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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

CounterexampleGeometry and topologyopenai/math #318

Chromatic splitting: filtrations and counterexamples

Disproves strong chromatic splitting at height three for primes p ≥ 5, and weak splitting for the derived p-completed sphere at heights p (p ≥ 5) and p+1p+1 (p ≥ 7). Nevertheless, for n ≥ 1 and p>n+1p\gt n+1, the overlap Ln−1LK(n)Sp∧L_{n-1}L_{K(n)}S_p^\wedge admits a 2n2^n-stage filtration by the predicted localized-sphere pieces. At height three and prime three, even finite assembly from such pieces fails in the category of E(2)E(2)-local modules over the derived completed sphere.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #317

Thomason model structures in all strict higher dimensions

Resolves the Ara–Maltsiniotis conjecture: for every n ≥ 1 and n = ω, small strict globular n-categories admit proper combinatorial Thomason model structures Quillen equivalent to simplicial sets. Thus strict higher categories model the homotopy theory of spaces in every stated dimension.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #316

Curtis’s conjecture

Proves Curtis’s conjecture: the positive-degree mod-two stable Hurewicz image of the sphere is spanned by the images of the Hopf-invariant-one classes η, ν, σ and the Kervaire-invariant-one classes that exist.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #315

The four-dimensional Singer conjecture

Proves that the L2-Betti numbers of the universal cover of every closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same conclusion holds for every finite connected aspherical integral Poincaré complex of formal dimension four, proving the four-dimensional Singer conjecture in this wider class.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #314

Cyclic length and chromatic fixed-point loss

Determines the optimal chromatic loss from geometric H-fixed points to geometric G-fixed points for every subgroup H of a finite p-group G. At every nonnegative height, the loss equals the shortest subnormal-chain length from H to G with cyclic quotients. Each quotient counts once regardless of order, and finite spectra witness sharpness.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #313

Finite generation for the K(n)K(n)-local sphere

Answers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the K(n)K(n)-local sphere is a finitely generated ℤp-module, for every prime, positive height and integer degree. The same conclusion holds after K(n)K(n)-localizing any finite p-local spectrum.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #312

The Grothendieck homotopy hypothesis

Proves the Grothendieck homotopy hypothesis for ∞-groupoids associated with every Grothendieck coherator in the Ara–Henry convention: these algebraic objects recover the homotopy theory of spaces.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #311

The Hovey–Strickland and Chai conjectures

Proves Chai's invariant-ideal conjecture for Lubin–Tate deformation rings over finite residue fields, at every prime and positive height n. Through the implication of Barthel–Heard–Naumann, this proves the Hovey–Strickland conjecture: dualizable K(n)K(n)-local spectra have exactly n+2n+2 thick tensor ideals, and their Balmer spectrum is a chain of n+1n+1 points.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #310

Quillen's conjecture in rational homology

Proves the rational-homology form of Quillen's conjecture for every finite group and every prime. If the largest normal p-subgroup of G is trivial, the poset of nontrivial elementary abelian p-subgroups has nonzero augmented reduced rational homology and is therefore not contractible.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #309

The Kervaire invariant problem at the prime three

Resolves the odd-primary Kervaire invariant problem at the prime three: exactly the standard classes with indices 0, 2, and 3 survive in the mod-three Adams spectral sequence, in stems 10, 106, and 322. Each surviving detection coset contains an element of exact additive order three.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #308

Finite Smith–Toda complexes at every height

For every n ≥ 0, constructs a finite Smith–Toda spectrum at a prime p depending on n, with Brown–Peterson homology BP∗/(p,v1,…,vn)BP_*/(p,v_1,\ldots,v_n) and the canonical comodule structure. Thus Smith–Toda complexes exist at every height when the prime may vary; an explicit example realizes V(4)V(4) at p = 1009.

Unreleased OpenAI model
LFCR
CounterexampleGeometry and topologyopenai/math #307

Failure of rational injectivity for maximal coarse assembly

Constructs a uniformly discrete bounded-geometry space whose maximal coarse assembly map is not rationally injective. The example is a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order kernel class. A companion gives the analogous failure for reduced coarse assembly, disproving the rational coarse Novikov conjecture.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #306

The purely cosmetic surgery conjecture

Distinct Dehn surgery slopes on a nontrivial smooth knot in S3 never produce orientation-preservingly homeomorphic manifolds, proving the purely cosmetic surgery conjecture. The statement includes the meridional slope.

Unreleased OpenAI model
LFCR
CounterexampleGeometry and topologyopenai/math #305

Four-dimensional disk embedding and Wall's conjecture

The unrestricted four-dimensional disk-embedding conjecture fails: framed algebraic dual spheres do not suffice to obtain disjoint locally flat spanning disks. In particular, the free group F2 is not good in the sense of Freedman–Quinn. Also constructs a finitely presented integral Poincaré duality group of dimension four with a finite classifying space but no realization as the fundamental group of a closed aspherical topological four-manifold, disproving Wall's conjecture.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #304

The Hilbert–Smith conjecture in every dimension

Every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected finite-dimensional topological manifold is a Lie group. This proves the Hilbert–Smith conjecture in all finite dimensions, for Hausdorff second-countable manifolds without boundary.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #303

Weak pure infiniteness and Cuntz-algebra absorption

Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex C∗-algebras. For exact algebras, proper infiniteness of one fixed finite amplification of every positive element also suffices. Consequently, every separable nuclear algebra with this property absorbs O∞\mathcal O_\infty, without unitality or simplicity assumptions.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #302

Radius of comparison equals half the mean dimension

For every minimal homeomorphism h of an infinite compact metrizable space X, the radius of comparison of C(X)⋊hZC(X)\rtimes_h\mathbb Z equals 12mdim(X,h)\tfrac12\mathrm{mdim}(X,h), including infinite values. Zero mean dimension is equivalent to the small boundary property, Jiang–Su stability and finite nuclear dimension; in this case nuclear dimension is at most one.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #301

Trace cones and Razak–Jacelon stabilization

Classifies separable nuclear complex C∗-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their full topological cones of extended lower-semicontinuous tracial weights. This answers Robert’s trace-cone question, including algebras with arbitrary ideal structure and both finite and infinite subquotients.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #300

Approximation and quadratic strong-operator paving

Proves that every self-adjoint element of a complex von Neumann algebra admits strong-operator paving relative to any maximal abelian subalgebra with O(ε−2)O(\varepsilon^{-2}) blocks. The norm bound holds after compression by a projection arbitrarily close to the identity in the strong topology, resolving the Popa–Vaes quadratic paving conjecture.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #299

The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability

A nonzero unital separable complex C∗-algebra is Jiang–Su stable exactly when its norm central-sequence algebra has no characters, for every free ultrafilter. This answers the Kirchberg–Rørdam character question. Also, the infinite minimal tensor power of every such algebra without characters is Jiang–Su stable, answering the Dadarlat–Toms question.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #298

Two notions of free entropy differ even when both are finite

Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy −∞<χ<χ∗<∞-\infty\lt \chi\lt \chi^*\lt \infty. This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite.

Unreleased OpenAI model
LFCR
CounterexampleAlgebraopenai/math #297

A ZFC counterexample to Naimark's problem

Gives an alternative to [Tanaka's ZFC construction](https://arxiv.org/abs/2609.26930v1) of a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state and exactly one nonzero irreducible representation up to unitary equivalence. Thus the unrestricted compact-operator characterization fails without additional set-theoretic assumptions; the counterexample is nonseparable.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #296

The generator problem for finite factors

Proves that every type II1 factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion P⊂MP\subset M of such factors, the unitaries u with M=W∗(P,u)M=W^*(P,u) form a dense Gδ subset in the trace 2-norm topology.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #295

The Kadison–Ringrose cohomology conjecture

Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive. Equivalently, all higher bounded Hochschild cohomology groups vanish, resolving the Kadison–Ringrose conjecture.

Unreleased OpenAI model
LFCR
CounterexampleAlgebraopenai/math #294

Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness

Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra admitting normalized 2-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite C∗-algebras can have a properly infinite minimal tensor product, with one factor Cr∗(F2)C_r^*(\mathbb F_2).

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #293

Invariant projections, hyperinvariant subspaces, and transitive algebras

Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator. The construction also gives operators with no nontrivial invariant projection in the hyperfinite type II1 factor.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #292

Kirchberg's O2\mathcal O_2 norm-ultrapower embedding problem

Constructs an explicit separable unital full group C∗-algebra that cannot embed unitally into the norm ultrapower of any fixed nonzero unital nuclear C∗-algebra, for any free ultrafilter on the natural numbers. Taking the target to be O2\mathcal O_2 answers Kirchberg's norm-ultrapower embedding problem negatively.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #291

Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability

Proves equivariant Jiang–Su stability for every countable discrete amenable group action on a simple separable unital infinite-dimensional nuclear stably finite Jiang–Su-stable C∗-algebra, resolving this case of Szabó's conjecture without restrictions on trace dynamics. The family also proves the unital Toms–Winter conjecture, equating strict comparison, finite nuclear dimension and Jiang–Su stability in the simple separable unital infinite-dimensional nuclear setting.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #290

Relative bicentralizers and modular spectral recovery

Proves Connes' bicentralizer conjecture for every type III1 factor with separable predual and every faithful normal state. More generally, for every inclusion N⊂MN\subset M of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra P⊂NP\subset N with P′∩c(M)=N′∩c(M)P'\cap c(M)=N'\cap c(M), resolving the relative bicentralizer conjecture.

Unreleased OpenAI model
LFCR
CounterexampleAlgebraopenai/math #289

Strong Kadison–Kastler stability and its spatial boundaries

Proves that sufficiently close unital von Neumann algebras on the same Hilbert space are conjugate by a unitary arbitrarily close to the identity, with a universal tolerance in the operator-norm distance between unit balls. Counterexamples show that near-identity conjugacy fails for one-sided near inclusions, and that arbitrarily close norm-separable C∗-algebras need not be ambiently unitarily conjugate.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #288

Kadison's similarity conjecture

Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra to operators on a Hilbert space becomes a ∗*-homomorphism after conjugation by a bounded invertible operator.

Unreleased OpenAI model
LFCR
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
CounterexampleAlgebraopenai/math #285

Counterexamples to Baum–Connes and Kadison–Kaplansky

Disproves the coefficient-free reduced Baum–Connes conjecture through a failure of rational injectivity and a separate failure of surjectivity of assembly. A finitely generated torsion-free example witnesses the injectivity failure. Separately, a torsion-free group has a nontrivial projection in its reduced group C∗-algebra, disproving the Kadison–Kaplansky conjecture.

Unreleased OpenAI model
LFCR
CounterexamplePDE and mathematical physicsopenai/math #284

The optimal quartic separation between randomized and quantum queries

Shows that the universal bound R(f)=O((1+Q(f))4)R(f)=O((1+Q(f))^4) for total Boolean functions is sharp in its exponent, ruling out every smaller power and disproving the conjectured cubic relation. Here R and Q are randomized and quantum worst-case bit-query complexities with error at most 1/3; computation between queries is unrestricted.

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
CounterexamplePDE and mathematical physicsopenai/math #278

Failure of Kohn–Sham ensemble representation

Constructs a three-electron Coulomb molecule with two equal positive-integer-charge nuclei whose absolute ground-state density has no noninteracting ground-state ensemble representation by a single real spin-independent local potential in L3/2(R3)+L∞(R3)L^{3/2}(\mathbb R^3)+L^\infty(\mathbb R^3). This disproves Kohn–Sham ensemble representability for that potential class; the required nuclear charge is specified nonnumerically.

Unreleased OpenAI model
LFCR