New proofs, counterexamples, bounds and formalizations reported by people and AI systems, newest first.
Proves Shelah's eventual categoricity conjecture in ZFC: for each bound on the Löwenheim–Skolem number, a uniform threshold makes categoricity of an abstract elementary class in one cardinal above that threshold imply categoricity throughout the same tail. Categoricity means uniqueness up to isomorphism at a given cardinality. Under the continuum hypothesis, a proposed specific Hanf threshold need not suffice.
Proves over ℂ, giving arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension na with a > 0.465 permits rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field.
Proves , resolving derandomization for bounded-error logarithmic-space computation. An effective compiler converts each randomized polynomial-time logarithmic-space machine deciding a language with one-sided or two-sided error into a deterministic logarithmic-space decider with explicit polynomial running-time bounds.
Resolves the interior regularity conjecture for reduced absolute planar Mumford–Shah minimizers with bounded fidelity data. Locally, the closed discontinuity set is a arc, a regular crack tip, or three arcs meeting at ; only finitely many global connected components meet any compact interior region.
Proves the sharp upper bound for nodal length on every fixed smooth closed surface, completing Yau's conjecture there. The upper bound fails for fixed smooth metrics in dimensions three and four, including metrics on S3 arbitrarily close to round. In dimension five, nodal measure can grow faster than for some fixed , ruling out even arbitrarily small power losses.
Proves that every stationary integral m-varifold in a Euclidean open set has singular set of Hausdorff dimension at most , a sharp bound in every positive dimension and codimension. On round spheres, the family also proves almost-everywhere regularity: the singular set has zero m-dimensional Hausdorff measure.
Proves for every closed connected oriented smooth n-manifold, n ≥ 3, and every smooth metric, with depending only on dimension. Here and is real simplicial volume. Also proves rational inessentiality under positive scalar curvature, resolving the Gromov–Lawson conjecture; every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat.
Proves that the undeformed relative BFSS model has exactly one normalizable zero-energy state for every finite N ≥ 2, resolving the threshold-bound-state conjecture. For , a companion proves infinitely many normalizable positive-energy eigenstates with unbounded energies, contradicting the original BFSS paper's exclusion of additional bound states at N = 2.
Proves the second Kahn–Kalai conjecture: for every finite simple graph H with h ≥ 1 edges and at most n vertices, its appearance threshold in is at most , with universal C. Here is the least density at which every subgraph of H has expected copy count at least 1/2.
Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but points of any n-point planar set determine at least distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by for an absolute δ > 0.
Proves Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. Quantitatively, for each fixed k ≥ 3, every subset of with no nonconstant k-term progression has size at most , with positive constants depending only on k.
For fixed A > 0, a one-pass learner with persistent bits needs noiseless Gaussian samples to recover a unit vector to angular error with probability 2/3, uniformly in accuracy for large d. Computation and randomized updates are unrestricted, but output uses only the terminal state, stopping index and fresh randomness.
Over every characteristic-zero field, the entry of a product of n independent variable matrices requires gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables.
At every fixed deletion probability in , reconstructing an arbitrary length-n binary string requires independent traces, ruling out polynomial-sample reconstruction. A uniform decoder achieves quasipolynomial sample and running-time bounds for known fixed rational retention probabilities. When the deletion probability is at most for fixed ε > 0, both bounds become polynomial in the input and parameter encoding.
Proves an lower bound for the border determinantal complexity of the permanent over ℂ. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least , for an absolute c > 0 and all sufficiently large n; the same bound therefore holds for exact representations.
Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within , independently of length. Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums. A matching lower bound gives the optimal order .
Proves that the principal-value trilinear Hilbert transform with shifts , , is bounded from to . This resolves the L3 exponent case of the standard conjecture for slopes 1, 2, 3.
Proves maximal and annular r-variation bounds, for every r > 2, from complex to . The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric point.
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.
Two October preprints report dimension-independent bounds; their arguments and version histories are distinct.
arXiv: An exponential lower bound for the bit pigeonhole principle in resolution over parities
arXiv: An Improved Lower Bound for the Complex Grothendieck Constant
A paper proves an upper bound of 212. The Lean certificate verifies a deduction from stated distribution estimates and other analytic inputs.
Manuscript: Erdős_4_GPT_5.6_Sol.pdf
arXiv: Sub-optimality of Marton's Inner Bound for the Two-Receiver Broadcast Channel
arXiv: A dimension-free weak-type (1,1) bound for the vector Riesz transform
A lower bound about 67.250% for simple critical-line zeros; a related distinct-zero bound about 83.625% is also reported.
A sharp finite-n bound, and bounds for general δ, are reported; the δ≥1 conclusion settles the stated Feige assertion if the proof is valid.
A near-quadratic lower bound is reported, with formal proof artifacts.