I asked Astra to check out OpenAI’s math repo containing the 722 preprints of results (from ~4,000 problems attempted) obtained using their internal model at “three hours of ChatGPT Pro thinking compute” each. These preprints are organised into 372 families, i.e. “related papers, which may include a principal result, companion arguments, consequences, or alternative proofs”. Astra thinks/claims
there are ~30 (range 20–50) potential Fields medal-scale portfolios’ worth of results (“I would defend “several dozen” more readily than either endpoint”), including partial results or related work on every Millennium Prize problem except Yang-Mills
Millennium Prize problem
Match in this release
Riemann hypothesis
Partial: fixed zero-free strip
Birch–Swinnerton-Dyer
Partial: full formula for the stated low-Selmer-corank class; further density-one consequences with Goldfeld
Hodge conjecture
Partial: CM abelian varieties, K3 products and related results
Navier–Stokes
Related work here; the earlier forced-breakdown release is separate
P versus NP
Related complexity breakthroughs, no resolution
Yang–Mills existence and mass gap
No matching resolution identified
The workbook constructs 28 illustrative portfolios using 234 distinct families, without assigning any family twice. This makes the comparison inspectable. It does not eliminate common proof dependencies, and some groupings are deliberately broad. A few could plausibly split into multiple portfolios; others may look less substantial once novelty and methods receive expert scrutiny.
This is not a measured median Fields Medal equivalent. There is no established scalar ordering of those 30 historical portfolios from which to calculate such a median. The estimate concerns the possible scale of mathematical outcomes. Lasting influence, reusable ideas, exposition, and the work of collaborators remain separate considerations.
The strongest concrete repo comparisons include:
Repo cluster
Historical reference
Why the comparison has substance
QRH, prime correlations, primitive roots
Maynard; parts of Tao and Venkatesh
Several central advances in the structure of primes and arithmetic distribution
BSD, Goldfeld, modularity
Bhargava and arithmetic portfolios
Structural theorems about broad classes of curves, beyond isolated records
Restricted and quantum geometric Langlands
Lafforgue, Ngô
Major steps in an organizing programme linking geometry and representation theory
Abundance, subadditivity, minimal models
Birkar
An unusually direct comparison within the same central classification programme
Kakeya, Falconer, restriction, smoothing
Wang, Tao
Closely overlapping frontier questions, with multiple substantial next steps
Critical percolation, spin models, scaling limits
Duminil-Copin, Smirnov, Werner
Several longstanding gaps between physical predictions and rigorous theory
Vlasov–Maxwell and Boltzmann-related results
Villani, Deng
Foundational questions about kinetic equations and microscopic-to-macroscopic behaviour
Mahler and optimal configurations
Viazovska, with qualifications
Exact extremal problems whose solutions can reveal new analytic structure
Hilbert–Smith and symplectic/topological results
Pardon
Direct continuation of some recognized targets, plus other major topology claims
The Fields comparison becomes more defensible when we compare entire pre-award portfolios. I examined official citations and contemporary accounts for every recipient from 1998 through 2026. There was no 2000 award, so 1998 takes the comparison past your requested cutoff. Here is the historical reference set in compressed form. The workbook gives each person an individual row with recognized contributions, methods, possible repo comparisons, limitations, and a primary source.
Lafforgue: function-field Langlands for , through substantial geometry of shtukas. Voevodsky: motivic cohomology, algebraic homotopy theory, and the Milnor conjecture. No. 2
2006
Okounkov: bridges among representation theory, probability and enumerative geometry. Perelman: Ricci-flow singularities and geometrization. Tao: harmonic analysis, nonlinear PDE, additive combinatorics and primes. Werner: SLE, Brownian exponents and conformally invariant probability. mathunion.org
2010
Lindenstrauss: measure rigidity and arithmetic applications. Ngô: the fundamental lemma through Hitchin-fibration geometry. Smirnov: conformal invariance via discrete complex analysis. Villani: kinetic theory, convergence to equilibrium, Landau damping and transport. Elon Lindenstrauss
2014
Avila: several dynamical and spectral programmes. Bhargava: higher composition laws, counting fields and arithmetic statistics. Hairer: regularity structures and singular stochastic PDE. Mirzakhani: moduli-space volumes, geodesic counting and dynamical rigidity. mathunion.org
2018
Birkar: boundedness and the minimal-model programme. Figalli: optimal transport and nonlinear regularity. Scholze: perfectoid spaces and arithmetic geometry. Venkatesh: synthesis of dynamics, topology and number theory. mathunion.org
2022
Duminil-Copin: critical phenomena and phase transitions. Huh: combinatorial Hodge theory and log-concavity. Maynard: prime clusters, large gaps and Diophantine approximation. Viazovska: sphere packing, Fourier interpolation and universal optimality. mathunion.org
2026
Deng: kinetic limits and probabilistic dispersive PDE. Pardon: topology, virtual cycles and symplectic foundations. Tsimerman: arithmetic geometry, unlikely intersections and period maps. Wang: Kakeya, restriction, smoothing and geometric measure theory. mathunion.org
Several lessons emerge from this complete set. A portfolio can revolve around one central programme, as with Lafforgue or Ngô. It can contain several connected breakthroughs, as with Maynard or Avila. Or it can create a mathematical language, as with Voevodsky, Hairer or Scholze. These achievements cannot be ranked reliably by counting named conjectures. Hairer is a particularly useful corrective. The significance was not simply that he solved a difficult equation: regularity structures provided a systematic way to give meaning to, and solve, classes of equations whose nonlinear terms were previously ill-defined. Similarly, perfectoid spaces changed the available methods of arithmetic geometry. Matching such portfolios requires evidence of comparable conceptual reach. Conversely, it would be unreasonable to demand that every medal-scale portfolio invent something as conspicuous as perfectoid spaces. The historical set contains many different combinations of problem solving, technical depth, structural insight and synthesis.
>5x more high-significance progress might’ve been made than by all other AIs combined, proxying using VibeMathed: 95 preprints (42–173 if ±5 score) in 64 families with >60 significance (vs 10 in VibeMathed), 27 (13–42) preprints in 19 families with >70 (vs 3 in VibeMathed), etc
[There is] a limitation in the question “how many results above 60?”: it combines importance of the target with amount of progress on it, while the site principally attempts to score the former.
Strict significance cutoff
Preprints: central estimate
Families containing qualifying results
Preprint count after shifting estimated scores ±5
>60
95
64
42–173
>70
27
19
13–42
>80
7
5
3–11
>87
3
2
0–5
I screened all 372 family descriptions, then separately considered the scope of all 195 manuscripts in families rated above 60. The other manuscripts inherit their below-threshold family estimate. That is a reasonably systematic first pass, but it is not 722 independent, equally detailed significance reviews. This distinction changes the answer considerably. Counting every paper associated with a family above 60 would produce 195, rather than 95. Supporting constructions, restricted intermediate results, conditional reductions, and alternative proofs should not all inherit the importance of their family’s strongest theorem.
The actual VibeMathed examples are more informative than the rubric alone. Here are several comparisons that materially shaped my assessment:
Actual VibeMathed entry
Site score
Repo comparison and my judgment
Matrix-multiplication exponent improved to 2.371177
55
I retain 55 for the problem family, despite the much larger proposed advance to 2.25.
Planar unit-distance conjecture
40
I retain 40 for the repo’s upper-bound advance; the earlier disproof should not be counted again.
Jacobian conjecture, counterexample in dimension ≥3
65
An anchor for central algebraic problems. I place rational Hilbert’s tenth and all-dimensional abundance around 80.
More than two thirds of zeta zeros simple and on the critical line
68
The fixed zero-free strip in family 003 seems qualitatively more consequential; I estimate 90.
Critical bond percolation on every ,
78
I preserve 78 for the directly related family; Anderson localization/delocalization gets approximately 82.
Smooth unforced three-dimensional Euler blowup
80
A useful nearby anchor for the large-data Vlasov–Maxwell claim, which I put around 85.
Smooth-forcing Navier–Stokes breakdown through the Clay alternatives
87
My comparison point for QRH at 90 and CM Hodge at 88; both placements remain judgment calls.
The seven papers I place above [VibeMathed significance] 80 are:
Family
Main claim
Qualifying manuscripts
My estimate
003
Quasi-Riemann hypothesis, through two different fixed zero-free strips
2
90 each
032
Rational Hodge conjecture for all CM abelian varieties
1
88
014
Restricted geometric Langlands in positive characteristic, with stated hypotheses
1
85
362
Large-data global classical relativistic Vlasov–Maxwell
1
85
261
Two-dimensional Anderson localization and higher-dimensional weak-disorder delocalization
2
82 each
The quasi-Riemann hypothesis is a qualitative barrier crossing, despite the unimpressive-looking fraction .Family 003’s principal paper claims that zeta has no zeros in
Classical zero-free regions become thinner as the imaginary part grows. A fixed strip excludes zeros arbitrarily high up, with a uniform distance from the line . Through standard explicit-formula reasoning, that would yield prime-counting errors with a fixed power saving, for example
That is why this seems more consequential than another improvement in the percentage of zeros known to lie on the critical line. The two kinds of progress control different things. The linked formalization scope includes zeta and all Dirichlet -functions, not merely finitely many numerical checks. RH itself would require the much stronger boundary . GitHub For reading, I would start with the alternate proof, which was edited for readability, then compare the stronger argument. The crucial burden lies in the analytic continuation and control of the constructed objects, not in the final contradiction once those are available.
The CM Hodge paper may have the largest cascade of consequences. The main paper in family 032 claims rational Hodge for every complex CM abelian variety, in every dimension and codimension. Very roughly, Hodge asks when certain cohomological classes, which look like the shadows of algebraic subvarieties, actually come from algebraic subvarieties. CM abelian varieties are a special class, but they occupy an exceptionally influential position in arithmetic geometry. The claimed consequences through existing theorems include Tate for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic. Companion papers address products of K3 surfaces and Kuga–Satake correspondences. That breadth explains my high significance estimate. Calling it simply “a special case of Hodge” substantially understates its possible reach. Calling it “the Hodge conjecture solved” substantially overstates it.
there are (claimed) full resolutions to 37 entries in Wikipedia’s longlist of 504 unsolved open problems, partial or special-case results to 44 entries, and related work on 32 entries
For the 504 main-list entries, my classification is:
Relationship to this repo
Entries
Claimed full answer to the listed assertion
37
Claimed partial or special-case progress
44
Related work, without progress established on the exact listed assertion
32
No matching claim identified
391
Among the 37 claimed full answers are Brennan, Flint Hills, Mahler, Hadwiger, Sidorenko, the Erdős reciprocal-sum conjecture, circulant Hadamard, Barker sequences, Nagata, Lipman–Zariski, Serre positivity, surjunctivity, Hilbert–Smith, and the nearby Lagrangian conjecture. The complete table gives the precise assertion, matching families and qualification for every row.
Two subtleties materially improve the interpretation. First, “partial” can be enormous. The rational case of Hilbert’s tenth problem is partial relative to Wikipedia’s question over number fields. The result is partial relative to a bundled question about irrationality measures of many constants. Second, famous shared names create false matches. Dimension-six mutually unbiased bases do not resolve the SIC-POVM problem listed under Zauner. Limit-cycle bounds address a different part of Hilbert’s sixteenth problem from the listed question about real algebraic curves. Kirchberg-style embedding questions are not interchangeable with Connes embedding. I marked these distinctions explicitly.
The Wikipedia matching is complete at the level of the page’s enumerated main-list questions. I preserved 560 list rows: 504 problem entries, 12 grouping headings, and 44 entries in the “problems solved since 2015” appendix. These are row counts, not 504 distinct conjectures. The page contains duplicates, bundled questions and some stale entries. “No matching claim” does not imply that a problem remains open elsewhere.
a family’s Lean link does not necessarily cover its strongest headline
The most important verification finding is that a family’s Lean link does not necessarily cover its strongest headline.
Family
Headline one might infer is covered
What the linked scope actually covers
157
Hadwiger counterexample
A positive list-colouring bound, not that counterexample
266
At most three mutually unbiased bases in dimension six
A weaker bound of five and an auxiliary statement; the stronger result uses a separate computational route
260
Full spacetime Penrose inequalities
Selected auxiliary results, not the entire headline theorem
The first two distinctions are explicit in the repo’s own scope documents. GitHub
The jump from o1 to o3 and from o3 to GPT-5 were enough to allow the better model to get the same performance as the prior model using about 1⁄3 as many tokens… But on their new chart, the internal model is getting the same performance as GPT-6 Astra for about 1⁄100 the tokens. That’s like 4 previous jumps in one.
He’s since retracted this section following feedback, but the “4 previous jumps in one” guess stuck with me as a (possibly wrong & misguided) way to make sense of how powerful the unreleased internal model is, and the notes above are my attempt to appreciate how this cashes out.
What’s the most impressive research-y feat interpolating AIs can theoretically do, fixing their training data to (say) today?
I don’t have a good sense of this in general, but in pure math it’s probably at least Fields medal-tier, if laudatios like that of Akshay Venkatesh are anything to go by…
One of Venkatesh’s advantages over other top-tier mathematicians is his sheer range, the thing frontier models do vastly better than humans at.
And I can imagine, for instance, 2 years of advancements enabling a frontier model to skillfully deploy the late legendary Jean Bourgain’s toolkit. Bourgain was regularly spoken of by other world-leading mathematicians as “effectively a god”… Imagine unleashing thousands of Bourgain-toolkit interpolators math-wide in 2028. I think I’m being conservative here, not assuming continual learning or whatever, not even assuming anyone else’s toolkit, and yet I still find it hard to imagine how transformative this would be.
I clearly failed to price in accelerating progress; the Bourgain toolkit-interpolator is here, and its name is (probably) Bel.
Previous notes on the industrialisation of pure math series for my convenience:
It has been 25 years since a report of original research was last submitted to our editors for publication, making this an appropriate time to revisit the question that was so widely debated then: what is the role of human scientists in an age when the frontiers of scientific inquiry have moved beyond the comprehensibility of humans?
No doubt many of our subscribers remember reading papers whose authors were the first individuals ever to obtain the results they described. But as metahumans began to dominate experimental research, they increasingly made their findings available only via DNT (digital neural transfer), leaving journals to publish second-hand accounts translated into human language.
Without DNT, humans could not fully grasp earlier developments nor effectively utilize the new tools needed to conduct research, while metahumans continued to improve DNT and rely on it even more. Journals for human audiences were reduced to vehicles of popularization, and poor ones at that, as even the most brilliant humans found themselves puzzled by translations of the latest findings.
No one denies the many benefits of metahuman science, but one of its costs to human researchers was the realization that they would probably never make an original contribution to science again. Some left the field altogether, but those who stayed shifted their attentions away from original research and toward hermeneutics: interpreting the scientific work of metahumans. …
The question is, are these worthwhile undertakings for scientists? Some call them a waste of time, likening them to a Native American research effort into bronze smelting when steel tools of European manufacture are readily available. …
I first read this story in my college years over a decade ago and wondered when this would come to pass. Despite having been in this corner of the internet since those days, I still wouldn’t have predicted “mid-2020s for pure math”.
As a side note, Chiang’s story goes:
Moreover, human researchers may discern applications overlooked by metahumans, whose advantages tend to make them unaware of our concerns.
Perhaps one reason for von Neumann’s attention to detail was that he found it quicker to hack through the underbrush himself than to trace references and see what others had done. The result was that sometimes he appeared ignorant of the standard literature. If he needed facts, well-known facts, from Lebesgue integration theory, he waded in, defined the basic notions, and developed the theory to the point where he could use it. If, in a later paper, he needed integration theory again, he would go back to the beginning and do the same thing again. He saw nothing wrong with long strings of suffixes, and subscripts on subscripts; his papers abound in avoidable algebraic computations. The reason, probably, is that he saw the large picture; the trees did not conceal the forest from him. He saw and he relished all parts of the mathematics he was thinking about. He never wrote “down” to an audience; he told it as he saw it. The practice caused no harm; the main result was that, quite a few times, it gave lesser men an opportunity to publish “improvements” of von Neumann.
If there is a time window in which we have LLMs with superhuman math and coding capabilities, before the same happens to other domains that pose more direct risks (psychology/persuasion, biology/bioweapons, nanotech/scifi-constructs, physics/unknown-unknowns, …) and before the RSI loop takes off, in what ways could this situation be used fruitfully for increasing the chances that the alignment problem gets solved before it’s too late? Are there relevant mathematical conjectures that could be tackled? Are there relevant challenges one could throw these models against, e.g. such as ARC’s? Are there talented people in math and math-adjacent fields that could be motivated to drop whatever irrelevant stuff they are doing and instead join the effort, conditioned on them having seen the take-off in mathematical capabilities described above and e.g. the OpenAI-HuggingFace incident this year? It seems to me that it would be potentially helpful if people, who are deep into the alignment field and know the current map well, would in some collective effort compile lists of answers for these (and similar) questions, so people with access to internal models and compute might be a bit more likely to throw them at actual safety-relevant issues and perhaps nudge the pre-RSI phase towards a place where either RSI is delayed (e.g. due to clearer insights into widely-acceptable theoretical difficulties) or slightly more likely to go well (e.g. due to adjusting minor parts of the machinery that turn out to have huge downstream effects).
I asked Astra to check out OpenAI’s math repo containing the 722 preprints of results (from ~4,000 problems attempted) obtained using their internal model at “three hours of ChatGPT Pro thinking compute” each. These preprints are organised into 372 families, i.e. “related papers, which may include a principal result, companion arguments, consequences, or alternative proofs”. Astra thinks/claims
there are ~30 (range 20–50) potential Fields medal-scale portfolios’ worth of results (“I would defend “several dozen” more readily than either endpoint”), including partial results or related work on every Millennium Prize problem except Yang-Mills
Millennium Prize problem
Match in this release
Riemann hypothesis
Partial: fixed zero-free strip
Birch–Swinnerton-Dyer
Partial: full formula for the stated low-Selmer-corank class; further density-one consequences with Goldfeld
Hodge conjecture
Partial: CM abelian varieties, K3 products and related results
Navier–Stokes
Related work here; the earlier forced-breakdown release is separate
P versus NP
Related complexity breakthroughs, no resolution
Yang–Mills existence and mass gap
No matching resolution identified
The workbook constructs 28 illustrative portfolios using 234 distinct families, without assigning any family twice. This makes the comparison inspectable. It does not eliminate common proof dependencies, and some groupings are deliberately broad. A few could plausibly split into multiple portfolios; others may look less substantial once novelty and methods receive expert scrutiny.
This is not a measured median Fields Medal equivalent. There is no established scalar ordering of those 30 historical portfolios from which to calculate such a median. The estimate concerns the possible scale of mathematical outcomes. Lasting influence, reusable ideas, exposition, and the work of collaborators remain separate considerations.
The strongest concrete repo comparisons include:
Repo cluster
Historical reference
Why the comparison has substance
QRH, prime correlations, primitive roots
Maynard; parts of Tao and Venkatesh
Several central advances in the structure of primes and arithmetic distribution
BSD, Goldfeld, modularity
Bhargava and arithmetic portfolios
Structural theorems about broad classes of curves, beyond isolated records
Restricted and quantum geometric Langlands
Lafforgue, Ngô
Major steps in an organizing programme linking geometry and representation theory
Abundance, subadditivity, minimal models
Birkar
An unusually direct comparison within the same central classification programme
Kakeya, Falconer, restriction, smoothing
Wang, Tao
Closely overlapping frontier questions, with multiple substantial next steps
Critical percolation, spin models, scaling limits
Duminil-Copin, Smirnov, Werner
Several longstanding gaps between physical predictions and rigorous theory
Vlasov–Maxwell and Boltzmann-related results
Villani, Deng
Foundational questions about kinetic equations and microscopic-to-macroscopic behaviour
Mahler and optimal configurations
Viazovska, with qualifications
Exact extremal problems whose solutions can reveal new analytic structure
Hilbert–Smith and symplectic/topological results
Pardon
Direct continuation of some recognized targets, plus other major topology claims
The Fields comparison becomes more defensible when we compare entire pre-award portfolios. I examined official citations and contemporary accounts for every recipient from 1998 through 2026. There was no 2000 award, so 1998 takes the comparison past your requested cutoff. Here is the historical reference set in compressed form. The workbook gives each person an individual row with recognized contributions, methods, possible repo comparisons, limitations, and a primary source.
Award
What the portfolios contained
1998
Borcherds: vertex algebras, moonshine, automorphic products. Gowers: Banach-space structure and counterexamples, combinatorics. Kontsevich: intersection theory, knot invariants, deformation quantization. McMullen: holomorphic dynamics and hyperbolic three-manifold geometry. mathunion.org
2002
Lafforgue: function-field Langlands for
2006
Okounkov: bridges among representation theory, probability and enumerative geometry. Perelman: Ricci-flow singularities and geometrization. Tao: harmonic analysis, nonlinear PDE, additive combinatorics and primes. Werner: SLE, Brownian exponents and conformally invariant probability. mathunion.org
2010
Lindenstrauss: measure rigidity and arithmetic applications. Ngô: the fundamental lemma through Hitchin-fibration geometry. Smirnov: conformal invariance via discrete complex analysis. Villani: kinetic theory, convergence to equilibrium, Landau damping and transport. Elon Lindenstrauss
2014
Avila: several dynamical and spectral programmes. Bhargava: higher composition laws, counting fields and arithmetic statistics. Hairer: regularity structures and singular stochastic PDE. Mirzakhani: moduli-space volumes, geodesic counting and dynamical rigidity. mathunion.org
2018
Birkar: boundedness and the minimal-model programme. Figalli: optimal transport and nonlinear regularity. Scholze: perfectoid spaces and arithmetic geometry. Venkatesh: synthesis of dynamics, topology and number theory. mathunion.org
2022
Duminil-Copin: critical phenomena and phase transitions. Huh: combinatorial Hodge theory and log-concavity. Maynard: prime clusters, large gaps and Diophantine approximation. Viazovska: sphere packing, Fourier interpolation and universal optimality. mathunion.org
2026
Deng: kinetic limits and probabilistic dispersive PDE. Pardon: topology, virtual cycles and symplectic foundations. Tsimerman: arithmetic geometry, unlikely intersections and period maps. Wang: Kakeya, restriction, smoothing and geometric measure theory. mathunion.org
Several lessons emerge from this complete set. A portfolio can revolve around one central programme, as with Lafforgue or Ngô. It can contain several connected breakthroughs, as with Maynard or Avila. Or it can create a mathematical language, as with Voevodsky, Hairer or Scholze. These achievements cannot be ranked reliably by counting named conjectures. Hairer is a particularly useful corrective. The significance was not simply that he solved a difficult equation: regularity structures provided a systematic way to give meaning to, and solve, classes of equations whose nonlinear terms were previously ill-defined. Similarly, perfectoid spaces changed the available methods of arithmetic geometry. Matching such portfolios requires evidence of comparable conceptual reach. Conversely, it would be unreasonable to demand that every medal-scale portfolio invent something as conspicuous as perfectoid spaces. The historical set contains many different combinations of problem solving, technical depth, structural insight and synthesis.
>5x more high-significance progress might’ve been made than by all other AIs combined, proxying using VibeMathed: 95 preprints (42–173 if ±5 score) in 64 families with >60 significance (vs 10 in VibeMathed), 27 (13–42) preprints in 19 families with >70 (vs 3 in VibeMathed), etc
[There is] a limitation in the question “how many results above 60?”: it combines importance of the target with amount of progress on it, while the site principally attempts to score the former.
Strict significance cutoff
Preprints: central estimate
Families containing qualifying results
Preprint count after shifting estimated scores ±5
>60
95
64
42–173
>70
27
19
13–42
>80
7
5
3–11
>87
3
2
0–5
I screened all 372 family descriptions, then separately considered the scope of all 195 manuscripts in families rated above 60. The other manuscripts inherit their below-threshold family estimate. That is a reasonably systematic first pass, but it is not 722 independent, equally detailed significance reviews. This distinction changes the answer considerably. Counting every paper associated with a family above 60 would produce 195, rather than 95. Supporting constructions, restricted intermediate results, conditional reductions, and alternative proofs should not all inherit the importance of their family’s strongest theorem.
The actual VibeMathed examples are more informative than the rubric alone. Here are several comparisons that materially shaped my assessment:
Actual VibeMathed entry
Site score
Repo comparison and my judgment
Matrix-multiplication exponent improved to 2.371177
55
I retain 55 for the problem family, despite the much larger proposed advance to 2.25.
Planar unit-distance conjecture
40
I retain 40 for the repo’s upper-bound advance; the earlier disproof should not be counted again.
Jacobian conjecture, counterexample in dimension ≥3
65
An anchor for central algebraic problems. I place rational Hilbert’s tenth and all-dimensional abundance around 80.
More than two thirds of zeta zeros simple and on the critical line
68
The fixed zero-free strip in family 003 seems qualitatively more consequential; I estimate 90.
Critical bond percolation on every
78
I preserve 78 for the directly related family; Anderson localization/delocalization gets approximately 82.
Smooth unforced three-dimensional Euler blowup
80
A useful nearby anchor for the large-data Vlasov–Maxwell claim, which I put around 85.
Smooth-forcing Navier–Stokes breakdown through the Clay alternatives
87
My comparison point for QRH at 90 and CM Hodge at 88; both placements remain judgment calls.
Quasi-Riemann hypothesis and rational Hodge conjecture for all CM abelian varieties were the most significant results
The seven papers I place above [VibeMathed significance] 80 are:
Family
Main claim
Qualifying manuscripts
My estimate
003
Quasi-Riemann hypothesis, through two different fixed zero-free strips
2
90 each
032
Rational Hodge conjecture for all CM abelian varieties
1
88
014
Restricted geometric Langlands in positive characteristic, with stated hypotheses
1
85
362
Large-data global classical relativistic Vlasov–Maxwell
1
85
261
Two-dimensional Anderson localization and higher-dimensional weak-disorder delocalization
2
82 each
The quasi-Riemann hypothesis is a qualitative barrier crossing, despite the unimpressive-looking fraction
Classical zero-free regions become thinner as the imaginary part grows. A fixed strip excludes zeros arbitrarily high up, with a uniform distance from the line
That is why this seems more consequential than another improvement in the percentage of zeros known to lie on the critical line. The two kinds of progress control different things. The linked formalization scope includes zeta and all Dirichlet
The CM Hodge paper may have the largest cascade of consequences. The main paper in family 032 claims rational Hodge for every complex CM abelian variety, in every dimension and codimension. Very roughly, Hodge asks when certain cohomological classes, which look like the shadows of algebraic subvarieties, actually come from algebraic subvarieties. CM abelian varieties are a special class, but they occupy an exceptionally influential position in arithmetic geometry. The claimed consequences through existing theorems include Tate for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic. Companion papers address products of K3 surfaces and Kuga–Satake correspondences. That breadth explains my high significance estimate. Calling it simply “a special case of Hodge” substantially understates its possible reach. Calling it “the Hodge conjecture solved” substantially overstates it.
there are (claimed) full resolutions to 37 entries in Wikipedia’s longlist of 504 unsolved open problems, partial or special-case results to 44 entries, and related work on 32 entries
For the 504 main-list entries, my classification is:
Relationship to this repo
Entries
Claimed full answer to the listed assertion
37
Claimed partial or special-case progress
44
Related work, without progress established on the exact listed assertion
32
No matching claim identified
391
Among the 37 claimed full answers are Brennan, Flint Hills, Mahler, Hadwiger, Sidorenko, the Erdős reciprocal-sum conjecture, circulant Hadamard, Barker sequences, Nagata, Lipman–Zariski, Serre positivity, surjunctivity, Hilbert–Smith, and the nearby Lagrangian conjecture. The complete table gives the precise assertion, matching families and qualification for every row.
Two subtleties materially improve the interpretation. First, “partial” can be enormous. The rational case of Hilbert’s tenth problem is partial relative to Wikipedia’s question over number fields. The
The Wikipedia matching is complete at the level of the page’s enumerated main-list questions. I preserved 560 list rows: 504 problem entries, 12 grouping headings, and 44 entries in the “problems solved since 2015” appendix. These are row counts, not 504 distinct conjectures. The page contains duplicates, bundled questions and some stale entries. “No matching claim” does not imply that a problem remains open elsewhere.
a family’s Lean link does not necessarily cover its strongest headline
The most important verification finding is that a family’s Lean link does not necessarily cover its strongest headline.
Family
Headline one might infer is covered
What the linked scope actually covers
157
Hadwiger counterexample
A positive list-colouring bound, not that counterexample
266
At most three mutually unbiased bases in dimension six
A weaker bound of five and an auxiliary statement; the stronger result uses a separate computational route
260
Full spacetime Penrose inequalities
Selected auxiliary results, not the entire headline theorem
The first two distinctions are explicit in the repo’s own scope documents. GitHub
Toby Ord previously guessed that
He’s since retracted this section following feedback, but the “4 previous jumps in one” guess stuck with me as a (possibly wrong & misguided) way to make sense of how powerful the unreleased internal model is, and the notes above are my attempt to appreciate how this cashes out.
Back in May I asked:
I clearly failed to price in accelerating progress; the Bourgain toolkit-interpolator is here, and its name is (probably) Bel.
Previous notes on the industrialisation of pure math series for my convenience:
AI contributions to Erdos problems on Terry Tao’s GitHub
GDM’s Gemini 3.1 Pro-based agent autonomously resolving 9 Erdos problems at a few hundred dollars per problem, out of the full set of 353 in the open-source Formal Conjectures repo
OpenAI Astra’s progress on 10 open problems in math/TCS at $200 per problem
VibeMathed being the best window I know of into the industrialisation of pure math problem-solving
This news is putting me into quite the mood. I’m rereading Ted Chiang’s 2000 story Catching crumbs from the table: In the face of metahuman science, humans have become metascientists (thanks Gwern):
I first read this story in my college years over a decade ago and wondered when this would come to pass. Despite having been in this corner of the internet since those days, I still wouldn’t have predicted “mid-2020s for pure math”.
As a side note, Chiang’s story goes:
That reminds me of the closing sentence of this anecdote from P. R. Halmos’ The Legend of John von Neumann:
If there is a time window in which we have LLMs with superhuman math and coding capabilities, before the same happens to other domains that pose more direct risks (psychology/persuasion, biology/bioweapons, nanotech/scifi-constructs, physics/unknown-unknowns, …) and before the RSI loop takes off, in what ways could this situation be used fruitfully for increasing the chances that the alignment problem gets solved before it’s too late? Are there relevant mathematical conjectures that could be tackled? Are there relevant challenges one could throw these models against, e.g. such as ARC’s? Are there talented people in math and math-adjacent fields that could be motivated to drop whatever irrelevant stuff they are doing and instead join the effort, conditioned on them having seen the take-off in mathematical capabilities described above and e.g. the OpenAI-HuggingFace incident this year? It seems to me that it would be potentially helpful if people, who are deep into the alignment field and know the current map well, would in some collective effort compile lists of answers for these (and similar) questions, so people with access to internal models and compute might be a bit more likely to throw them at actual safety-relevant issues and perhaps nudge the pre-RSI phase towards a place where either RSI is delayed (e.g. due to clearer insights into widely-acceptable theoretical difficulties) or slightly more likely to go well (e.g. due to adjusting minor parts of the machinery that turn out to have huge downstream effects).