Perplexity
Perplexity develops AI tools for search, browsing, research, content creation, organization, and workflow automation, including perplexity.ai, Comet, and Computer.
A Symmetric Algebra Has a Nonprojective Module With No Positive Self-Extensions
An OpenAI preprint claims to disprove Tachikawa’s second conjecture, which asks whether a finite-dimensional module with vanishing self-extensions in every positive degree must be projective. The authors construct a nonprojective module with that vanishing property over a finite-dimensional symmetric algebra, using a complete resolution whose negative-degree cohomology enables the final transfer to the symmetric setting.
A Torsion-Free Group Algebra Fails Direct Finiteness Over F₂
An October 2026 OpenAI manuscript claims a counterexample to Kaplansky’s conjecture: a finitely presented, torsion-free group whose group algebra over the two-element field is not directly finite. The authors reduce the claim to constructing three finite sums with \(ab=1\), \(ac=0\), and \(c\ne0\), using labeled graphs and parity to control which terms cancel. The construction selects a finite matching outcome, but the manuscript does not print the witness’s edge list.
Bounded Defect Groups Yield Finitely Many Morita Types of Blocks
An OpenAI manuscript dated September 24, 2026, claims a finiteness result for blocks of finite groups: with the prime \(p\) and algebraically closed field \(K\) fixed, a bound on the order of a block’s defect group leaves only finitely many block categories up to \(K\)-linear Morita equivalence. The claim does not bound the groups themselves; it says their blocks can belong to only finitely many such equivalence classes.
Lex-Plus-Powers Ideals Bound Every Graded Betti Number
The paper states that, for Artinian homogeneous ideals containing a regular sequence, a lex plus powers ideal can preserve the Hilbert function while having at least as large a graded Betti number at every relation level and degree. Its authors extend the claim to characteristic-zero fields and regular sequences shorter than the number of variables, while making no corresponding assertion in positive characteristic. The result makes the lex plus powers ideal a bound on the relations among equations, not just on the number of surviving terms in each degree.
One Irreducible Tensor Square Covers Every Type of the Symmetric Group
A manuscript on universal tensor squares claims that, for every positive integer \(n\) other than 2, 4 and 9, one irreducible representation of the symmetric group \(S_n\) has a tensor square containing every irreducible representation of \(S_n\). Its construction starts from a staircase-shaped partition and adds boxes symmetrically; the proof combines a companion staircase result, two sufficient tests for coverage, an area bound and finite checks. Perplexity Computer adapted the OpenAI preprint into an animated video, but the explainer does not reproduce several technical parts of the proof.
Critical Percolation on the Cubic Lattice Has No Infinite Cluster
An OpenAI manuscript proves that, for bond and site percolation on the cubic lattice, almost surely there is no infinite open cluster at each model’s own critical parameter. Its contradiction argument assumes an infinite cluster at criticality, preserves reliable connections across finite regions after slightly lowering the opening probability, and uses an adaptive exploration to produce an infinite cluster below the threshold. The proof controls dependence between overlapping regions rather than treating their connections as independent.
One Finite-Dimensional Algebra Has No Bound on Finite Projective Dimensions
An OpenAI preprint constructs a single finite-dimensional complex algebra for which finitely generated modules of finite projective dimension have no common upper bound on their resolution lengths. For every positive integer \(m\), it produces a module with finite projective dimension at least \(2m-2\), showing that the algebra’s little finitistic dimension is infinite. The proof builds a mechanism that keeps selected modules nonzero for a controlled number of steps, then makes them vanish.
A Cyclic Polytabloid Proof Establishes Saxl’s Conjecture for Staircase Shapes
An OpenAI preprint claims a proof of Saxl’s conjecture for every staircase-shaped partition: the tensor square of each corresponding symmetric-group representation contains every irreducible representation of that group. The manuscript makes a stronger claim, that every irreducible type already occurs in the orbit of one specifically constructed vector, built from row- and column-based polytabloids. Its proof reduces the claim by induction, using projections onto smaller staircases and a calculation for the intervening bands.
Each Block’s Simple-Module Count Equals Its Local Weight Count
An OpenAI manuscript claims a blockwise form of the Alperin Weight Conjecture: for every block of a finite group in characteristic \(p\), the number of simple modules equals the number of associated local weights, counted up to conjugacy. Its proof reduces that equality to a uniform divisibility claim for counts of group-generated tuples, then uses covers of punctured tori to establish the claim. The argument is presented as a proof in the manuscript; the explainer says it was not independently certified.
Exponential First-Passage Limit Shapes Have No Corners or Flat Sides
An OpenAI preprint dated September 24, 2026, claims that the planar exponential first-passage percolation limit shape has neither flat sides nor corners: it is strictly convex and has a continuously differentiable boundary at every positive rate. The authors use separate arguments to rule out corners and flat segments, and say differentiability extends to gamma-distributed edge times, though strict convexity is not claimed across that broader family. The result is qualitative; the manuscript provides no formula for the shape or bound on its curvature.
Microscopic Pin Cancellation Establishes the Critical XY Field Limit
An OpenAI preprint proves that, under fixed boundary spins, the normalized spin fields of the critical planar XY model converge as the square lattice is refined to a corrected imaginary exponential of the Gaussian free field. The authors’ main challenge is connecting continuum height-field estimates to spin insertions at the scale of a single lattice spacing, while controlling collisions between insertions. Their proof uses cancellation against a center reference pin and uniform correlation bounds to establish convergence of the fields, not of pointwise spin values.
Planar XY Correlations Acquire a Critical (log r)¹⁄⁸ Correction
An OpenAI preprint on the critical planar XY model claims that its spin correlation decays as \(r^{-1/4}(\log r)^{1/8}\), with a positive, finite amplitude. The result sharpens the familiar power-law prediction: the logarithmic factor captures a correction accumulated across scales, while the proof’s control of the remaining terms establishes that the rescaled correlation converges rather than drifting further.
The Planar XY Correlation Length Diverges as an Essential Singularity
An OpenAI preprint claims that the planar XY model’s correlation length has an essential singularity at its critical temperature: as the inverse temperature approaches the critical value from below, the logarithm of the length grows in proportion to the inverse square root of the distance to criticality. The manuscript derives a finite passage time from a rescaled flow, then uses separate variance and correlation estimates to argue that this scale is the spin correlation length. Its constant is positive and finite, but not evaluated in closed form.
Height Fluctuations Reach the Roughening Threshold in a Gaussian Free Field Limit
An OpenAI preprint proves Gaussian free field limits for two-dimensional lattice height and planar-spin models, but the results reach different temperature regimes. For heights, the limit extends to the roughening threshold, where the effective parameter \(a\) reaches \(8\pi\); for planar spins, the theorem applies at sufficiently low fixed temperature and does not establish a limit at the spin transition. The paper’s central result is thus a pair of Gaussian free field limits with unequal scope.
A Degree-Five K-Theory Class Vanishes Over the Fraction Field but Not the Ring
An OpenAI preprint gives a degree-5 counterexample to Gersten’s conjecture, constructing a nonzero class in the algebraic K-theory of a two-dimensional regular local ring that vanishes over its fraction field. The authors establish the two sides of the result separately: support on a divisor forces the class to disappear after passing to the fraction field, while an arithmetic and geometric detector shows it remains nonzero in the ring. The example is specific to a ramified mixed-characteristic setting; the paper does not determine the class’s order or whether the failure occurs in lower degrees.
Even Spin Groups Have Saturation Factor One Under a Root-Lattice Condition
An OpenAI preprint claims that for the even spin groups \(\mathrm{Spin}(2n)\), an invariant at any positive scale implies an invariant at scale one, provided the three dominant weights sum to an element of the \(D_n\) root lattice. The result includes spin weights, but the root-lattice condition is essential: without it, scaling can change whether an invariant exists. The proof’s central challenge is converting real geometric data into an integral schedule while preserving the constraints needed to recover an invariant.
Finite Symmetric Tensor Categories Embed in a Higher Verlinde Level
An OpenAI preprint proves that every finite symmetric tensor category over an algebraically closed field of positive characteristic admits an exact, faithful, symmetric monoidal functor into some finite level of the Verlinde category tower. The required level can depend on the category; the result does not assert that one fixed level works for all categories. The proof builds an intermediate functor using a tilting-module result, then uses projectives and a finite graph of simple objects to establish exactness and reach the claimed target.
Tree-Weighted Planar-Map Walks Converge After Exactly n-Fold Acceleration
An OpenAI preprint on random walks on tree-weighted planar maps identifies an exact clock: on a map with \(n\) edges, accelerating the rate-one walk by \(n\) yields convergence to stationary Liouville Brownian motion on the limiting sphere, with no additional time constant. The paper argues that this factor follows from an exact energy identity, then establishes that the maps preserve the limiting energy and that their area measures give the walk the correct speed.
The Critical Planar XY Model Has a One-Quarter Correlation Exponent
An OpenAI preprint claims that spin correlations at the critical point of the square-lattice planar XY model decay with exponent \(1/4\). The authors derive the exponent by relating spin correlations to fluctuations in a height representation, then show that the critical fluctuation coefficient must equal \(8\pi\): lower bounds rule out zero, while stability rules out larger values. The result fixes the asymptotic exponent, not the full correlation formula; the preprint leaves the prefactor and possible logarithmic corrections undetermined.
A Quadratic Bound Guarantees Surjectivity of the Foulkes–Howe Map
An OpenAI preprint proves that the canonical Foulkes–Howe map from Symᵇ(Symᵃ V) to Symᵃ(Symᵇ V) is surjective over finite-dimensional complex vector spaces whenever a ≥ 2 and b ≥ a(a − 1). The bound is sufficient, not asserted to be sharp. The proof rules out any nonzero linear functional that vanishes on the map’s image, using a derivative argument whose injectivity depends on a strict degree inequality.
Planar First-Passage Percolation Has No Bigeodesics Under a Moment Bound
An OpenAI manuscript claims that planar first-passage percolation has no bigeodesics almost surely when edge times are independent, identically distributed, nonnegative and atomless, and the expected square of the minimum of four independent edge times is finite. Its proof addresses a limitation of fixed-label exclusions: a bigeodesic’s large-scale label could depend on the random environment. The authors argue that a compact range of labels cannot support the arbitrarily large extra costs a hypothetical bigeodesic would incur.
An Explicit Nonprojective Module Defies the Auslander–Reiten Conjecture
An OpenAI preprint presents an explicit counterexample to the Auslander–Reiten conjecture: a nonprojective module over a noncommutative Artin algebra whose positive-degree extensions to itself and to the algebra all vanish. The construction uses two independent twists to eliminate the relevant homological obstructions in every positive degree while preserving a nonzero degree-zero stable endomorphism, which certifies that the module is not projective.
Two-Dimensional O(n) Spin Correlations Decay Exponentially for n ≥ 3
A preprint on two-dimensional classical \(O(n)\) models claims that, for \(n\ge3\), spin correlations decay exponentially at every fixed positive temperature. Its proof bounds correlations by the probability that two sites are joined in an auxiliary random-bond model, then uses rare crossings of large annuli to show that long connections are exponentially unlikely. The stated bound is uniform over finite square-lattice subgraphs and allowed nonnegative edge strengths, but not uniform as temperature approaches zero.
The Two-Dimensional O(3) Spin Field Has an Isolated Lowest-Mass Particle
An OpenAI preprint proves that, under a specified continuum-limit construction, the two-dimensional O(3) spin field has a lowest-mass contribution with positive spectral weight, separated from the rest of the spectrum. The authors’ argument goes beyond establishing a mass gap: it bounds the number of low-energy states in finite volume, then uses that bound to show the field’s spectral measure contains an isolated atom at its lowest mass. The theorem does not give the mass or weight numerically, or claim to cover every continuum limit.