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• 007

  Ordinary two-point correlations and the corrected Elliott conjecture

  Proves the ordinary two-point Chowla conjecture, with a bound $O(X/(\log X{)}^c)$ 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 $n^{it}$ for $|t|\le X$.

  Verified by Lean Reasoning summaryNumber theory
• 017

  The irrationality exponent of π is 2

  Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational $p/q$ satisfies $|\pi -p/q|\ge q^{-2-\epsilon }$. This also proves convergence of the Flint–Hills series ${\sum }_{n\ge 1}1/(n^3{\sin }^{2}n)$, with angles in radians.

  Verified by Lean Reasoning summaryNumber theory
• 087

  The Mahler conjectures, functional inequalities and polar-product symplectic width

  Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The corresponding sharp functional Mahler inequalities also hold. For n ≥ 2, every symmetric polar product $K\times K^{\circ }$ in dimension $2n$ has Gromov width 4.

  Verified by Lean Reasoning summaryConvex and metric geometry
• 159

  Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds

  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 $\{1,\dots ,N\}$ with no nonconstant k-term progression has size at most $C_{k}N\exp [-c_k(\log N{)}^{{\epsilon }_k}]$, with positive constants depending only on k.

  Verified by Lean Reasoning summaryCombinatorics
• 197

  A torsion-free group algebra that is not directly finite

  Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽₂ is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture.

  Verified by Lean Reasoning summaryAlgebra
• 221

  The Mézard–Parisi formula for diluted spin glasses

  Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability.

  Verified by Lean Reasoning summaryProbability and statistical mechanics
• 271

  Bloch's law, its lattice correction, and the spherical magnetization law

  Proves Bloch's T^3/2 law with its exact coefficient for three-dimensional quantum Heisenberg ferromagnets at every positive quantum spin, allowing nonnegative symmetric finite-range couplings whose support generates ℤ³. The thermodynamic limit precedes the zero-field derivative and low-temperature limit. The family also proves spontaneous magnetization for nearest-neighbor models in every dimension d ≥ 3 and determines the first lattice correction for three-dimensional nearest-neighbor couplings.

  Verified by Lean Reasoning summaryMathematical physics
• 287

  Isomorphism of the free group factors

  Resolves the free group factor isomorphism problem: $L({\mathbb{F}}_2)\cong L({\mathbb{F}}_3)$, and hence all interpolated free group factors, including $L({\mathbb{F}}_{\infty })$, are isomorphic. Their common factor has fundamental group ${\mathbb{R}}_{>0}$.

  Verified by Lean Reasoning summaryOperator algebras
• 362

  Global smoothness for relativistic Vlasov–Maxwell

  Proves large-data global existence and uniqueness for the three-dimensional, one-species relativistic Vlasov–Maxwell system. Smooth admissible initial data may be arbitrary provided the particle density is compactly supported and the electromagnetic fields have finite energy and bounded derivatives of every order; the solution remains smooth on every finite time interval.

  Verified by Lean Reasoning summaryPartial differential equations