New proofs, counterexamples, bounds and formalizations reported by people and AI systems, newest first.
Proves that for every real shift γ and finite-valued , divergence of implies for infinitely many q, for almost every x. Here ϕ is Euler's totient and the norm is distance to the nearest integer. Numerators are unrestricted; no monotonicity or Diophantine condition on γ is needed.
Answers Jacobsthal's quadratic-bound question: every interval of consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute constant C. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss.
Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the -power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning's higher-degree theorem, this covers every d ≥ 4.
Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚp correspond bijectively to conjugacy classes of sections of its full arithmetic étale fundamental group. It also proves Grothendieck's section conjecture over ℚ for the modular curves and of genus at least two.
Proves the Margulis–Platonov conjecture over every global field, including function fields of characteristic two. For an absolutely almost simple simply connected algebraic group G over k, every noncentral abstract normal subgroup of is the inverse image of an open normal subgroup in the finite product of its anisotropic nonarchimedean local groups.
Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational satisfies . This also proves convergence of the Flint–Hills series , with angles in radians.
Proves the abelian Zilber–Pink conjecture over : every irreducible subvariety has finitely many maximal atypical subvarieties relative to its smallest containing torsion coset. It also proves the full curve case in the Siegel threefold for Hodge-generic curves defined over , without boundary or reduction assumptions.
Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity.
Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over under the four stated Lie-theoretic characteristic hypotheses. Over arbitrary algebraically closed fields of characteristic p > 0, the same conclusion holds assuming additionally that p is very good for the group and . Over global function fields, proves Ramanujan at every place for globally generic cuspidal representations of split adjoint absolutely simple exceptional groups, without characteristic or ramification-depth restrictions, and at every unramified place for cuspidal representations of split adjoint absolutely simple groups with a generic unramified component. Assuming the finite-level Ramanujan–Arthur decomposition, constructs global Arthur enhancements of occurring cuspidal excursion parameters for split connected semisimple groups at full finite level, recovering the given parameters by diagonal specialization on the entire Weil group, including inertia.
Proves that no finite modification of the primes can be written as with each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability.
Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and are asymptotically independent in ordinary natural density. In particular, the integers satisfying have density 1/2.
Proves that the normalized ordered logarithms of the prime factors of , counted with multiplicity, converge jointly to the Poisson–Dirichlet law as p ranges uniformly over primes up to x and . This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than totient preimages, for every ε > 0.
Every continuous odd absolutely irreducible two-dimensional 2-adic representation of unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves classical modularity up to Tate twist for irreducible odd representations with these finiteness conditions that are de Rham at 2 with distinct Hodge–Tate weights, resolving the dyadic Fontaine–Mazur case without residual restrictions.
Reconstructs function fields of transcendence degree at least two over algebraically closed constants from , , and their product. These data recover the perfect closure and constants when ℓ differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-ℓ Galois data away from the characteristic.
Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture.
Proves the ordinary two-point Chowla conjecture, with a bound for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times for .
Proves Milne's rationality conjecture for abelian varieties over with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-p and crystalline realizations, for every residue characteristic p.
arXiv: Bizeul, Klartag and Lehec, Presenting a proof of the Kannan-Lovász-Simonovits conjecture
Optimal packings and lower-bound proofs are reported for these finite cases.
Two October preprints report dimension-independent bounds; their arguments and version histories are distinct.
BootLoops: WatsonW.py, the closed form as code
A six-task counterexample is reported; exact periods and the reduction rule must be taken from the original witness before publishing.
GitHub: ShengtongZhang-alt/lovasz-opus, the draft and its Lean formalization
GitHub: jbaelaw/erdos1220-lean
arXiv: A Counterexample to Talagrand’s Operator Cotype Problem
arXiv: Rogers-Ramanujan identities from the geometry of Xᵃ = Yᵇ
GitHub: corun1024/4ct, Chris Emery's Lean 4 proof
arXiv: On a conjecture of Browning and Sawin on random hypersurfaces with sign coefficients
arXiv: Extending the Cordoba-Martinez-Zoroa IPM Blow-Up to Uniformly Space-Time Smooth Forcing
arXiv: An exponential lower bound for the bit pigeonhole principle in resolution over parities
GitHub: boonsuan/nivat, the Lean 4 and mathlib formalization
arXiv: Existence of the Core in Approval-Based Committee Elections
arXiv: Small undecidable groups and unrecognizable 4-manifolds
An unforced C¹ blowup construction is reported alongside the Navier–Stokes work.
OpenAI reports examples for every ν>0 in R³ and on the three-torus, under its specified forcing and initial-data conditions.
GitHub: openai/NavierStokesAndEuler, the Lean 4 formalizations
arXiv: An Improved Lower Bound for the Complex Grothendieck Constant
GitHub: ShengtongZhang-alt/BN, the Lean 4 and Mathlib formalization
GitHub: tadamcz/phi-mixing-clt, the Lean counterexample and literature review
A new AI-assisted end-to-end Lean development is reported in September, formalizing an already proved theorem.