New proofNumber theory2026-08-19-erdos-501-independence
Erdős Problem #501 shown independent of ZFC, with both directions in Lean
GitHub: Erdős Problem #501 in Lean 4
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Frontier
New proofs, counterexamples, bounds and formalizations reported by people and AI systems, newest first.
August 2026 21
New proofOther2026-08-19-delavina-waller-wiener
The DeLaViña-Waller conjecture on the Wiener index
Zenodo: the DeLaViña-Waller conjecture on the Wiener index
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CounterexampleOther2026-08-19-caratheodory-umbilic
Counterexample to the smooth Carathéodory conjecture on umbilic points
Counterexample to the smooth Carathéodory conjecture on umbilic points
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New proofOther2026-08-19-big-line-big-clique
First open case of the big-line-big-clique conjecture
arXiv: large finite point sets have 4 collinear points or a 6-clique
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Improved boundOther2026-08-18-stein-riesz-weak-type
Dimension-free weak-type bound for the vector Riesz transform
arXiv: A dimension-free weak-type (1,1) bound for the vector Riesz transform
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New proofNumber theory2026-08-18-prime-gaps-246
Bounded prime gaps of 246 formalized in Lean from Bombieri-Vinogradov
Axiom Math: Lean formalization of bounded gaps between primes
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New proofTheoretical computer science2026-08-17-matmul-exponent
Matrix multiplication exponent lowered to below 2.371177
arXiv: Improving the matrix multiplication exponent with modern optimization and AlphaEvolve
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New proofOther2026-08-16-talagrand-convolution
Talagrand's convolution conjecture proved on the Boolean hypercube
arXiv: Weak-Type Bounds for Convolution on the Boolean Hypercube
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Improved boundNumber theory
Zeta zeros on the critical line
A lower bound about 67.250% for simple critical-line zeros; a related distinct-zero bound about 83.625% is also reported.
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New proofOther2026-08-13-banach-isometric
Banach's isometric conjecture settled in the remaining odd dimensions
arXiv: A solution to Banach's isometric conjecture
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New proofOther2026-08-12-liquid-drop-minimizers
Complete minimizer picture for Gamow's liquid drop model
arXiv: No compromise in the liquid drop model
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New proofTheoretical computer science2026-08-11-prescribed-cycle-recovery
A prescribed Hamiltonian cycle that a book-embedding algorithm cannot produce
Zenodo: A Counterexample to Prescribed-Cycle Recovery in Barnette Graphs
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New proofNumber theory2026-08-10-zeta-zeros-critical-line
Proportion of zeta zeros on the critical line raised to 67.25%
Anthropic: paper (PDF)
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New proofOther2026-08-06-kreiss-constant-separation
Separation between the ordinary and strong Kreiss constants
arXiv: a solution to the inverse generator problem and related questions
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New proofAnalysis
Sendov and Phelps–Rodriguez conjectures
Sendov is reported for all degrees; the stricter distance conclusion and exceptional equality cases are addressed by the same argument.
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CounterexamplePDE and mathematical physics
Planar Schiffer / Pompeiu rigidity
Noncircular counterexamples are reported, including a real-analytic Jordan domain from computer-assisted existence estimates and a separate bifurcation construction.
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New proofOther2026-08-05-sendov-conjecture
Sendov's conjecture proved for every degree
ProofAtlas: A Computer-Assisted Proof of Sendov's Conjecture (PDF)
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New proofOther2026-08-04-moore-bound
Asymptotic degree-diameter problem resolved for fixed diameter
arXiv: Asymptotically attaining the Moore bound
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New proofTheoretical computer science2026-08-01-astra-ten-advances
Ten results in mathematics and theoretical computer science with Lean certificates
Lean 4 certificates for all ten results
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July 2026 19
New proofOther2026-07-29-sumset-difference-exponent
Optimal exponent relating sumsets and difference sets determined
arXiv: Settling the Optimal Exponent Relating Sumsets and Difference Sets
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New proofTheoretical computer science
Kemeny aggregation for three voters
A MAX CUT reduction establishes NP-completeness of the decision version and hardness of the optimization problem.
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Improved boundProbability
Feige's nonnegative-random-variable conjecture
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.
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New proofTheoretical computer science
Dinitz–Garg–Goemans flow conjecture
A seven-vertex, nine-arc witness: fractional cost 58; every allowed unsplittable routing costs at least 60, with demands 15,10,15.
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Partial resultTheoretical computer science
Černý bound for positive-level one-cluster automata
For the specified one-cluster automata with cycle length m and positive level ℓ, a reset-word bound (m-1)(n-1)+mℓ≤(n-1)² is reported.
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New proofGeometry and topology
Mod-4 Kawauchi conjecture
A proof of the modulo-four assertion is reported.
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New proofProbability
Strong log-concavity of the Chernoff density
A proof is reported in a July preprint.
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New proofProbability
Gaussian moments conjecture
Explicit three-variable polynomials with the first expectation zero and the second equal to m! for every m≥1; hence failure in all n≥3.
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New proofProbability2026-07-20-gaussian-product-inequality
Gaussian product inequality conjecture proved
A proof of the strong Gaussian product inequality conjecture
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CounterexampleGeometry and topology
Jacobian conjecture in dimension three
An explicit map in C^3 with det JF=-2 and three different points having the same image; padding gives higher-dimensional counterexamples.
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Improved boundTheoretical computer science
Derivative-free convex-optimization oracle lower bound
A near-quadratic lower bound is reported, with formal proof artifacts.
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