Perplexity
Perplexity develops AI tools for search, browsing, research, content creation, organization, and workflow automation, including perplexity.ai, Comet, and Computer.
A Group-Ring Matrix Has Fuglede–Kadison Determinant Below One
An OpenAI preprint presents a counterexample to the Group-Ring Determinant Conjecture: an integer matrix over a group ring can have an inverse over the rational group ring while its Fuglede–Kadison determinant is strictly between zero and one. The construction turns a one-sided inverse over a field of characteristic two into a projection with trace greater than its factorization dimension, then carries that trace gap into an integer matrix. The resulting operator is invertible, so the failure does not arise from a divergence at zero.
A Three-Dimensional Complete Local Domain Has No Finite Maximal Cohen–Macaulay Module
An OpenAI manuscript constructs a three-dimensional complete local domain over the complex numbers with no nonzero finitely generated module of full depth. The authors derive a numerical inequality that any such module would impose on a chosen projective surface, then construct a surface that violates it. The obstruction applies to arbitrary finite modules, not only graded modules or the ring itself.
A Trace Theorem Proves the Characteristic-Zero Idempotent Conjecture for Torsion-Free Groups
An OpenAI manuscript dated September 24, 2026, proves that the Hattori–Stallings trace of an idempotent over a complex group ring vanishes on every conjugacy class of infinite-order elements. The result does not eliminate contributions from finite-order classes, but for torsion-free groups it leaves only the identity class; the authors use that trace constraint to conclude that, over a commutative characteristic-zero domain, every scalar group-ring idempotent is zero or one. The argument does not settle the corresponding questions in positive characteristic or for idempotent matrices.
Subcritical FK and Spanning-Tree Maps Converge to the Liouville Quantum Gravity Sphere
An OpenAI preprint argues that two ensembles of decorated random planar maps converge, after deterministic distance rescaling, to the unit-area Liouville quantum gravity sphere: maps sampled with an FK configuration for \(0<q<4\), whose associated LQG parameter \(\gamma\) is below 2, and maps sampled uniformly as map–spanning-tree pairs. The claim uses shortest-path distance across all map edges, not only edges in the decoration, and carries the degree-weighted measure into the limit. The authors say proving this joint metric-and-measure convergence requires separate strategies for the two ensembles.
At Criticality, Every Cluster Is Finite on Quasi-Transitive Graphs
An OpenAI preprint proves that Bernoulli bond percolation on an infinite, connected, locally finite quasi-transitive graph has no infinite clusters at the critical probability, provided the critical probability is below one. The authors argue that assuming an infinite critical cluster exists would force large finite clusters to remain common at parameters below criticality; separate arguments for different growth regimes then turn that abundance into a contradiction. The result establishes almost-sure finiteness at criticality, not a uniform bound on cluster sizes.
The ℓ¹-Bass Conjecture Restricts Idempotent Traces to Finite-Order Classes
An OpenAI preprint claims that for an idempotent matrix over the complex ℓ¹ algebra of any discrete group, its Hattori–Stallings trace has only finitely many nonzero conjugacy-class coefficients, all on classes of finite-order elements. The proposed proof uses curvature estimates and a cutoff argument to force coefficients outside a finite set to vanish; the finite-support conclusion applies to the trace, not to the idempotent’s coefficients. The article notes that the global analytic proof was not certified.
The O(4) Mass Gap Scales as √βe⁻πβ at Strong Coupling
An OpenAI preprint on the two-dimensional O(4) model argues that, at strong coupling, the full mass gap is bounded above and below by fixed multiples of \(\sqrt{\beta}e^{-\pi\beta}\). The bound applies across local observables, including rotation-invariant ones, rather than only to correlations of individual spin components. The authors also claim a positive full gap at every finite positive coupling and a unique local limit of periodic square lattices.
Subcritical FK Maps Converge Jointly in a Canonical Quantum-Sphere Coordinate
An OpenAI manuscript on subcritical FK planar maps claims that their conformal embeddings converge jointly: area, rescaled graph distances and nested cluster-boundary loops all approach continuum objects in one prescribed coordinate on a random quantum sphere. The authors’ emphasis is that the limit identifies not only an abstract surface and its geometry, but also where those objects sit in the conformal drawing, with the loops retaining their order and nesting.
A Group Algebra Has a One-Sided Inverse That Is Not Two-Sided
An OpenAI preprint presents a counterexample to Kaplansky’s direct-finiteness conjecture: over a finite field of characteristic two, it constructs a finitely presented group whose group algebra contains elements \(a\) and \(b\) with \(ab=1\) but \(ba\ne1\). The result also yields an injective but nonsurjective cellular automaton and, by the paper’s stated reasoning, a nonsofic group. The preprint gives a terminating search for the witnesses, but says that search has not been run.
No Algorithm Can Decide Which Finite Lattices Are Full Congruence Lattices
An OpenAI preprint claims that no algorithm can decide whether a given finite lattice is the full congruence lattice of some finite algebra. It characterizes representable lattices through finite colored graphs, where admissible maps become algebraic operations, then argues that a decision procedure would imply a computable bound on the size of representation witnesses. The authors use a reduction from circuit solvability to show that no such bound—and therefore no decision procedure—can exist.
A Torsion-Free Group Algebra Contains Zero Divisors Over F₂
An OpenAI manuscript dated September 2026 claims a counterexample to Kaplansky’s Zero Divisor Conjecture, constructing a finitely presented torsion-free group whose group algebra over the two-element field contains nonzero elements with product zero. The construction uses parity in a graph-based argument to force cancellation; separate random and topological arguments are used to keep both elements nonzero and establish that the group is torsion-free.
Infinite Clusters Precede Uniqueness on Expanding Graphs
An OpenAI preprint on percolation in nonamenable quasi-transitive graphs proves that the threshold for infinite clusters is strictly below the threshold for a unique infinite cluster. Its argument shows that the connection operator is bounded at the existence threshold and remains bounded at some parameter strictly above it, while uniqueness would force the operator to be unbounded. As a result, at every fixed parameter between the two thresholds, there are infinitely many infinite clusters.
A Fixed 66-by-66 Pattern Defeats Polynomial Removal Bounds
An OpenAI preprint constructs a fixed 66-by-66 ordered binary pattern for which no polynomial removal bound holds. Its host matrices require at least a depth-dependent fraction of entries to be changed to eliminate all copies, while their normalized copy counts fall below every fixed power of that fraction. The result does not overturn qualitative removal: matrices far from pattern-free still contain copies, but the paper shows those copies can be sparser than any polynomial guarantee would require.
A Direct-Finiteness Counterexample Emerges in One Odd Characteristic
An OpenAI preprint claims a counterexample to Kaplansky’s direct-finiteness conjecture: in a group algebra over a field of odd characteristic, it constructs elements \(a\) and \(b\) with \(ab=1\) but \(ba\ne1\). The result is specific to a prime defined by the construction and a finitely generated group containing torsion; it does not establish a counterexample for every odd characteristic or for torsion-free groups. The defect also yields an injective update on configurations that is not surjective.
Critical FK–Ising Maps Converge Jointly in Geometry, Spectrum, and Heat Flow
An OpenAI manuscript on critical FK–Ising planar maps claims that their geometry and random-walk spectrum converge together to those of a unit-area Liouville quantum gravity sphere with \(\gamma=\sqrt{3}\). With the walk sped up by the number of edges, the paper establishes joint convergence of the rescaled metric and area measure, each fixed-index diffusion eigenvalue, and the heat trace uniformly over time intervals bounded away from zero. The result rules out a mismatch in which the surfaces converge but hidden slow modes prevent the walks from having the corresponding continuum limit.
A Nonzero K₃ Class Dies in the Fraction Field
An OpenAI preprint constructs a counterexample to the integral degree-three Gersten conjecture: a nonzero class in \(K_3(A)\) for a two-dimensional regular local ring that becomes zero in the ring’s fraction field. The construction uses a ramified mixed-characteristic ring and a class supported on a divisor; a residue-degree argument then proves the class survives in \(K_3(A)\), even though inverting a parameter kills it. The manuscript does not determine the class’s order.
Critical FK–Ising Map Walks Converge to Liouville Brownian Motion
A manuscript on critical FK–Ising maps argues that convergence of the random surfaces is not enough to establish a limit for walks on them: the walk’s time scale and energy must also match the continuum diffusion. With the stated distance scaling and time acceleration by \(n\), it proves joint convergence of the measured maps and any fixed finite number of stationary-start walks to a unit-area quantum sphere and conditionally independent Liouville Brownian motions on that shared surface. The proof adds whole-path control to the geometric and energy results supplied by a companion paper.
Fortuin–Kasteleyn Maps Above Four Converge to the Brownian Continuum Random Tree
An OpenAI manuscript claims that, for each fixed \(q>4\), finite planar maps sampled under a Fortuin–Kasteleyn weight converge to the Brownian continuum random tree when distances are scaled by \(c(q)/\sqrt n\), for a positive, finite constant \(c(q)\). The maps retain their cycles, but their rescaled distances and degree-weighted mass converge to those of the tree.
Bounded Defect Yields Finitely Many Integral Morita Types
The preprint *Integral Donovan Finiteness over Witt Vectors* claims that, for a fixed prime \(p\) and defect-group size bound \(M\), only finitely many Morita equivalence classes of blocks exist over the Witt ring \(W(\overline{\mathbb F}_p)\), even when the ambient finite groups have unbounded order. Its argument addresses the gap between a finite-field bound and an integral result by controlling scalar errors exactly, preserving the blocks’ labeled structure, and reducing the effect of an uncontrolled kernel to matrix factors.
The Two-Dimensional O(4) Lattice Mass Has an Exact Leading Prefactor
An OpenAI preprint claims an exact leading asymptotic for the mass gap of the two-dimensional O(4) lattice model at low temperature: \(m_{\mathrm{lat}}(\beta)\sim 32e^{\pi/4-1/2}\sqrt{\beta}\,e^{-\pi\beta}\). The result goes beyond identifying the exponential decay scale: the paper argues that its coefficient follows from isolating odd states with a boundary twist, ruling out slower even-sector modes, and carefully matching an auxiliary regulated model to the lattice model.
A Finite Lattice Cannot Be the Full Congruence Lattice of Any Finite Algebra
An OpenAI preprint claims that some finite lattice cannot be the full congruence lattice of any finite algebra, regardless of the algebra’s operations or signature. The authors reduce the problem, using a theorem of Pálfy and Pudlák, to finding a finite lattice that cannot occur as a full interval of subgroups in any finite group; they then construct such a lattice by combining local restrictions on subgroup intervals with a uniform bound on certain chains.
The Two-Dimensional O(3) Model Has a Canonical Massive Continuum Limit
An OpenAI preprint proves that a specific two-dimensional lattice O(3) model has a canonical massive continuum limit as its coupling tends to infinity: after normalizing by the model’s own correlation scales, the limit is the same along every route to that regime. The authors also show that the resulting field is non-Gaussian and reconstructs a relativistic theory with a positive energy gap above its unique vacuum. The result is specific to the lattice model and normalization studied; refining the lattice alone does not establish it.
One Monomial Ideal Matches the Entire Hilbert Function
A September 2026 OpenAI manuscript claims that, for a homogeneous regular sequence of length \(c\) in \(n\) variables, with degrees \(2 \le a_1 \le \cdots \le a_c\) and \(1 \le c \le n\), every ideal containing the sequence has the same Hilbert function as a monomial ideal containing the corresponding pure powers. The same monomial ideal matches the count of surviving classes in every degree. The manuscript gives its construction over the complex numbers, using commuting linear forms over a division algebra; a separate extension establishes the broader characteristic-zero conclusion.
FK Maps at q=4 Converge Jointly in Area, Distance, and CLE4 Loops
An OpenAI preprint claims that critical Fortuin–Kasteleyn maps at \(q=4\), with distances rescaled by a deterministic factor, converge jointly in area, distance and loop structure to a critical Liouville quantum sphere decorated by nested CLE4 loops. The theorem treats these as features of one limiting surface: it claims convergence of area and all vertex-pair distances, alongside a precise matching of large loops.