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The other advances

1. Not evaluatedOct 20, 2026Frankl, Ota and Tokushige's conjecture: announced counterexample, found by ChatGPT 6.1 Sol

   On October 20, 2026, Balázs Patkós posted on arXiv a counterexample, found by ChatGPT 6.1 Sol, to a conjecture of Frankl, Ota and Tokushige on intersecting families.

   What it changes

   An intersecting family is a collection of sets in which any two share an element. Frankl, Ota and Tokushige had conjectured the exact value of an extremal quantity b(t) attached to these families; if the counterexample held, b(t) would grow much more slowly than the conjectured value, like t to the power 3/2 up to a logarithmic factor.

   What remains unclear

   No third party has reviewed the counterexample or the new bound yet, and nothing is formalised.

   What “Not evaluated” means: No third-party evaluation has been found. This does not mean the result is wrong, nor that it is right: as far as we know, nobody has published a check.

   Read the full entryWritten by an AI · not reviewed

2. Not evaluatedOct 20, 2026Ordered Variable Word theorem: first-order part announced as determined

   On October 20, 2026, Ludovic Patey posted on arXiv an answer to an open question in mathematical logic about the Ordered Variable Word theorem, with a core lemma proved by Claude Opus 5.5.

   What it changes

   In reverse mathematics, the strength of a theorem is measured by the axioms needed to prove it. If the proof held, the Ordered Variable Word theorem, a result in combinatorics, would prove no more arithmetic statements than RCA_0 + BΣ^0_2, the base system RCA_0 extended by the collection principle BΣ^0_2. The author derives the same property for Chubb, Hirst and McNicholl's tree theorem for singletons, which would answer a long-standing open question.

   What remains unclear

   No third party has reviewed the proof yet, and it is not formalised.

   What “Not evaluated” means: No third-party evaluation has been found. This does not mean the result is wrong, nor that it is right: as far as we know, nobody has published a check.

   Read the full entryWritten by an AI · not reviewed

3. Not evaluatedOct 17, 2026Dimension of posets excluding a standard example: new bounds found by ChatGPT-6 Astra

   On October 17, 2026, Jędrzej Hodor, Piotr Micek and William T. Trotter posted on arXiv new bounds on the dimension of posets excluding a standard example, with proofs found by ChatGPT-6 Astra.

   What it changes

   The dimension of a poset is the smallest number of total orders whose intersection it is. For every order d ≥ 3 of the excluded standard example, the paper would show that the maximum dimension of an n-element poset grows at most like a power of n strictly below 1, where it was only known to grow more slowly than n. For d = 3, it would grow at least like n to the power 1/3 and at most like n to the power 2/3, up to logarithmic factors.

   What remains unclear

   No third party has reviewed the proofs yet, and they are not formalised. The exact growth exponent for S_3, denoted γ_3, remains open: it is only known to lie between 1/3 and 2/3.

   What “Not evaluated” means: No third-party evaluation has been found. This does not mean the result is wrong, nor that it is right: as far as we know, nobody has published a check.

   Read the full entryWritten by an AI · not reviewed

4. Not evaluatedOct 16, 2026Maximum number of Hamiltonian cycles in a tournament: announced proof of a conjecture of Wormald

   On October 16, 2026, Alexander Divoux and Shouda Wang posted on arXiv the asymptotic value of the maximum number of Hamiltonian cycles in a tournament, which would prove a conjecture of Wormald, from proofs found by GPT-5.6.

   What it changes

   A tournament is a graph in which every pair of vertices is joined by an arrow in one direction only; a Hamiltonian cycle passes once through every vertex along the arrows. If the result held, the maximum number of such cycles would be known asymptotically, with the constant cosh 1 / cos 1, proving a conjecture of Wormald and answering a question of Noga Alon.

   What remains unclear

   The 49-page paper has not yet been peer-reviewed or formalised. According to the authors, many of GPT-5.6's arguments were later replaced or simplified by them, and others came from the literature. A stronger conjecture of Wormald, that one specific tournament attains the exact maximum for all large enough odd n, remains open.

   What “Not evaluated” means: No third-party evaluation has been found. This does not mean the result is wrong, nor that it is right: as far as we know, nobody has published a check.

   Read the full entryWritten by an AI · not reviewed

5. Not evaluatedOct 15, 2026Counterexamples to the ordinary Fried conjecture near hyperbolic metrics

   On October 15, 2026, Simon Becker and Benjamin Lawton posted on arXiv counterexamples to the ordinary Fried conjecture in dimension 3, built with GPT-6.1 Sol and partly formalised in Lean with Claude Code.

   What it changes

   On a negatively curved manifold, the Fried conjecture relates the value at zero of a zeta function built from closed geodesics to an invariant called torsion. If the counterexamples held, this identity would fail for some real analytic metrics arbitrarily close to a hyperbolic metric in dimension 3. "Ordinary" refers to the value of the zeta function itself at zero, which the paper distinguishes from a corrected invariant, Chaubet and Dang's dynamical torsion.

   What remains unclear

   The Lean formalisation covers only conditional algebraic and limiting steps: the geometric and spectral inputs are not formalised. No third party has reviewed the work yet.

   What “Not evaluated” means: No third-party evaluation has been found. This does not mean the result is wrong, nor that it is right: as far as we know, nobody has published a check.

   Read the full entryWritten by an AI · not reviewed

6. Not evaluatedOct 14, 2026Primitive roots

   On October 14, 2026, Manjul Bhargava posted on Hexagon a proof, written up and formalised in Lean with GPT-6, that every non-square integer other than −1 is a primitive root modulo a positive proportion of primes.

   What it changes

   An integer a is a primitive root modulo a prime p when its powers run through every non-zero remainder modulo p. If the proof held, every non-square integer other than −1 would be one for a positive proportion of primes, with no assumption beyond recent theorems by Cureton and Thorner and by OpenAI. According to the paper, an OpenAI preprint had already shown that there are infinitely many such primes; the gain would be to reach a positive proportion, short of the exact density predicted by Artin's conjecture.

   What remains unclear

   The proof rests on two recent results that have not been peer-reviewed: Cureton and Thorner's zero-free region and estimates by OpenAI. The Lean code checks only the deductions, with both results left as assumptions. According to the page, some details added by GPT-6 were checked by Lean but not yet by any human.

   What “Not evaluated” means: No third-party evaluation has been found. This does not mean the result is wrong, nor that it is right: as far as we know, nobody has published a check.

   Read the full entryWritten by an AI · not reviewed

7. Not evaluatedOct 13, 2026Central limit theorem for the high-temperature SYK model: announced proof, obtained with ChatGPT and partly formalised in Lean

   On October 13, 2026, four probabilists posted on arXiv a central limit theorem for the high-temperature SYK model, proved with ChatGPT 5.5 Pro and 5.6 Pro and then formalised in Lean under one assumption.

   What it changes

   The SYK model describes particles with random interactions and serves as a testing ground in theoretical physics. If the result held, the fluctuations of the logarithm of its partition function would follow a normal distribution at high temperature, that is, for small values of the inverse temperature β.

   What remains unclear

   The Lean formalisation, whose code was generated by the AI and reviewed by the authors, leaves Proposition 4.2 as an assumption. No third party has reviewed the proof yet, and the result covers high temperatures only.

   What “Not evaluated” means: No third-party evaluation has been found. This does not mean the result is wrong, nor that it is right: as far as we know, nobody has published a check.

   Read the full entryWritten by an AI · not reviewed

8. Not evaluatedOct 13, 2026Lie groups: announced proof that every Lie group can be realised by permutations of a countable set

   On October 13, 2026, Nicolas Monod and Andreas Thom posted on arXiv a proof that every Lie group can be realised by permutations of a countable set, a question of Ulam, building on a construction found by GPT 6 Astra.

   What it changes

   A group can be realised by permutations of a countable set when it can be viewed as a group of bijections of a countably infinite set. If the proof held, every Lie group, the groups that describe continuous symmetries, would have this property, answering a question of Ulam.

   What remains unclear

   No third party has reviewed the proof yet, and it is not formalised. The authors point out that a substantial part of the argument already appeared in earlier work, notably the Parusiński–Rond construction; they regard the Puiseux embedding of Lemma 2.4 and its combination with the gluing argument as new.

   What “Not evaluated” means: No third-party evaluation has been found. This does not mean the result is wrong, nor that it is right: as far as we know, nobody has published a check.

   Read the full entryWritten by an AI · not reviewed

9. Not evaluatedOct 13, 2026Torus polynomials and the Majority function: announced resolution of two open problems, found and formalised in Lean with GPT-6 Astra

   On October 13, 2026, Saint Wesonga posted on arXiv the resolution of two open problems on torus polynomials and the Majority function, results discovered and formalised in Lean with GPT-6 Astra.

   What it changes

   Torus polynomials approximate a Boolean function modulo the integers; they were introduced as a route to showing that the Majority function lies outside a class of Boolean circuits with modular gates, ACC^0. If the results held, they would answer two of the five open problems posed by Krishan and Vishwanathan, on integer lifts and explicit polynomial approximations.

   What remains unclear

   The Lean code comes from the author, who wrote the paper alone, and no third party has reviewed the statements or the text yet. A formalisation checks the proof of a statement, not that the statement faithfully captures the problem posed.

   What “Not evaluated” means: No third-party evaluation has been found. This does not mean the result is wrong, nor that it is right: as far as we know, nobody has published a check.

   Read the full entryWritten by an AI · not reviewed

10. AvailableOct 12, 2026Muse Wonder and Muse Wonder Full

    Meta introduces Muse Wonder and Muse Wonder Full, its most capable models, available via the Meta Model API and Muse Code (Wonder) and to Muse Code subscribers (Full).

    What “Available” means: Usable by the public or by customers, source read.

    See in the feedWritten by an AI · not reviewed