L. Toscani De Col, T. Riebner, I. Morawetz and collaborators; Beichen Huang, Gaowei Yan, Qi Xiao and collaborators

Institutions

TU Wien-led European collaboration; Tsinghua University-led Chinese collaboration

Result date

Both Nature papers published October 7, 2026

Why it matters: The paired demonstrations turn a long-standing nuclear-clock proposal into a reproducible clock architecture and create a new platform for precision tests of changing constants and ultralight dark matter.

Limits: Neither prototype is yet more precise than the best optical atomic clocks. Proposed orders-of-magnitude gains, compact deployment and stronger new-physics limits are future prospects rather than demonstrated capabilities.

BASE-STEP transported 92 trapped antiprotons by road without particle loss

The BASE-STEP collaboration transported 92 trapped antiprotons 7.5 kilometres across CERN in 24 minutes without losing particles or measurably degrading the trap vacuum. Monitoring for 33 days set a 7.27-year storage-lifetime lower bound at 68% confidence.

Problem or result

Portable confinement and transport of trapped antiprotons

Authors

M. Leonhardt, J. Morgner, F. Abbass and the BASE-STEP collaboration

Institutions

CERN BASE-STEP; Heinrich Heine University Düsseldorf; CERN; Mainz; GSI; MPIK; RIKEN; University of Tokyo; Leibniz University Hannover

Result date

Transport March 24; published September 16, 2026

Why it matters: Portable antiproton traps can move precision experiments away from accelerator noise; the collaboration estimates that off-site CPT comparisons could eventually improve by at least two orders of magnitude.

Limits: This was a proof-of-feasibility trip within CERN, not intercity delivery or a new CPT measurement. The demonstrated autonomous interval was 2.72 hours, and the projected precision gain has not yet been achieved.

LZ reports one unexplained high-energy nuclear-recoil candidate, below discovery significance

The LUX-ZEPLIN collaboration reports one event consistent with a 248±23(stat)±23(sys) keV nuclear recoil in 2.84 tonne-years of exposure. Across the tested dark-matter models, the background-only tension is 2.6σ globally after look-elsewhere effects, with a maximum local significance of 3.4σ.

Problem or result

Search for high-energy nuclear recoils from effective-field-theory and inelastic dark-matter models

Authors

LUX-ZEPLIN Collaboration

Institutions

About 250 researchers across 39 institutions; detector at Sanford Underground Research Facility

Result date

Result presented September 1 and preprint submitted September 2, 2026

Why it matters: The extended recoil window opens dark-matter parameter space that standard low-energy analyses sample poorly. The event is unusual enough to motivate scrutiny and additional exposure without constituting a detection.

Limits: This is one event in a non-peer-reviewed analysis. Its 2.6σ global significance is far below the usual 5σ discovery threshold, the analysis was not blind above 55 keV, and no particle identity can be inferred from a single unexplained recoil candidate.

A cavity vacuum field measurably enhances superconductivity in thin NbSe₂

Experiments with thin niobium diselenide inside a split-ring cavity report increases in the superconducting transition temperature, critical current and critical magnetic field. The authors’ model attributes the effect to hybridization with the cavity’s vacuum electromagnetic mode.

Problem or result

Vacuum-field modification of superconducting properties in layered NbSe₂

Authors

Zheyan Wang and collaborators

Institutions

University of Science and Technology of China and an international collaboration

Result date

Published online August 19, 2026

Why it matters: The work supplies direct experimental evidence that an unpopulated cavity mode can tune a material’s equilibrium superconducting state, advancing cavity control of quantum materials.

Limits: This is a proof-of-principle in one layered material and a specific split-ring cavity, not a general route to room-temperature superconductivity. The institutional release reports a 5.4% transition-temperature increase in a six-layer sample; the full Nature text is access-limited.

A Rydberg quantum simulator directly resolves conformal-field-theory spectra

A programmable chain of Rydberg atoms directly measures the low-energy spectra and scaling dimensions associated with the Ising and tricritical Ising conformal field theories. The experiment identifies both primary fields and descendants from finite-system spectroscopy.

Problem or result

Direct spectroscopy of Ising and tricritical Ising conformal-field-theory operator content

Authors

Jingxiang Sun, Martin Le and collaborators

Institutions

Caltech, Paris-Saclay and CNRS, Technical University of Munich and collaborators

Result date

Published online August 19, 2026

Why it matters: Conformal field theory organizes universal critical behaviour across many systems. Reading its operator spectrum directly from a controllable quantum device turns an abstract classification tool into an experimental observable.

Limits: The demonstration uses one-dimensional Rydberg chains of up to 35 atoms and two well-understood universality classes. It is not a generic solver for arbitrary conformal field theories and does not by itself test quantum-gravity models.

Magnetar X-ray polarization gave strong evidence for vacuum birefringence

A coordinated IXPE, NICER and Parkes analysis of the radio magnetar 1E 1547.0−5408 measured very high X-ray polarization whose energy-dependent behaviour is naturally explained by quantum-electrodynamic vacuum birefringence in the star’s extreme magnetic field.

Problem or result

Astrophysical evidence for QED vacuum birefringence in an ultrastrong magnetic field

Authors

Rachael E. Stewart, Hoa Dinh Thi, George Younes, Marcus E. Lower, Matthew G. Baring, Michela Negro, Fernando Camilo, Joel B. Coley, Teruaki Enoto, Alice K. Harding, Wynn C. G. Ho, Chin-Ping Hu, Philip Kaaret, Paul Scholz, Alex Van Kooten and Zorawar Wadiasingh

Institutions

George Washington University, Rice University, NASA Goddard Space Flight Center, Swinburne University of Technology and collaborating institutions

Result date

Published in Nature on August 5, 2026

Why it matters: Vacuum birefringence is a non-linear prediction of quantum electrodynamics that has remained difficult to measure directly. Magnetars provide magnetic fields far beyond laboratory reach, and the combined polarization signal sharply constrains models of photon propagation there.

Limits: This is model-dependent astrophysical evidence, not a controlled laboratory detection or an exclusive proof of vacuum birefringence. Earlier magnetar and neutron-star observations supplied suggestive evidence, so the result should not be described as the first confirmation.

A cold-atom quantum simulator mapped the Hubbard-model pseudogap

A two-dimensional ultracold-atom implementation of the repulsive Fermi–Hubbard model observed the crossover from a normal metal to a pseudogapped metal and mapped the pseudogap across interaction, temperature and doping using single- and two-particle response measurements.

Problem or result

Experimental phase mapping of the pseudogap in the two-dimensional Fermi–Hubbard model

Authors

Lev Haldar Kendrick, Anant Kale, Youqi Gang, Alexander Dennisovich Deters, Martin Lebrat, Aaron W. Young and Markus Greiner

Institutions

Harvard University and JILA, University of Colorado Boulder

Result date

Published in Nature on August 5, 2026

Why it matters: The Hubbard model is a central minimal model for strongly correlated electrons, but its doped two-dimensional regime is hard to calculate. A controlled quantum simulator provides a direct experimental map of the pseudogap and a benchmark for theories connected to high-temperature superconductivity.

Limits: This is an analogue quantum simulation of the two-dimensional Hubbard model, not a direct observation in a cuprate material. Mapping the normal-state pseudogap does not by itself establish the mechanism of high-temperature superconductivity.

A discrete time quasicrystal was observed in a driven Rydberg-atom ensemble

A quasiperiodically driven, dissipative ensemble of strongly interacting Rydberg atoms produced robust subharmonic responses at multiple incommensurate frequencies. The experiment mapped the resulting discrete-time-quasicrystal phase and tested its rigidity against changes in radio-frequency intensity and laser detuning.

Problem or result

Experimental realization and phase mapping of discrete time-quasicrystalline order in a Rydberg-atom platform

Authors

Dong-Yang Zhu, Zheng-Yuan Zhang, Qi-Feng Wang, Yu Ma, Tian-Yu Han, Chao Yu, Qiao-Qiao Fang, Shi-Yao Shao, Qing Li, Ya-Jun Wang, Jun Zhang, Han-Chao Chen, Xin Liu, Jia-Dou Nan, Yi-Ming Yin, Li-Hua Zhang, Guang-Can Guo, Bang Liu, Dong-Sheng Ding and Bao-Sen Shi

Institutions

Laboratory of Quantum Information and Anhui Province Key Laboratory of Quantum Network, University of Science and Technology of China

Result date

Published July 29, 2026; preprint first posted September 25, 2025

Why it matters: The controlled atomic platform connects quasiperiodic driving, finite Abelian symmetry and rigidity in a directly mapped non-equilibrium phase, expanding the experimental toolkit for studying time-translation symmetry breaking.

Limits: This is not the first reported time quasicrystal, nor an equilibrium ground-state crystal. It is a platform-specific driven-dissipative result, and the journal currently labels the article an unedited early-access version that may still receive copy-editing corrections.

Hidden spin-valley locking stabilized room-temperature spin polarization

Time-resolved circular-dichroism measurements found optically generated spin-polarized carriers lasting 686 picoseconds in a lead iodide perovskite and 4.7 nanoseconds in its lead-free tin analogue at room temperature, without an external magnetic field. First-principles calculations support hidden spin valleys as part of the mechanism.

Problem or result

Room-temperature spin stabilization through hidden spin-valley locking in inversion-symmetric semiconductors

Authors

Rayan Chakraborty, Yifan Dong, Jacob L. Shelton, Kelsey Garden, Leo Romanetz, Matthew P. Hautzinger, Matthew C. Beard, Volker Blum and David B. Mitzi

Institutions

Duke University; National Laboratory of the Rockies; University of California, Riverside; University of Colorado Boulder

Result date

Published July 27, 2026

Why it matters: Strong spin dephasing at higher temperatures limits semiconductor spin control. The reported nanosecond-scale lifetime in a lead-free two-dimensional material broadens the experimentally demonstrated design space for room-temperature spin-optoelectronics.

Limits: This is a materials and spectroscopy result, not a functioning spintronic device. The mechanism combines experiment with first-principles interpretation, and durability, integration, switching performance and manufacturing scale were not demonstrated.

Phonon focusing observed at room temperature

Tip-enhanced Raman temperature maps revealed ray-like heat propagation in boron arsenide at room temperature. The patterns, their time evolution and their orientation-dependent symmetry matched first-principles Boltzmann-transport calculations across multiple samples.

Problem or result

Experimental observation of phonon focusing at room temperature

Authors

Man Li, Huan Wu, Zihao Qin, Chuanjin Su, Huu Duy Nguyen and Yongjie Hu

Institutions

University of California, Los Angeles, including Mechanical and Aerospace Engineering and the California NanoSystems Institute

Result date

Published July 23, 2026

Why it matters: Phonon focusing had previously been observed only at cryogenic temperatures. Room-temperature directional heat flow creates a new experimental regime for nanoscale thermal transport and possible future thermal or quantum devices.

Limits: The demonstration is in high-thermal-conductivity boron arsenide and the device applications are prospective. The extended data show three principal experimental samples; the article states that data and code are available from the corresponding author on reasonable request rather than through an open repository.

Hybrid calculation of hadronic vacuum polarization in muon g − 2 to 0.48%

A hybrid lattice-QCD and experimental calculation determined the leading hadronic-vacuum-polarization contribution to 0.48% precision. With other Standard Model terms included, the prediction differs from the final experimental value by only 0.5 standard deviations.

Problem or result

The discrepancy between measured and predicted values of the muon's anomalous magnetic moment

Authors

A. Boccaletti, Sz. Borsanyi, A. Cotellucci, M. Davier, Z. Fodor and collaborators

Institutions

University of Wuppertal, Forschungszentrum Jülich, Université Paris-Saclay/CNRS, Aix-Marseille University/CNRS and partners

Result date

April 22, 2026

Why it matters: The calculation strongly weakens the former interpretation of muon g − 2 as evidence for undiscovered particles.

Limits: It does not remove every theory-level disagreement: tensions remain among some electron–positron datasets and between certain dispersive and lattice determinations. The careful conclusion is that the former anomaly is largely reconciled under the updated lattice-led prediction.

Direct observation of the Migdal effect induced by neutron bombardment

A purpose-built detector recorded six correlated nuclear-recoil and electron-track candidates among almost one million events, producing a reported significance above 5σ. The measured cross-section ratio agrees within uncertainty with theory.

Problem or result

Direct laboratory validation of the Migdal effect in neutral-particle collisions

Authors

Difan Yi, Qian Liu, Shi Chen and collaborators

Institutions

University of Chinese Academy of Sciences, Guangxi University, Central China Normal University, Lanzhou University and partners

Result date

January 14, 2026

Why it matters: The observation supplies an empirical anchor for light-dark-matter searches that use the Migdal effect to extend their sensitivity.

Limits: This was a controlled neutron–nucleus experiment, not a dark-matter detection. The signal rests on six candidate events, so replication in other target materials and recoil regimes remains important.

Observation of charge–parity symmetry breaking in baryon decays

LHCb measured significant asymmetries between a beauty baryon's decay rate and that of its antibaryon. This is the first observation of CP violation in baryon decay.

Problem or result

Whether CP violation, previously observed in mesons, can be observed in baryon decays

Authors

LHCb Collaboration

Institutions

International LHCb collaboration at CERN

Result date

March 25, 2025; published July 16, 2025

Why it matters: The result opens a new class of matter–antimatter asymmetry measurements and a new test bed for the Standard Model.

Limits: CP violation is already part of the Standard Model, and this result alone does not explain the universe's matter excess. It is not by itself evidence for a new force or particle.

Direct observation of coherent elastic antineutrino–nucleus scattering

CONUS+ inferred 395 ± 106 reactor-antineutrino signal events, consistent with the Standard Model expectation of 347 ± 59. The reported significance was 3.7σ.

Problem or result

Detecting coherent elastic neutrino–nucleus scattering with low-energy reactor antineutrinos

Authors

N. Ackermann, H. Bonet, A. Bonhomme, C. Buck and the CONUS Collaboration

Institutions

Max Planck Institute for Nuclear Physics and the Brokdorf and Leibstadt nuclear plants

Result date

Preprint April 1, 2025; published July 30, 2025

Why it matters: Reactor CEνNS enables compact neutrino detectors and a clean low-energy test of the Standard Model.

Limits: CEνNS was first observed by COHERENT in 2017 with higher-energy accelerator neutrinos; the novelty is the reactor-antineutrino signal in the fully coherent regime. Its 3.7σ strength is below the conventional 5σ particle-physics discovery threshold.

Observation of a B_c*+ meson with the ATLAS detector

ATLAS reported a new state above 8σ significance with a mass and mass gap matching expectations for the long-predicted B_c*+ meson. The collaboration submitted the result to Physical Review Letters.

Problem or result

Completing the lowest-lying spectrum of the meson made from a bottom antiquark and a charm quark

Authors

ATLAS Collaboration

Institutions

International ATLAS collaboration at CERN

Result date

May 15, 2026

Why it matters: The state fills a basic gap in the heavy-meson spectrum and tests calculations of how the strong force binds two different heavy quark flavours.

Limits: This is a new composite hadron, not a new elementary particle or evidence beyond the Standard Model. At the cutoff date it remained an official collaboration preprint submitted to a journal.

An M31 superbubble census finds clustered supernovae can sustain galactic turbulence

A combined FAST and Jansky VLA survey identified 118 superbubbles across M31. Their inferred kinetic-energy injection rates match independently estimated turbulence-dissipation rates in magnitude and spatial distribution.

Problem or result

Whether clustered supernova feedback can sustain turbulence across a spiral-galaxy disk

Authors

Fanyi Meng, Chao-Wei Tsai, Jingwen Wu, Sihan Jiao and collaborators

Institutions

Tsinghua University; Chinese Academy of Sciences; Beijing Normal University; Max Planck institutes; AMNH; Nanjing University; KASI; Peking University

Result date

Published September 17, 2026

Why it matters: Galactic turbulence regulates gas structure and star formation. This galaxy-wide census provides unusually direct observational energy accounting across a nearby spiral galaxy.

Limits: This is one galaxy, and bubble properties are inferred. Matching injection and dissipation establishes energetic sufficiency, not that supernovae are the unique driver; gravity, magnetic effects and other processes may contribute.

Chandra observations reveal a class of 84 hypersoft extragalactic X-ray sources

Mustafa Muhibullah, Jimmy Irwin and Rosanne Di Stefano identify 84 luminous, non-nuclear point sources across six galaxies that appear mainly or only below 0.3 keV, defining a previously missed hypersoft X-ray-source class whose spectra peak in the extreme ultraviolet.

Problem or result

A low-energy class of luminous compact extragalactic X-ray emitters

Authors

Mustafa Muhibullah, Jimmy A. Irwin and Rosanne Di Stefano

Institutions

University of Alabama; Center for Astrophysics | Harvard & Smithsonian

Result date

Published September 9, 2026

Why it matters: The objects occupy an observational blind spot and may expose previously hidden compact binaries. Some could be Type Ia supernova progenitors or important sources of ionizing radiation in galaxies.

Limits: The class is observational; the physical systems behind it are not yet identified. White dwarfs, post-nova systems, neutron stars and black-hole binaries remain candidate explanations, so the proposed links to supernova progenitors and galactic ionization are hypotheses rather than established functions.

JWST timing variations point to a possible third planet around LP 890-9

Kevin Stevenson and collaborators report statistically significant timing variations across 20 JWST/NIRSpec transits of LP 890-9 b and c. The known two-planet model cannot reproduce the signal, while three-planet models fit substantially better and favor a possible interior planet near 4.4 days.

Problem or result

Evidence for an additional non-transiting planet in the LP 890-9 system

Authors

Kevin B. Stevenson and twelve collaborators

Institutions

Johns Hopkins APL and University; University of Bristol; UNLV; University of Washington; Space Telescope Science Institute

Result date

Preprint submitted September 4, 2026; submitted to ApJ Letters

Why it matters: A third rocky-system companion would revise the architecture of a nearby temperate M-dwarf system and demonstrates that repeated atmospheric-transit programs can also reveal unseen planets dynamically.

Limits: No transit of LP 890-9 d was detected. The present data do not uniquely determine the architecture, periods from 4.0 to 6.9 days remain plausible, and more high-precision transit times are required.

Zhurong observations support a primary gypsum evaporite on geologically recent Mars

Zhurong rover imagery and spectroscopy of a 5–10-centimetre surface layer in southern Utopia Planitia revealed decimetre-scale platy crystal forms and hydrous-mineral signatures that the authors interpret as euhedral selenite gypsum formed as a primary evaporite about 760 million years ago.

Problem or result

Evidence for surface-water evaporation and primary gypsum formation in Amazonian-aged southern Utopia Planitia

Authors

Jiacheng Liu, Bo Wu, Wenyuan Gao, James W. Head III, Joseph R. Michalski, Mei-Fu Zhou, Anthony E. Williams-Jones, Xingao Hou, Zongcheng Ling, Zhongchen Wu, Changqing Liu, Yuqi Qian, Honglei Lin, Sen Hu, Yu Sun, Wei Tan, Xiao Wang and Guochun Zhao

Institutions

The University of Hong Kong and NWU–HKU Joint Centre of Earth and Planetary Sciences; Hong Kong Polytechnic University; Brown University; Northeastern University, China; Chinese University of Hong Kong; McGill University; Shandong University; Chinese Academy of Sciences; Northwest University; collaborating institutions

Result date

Published July 29, 2026

Why it matters: If the mineral and depositional interpretation holds, it records much more recent surface-associated aqueous activity than Mars’s ancient valley networks and would constrain late groundwater, brine and ice-covered water environments.

Limits: The mineral identity and formation history are remote interpretations from rover imagery and spectroscopic measurements, not confirmation by a returned sample or in-situ X-ray diffraction. The estimated 6.25–25-metre water-equivalent column is cumulative over time; the authors estimate an actual liquid layer of about one metre or less at any moment.

XRISM measured unusually vigorous gas motions around a radio-quiet quasar

XRISM spectroscopy of the H1821+643 cluster measured iron-line broadening corresponding to gas motions of about 300 kilometres per second, mostly 20–100 kiloparsecs from its central radio-quiet quasar. Under the authors’ shock-driven interpretation, the quasar coupled at least roughly 1–10% of its radiative energy into the surrounding intracluster medium.

Problem or result

Direct constraints on the reach and efficiency of quasar-mode feedback in a galaxy-cluster core

Authors

Satoshi Yamada, Shutaro Ueda, Hirofumi Noda, Yutaka Fujita, Misaki Mizumoto, Kentaro Nagamine, Claudio Ricci, Shoji Ogawa, Taiki Kawamuro, Shinya Yamada, Yuichi Terashima and Yoshihiro Ueda

Institutions

Tohoku University; University of Geneva; RIKEN; Kanazawa University; The University of Osaka; ISAS/JAXA; Kyoto University; collaborating institutions

Result date

Published July 28, 2026

Why it matters: Cosmological simulations require powerful black holes to inject substantial energy beyond their host galaxies, but direct constraints on quasar-mode coupling at cluster scales have been sparse. The measurement reaches the efficiency range used in leading simulations.

Limits: The large line width is directly measured, but its division among turbulence, velocity shear, sloshing and radial flows is not unique. The 1–10% feedback efficiency depends strongly on a quasar-driven-shock interpretation and assumed shock radius; it is not a model-independent measurement.

Discovery of an Exterior Third Planet Orbiting β Pictoris

JWST spectroscopy revealed the carbon-monoxide fingerprint, position and motion of a third giant planet in the Beta Pictoris system. A separate VLT/JWST imaging team independently confirmed the object.

Problem or result

Finding planets hidden within the bright debris disk around the young star Beta Pictoris

Authors

Aidan Gibbs, Jean-Baptiste Ruffio, Alexis Bidot and collaborators

Institutions

University of California San Diego, Space Telescope Science Institute and partners

Result date

July 15, 2026

Why it matters: The discovery demonstrates a way to find planets amid bright circumstellar dust and creates only the second known system with at least three directly imaged planets.

Limits: The planet's mass, orbit, temperature and atmospheric abundances remain model dependent. It is a young giant planet, not an Earth analogue or a biosignature result.

Water D/H in 3I/ATLAS as a probe of formation conditions in another planetary system

ALMA constrained the water deuterium-to-hydrogen ratio in interstellar comet 3I/ATLAS to more than 6.6 × 10^-3. That lower bound is over 40 times Earth's ocean value and over 30 times typical Solar System comet values.

Problem or result

Measuring an isotopic formation tracer in material from outside the Solar System

Authors

Luis E. Salazar Manzano, Teresa Paneque-Carreño, Martin A. Cordiner and collaborators

Institutions

University of Michigan, NASA Goddard, JPL/Caltech, UC Berkeley, Observatoire de Paris/CNRS, NRAO and partners

Result date

April 23, 2026

Why it matters: The result provides a direct chemical constraint on planetary material formed around another star and points to unusually cold or weakly processed formation conditions.

Limits: The result is a lower bound derived from HDO detection and H2O non-detection, not a precise ratio. The birth-environment inference depends on chemical and excitation modelling and does not identify the comet's home star.

Observation of an ultra-high-energy cosmic neutrino with KM3NeT

KM3NeT detected KM3-230213A, a through-going muon event consistent with a cosmic neutrino whose estimated energy is about 220 PeV. It is the highest-energy cosmic neutrino observed to date.

Problem or result

Extending neutrino observations into the ultra-high-energy regime and identifying their cosmic sources

Authors

KM3NeT Collaboration

Institutions

International KM3NeT collaboration

Result date

Observed February 13, 2023; published February 12, 2025

Why it matters: The event opens an observational window in which cosmogenic neutrinos and extreme cosmic accelerators may become distinguishable.

Limits: This is one event, its source has not been identified and origin scenarios remain speculative. A March 2025 author correction fixed a surname only and did not alter the scientific result.

GW231123: a Binary Black Hole Merger with Total Mass 190–265 M⊙

The collaboration interpreted GW231123 as a black-hole merger with a total source-frame mass of 190–265 solar masses, the most massive such merger it had reported. Later peer-reviewed analyses found that component masses and spins depend strongly on waveform and possible signal-overlap assumptions.

Problem or result

Characterizing a short, high-mass gravitational-wave signal with strongly model-dependent source properties

Authors

LIGO Scientific, Virgo and KAGRA Collaborations

Institutions

International LIGO–Virgo–KAGRA network

Result date

Primary preprint July 10, 2025; interpretation updated May 26, 2026

Why it matters: A secure characterization would probe how very massive black holes form and grow through hierarchical mergers or other channels.

Limits: The detection is secure, but the event is extremely short and its exact source properties remain model dependent. A two-signal model is favoured in one likelihood comparison but has low prior probability and only modest background significance, while other studies attribute the discrepancies to waveform systematics.

A planetary-mass exosatellite candidate around CD-35 2722 B

High-resolution radial-velocity monitoring with VLT/CRIRES+ found an approximately 170-day wobble in the young brown dwarf CD-35 2722 B. The preferred interpretation includes an orbiting companion with a minimum mass of about 0.9 Jupiter masses.

Problem or result

Detecting a natural satellite outside the Solar System

Authors

Kevin Hoy, Alice Zurlo, P. A. Peña R. et al.

Institutions

European Southern Observatory; Universidad Diego Portales; Millennium Nucleus on Young Exoplanets and their Moons; collaborating observatories

Result date

Published July 22, 2026

Why it matters: The result is strong peer-reviewed evidence for a planetary-mass satellite around an extrasolar substellar object and demonstrates a radial-velocity route to satellite systems beyond the Solar System.

Limits: The host is a brown dwarf orbiting a star, not a planet, so calling the object an exomoon is partly definitional. The satellite interpretation still needs independent or astrometric confirmation, and radial velocity supplies a minimum rather than a true mass.

The first true sugar detected in interstellar space

Broadband spectral surveys with the Yebes 40 m and IRAM 30 m telescopes identified 17 transitions consistent with erythrulose in the Galactic-centre molecular cloud G+0.693−0.027. Erythrulose is a chiral four-carbon ketose and the first chemically defined sugar reported in the interstellar medium.

Problem or result

Detecting chemically defined sugars in interstellar space and testing whether complex prebiotic feedstocks can form before planets

Authors

Izaskun Jiménez-Serra, Juan García de la Concepción, Herma M. Cuppen and collaborators

Institutions

CAB/CSIC-INTA, University of Extremadura, Radboud University, Complutense University of Madrid, Max Planck Institute for Extraterrestrial Physics, ESO, Towson University, Yebes Observatory, University of the Basque Country and partners

Result date

Published July 13, 2026

Why it matters: Sugars are central to metabolism and nucleic-acid chemistry. The peer-reviewed detection shows that a comparatively complex sugar can form before planets, strengthening the empirical basis for studying interstellar contributions to prebiotic chemistry.

Limits: The detection does not establish life, a biosignature or delivery of erythrulose to early Earth. Some transitions occur in a line-rich source where blending is a challenge, and the proposed grain-surface formation pathway and early-Earth contribution rely on chemical models and later delivery scenarios.

Disease-specific human tau folds were copied by endogenous mouse tau

Lövestam and collaborators seeded wild-type mouse brains with human Alzheimer’s-disease or corticobasal-degeneration tau. Later cryo-electron microscopy found that endogenous mouse tau retained the seed-specific folds, providing direct structural evidence for prion-like templated transmission in a mouse model.

Problem or result

Structural retention of disease-specific tau folds after templated propagation in vivo

Authors

Sofia Lövestam, Aki Shimozawa, Airi Tarutani, Reiko Ohtani and collaborators

Institutions

MRC Laboratory of Molecular Biology; Tokyo Metropolitan Institute of Medical Science; collaborating neuropathology centres

Result date

Published online September 30, 2026

Why it matters: The work directly links disease-specific tau conformations with templated propagation in an intact brain and strengthens the mechanistic basis for treating distinct tau folds as biologically meaningful strains.

Limits: The mice did not develop the corresponding human clinical diseases. Seeds were injected directly into the brain, adult mouse tau differs from human tau, pooled material was used for structural preparations, and the result does not imply person-to-person contagiousness, prevention or treatment.

Human pituitary organoids produced transient hormone responses in a primate model

Kondo and collaborators transplanted human pluripotent-stem-cell-derived pituitary–hypothalamus organoids into mice and one cynomolgus monkey. The primate showed increased ACTH and cortisol for about six weeks before function waned as inflammatory rejection emerged.

Problem or result

Functional pituitary-cell replacement beyond rodents

Authors

Tatsuma Kondo, Hidetaka Suga, Kenji Watari, Shiori Taga and collaborators

Institutions

Nagoya University with collaborating academic and industry researchers

Result date

Published October 2, 2026

Why it matters: The study moves pituitary replacement from rodents to a functional large-animal feasibility test and shows that the graft can temporarily engage an intact adrenal axis.

Limits: The primate evidence is one uncontrolled xenograft with short follow-up, intensive immunosuppression and eventual rejection. It establishes neither human safety nor durable efficacy, dosing, tumour safety or a clinical treatment.

One anomalous trial materially distorted meta-analytic support for paediatric fluoxetine

Lyus and collaborators trace conflicting network meta-analyses to a 40-participant active-comparator trial with an implausibly large, unusually orderly effect and missing methodological details. Reanalysis without it removed the outsized fluoxetine advantage.

Problem or result

Integrity and sensitivity of evidence for fluoxetine in paediatric depression

Authors

Richard Lyus, Florian Naudet, Gert van Valkenhoef and Martin Plöderl

Institutions

Affiliations in the version of record span UK child and adolescent psychiatry, University of Rennes, University of Groningen and Paracelsus Medical Private University

Result date

Peer-reviewed online article and independent specialist reporting in October 2026

Why it matters: The analysis shows how one small, apparently unreliable trial can materially change a high-level evidence network used in paediatric treatment discussions, separating data-integrity risk from an ordinary statistical disagreement.

Limits: This is not a retraction, regulator decision, proof that fluoxetine is ineffective or advice to stop treatment. The questioned trial remained published at the cutoff, earlier reviews had already rated confidence very low, and total benefit–risk decisions require the full evidence base and individualized clinical care.

FDA approves the first direct muscle-targeted add-on therapy for spinal muscular atrophy

FDA approved apitegromab for adults and children aged two years or older with spinal muscular atrophy who already receive an SMN2-targeted treatment. The pivotal 52-week trial found a 1.8-point adjusted motor-scale benefit at the approved dose.

Problem or result

Add-on muscle-targeted treatment for spinal muscular atrophy

Authors

Thomas O. Crawford, Laurent Servais, Eugenio Mercuri and the SAPPHIRE Study Group

Institutions

International SAPPHIRE trial sites; Scholar Rock; neuromuscular centres in the United States and Europe

Result date

FDA approval September 11, 2026

Why it matters: Existing disease-modifying SMA therapies increase survival motor neuron protein. Apitegromab is the first approval to complement them by directly inhibiting latent myostatin in muscle.

Limits: It is an add-on, not a cure or replacement for SMN-directed therapy. The mean gain was modest, the pivotal primary population was non-ambulatory children aged 2–12, and longer-term functional durability remains uncertain.

FDA approves the first treatment for ataxia in ataxia-telangiectasia

FDA approved levacetylleucine for ataxia in people with ataxia-telangiectasia weighing at least 15 kilograms. In a 73-participant randomized crossover trial, 12 weeks of treatment improved SARA scores by an estimated 1.88 points versus placebo.

Problem or result

Symptomatic treatment of ataxia in ataxia-telangiectasia

Authors

Kyriakos Martakis, Tatiana Bremova-Ertl, Clare Bolton and the multinational phase 3 investigators

Institutions

Ten multinational trial hospitals; University of Cologne; Bern; UCLA; Massachusetts General Hospital; UTHealth Houston; IntraBio

Result date

FDA approval September 18, 2026

Why it matters: Ataxia-telangiectasia causes progressive childhood-onset coordination loss, immune defects and cancer susceptibility. This is the first FDA-approved treatment for its disabling ataxia.

Limits: The rare-disease trial was small, sponsor-funded and limited to 12 weeks per period. The approval concerns symptomatic improvement; long-term neuroprotection or disease modification is unproven.

FDA approves the first disease-modifying treatment for Alexander disease

The U.S. Food and Drug Administration approved intrathecal zilganersen (Zanvastro) for pediatric and adult Alexander disease. The antisense drug reduces production of abnormal GFAP and is the first approved treatment to target the disease’s underlying protein accumulation.

Problem or result

First approved treatment for the ultra-rare GFAP-driven neurological disorder Alexander disease

Authors

FDA review; pivotal study led by Amy Waldman and collaborators

Institutions

U.S. Food and Drug Administration; Ionis Pharmaceuticals; international trial sites

Result date

FDA approval announced September 3, 2026

Why it matters: Patients previously had only supportive care. In the 49-person randomized study, the 50-mg group met the walking-speed primary endpoint at week 61, and the approval extends from infancy through adulthood.

Limits: Alexander disease affects fewer than one person per million, so the evidence base is small. The randomized study enrolled 49 patients aged two or older; the under-two indication relies on four open-label patients plus pharmacokinetic extrapolation. Aseptic meningitis has been reported, and the pivotal results located for this cycle were a conference-supplement abstract rather than a full article.

The FDA approved the first treatment that directly restores orexin signaling in narcolepsy type 1

The FDA approved Orzeyful (oveporexton) for adults with narcolepsy type 1 after two randomized, double-blind, placebo-controlled 12-week studies enrolling 273 adults. It is the first approved medicine to act directly on the lost orexin-signaling pathway and address the disorder’s full symptom range.

Problem or result

Disease-mechanism-directed treatment for adult narcolepsy type 1

Authors

FDA review teams; FirstLight and RadiantLight investigators

Institutions

U.S. Food and Drug Administration; Takeda; multinational phase 3 sites

Result date

FDA approval August 5, 2026

Why it matters: Existing medicines treated separate symptoms. Oveporexton is the first approved oral orexin-2 receptor agonist intended to restore the missing wakefulness signal underlying narcolepsy type 1.

Limits: Approval is for adults with narcolepsy type 1, not narcolepsy type 2 or a cure. The pivotal studies lasted 12 weeks; insomnia and urinary effects were common, strong CYP3A inhibitors are contraindicated, pediatric safety is unestablished, and marketing awaits DEA scheduling.

Mitochondrial plaques were described in Alzheimer’s models and postmortem human brains

Researchers found large accumulations of acidic and neutral mitochondria inside neuronal processes in two Alzheimer’s mouse models and in postmortem human tissue. They named the structures mitochondrial plaques and linked their persistence to delayed lysosomal recruitment and incomplete mitochondrial degradation.

Problem or result

A newly described mitochondrial-rich pathological structure associated with impaired mitophagy and lysosomal function in Alzheimer’s disease

Authors

Xiuli Dan, Deborah L. Croteau, Wenlong Liu, Xixia Chu, Ross A. McDevitt, Paul D. Robbins and Vilhelm A. Bohr

Institutions

National Institute on Aging, NIH; National Cancer Institute, NIH; University of Minnesota; University of Copenhagen

Result date

Published July 29, 2026

Why it matters: The structures sometimes appeared before or apart from amyloid plaques in the mouse models, offering a concrete tissue-level feature through which mitochondrial and lysosomal dysfunction can be studied alongside established Alzheimer’s pathology.

Limits: Most mechanistic evidence comes from transgenic mouse models. Human validation used postmortem tissue from four Alzheimer’s cases and four controls, so independent replication, disease-specificity and clinical relevance remain unresolved. The study does not show that the structures cause cognitive decline or provide a validated diagnostic or treatment target.

A one-time in-vivo CRISPR treatment cut hereditary-angioedema attacks by 87% in a phase 3 trial

In the randomized phase 3 HAELO trial, 80 participants received one 50-mg intravenous dose of lonvoguran ziclumeran or placebo. From weeks 5 through 28, mean monthly attack rates were 0.26 versus 2.10, an estimated 87% relative reduction.

Problem or result

Hereditary angioedema causes unpredictable, potentially life-threatening swelling attacks and usually requires continuing preventive treatment.

Authors

Danny M. Cohn et al.; HAELO Investigators

Institutions

International HAELO clinical sites; Intellia Therapeutics

Result date

June 13, 2026

Why it matters: It is the first placebo-controlled phase 3 result to show a large clinical benefit from a single systemic CRISPR treatment that edits cells inside the body.

Limits: The treatment remains investigational, the rare-disease cohort was small, and median follow-up was about 7.5 months. Long-term durability, off-target editing, late adverse effects and permanent kallikrein-suppression risks remain unresolved; Intellia funded the study.

The first bespoke in-vivo base-editing medicine was designed and given to one infant

Researchers created a lipid-nanoparticle-delivered adenine base editor for one infant's pathogenic CPS1 variant. After two infusions, the child tolerated more dietary protein and required about half the baseline nitrogen-scavenger dose during the short observation reported in the paper.

Problem or result

Carbamoyl-phosphate synthetase 1 deficiency is an ultra-rare genetic disorder that can cause dangerous ammonia accumulation, brain injury and death.

Authors

Kiran Musunuru, Rebecca C. Ahrens-Nicklas et al.

Institutions

Children's Hospital of Philadelphia; University of Pennsylvania; Innovative Genomics Institute; Broad Institute and collaborators

Result date

May 15, 2025

Why it matters: The case proves that an n-of-1 programmable genetic medicine can be designed, manufactured, reviewed and delivered on a clinically meaningful timescale.

Limits: This was one infant with no control group and only seven weeks of reported post-treatment observation. The fraction of liver cells edited was not directly measured, so the result does not establish a cure, durable efficacy, scalability or long-term safety.

The first gene therapy for OTOF-related genetic hearing loss won accelerated FDA approval

The peer-reviewed CHORD report found that nine of 12 children met the week-24 auditory endpoint, six could hear soft speech without assistive devices and three reached average normal hearing sensitivity. The FDA later reviewed an ongoing single-arm study with 24 treated children, 20 of whom were efficacy-evaluable, and granted accelerated approval to Otarmeni after 80% of evaluable participants improved.

Problem or result

People with biallelic OTOF variants can have severe-to-profound congenital hearing loss because inner hair cells cannot transmit auditory signals normally.

Authors

Vassili Valayannopoulos et al.; CHORD Study Group

Institutions

Multicentre CHORD trial sites in North America and Europe; Regeneron Pharmaceuticals

Result date

April 23, 2026

Why it matters: Otarmeni is the first approved disease-modifying treatment for OTOF-related deafness and the first FDA-approved dual-AAV gene therapy.

Limits: Accelerated approval rests on a single-arm rare-disease trial and a hearing-sensitivity surrogate; continued approval may depend on durability and confirmation of speech-development and quality-of-life benefits. The indication is limited to molecularly confirmed biallelic OTOF variants with preserved outer hair-cell function and no prior implant in the treated ear.

Twice-yearly lenacapavir prevented nearly all HIV infections in two phase 3 trials and became a WHO-recommended PrEP option

PURPOSE 1 recorded zero HIV infections among 2,134 participants assigned twice-yearly lenacapavir, compared with infections in both daily-oral-PrEP groups; PURPOSE 2 showed similarly high prevention efficacy in a broader gender population. The FDA approved injectable lenacapavir for prevention in June 2025, and WHO recommended it as an additional PrEP choice in July 2025.

Problem or result

Daily oral HIV pre-exposure prophylaxis is highly effective when taken consistently, but adherence and access barriers reduce its real-world impact.

Authors

Linda-Gail Bekker et al., PURPOSE 1 Study Team; Colleen F. Kelley et al., PURPOSE 2 Study Team

Institutions

Multicountry PURPOSE trial networks; Gilead Sciences

Result date

July 14, 2025

Why it matters: A six-month injection can remove much of the day-to-day adherence burden that limits oral PrEP.

Limits: Public-health impact depends on price, equitable access, clinic capacity and timely HIV testing. Injection-site reactions occur, infection during the long pharmacologic tail can select resistance, and both phase 3 trials were manufacturer-sponsored.

Gene-edited donor islet cells survived and secreted insulin without immunosuppression for 14 months in one patient

Gene-edited allogeneic pancreatic islets transplanted into the forearm muscle of one man with longstanding type 1 diabetes continued to show beta-cell function through 14 months without systemic immunosuppression. The investigators reported no detected immune response against the allograft.

Problem or result

Donor islet replacement can restore insulin production, but immune rejection normally requires chronic immunosuppression.

Authors

Per-Ola Carlsson, Xiaomeng Hu, Hanne Scholz et al.

Institutions

Uppsala University; UCSF; University of Oslo and Oslo University Hospital; Karolinska Institutet; Sana Biotechnology

Result date

Initial report August 4, 2025; follow-up July 10, 2026

Why it matters: Functional donor-cell survival without immunosuppressive drugs addresses the central barrier to off-the-shelf islet replacement.

Limits: This remains a one-person, uncontrolled early-phase result using a low cell dose; the participant did not become insulin-independent. Fourteen months does not establish population-level efficacy, lifetime immune evasion, malignancy or infection safety, or performance at a therapeutic dose. The registry still lists two planned participants and has stale completion dates.

Natural-language protein designs show four intended functions in laboratory tests

Dai, You, Yuan and collaborators report that Pinal, a 16-billion-parameter text-to-protein model, generated candidates with experimentally observed function in four tasks: fluorescence, PET hydrolysis, alcohol dehydrogenation and metabolic H-protein activity.

Problem or result

Functional protein candidate generation from natural-language specifications

Authors

Fengyuan Dai, Shiyang You, Yudian Zhu, Fajie Yuan and collaborators

Institutions

Westlake University-led multi-institution collaboration; complete affiliations are listed in the Nature record

Result date

Nature publication October 21, 2026; underlying preprint revised before publication

Why it matters: The work moves text-conditioned protein design beyond predicted structure by testing distinct functions in the laboratory. A designed H-protein increased product titre 1.7-fold over the native Escherichia coli H-protein in a multi-enzyme carbon-fixation pathway without iterative experimental optimization.

Limits: Four successful task classes do not establish general prompt-to-protein reliability, safety or superiority to specialist design pipelines. Most candidate screening remained computational, and the reported H-protein gain is a pathway-specific laboratory result rather than an organism-level or industrial production result.

Designed proteins cleaved peptide bonds through two catalytic strategies

Two related Nature studies report wet-lab-validated de novo cysteine proteases and zinc metalloproteases, including sequence-selective cleavage, high rate enhancements and crystal structures close to the intended designs.

Problem or result

De novo design of sequence-selective cysteine proteases and zinc metalloproteases

Authors

Hojae Choi, Magnus S. Bauer, Samuel J. Pellock, Donald Hilvert, David Baker and collaborators; Anqi Chen, Kejia Wu, Hojae Choi and collaborators

Institutions

University of Washington Institute for Protein Design with ETH Zurich, HHMI, LBNL and international collaborators

Result date

Both Nature papers published October 14, 2026

Why it matters: Cleaving unactivated peptide bonds has been a difficult de novo enzyme-design target. The paired studies demonstrate two experimentally and structurally tested catalytic routes rather than only binding predictions or activated-substrate chemistry.

Limits: These are laboratory and cultured-cell results, not therapies. The designs remain less efficient than the best natural proteases, and delivery, off-target cleavage, immunogenicity, organism-level efficacy and safety are untested. The two papers substantially share a research ecosystem and are not independent replication.

A Jurassic avialan broadens the earliest known flight ecology

Xu and collaborators describe Yuezhuornis yangyuanensis from a partial Upper Jurassic skeleton and infer from its short forelimbs, long hindlimbs and related anatomy that it was a terrestrial, non-volant early-diverging avialan.

Problem or result

Ecological diversity and loss or absence of flight in early avialans

Authors

Liming Xu, Min Wang, Ganmin Lin, Liping Dong, Ke Deng, Xing Xu, Zhonghe Zhou and collaborators

Institutions

Institute of Vertebrate Paleontology and Paleoanthropology, Chinese Academy of Sciences; Fujian Institute of Geological Sciences; collaborators

Result date

Published October 14, 2026

Why it matters: A non-volant avialan contemporary with Archaeopteryx expands the known ecological range near the origin of birds and argues against a simple progression toward steadily improved flight.

Limits: The taxon is known from one incomplete specimen. Flight ability and phylogenetic position are anatomical and model-based inferences that further fossils may revise; the result does not imply multiple origins of modern birds.

Norellraptor supports convergent assembly of flight-related anatomy

Wang and collaborators describe Norellraptor barsboldi and infer that microraptorines and early birds acquired a similar flight apparatus through different evolutionary sequences, supporting convergence rather than inheritance of one fully assembled package.

Problem or result

Whether microraptorine and avialan flight apparatuses were assembled in the same evolutionary sequence

Authors

Wang Xuri and collaborators

Institutions

Chinese Academy of Geological Sciences; Hebei GEO University; OPHIS Museo Paleontologico; collaborating institutions

Result date

Published September 29; specialist coverage October 2, 2026

Why it matters: The unusually complete fossil provides a stronger basis for separating convergence from shared inheritance and complicates linear accounts of how the avian flight package accumulated.

Limits: The inference rests on one fossil and model-dependent anatomical coding and phylogeny. Flight-related anatomy and plumage do not by themselves demonstrate sustained powered flight, multiple origins of modern birds or repeated evolution of every flight component.

Human cortical organoids integrate extensively into a developing mouse brain

After genetically depleting glutamatergic neurons in mouse cortex and hippocampus, Stanford researchers transplanted human cortical organoids into neonatal mice. Human-derived tissue expanded through the vacant niche, formed several cell types, integrated anatomically and showed organized developing activity.

Problem or result

Extensive developmental integration of human cortical organoid tissue in vivo

Authors

Konstantin Kaganovsky, Kevin W. Kelley, Tilo Gschwind and collaborators led by Sergiu P. Pașca

Institutions

Stanford University and Stanford Medicine

Result date

Published September 16, 2026

Why it matters: Earlier organoid grafts remained relatively localized. This method creates a broader in-vivo platform for studying human cortical development, injury and disease than an isolated organoid dish can provide.

Limits: This artificial chimera is not a human brain in a mouse. The cells remained immature and lacked normal cortical lamination, areal organization and full interneuron diversity; connectivity and activity do not establish human-like cognition. Ethical oversight and long-term safety remain central.

A functional atlas maps human-specific skeletal gene regulation

Researchers assayed 561,410 human-derived substitutions in candidate regulatory elements in cartilage-forming cells and identified 15,077 loci with human-specific activity, revealing broad rewiring of extracellular-matrix and skeletal-development programs.

Problem or result

Genome-scale functional mapping of human-specific skeletal regulatory substitutions

Authors

Yizhi Yan, Nadav Mishol, Katharina Lange, Zicong Zhang and collaborators led by David Gokhman

Institutions

Kyoto University; McGill University; Weizmann Institute; Technion; EMBL Barcelona; Pompeu Fabra; CRG; CNAG and collaborators

Result date

Published September 23, 2026

Why it matters: The study links large-scale regulatory change with the evolution of human skeletal traits and supplies a reusable functional map rather than relying only on comparative sequence annotation.

Limits: The trait and disease connections are associations, not proof that any one regulatory substitution caused a human anatomical feature or osteoarthritis. Hybrid-cell and in-vitro systems do not reproduce whole-organism development.

Bianfu Cave fossils and artefacts expand the Denisovan body and behavioural record

Two coordinated Nature studies identify five Denisovan fossils from Bianfu Cave in southwestern China, including the first attributed radius from the group, and reconstruct a roughly 190,000-to-70,000-year cultural sequence with specialized hunting, stone tools and bone-tool use.

Problem or result

Denisovan anatomy, geographic range and material culture at Bianfu Cave

Authors

Qiaomei Fu, Qianjun Ruan and collaborating teams

Institutions

Chinese Academy of Sciences and collaborating Chinese research institutes

Result date

Published September 9, 2026