AI-generated results not verified by a human

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In this paper we study the Frobenius volumes, an analogue of local volumes for strongly F-regular singularities over an algebraically closed field of characteristic p > 0. We prove the finite degree formula for Frobenius volumes, and show that the Hilbert–Kunz Frobenius volume equals the Frobenius volume divided by n!, where n is the dimension of the ambient variety. This gives a positive answer to Problem 49 in Xu–Zhuang’s open problem list in K-stability of Fano varieties.

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Disclaimer: This is an AI-generated manuscript. The proof has been checked by machines (AI reviews of the manuscript), but it has not been verified by a human, and none of this is formal verification. Readers should independently check every mathematical claim before relying on or citing it.

We prove Shokurov’s local index conjecture in dimension three: for every positive real number a there is a positive integer R₃(a) such that, for every klt threefold germ X ∋ x with mld(X ∋ x) = a, the divisor R₃(a)K_X is Cartier near x. More generally, for every finite set Γ of rational numbers in [0,1] there is a positive integer R₃(a,Γ) such that R₃(a,Γ)(K_X + B) is Cartier near x for every klt threefold germ (X ∋ x, B) with the coefficients of B in Γ and mld(X ∋ x, B) = a.

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Disclaimer: This is an AI-generated manuscript. The proof has been checked by machines (an internal AI verifier and AI reviews of the rewritten manuscript), but it has not been verified by a human, and none of this is formal verification. Readers should independently check every mathematical claim before relying on or citing it.

We prove that the orbifold fundamental group of a special smooth projective orbifold pair is virtually abelian. This answers a conjecture of Campana and extends a recent result of OpenAI to the orbifold projective case.

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Disclaimer: This is an AI-generated manuscript. The proof has been checked by machines (an internal AI verifier and AI reviews of the rewritten manuscript), but it has not been verified by a human, and none of this is formal verification. Readers should independently check every mathematical claim before relying on or citing it.

We prove that every real planar polynomial vector field has only finitely many isolated periodic orbits. The proof follows Écalle’s formalization and actualization architecture by reconstructing complete physical return maps, including divergent normalizers and compatible nonreal transition data, and controlling the entire remainder after all cancellations. The theorem is individual finiteness: no bound in terms of the degree is obtained.

Relation to Écalle’s route. Écalle’s 1992 monograph proves Dulac’s theorem in two stages: formalization, which attaches to the return map of a polycycle a formal object (a transseries), and actualization, which shows that the formal object determines the actual map, so that a nonzero formal displacement forces a sign of the actual displacement near the polycycle. The actualization is carried out there through accelero-summation in the class of analysable functions, with median averages at the saddle-nodes. This manuscript keeps the two-stage architecture but changes the devices. There are no acceleration operators, no median or well-behaved averages, and no transseries class. The formal side is carried by normed spaces of the original analytic data (coefficients, integration data and inverse families, called source presentations) with bounded comparison maps that read off all terms of a given action; the quasi-analyticity statement, namely that the common kernel of these maps equals the analytic kernel, is proved by reconstruction theorems resting on a finite-sector flatness lemma of Phragmén–Lindelöf and Ramis–Sibuya type on the actual complex domains, along an induction over finite physical words. Saddle-node passages use Borel–Laplace sectorial normalizers along a fixed non-real direction, and reality is proved for the matched physical return rather than for the individual normalizers; this is the point at which Ilyashenko’s 1991 proof has an acknowledged gap (Yeung, Qual. Theory Dyn. Syst. 2025). Hyperbolic saddle passages are represented by convergent measure-valued Laplace integrals whose distributional record is the Dulac series. The geometric reduction uses compactification, elementary resolution and Roussarie’s limit-graphic theorem.

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Disclaimer: This is an AI-generated manuscript. The proof has been checked by machines (the internal verifier of Danus, the writer–verifier reviews of the manuscript, three independent AI cold reviews in September 2026, which found no mathematical error, and an independent review by a second AI system whose two statement-level findings were repaired in this version), but it has not been verified by a human, and none of this is formal verification. Readers should independently check every mathematical claim before relying on or citing it.

On a flat integral affine torus V_ℝ/Γ with an arbitrary full-rank real period lattice Γ, in every dimension and every cycle dimension, the rational classes in the kernel of the Mikhalkin–Zharkov eigenwave are exactly the classes of finite balanced straight tropical cycles with rational, possibly negative, weights. In particular every rational tropical Hodge class on every tropical abelian variety is the class of such a cycle. Jell–Rau–Shaw had proved the codimension-one case with integral coefficients, and Amini–Piquerez the case of smooth projective tropical varieties admitting a rational triangulation, leaving open the tropical abelian varieties with irrational periods. The proof works with rational-faced polytopes with real offsets and needs no rational triangulation: complete endpoint moments and a Koszul lift over the period lattice, finite signed specialization and filling, a periodic polyhedral refinement, and a direct Stokes computation of the full marked class.

Why this is posted here. Kontsevich proposed to test the Hodge conjecture by finding, on a maximally degenerating abelian variety, a tropical Hodge class with no balanced tropical representative (see Zharkov, arXiv:2002.02347). Section 8 of the note explains why the theorem leaves no room for such a witness: every rational class in the eigenwave kernel has a balanced representative on the torus itself, for every real period lattice and every dimension, so no linear class invariant can separate a tropical Hodge class from the cycle classes. The note does not assert anything about the classical Hodge conjecture: an algebraic cycle specializes to a tropical representative, but a tropical representative is not shown to lift to an algebraic cycle, and finer data of a chosen degeneration are outside the theorem.

Disclaimer: This is an AI-generated manuscript. The proof has been checked by machines (the internal verifier of Danus, the writer–verifier reviews of the manuscript, and an independent AI review of the underlying facts, which found no gap), but it has not been verified by a human, and none of this is formal verification. Readers should independently check every mathematical claim before relying on or citing it. In June 2024 Omid Amini announced an outline of a proof of the tropical Hodge conjecture in joint work with Matthieu Piquerez; I have not seen a written version, and their result may already contain the theorem of this note.

This paper gives affirmative answers to both parts of Problem 23 in the list of Chenyang Xu and Ziquan Zhuang, Open problems in K-stability of Fano varieties, for every complex semisimple Lie algebra: the natural projective base of the normalization of a nonzero nilpotent orbit closure is K-polystable, and the natural projective base of each positive-dimensional component of a Slodowy slice of a nilpotent orbit closure, with the boundary given by the divisorial stabilizers of the Kazhdan action, is log K-polystable. Both statements come with weak Kähler–Einstein currents and with the equality case for all normal test configurations. Xu and Zhuang, with an appendix by Guenancia, had proved K-semistability of the orbit bases (arXiv:2606.02401) and recorded K-polystability for classical Lie algebras.

The proof constructs a homogeneous hyperkähler potential on the whole affine cone, not only on its smooth part. For matrix orbits and type-A slices the potential is the half-norm on a finite-dimensional hyperkähler quotient; a faithful representation pulls it back to an arbitrary semisimple Lie algebra, including the exceptional ones, and Kronheimer’s description of orbits and slices by trajectories makes the pullback compatible with all complex structures. A weighted descent then gives a weak Kähler–Einstein current of full mass on the projective base, and Berman’s theorem gives K-polystability.

Disclaimer: This is an AI-generated manuscript. The proof has been checked by machines (the internal verifier of Danus, the writer–verifier reviews of the manuscript, and an independent section-by-section AI review, which found no invalid inference), but it has not been verified by a human, and none of this is formal verification. Readers should independently check every mathematical claim before relying on or citing it.

Update (2026-09-24 17:35:31 UTC): independent work of Shilin Yu. Shilin Yu (East China Normal University) has informed me that he independently obtained the same results at around the same time, by a different method. His approach uses Nahm’s equations: he proves that the hyperkähler potential extends continuously to the closure of a nilpotent orbit and, more generally, to the relevant Slodowy slice intersections. His argument is uniform: it does not use ADHM constructions or quiver varieties, whereas my paper reduces the general case to type A. According to him, he had an argument for nilpotent orbit closures by the end of June 2026, whose main analytic input is an elementary ODE estimate, and the extension to general Slodowy slice intersections, which uses Bielawski’s work on Nahm’s equations on finite intervals, is written up in a note dated 10 August 2026. He plans to post his work as a preprint.

This paper proves Fujita’s base-point-freeness conjecture in dimension seven: K_X + 8L is globally generated for every smooth complex projective sevenfold X and every ample Cartier divisor L. It builds on the six-dimensional proof below. The proof constructs complete spaces of sections with prescribed order at a point and controls the entire sequence of log canonical centers arising from their thresholds. The finite inequalities used in the proof come with exact rational certificates and independently executable checkers, collected in a computational supplement. Danus obtained the sevenfold theorem 28 hours after it started working beyond dimension six; the conjecture in arbitrary dimension remains open.

Disclaimer: This is an AI-generated manuscript. The proof has been checked by machines (the internal verifier of Danus and the writer–verifier reviews of the manuscript; the thirteen certificate packages of the supplement pass a fresh full run of their checkers), but it has not been verified by a human, and none of this is formal verification. Readers should independently check every mathematical claim before relying on or citing it.

Update (2026-09-20 07:03:59 UTC): the PDF was replaced to add an acknowledgement to Zhixian Zhu for useful discussions; the mathematics is unchanged.

This paper proves Fujita’s base-point-freeness conjecture in dimension six: K_X + 7L is globally generated for every smooth complex projective sixfold X and every ample Cartier divisor L. The conjecture was previously known in dimensions at most five, by Reider, Ein–Lazarsfeld, Kawamata, and Ye–Zhu. At each point, the proof constructs an effective rational boundary of coefficient strictly less than seven with that point as its minimal log canonical center. It combines bounds for complete cutting chains with uniform local dimension estimates and amplification of complete spaces of sections. Danus ran with seven workers; from the creation of the project to the acceptance of the theorem took about 47 hours of calendar time, including an interruption.

Disclaimer: This is an AI-generated manuscript. The proof has been checked by machines (the internal verifier of Danus, and an independent AI cold review on 9 September 2026, which found one incomplete enumeration, completed in this version, and reported no other unresolved objection), but it has not been verified by a human. Readers should independently check every mathematical claim before relying on or citing it.

Update (2026-09-20 07:03:59 UTC): the PDF was replaced to add an acknowledgement to Zhixian Zhu for useful discussions; the mathematics is unchanged.

This paper constructs a countable discrete group that is not hyperlinear. Equivalently, its group von Neumann algebra, with its canonical trace, does not embed into an ultrapower of the hyperfinite II₁ factor, so Connes’ embedding problem has a negative answer for groups. The group is the generalized wreath product W = (⊕_G/Γ Z/2) ⋊ G, where Γ = EL₃(F₂[x₁, x₂, x₃]) and G = EL₃(F₂[x₁^±1, x₂^±1, x₃^±1]) ⋊ SL₃(Z). It is one of the groups that Kun and Thom (arXiv:2608.06222) proved to be nonsofic, building on the first nonsofic group, found by OpenAI.

The new ingredient is a theorem on tracial ultraproducts of matrix algebras: the relative commutant of a finitely generated Kazhdan group in such an ultraproduct is itself an ultraproduct of finite-dimensional subalgebras of the same matrix algebras (Theorem 1.2 of the paper). This is a positive answer to the centralizer problem, that is, Open Problem 6.2(a) of Alekseev and Thom (arXiv:2608.05362). The second step of the argument (Theorem 1.3 of the paper: a Kazhdan group generated by a subgroup together with finitely many elements that conjugate this subgroup into itself normalizes the relative commutant of the subgroup) was proved earlier by Andreas Thom, conditionally on the centralizer problem, in his preprint A conditional construction of a nonhyperlinear group and the centralizer problem; the credit for it belongs to him (see the update of 20 September below). Danus ran with seven proof workers and found the proof in 7 hours and 11 minutes.

Disclaimer: This is an AI-generated manuscript. The proof has been checked by machines (the internal verifier of Danus, and an independent line-by-line AI review that found no invalid inference), but it has not been verified by a human. Readers should independently check every mathematical claim before relying on or citing it.

Update (2026-09-20 08:07:35 UTC): the PDF was replaced. The introduction now cites OpenAI’s construction of the first nonsofic group, on whose compression criterion the work of Kun and Thom and this paper build; the mathematics is unchanged.

Update (2026-09-20 11:19:40 UTC): prior work of Andreas Thom. On 20 September 2026, after this paper had been posted here, Andreas Thom informed me of his preprint A conditional construction of a nonhyperlinear group and the centralizer problem. According to him, a first version was circulated on 10 August 2026 and submitted to arXiv on 17 August 2026, where it has been on hold; it has been publicly available on his homepage since 25 August 2026. His paper shows that a positive answer to the centralizer problem implies the normalization statement (his Theorem 1.2) and hence, conditionally, that the group doubles G ∗_H G of the Kun–Thom pairs are not hyperlinear (his Theorem 1.3). Theorem 1.3 of my paper overlaps with his Theorem 1.2, which has priority: the credit for it belongs to Thom. I was not aware of his preprint. The sources recorded during the Danus run do not contain it, but it was available online, and I cannot guarantee that the AI models involved had not come across it. What my paper contributes is Theorem 1.2, a positive answer to the centralizer problem; combined with Thom’s Theorem 1.3 it also shows unconditionally that those group doubles are not hyperlinear. The PDF was replaced: its introduction now states all of this and cites Thom’s preprint.

Update (2026-09-21 08:27:54 UTC): MIP* = RE and orthogonalization results. Mikael de la Salle pointed out that the paper gave no credit to two bodies of earlier work; the PDF was replaced and now does. First, the statement is equivalent to the existence of a group von Neumann algebra that does not embed into R^ω. The first tracial von Neumann algebras that do not embed into R^ω are due to Ji, Natarajan, Vidick, Wright and Yuen (MIP* = RE); what my paper adds is that one can take the von Neumann algebra of a group, with a proof that does not use MIP* = RE. Second, Theorem 4.3 (rounding an operator-valued partition to a projection-valued one) belongs to the line of orthogonalization results of Kempe–Vidick, of Ji–Natarajan–Vidick–Wright–Yuen, and of de la Salle (Orthogonalization of positive operator valued measures, C. R. Math. 2022). The distance estimate of Theorem 4.3 is de la Salle’s Theorem 2 with a worse constant, and its proof follows his strategy; the new part is the simultaneous control of the commutator energy, uniformly in the number of outcomes and in the dimension. The sources recorded during the Danus run do contain de la Salle’s paper; the first version of the manuscript omitted the reference. The introduction (the paragraphs on Connes’ embedding problem and “Relation to orthogonalization results”) and the beginning of Section 4 now state this, and nine references were added. The acknowledgements of the paper now thank Mikael de la Salle and Andreas Thom.

Update (2026-09-25 08:36:42 UTC): joint paper with Vadim Alekseev and Andreas Thom. I am now collaborating with Vadim Alekseev and Andreas Thom on a much better written paper. We have isolated the core strategy of the proof and put the ideas into a self-contained note which, together with Thom’s preprint A conditional construction of a nonhyperlinear group and the centralizer problem (arXiv:2609.27795), proves the existence of a nonhyperlinear group. The PDF posted here will in principle not be changed any further; it is kept as a historical record.

Update (2026-10-06 21:13:41 UTC): the human-verified paper is on arXiv. The joint paper with Vadim Alekseev and Andreas Thom announced in the previous update is now on arXiv: Spectral gap, internality of relative commutants, and non-hyperlinear groups, arXiv:2609.37761 (posted 29 September 2026; no. 75 in my list of publications). It is a complete draft, verified by humans, and it supersedes the manuscript posted here: the credit for the existence of a non-hyperlinear group goes to that paper. The PDF on this page is kept only as a historical reference, and this entry will not be updated after this note.