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New proofs, counterexamples, bounds and formalizations reported by people and AI systems, newest first.

Reported developments, attributed to their sources. Inclusion is not an OpenMath assessment; evidence slots show what each source reports. Updated Oct 13, 2026.

October 2026 40

New proofAlgebraopenai/math #286

Rigidity and arithmetic of lattice von Neumann algebras

Classifies finite-index bimodules between scalar-twisted group factors of ICC groups commensurable with property-(T)(T) lattices over characteristic-zero local fields and arbitrary ICC group factors. Every such bimodule is a summand of finite sums of models arising from finite-index subgroup isomorphisms and finite-dimensional projective representations. The classification also recovers the group, scalar cocycle and amplification scale up to the stated stable equivalence.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #283

Polynomial-time unitary synthesis from a Boolean oracle

Solves the constant-error Aaronson–Kuperberg unitary synthesis problem: a uniform polynomial-size quantum oracle circuit approximates every n-qubit unitary channel within diamond-norm error 1/2, after a suitable Boolean oracle is chosen. Gates, qubits, oracle calls and query length are polynomially bounded. The target-dependent oracle may have an unrestricted truth table; its efficient classical construction is not asserted.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #282

From scale symmetry to local conformal symmetry in four-dimensional QFT

Under the stated bounded-local-net and field-reconstruction hypotheses, proves that scale symmetry implies local conformal symmetry for four-dimensional unitary positive-energy theories with a discrete bounded-below scaling spectrum of finite multiplicity, finite scaling support and a physical local scale current. The stress tensor has a traceless improvement with unchanged spacetime charges. The conclusion concerns local Ward identities, not a global conformal action on the whole net.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #281

QAOA attains the SK optimum in the thermodynamic-first limit

Proves that QAOA approaches the ground-state energy of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity before circuit depth. For every accuracy, finite depth and deterministic angles independent of size and disorder achieve the required limiting expected energy per spin. This also yields leading-order optimal expected MaxCut values on large-degree random regular graphs, with size tending to infinity before degree.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #280

Unitary vertex operator algebras and conformal nets

Proves the strongly rational case of the strong-locality conjecture: every simple unitary strongly rational complex vertex operator algebra generates a completely rational conformal net. Its simple modules are unitarizable, and its representation category agrees with the net’s finite-index sectors as a braided unitary tensor category.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #279

Exact quantum factoring over a fixed finite gate set

Gives a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer with probability one. Both gate count and qubit count are polynomial in the input length, and one fixed finite gate set suffices.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #277

Threshold repetition for entangled games

Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is v < 1, the probability of winning at least a fraction v+δv+\delta of k independent repetitions decays exponentially in k, for 0<δ<1−v0\lt \delta\lt 1-v. Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #276

Classical capacity of generalized amplitude damping

Determines the unassisted classical capacity of every qubit generalized amplitude-damping channel, including all damping and thermal parameters. An explicit one-variable optimization gives the capacity, attained by independent two-state signal ensembles with collective decoding. Holevo capacity, minimum output entropy and regularized classical capacity are additive when tensoring with any finite-dimensional quantum channel.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #275

QMA-hardness of continuum Coulomb energy

Proves QMA-hardness of approximating the electronic Coulomb energy infimum in three dimensions, minimizing over the full spinful fermionic continuum space. Deterministic polynomial-time reductions work even with only unit-charge nuclei at distinct rational positions, polynomially many electrons and an energy-threshold separation of at least one.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #274

Parity is not in QAC0

Resolves Moore's parity conjecture in the measured-output model: constant-depth quantum circuits with arbitrary one-qubit gates, unbounded-arity Toffoli gates and polynomially many total qubits cannot compute parity with any fixed positive worst-case advantage. Ancillas start in zero, one output qubit is measured, and all other registers may be discarded. Xu–Li's reductions give the same bounded-error obstruction for strict majority.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #273

The entropy photon-number inequality

Proves the entropy photon-number inequality for beam-splitter mixing of two independent finite-energy bosonic inputs in any finite number of modes: the output's entropy photon number is at least the transmissivity-weighted average of the inputs'. Arbitrary entanglement within each input is allowed, and product thermal inputs attain equality even when their entropies differ.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #271

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

Proves Bloch's T3/2 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.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #269

Uniform Laughlin gap and stability under bounded scalar disorder

Proves the fermionic Laughlin spectral-gap conjecture for the full V1 interaction at filling 1/3 on the round sphere. The unique ground state remains uniformly gapped under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. Both the gap and disorder threshold are uniform over all sufficiently large particle numbers and all normalized potential profiles.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #268

The spin-one Haldane gap

Proves the spin-one Haldane gap conjecture for the pure antiferromagnetic Heisenberg chain on even periodic rings: the spectral gap stays uniformly positive as the chain grows. A companion establishes a gap for odd open chains with endpoint field h=3/5h=3/5 and gives boundary-selected infinite-volume states with topological index −1.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #267

Positive-temperature Bose–Einstein condensation and exact quantum depletion

Proves Bose–Einstein condensation for the exact canonical Gibbs state of the three-dimensional hard-sphere gas: each fixed exclusion distance and sufficiently small fixed density admit a strictly positive temperature, independent of volume, with positive condensate fraction in the thermodynamic limit. At zero temperature, proves the Bogoliubov leading quantum-depletion law for hard spheres and fixed bounded nonnegative radial finite-range potentials of positive scattering length, taking the thermodynamic limit before the dilute limit.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #266

Exactly three mutually unbiased bases in dimension six

Proves N(6)=3N(6)=3, resolving Zauner's dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ6, but four cannot. The exclusion is a complete certified computation under the stated binary64 arithmetic and compiler conditions. An independent companion proves the Matolcsi–Ruzsa–Weiner Fourier-vanishing conjecture for order-six complex Hadamard matrices outside Tao's cubic equivalence class.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #265

Area laws and tensor networks for two-dimensional gapped systems

Proves an entropy area law for unique ground states of finite-range Hamiltonians on arbitrary finite induced square-lattice domains, using only a uniform full-system spectral gap and bounds on the local interactions. On open L×LL\times L squares, uniformly gapped nearest-neighbor ground states also admit projected entangled-pair state approximations with polynomial bond dimension and global vector error at most L−1.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #264

Strong cosmic censorship near two-ended Kerr data

Proves local strong cosmic censorship near each fixed rotating subextremal Kerr bridge. A dense Gδ subset of a weighted smooth neighborhood of smooth complete two-ended asymptotically flat vacuum data has full maximal globally hyperbolic developments with no future continuous nondegenerate extension whose weak connection is locally square-integrable. No symmetry is imposed; extensions need not satisfy the vacuum equations.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #263

The ionization and generalized ionization conjectures

For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with M fixed nuclei of charges at least one and total charge Z strictly binds at most Z+CMZ+CM electrons. Neutral-atom first ionization energies and radii containing all but an expected half-electron have universal positive upper and lower bounds. The energy cost of removing m electrons has Thomas–Fermi asymptotics as m→∞m\to\infty and Z/m→∞Z/m\to\infty; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking Z→∞Z\to\infty first.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #262

Sharp finite-matrix Lieb–Thirring inequalities and all equality cases

Proves the sharp one-dimensional Lieb–Thirring inequality for 1/2<γ<3/21/2\lt \gamma\lt 3/2 and arbitrary finite-matrix potentials W ≥ 0 with ∫tr(Wγ+1/2)<∞\int\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty: the optimal constant is the scalar one-bound-state value, independent of matrix size. All equality cases are direct sums, in one constant unitary basis, of scalar sech2 solitons with independent scales and centers, and zero channels.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #261

Localization and delocalization in the Anderson model

Resolves the predicted spectral contrast for the lattice Anderson model with independent uniform site potentials. In dimension two, every positive disorder strength gives almost surely pure-point spectrum. In every fixed dimension d ≥ 3, sufficiently weak positive disorder gives purely absolutely continuous spectrum on a fixed open interval with nonzero spectral weight.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #260

Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions

Proves the sharp enclosing-area spacetime Penrose inequality for smooth one-ended asymptotically flat initial data in every spatial dimension n ≥ 3, under dominant energy, weak future trapping, positive enclosing area, and the stated decay assumptions. It bounds invariant ADM mass below using minimum enclosing area, with equality rigidity under additional horizon hypotheses. Charged upper-area bounds treat dyonic three-dimensional data; the higher-dimensional purely electric extension uses the matched neutral theorem.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #259

A group without fixed price

Constructs a finitely generated group with two essentially free probability-measure-preserving actions of different costs, answering the general fixed-price problem negatively. Its Bernoulli action has cost bounded away from one, while a sequence of finite height extensions has costs tending to one.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #258

Gersten’s conjecture and virtual compact specialness of one-relator groups

Proves Gersten's conjecture: every finitely generated one-relator group containing no Baumslag–Solitar subgroup BS(m,n)\mathrm{BS}(m,n), with m,n≠0m,n\ne0, is word-hyperbolic. It also proves that every word-hyperbolic one-relator group is virtually compact special.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #257

A hyperbolic group without a geometric CAT(0) action

Answers negatively whether every word-hyperbolic group is a CAT(0) group. Constructs one with a finite classifying space but no proper cocompact isometric action on any proper complete CAT(0)\mathop{\mathrm{CAT}}\nolimits (0) space, in any dimension.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #256

Nonsingular systems of equations over arbitrary groups

Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture. The coefficient group embeds in the presented quotient. A companion proves Kervaire's conjecture: adjoining one generator and one relation cannot trivialize a nontrivial group.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #255

Quasi-isometric recognition of virtually polycyclic groups

Proves that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic, resolving the Eskin–Fisher–Whyte lattice-recognition conjecture. Equivalently, a group quasi-isometric to a lattice in a connected simply connected solvable Lie group is virtually a uniform lattice in some such Lie group, possibly a different one.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #254

Classifying spaces and geometric obstructions for Artin groups

The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin K(π,1)K(\pi,1) conjecture. Arbitrary intersections of its parabolic subgroups are parabolic, proving the Parabolic Intersection Conjecture. An explicit Artin group admits no proper cocompact isometric action on any nonempty proper CAT(0)(0) space.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #253

An infinite finitely presented simple amenable group

Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #252

A torsion-free hyperbolic group that is neither residually finite nor linear over any field

Constructs a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question negatively. One fixed nonidentity element is killed by every finite-dimensional linear representation over every commutative field, so the group is not linear over any such field.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #251

Amenability, unitarizability, and strong Ulam stability

Resolves Dixmier's problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation. For countable discrete groups, amenability is also equivalent to strong Ulam stability: sufficiently accurate unitary approximate representations are uniformly close in operator norm to genuine representations on the same, possibly infinite-dimensional, Hilbert space.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #250

Boone–Higman embeddings with higher finiteness

A finitely generated group has decidable word problem exactly when it embeds in a finitely presented simple group, proving the Boone–Higman conjecture. The target can have type F∞: a classifying space with finitely many cells in each dimension. A single group of type F∞ can also contain every finitely presented group.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #248

Thompson's group F is nonamenable

Proves that Thompson's group F, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #247

An infinite finitely presented residually finite 2-group and a finitely presented nil algebra

Constructs an infinite finitely presented residually finite group whose elements all have finite 2-power order, answering the finitely presented Burnside problem negatively even in this class. The construction also yields an infinite-dimensional finitely presented nil associative 𝔽2-algebra and a finitely presented infinite-dimensional algebraic unitization, giving negative answers to the corresponding nilpotence and Kurosh finiteness questions.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #246

Cannon's conjecture

Every word-hyperbolic group with boundary homeomorphic to S2 admits a proper cocompact isometric action on hyperbolic three-space with finite kernel, proving Cannon's conjecture. Every torsion-free such group is therefore the fundamental group of a closed hyperbolic three-manifold.

Unreleased OpenAI model
LFCR
New proofLogicopenai/math #245

Weak normalization implies strong normalization in pure type systems

Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a β-normal form, every β-reduction sequence terminates. This resolves the β-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts.

Unreleased OpenAI model
LFCR
New proofLogicopenai/math #244

The Partition Principle does not imply Choice

Assuming ZF is consistent, constructs a model in which every surjective image of a set injects into that set, yet the axiom of choice fails. Choice for ordinal-indexed families still holds. From any countable transitive model of ZFC, a separate construction gives a transitive symmetric extension with these properties and no new countable sequences of ground-model elements.

Unreleased OpenAI model
LFCR
New proofLogicopenai/math #243

Separating choiceless counting from polynomial time and witnessed choice

Confirms the Blass–Gurevich–Shelah noncapture conjecture: consistency of a linear system over 𝔽3 defines a polynomial-time query on unordered finite structures that choiceless polynomial time with counting cannot express. A separate result shows that adding witnessed symmetric choice strictly increases expressive power. Both separations hold for the full counting formalism, allowing hereditarily finite sets of arbitrary finite rank.

Unreleased OpenAI model
LFCR
New proofLogicopenai/math #242

Single-fold Diophantine representations and undecidability under an at-most-one-solution promise

Every recursively enumerable set of tuples of natural numbers has a Diophantine representation with exactly one auxiliary solution for each member and none for nonmembers. This proves the single-fold conjecture and hence the finite-fold conjecture. Diophantine solvability over the nonnegative integers remains undecidable even with an at-most-one-solution promise.

Unreleased OpenAI model
LFCR
New proofLogicopenai/math #241

Rigidity of the Turing degrees

Every order automorphism of the Turing degrees is the identity, resolving their rigidity problem. Thus no nontrivial relabeling of degrees preserves the ordering by relative computability.

Unreleased OpenAI model
LFCR