Groups, rings, representations and the structure of algebraic systems.
There are a finite field K of characteristic two, a finitely presented group G containing odd-prime torsion, and scalar a,b in K[G] with ab=1 but ba≠1. The manuscript gives a terminating prescription for the witness; it does not assert K=F₂.
For even-order Latin squares, Babanskyy gives binary-rank and adjugate criteria for an extra factor of two in the centered determinant. An explicit family has an odd normalized determinant quotient for every n≡2 mod 4 with n≥6.
Babanskyy determines the complete p-primary Smith profile of the integer affine point–line incidence matrix over Z/p²Z for every prime p, including the nonunit factors p, p² and p³ and their multiplicities. The argument also gives a nonsplit transfer extension and a two-chart carry presentation.
Babanskyy reports every nonzero Smith factor of the integer affine point–line incidence matrix over Z/p³Z for every prime p. The profile reaches exponent p⁵ and has six multiplicities, with polynomial formulas for odd p and explicit p=2,3 rows. The new step resolves a q-Pascal common image through q-Newton minors and a cross-chart carry calculation.
For m>0 and N≥1, the formalized refinement says GL(N,ℤ) contains an element of order m with determinant −1 exactly when m is even and ψ(m)+1≤N, where ψ sums φ(p^k) over exact prime-power factors except 2. The same criterion holds over ℚ, and the least integral dimension is ψ(m)+1.
For every n≥2 over GF(2), the polynomial 1+X+⋯+X^n is irreducible exactly when its composition with X²+X+1 is irreducible. The formalization also checks the corresponding OEIS set statement and excludes n=1.
Claude agents devised an integer proof route for the s=1, index-two clauses of Theorem D in Jones’s Hadamard border note, replacing its rank and positive-semidefiniteness route. The formal development checks those Theorem D clauses.
For k≥3 and q≥k−1, let m=2kq+1. The numerical semigroup minimally generated by m,m+1,m+k,m+2k−1,m+2k has toric-ideal generator count 4k+2. The formal development also gives a nonredundant presentation.
For diag(1,1,1,−1,1,−1) and the local Hodge operator on coordinate plane 0123, fifteen orthogonal adjoint actions together with that operator generate sl(15,K) over every field of characteristic other than two. The same formal result gives dimension 224 and scalar rigidity for commuting endomorphisms, with determinant c²²⁴.
For a minimally four-generated numerical semigroup S=⟨n₀<n₁<n₂<n₃⟩, the minimal number of generators of its toric ideal is at most 2W over any field, where W=(n₃−n₀)/gcd(n₁−n₀,n₂−n₀,n₃−n₀). Consequently it is at most twice the ordinary width n₃−n₀.
For every discrete group G, the Hattori–Stallings trace of every idempotent matrix over the complex Banach algebra ℓ¹(G) has finite support, contained in conjugacy classes of finite-order elements.
A finite nonempty lattice is the full congruence lattice of some finite algebra exactly when it admits the finite colored-graph witness specified in the manuscript, whose edge colors encode the required congruence relations.
For every integer b≥6 and finite-dimensional complex vector space V, there is a GL(V)-equivariant injection Sym⁶(Symᵇ V)→Symᵇ(Sym⁶ V).
A three-dimensional complete Noetherian normal local domain containing ℂ, with residue field ℂ, has no nonzero finitely generated module of depth three.
Under every flat local homomorphism of nonzero Noetherian local rings, Hilbert–Samuel multiplicity of the source does not exceed that of the target.
There is a two-dimensional ramified regular local ring A of mixed characteristic (0,5) for which K₅(A)→K₅(Frac A) has a nonzero kernel.
Every finite symmetric tensor category over an algebraically closed field of positive characteristic has an exact faithful strong symmetric monoidal functor to some finite higher Verlinde category Ver_{p^n}(k).
For every algebraically closed field K of characteristic p>0 and every fixed bound on defect-group order, blocks of finite groups represent only finitely many K-linear Morita-equivalence classes.
For G=Spin(2n), n≥2, if three dominant integral weights sum to the root lattice, then a nonzero invariant in their tensor product after any positive integral dilation implies a nonzero invariant at the original weights.
For every finite group, every prime p, and every p-block B, the number of irreducible Brauer characters in B equals the number of conjugacy classes of B-weights.
A countable characteristic-zero division ring is algebraic over its center, generated by two elements as an algebra over that center, and infinite-dimensional over the center.
Over every characteristic-zero field, for a homogeneous ideal containing a regular sequence of any positive length with degrees at least two, a corresponding lex-plus-powers ideal has the same Hilbert function and at least as large a graded Betti number in every homological and internal degree.
For every positive integer n outside {2,4,9}, some irreducible complex representation of Sₙ has a tensor square containing every irreducible complex representation of Sₙ.
A finite-dimensional algebra over F₂(q,H₁,H₂) has a finite-dimensional nonprojective Gorenstein-projective module Z with Extⁱ(Z,Z)=Extⁱ(Z,Λ)=0 for every i>0.
Every order isomorphism between Bruhat intervals in arbitrary Coxeter systems preserves the corresponding equal-parameter Kazhdan–Lusztig polynomial.
For an absolutely almost simple simply connected algebraic group over any global function field, including characteristic two, every noncentral abstract normal subgroup of its rational points is the inverse image of an open normal subgroup of the finite product of its anisotropic local groups.
OpenAI reports that the rational Grothendieck–Teichmüller Lie algebra is free on one generator in each odd weight starting at three, with the corresponding weight-completed isomorphism.
OpenAI reports a finitely presented torsion-free group with a finite two-dimensional classifying space whose group algebra over the two-element field contains nonzero elements with product zero.
OpenAI reports a finitely presented torsion-free group with a finite two-dimensional classifying complex and elements a,b in its characteristic-two group algebra satisfying ab=1 but ba≠1.
OpenAI reports one fixed finite-dimensional complex algebra with finite-dimensional modules of unbounded finite projective dimension, refuting universal finiteness of little finitistic dimension.
OpenAI reports an infinite discrete group with an ordinary finite presentation that is residually finite and has finite two-power order for every element, with unbounded element orders.
OpenAI reports that Thompson’s group F, viewed as a discrete group, is nonamenable.
For globally generic cuspidal complex automorphic representations of split connected adjoint absolutely simple groups of types G₂, F₄, E₆, E₇ or E₈ over global function fields, every local component is tempered, including arbitrarily ramified places and without a characteristic restriction.
Oren Ben-Bassat reports that every polystable representation of a finite quiver over any complete non-Archimedean field admits a split HKKP harmonic norm, without spherical-completeness or value-group restrictions.
There exists a countable domain R that is left PCI and left V but neither right Ore, right PCI nor right V. Thus right PCI does not imply left PCI, and left V-domain does not imply right V-domain.
An explicit polynomial F in four complex variables has zero-fibre coordinate ring isomorphic to C[3], while its gradient vanishes at a point, so F is not a coordinate. Adding variables extends this counterexample to every ambient dimension at least four.
For every m≥1, the tensor square of the staircase Specht representation of the symmetric group S_{m(m+1)/2} contains every irreducible complex representation.
For every torsion-free discrete group G and commutative characteristic-zero domain R, the OpenAI Bass manuscript claims each scalar idempotent in R[G] is 0 or 1. The separate trace-rank assertion concerns algebraic K₀(C[G]).
For every order-six complex Hadamard matrix outside Tao’s cubic equivalence class, the normalized character sum vanishes at every permutation of (1,1,1,−1,−1,−1), allowing row and column permutations and arbitrary unit phases.
For every discrete group G, OpenAI claims that the Hattori–Stallings trace of every class in K₀(C G) vanishes on each infinite-order conjugacy class. The claim concerns the complex group ring; an integral identity-only consequence needs a separate known finite-order input.
For split connected semisimple groups over global function fields and every effective divisor, including arbitrary multiplicity, rational and ℓ-adic cuspidal-function spaces split into direct sums indexed by dual-group nilpotent orbits. The same orbit controls all unramified Hecke places.
Assuming the four stated characteristic conditions on the split connected semisimple group G over a finite field, every canonical unramified Arthur-filtration subspace on the full space of finitely supported automorphic functions is defined over Q and is independent of ℓ, including noncuspidal functions.
Over F₂(q,H₁,H₂), with three algebraically independent parameters, the construction gives a finite-dimensional symmetric algebra A and a finite-dimensional nonprojective A-module M with Ext_A^i(M,M)=0 for every i>0.
For a finite field K of odd characteristic and a finitely generated group G with torsion, the construction gives a linear cellular automaton on all K^G configurations that is injective but not surjective.
The September OpenAI manuscript claims an infinite-dimensional finitely presented nonunital nil associative algebra over F₂ that is Jacobson radical but not nilpotent. Its unitization is finitely presented, algebraic and infinite-dimensional.
OpenAI claims strict positivity of the Tor Euler characteristic for nonzero finite modules of complementary dimension over a regular local ring, when their tensor product has finite length.
Flake, Gruber and Heidersdorf report that the universal rigid monoidal category on one object has an abelian envelope without the quotient property.
Enomoto constructs a 36-dimensional algebra Λ(q) over an algebraically closed field with an element q of infinite multiplicative order. Every simple right module has period four and the algebra is twisted periodic, yet a two-dimensional module has pairwise nonisomorphic even syzygies, so the algebra is not periodic.
GPT-6 Astra constructed a nil algebra over the algebraic closure of F₂ with a nonnilpotent 2×2 matrix. In its unitization, the algebra is a nil two-sided ideal, refuting Krempa’s matrix form of Köthe’s conjecture.
For cut-defined omnific integers a,b,c,d represented as numeric games, a Lean theorem says ab=cd implies omnific e,f,g,h with a=ef, b=gh, c=eg and d=fh, including zero cases. The theorem is stated at each Lean universe, rather than putting all surreal numbers into one type.
For even N=2m≥2, pairwise distinct real numbers x₁,…,x₂ₘ and odd D=2r+1≥N−1, Ma proves that the matrix [(xⱼ−xᵢ)^D] is nonsingular. After sorting the nodes, the proof gives the stronger sign (−1)^(m choose 2) Pf[((xⱼ−xᵢ)^D)]>0.
For n ≥ 2 and 1 ≤ v ≤ π(n), the count C_{n,v} of minimal generators of the truncation ideal I_n with least support index v equals Legendre's sifting function Φ(n,p_v).
Over any characteristic-zero field, Wang and Zhang prove Fröberg’s predicted generic Hilbert series for equal-degree forms in four variables when the degree is 5 or 7 and the generator count is any r≥1. Exact rank certificates cover the previously open middle ranges r=6–11 and r=6–21, respectively; propagation covers the remaining counts.
For an integral domain k, the commutative ring R=k[X₁,X₂,…]/(Xᵢ(1−Xⱼ):i<j) has exactly the pure ideals 0, I=(x₁,x₂,…), and R. Thus 0 is purely-prime but not purely-maximal.
For any field extension K⊂L, associative L-algebra A and A-module M, Greenfeld’s manuscript states that amenability of M over A viewed as a K-algebra is equivalent to amenability over L. This also applies to amenable and fully amenable algebras.
Bhatt and Ma construct a five-dimensional finitely generated flat normal domain over Z or any field with a codimension-one prime J for which H²_J has infinitely many minimal primes. The characteristic-two case uses a separate construction; a cone gives a six-dimensional normal local example.
For every fixed power k≥1, sufficiently many variables give a lattice monomial in the Newton polytope of the k-th Vandermonde power with zero coefficient. The new step is an explicit even-power seed for k≥4; earlier cases and propagation complete the result.
For G=SmallGroup(128,859) over the algebraic closure of F₂, depth H*(G;k)=2 while every associated-prime quotient has dimension at least 3. The group-specific certificate enumerates 75 rank-two elementary abelian subgroups and checks their centralizer types.
Long gives explicit rational-coefficient polynomials in four variables defining a Poisson endomorphism of the canonical algebra P₂ in characteristic zero that is not an automorphism. An exact three-point fiber proves noninjectivity; extending the map by identity gives examples for every Pₙ with n≥2.
The Laurent polynomial f(x,z)=(1−z⁻¹)((1−x)+xz) has zero pure product-integral moments for every positive n but marked moment (−1)^(n−1)/(n+1). It refutes xz(1,1), with padding for all mixed k,l≥1. Its SU(2) lift F=(1+c)(ad+b), G=−c has the same pure and marked Haar moments and refutes the SU(2) Mathieu conjecture.
Larson gives a rank-nine binary matroid on 18 elements whose complementary basis pairs occupy disconnected symmetric-exchange classes. It refutes Farber’s two-base conjecture and White’s toric-ideal symmetric-exchange generation conjecture.
For n=3, K=2 and d=4, Peng gives positive definite symmetric matrices arbitrarily close to the identity with ||W_SS|| > ||W_RS||, refuting the SS–RS half. For all n≥2, K≥1 and d≥1, the RS–GD half holds when (1−1/(4n²+1))I ≤ Aᵢ ≤ I.
Every pure O-sequence of codimension three and type two is log-concave: h(i−1)h(i+1)≤h(i)^2 at every interior degree.
For semifree noncommutative differential graded algebras over a nontrivial computable unital commutative ring, stable tame isomorphism, quasi-isomorphism and derived Morita equivalence are undecidable.
There exists a weakly quasi-complete Noetherian local ring that is not quasi-complete.
For the specified row-stochastic 4×4 matrices supported on a 4-cycle, every nonreal eigenvalue a+ib has 0≤a≤1, a+|b|≤1 and (b²+a²+a)²+2a²−b²≥0. Conversely, upper-half-plane points with b>0, 0≤a≤1, a+b<1 and the polynomial nonnegative occur; the stated boundary arcs are attained. This includes Ran and Teng’s Conjecture 20.
For every numerical semigroup S and integer p≥0, AxiomProver’s Lean proof establishes Fel’s formula for the normalized alternating syzygy quantity Kₚ(S) in terms of universal symmetric polynomials, gap power sums and a final polynomial term.
DeepMind reports Lean proofs for IMO 2024 problems 1, 2, and 6, and publishes the specific proof artifacts.
Groups, rings, representations and the structure of algebraic systems.
Publication dates and editorial changes.
AlphaProof Nexus proves log-concavity for type-two pure O-sequences
OpenAI reports a counterexample to Tachikawa’s second conjecture
OpenAI reports a four-variable Abhyankar–Sathaye counterexample
OpenAI reports an odd-characteristic counterexample to surjunctivity
OpenAI reports a characteristic-zero Kaplansky idempotent result
AI-assisted work finds a Latin-square determinant parity criterion
AI-assisted work reports the depth-two affine incidence Smith profile
AI-built Lean proof reports Conway’s omnific refinement property
AI-assisted work reports the depth-three affine incidence Smith profile
AI-derived determinant restriction for finite-order matrices
Astra finds a five-generator semigroup family with exact Betti count
ChatGPT-assisted map challenges the rank-two Poisson conjecture
OpenAI reports Fourier vanishing for order-six Hadamard matrices
ChatGPT-assisted proof makes algebra amenability field-independent
ChatGPT-found matroid refutes White’s symmetric-exchange claim
Codex-assisted proof establishes eventual Vandermonde non-SNP
AI-assisted proof characterizes a four-cycle matrix spectrum
OpenAI reports the Ramanujan–Arthur cusp decomposition at full finite level
OpenAI reports rationality of the canonical Arthur filtration
OpenAI reports temperedness at ramified exceptional-group places
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