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Advancing Geometry with AI: Multi-agent Generation of Polytopes

T0 review · 3 major / 3 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A multi-agent AI system generated millions of new counterexamples to the Hirsch conjecture, including a 24-vertex prismatoid that yields the smallest known non-Hirsch polytope, in dimension 19.

desk verdict The 24-vertex prismatoid is either a major breakthrough or a direct contradiction of the Matschke–Santos–Weibel lower bound, and the paper does not give us enough to tell which. read the letter →

arxiv 2502.05199 v1 pith:SWLI7VOY submitted 2025-01-30 math.CO cs.CGmath.MG

Grzegorz Swirszcz , Adam Zsolt Wagner , Geordie Williamson , Sam Blackwell , Bogdan Georgiev , Alex Davies , Ali Eslami , Sebastien Racaniere
show 2 more authors
Theophane Weber Pushmeet Kohli
This is my paper · ORCID
classification math.COcs.CGmath.MG MSC 52B0552B11
keywords polytopesHirschconjectureprismatoidswidthmulti-agentsearchreinforcementlearningneighbourlymonotonepaths
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that a multi-agent, neural-network-guided search algorithm called Hopper can generate extremal polytopes that have resisted human construction. Its headline evidence is a new counterexample to the Hirsch conjecture: a 5-dimensional prismatoid with 24 vertices and width 6, which the authors say maps to a non-Hirsch polytope in dimension 19, the smallest known so far. The system also produced millions of width-6 prismatoids, improved known lower bounds for longest monotone paths, and found new neighbourly but non-cyclic polytopes. The authors present these constructions as evidence that automated search can create the hard examples that drive progress in polytope theory.

What carries the argument

The load-bearing machinery is the Hopper algorithm combined with the prismatoid reduction used by previous counterexample constructions. Hopper treats polytope generation as a one-player game: an agent selects a polytope from a shared repository, computes all hyperplanes spanned by d of its vertices, scores them with a transformer-based neural network trained online on which hops improve a fitness function, samples a cell bounded by high-scoring hyperplanes, and 'hops' a vertex to the center of the largest inscribed ball in that cell, keeping only candidates with better fitness. The reduction converts a 5-dimensional prismatoid with n vertices and base-facet distance (width) w into a polytope of dimension n−5 whose diameter is at least w + n − 10; hence a width-6 prismatoid gives a non-Hirsch polytope, and n=24 yields dimension 19.

What would settle it

Recompute the facet-ridge graph of the 24-vertex prismatoid in Section A.13 in independent exact-rational arithmetic and measure the distance between its two base facets. Any value other than 6 would falsify the paper's record claim; a value of 6 would corroborate it.

Watch

Extended reading notes

Core claim

The central claim is that Hopper—a population of agents that modify polytopes by 'hopping' vertices, with a shared transformer-based policy trained online on which hops succeed—discovered genuinely new extremal polytopes for three classical problems. For the Hirsch conjecture, the paper reports finding a 24-vertex prismatoid of width 6 with 12 points in each base facet; applying the prismatoid-to-polytope reduction gives a non-Hirsch polytope in dimension 19, improving on the previous best of 25 vertices (dimension 20). The authors state that their constructions are qualitatively different from earlier human ones: a sample of 100 Hopper-built non-Hirsch polytopes is not explained by the directed-2-cycle criterion that covers all previously known examples, and the coordinate scales differ by many orders of magnitude. For the monotone-path problem the system matched or improved the best known lower bounds for several parameter pairs, and for the neighbourly problem it found new non-cyclic neighbourly polytopes. The paper presents these results as evidence that AI can address geometry problems long considered extremely hard.

Load-bearing premise

The load-bearing premise is that the 24-vertex prismatoid listed in Section A.13 really has width 6; the paper verifies this in exact arithmetic but provides no code, certificate, or independent check, so the record claim collapses if that computation is wrong.

Editorial extensions

If this is right

  • If correct, the new record is a 24-vertex, width-6 prismatoid giving a non-Hirsch polytope in dimension 19, one dimension below the previous state of the art.
  • The reported production of millions of width-6 prismatoids would imply that non-Hirsch polytopes are abundant, not rare, and that most of them fall outside the directed-2-cycle explanation that covers earlier human examples.
  • The monotone-path results would update the state of the art to f(9,5)=30 and f(9,6)=29, and narrow the intervals for (10,5) and (11,5).
  • The authors' switch from rigid to flexible mode—allowing vertex addition and deletion—cut the time to the first counterexample from weeks to hours, making flexible mutation a practical accelerator rather than a cosmetic option.
  • Newly found non-cyclic neighbourly polytopes for pairs like (10,6) to (13,8) would enlarge the known range in which the cyclic-polytope conjecture fails.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A consequence the authors leave implicit: if the 24-vertex prismatoid is independently confirmed, nothing in the method suggests a barrier to a 23-vertex example, so a focused search could lower the dimension record by one more step.
  • Because the paper's defect-based fitness functions are what made prismatoid widths optimizable, a testable extension is to apply the same continuity trick to other diameter-type conjectures, such as the polynomial Hirsch conjecture.
  • The reported sample of Hopper-built prismatoids that escape the directed-2-cycle explanation points to a concrete next experiment: search those examples for a new sufficient condition for non-Hirsch prismatoids, a step the paper does not take.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper introduces Hopper, a multi-agent reinforcement-learning system that iteratively modifies the vertices of a polytope to optimize fitness functions encoding desired extremal properties. It reports applications to three problems: the Hirsch conjecture, longest monotone paths, and k-neighbourly polytopes. For the Hirsch conjecture the paper claims to have generated over a million width-6 five-dimensional prismatoids and, in particular, a 24-vertex width-6 prismatoid whose coordinates are listed in A.13, which would yield a 19-dimensional non-Hirsch polytope and beat the previous 'state of the art' of 25 vertices from [20]. For monotone paths, Table 1 reports new lower bounds, and for neighbourly polytopes it reports many new non-cyclic examples. The central evidence for the main record claim is the A.13 coordinate list, with final evaluation said to be done in exact arithmetic (A.6).

Significance. If the 24-vertex width-6 prismatoid existed and were verified, it would constitute a new record: a 19-dimensional non-Hirsch polytope, improving on the dimension-20 examples derived from the 25-vertex prismatoids of [20], and it would demonstrate that an AI-guided search can produce novel extremal polytopes. The paper also provides a reasonably detailed architectural description of the search system, states that final evaluations use exact rational arithmetic, and gives an explicit (though rounded) coordinate list for its main example. These are strengths that make the central claim in principle checkable. However, the main claim appears to conflict with a known theorem in the cited reference [20], and none of the reported computational discoveries is accompanied by verifiable exact data or code. As it stands, the record claims are not supported.

major comments (3)
  1. [§4.1, A.13, Reference [20]] The central claim of a 24-vertex width-6 five-dimensional prismatoid appears to contradict the main theorem of the cited paper [20], Matschke–Santos–Weibel, 'The width of five-dimensional prismatoids', which proves that the minimum number of vertices of a 5-prismatoid of width greater than 5 is 25. The manuscript cites [20] only as the source of the previous 25-vertex 'state of the art' and does not acknowledge that [20] establishes a lower bound. If the theorem in [20] is as stated, then no 24-vertex example with width 6 can exist, and the claimed 19-dimensional non-Hirsch polytope does not follow. The authors must state the exact theorem from [20] and explain how their 24-vertex example is compatible with it, or otherwise the main record claim collapses.
  2. [A.13, A.6] The coordinates in A.13 are printed to three decimal places, and the width of a prismatoid is a discrete invariant that can change under arbitrarily small perturbations. The statement in A.6 that final evaluation of candidates uses exact rational arithmetic cannot compensate for the absence of the exact coordinates from the paper. Without exact rational coordinates, a machine-checkable certificate, or a verification script, the width-6 computation for the A.13 example cannot be independently reproduced, and the rounded listing is not acceptable evidence for the record claim.
  3. [§4.2, §4.3] The monotone-path lower bounds in Table 1 (e.g., f(9,5)=30, 41≤f(10,5)≤42, f(9,6)=29) and the claimed new non-cyclic neighbourly polytopes in §4.3 are asserted with no polytope data, coordinates, or verification procedures. Since these results are part of the paper's abstract-level claim of success on three problems, at least one explicit, exact construction for each reported bound is needed for the claims to be checkable. As written, these sections are unverifiable and therefore do not support the stated conclusions.
minor comments (3)
  1. [§4.1.1, Figure 3] The name 'Maschke' is spelled inconsistently with the reference list, which uses 'Matschke'; this should be corrected throughout.
  2. [A.11.1] There is a typo: 'embelished' should be 'embellished'.
  3. [A.11.6] The heading 'T otal loss and the training' contains a spacing typo; it should read 'Total loss and the training'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the polytope examples are independently verified constructions, and no load-bearing claim reduces to the model's own fitted inputs or to self-citation.

full rationale

The central mathematical claims are existence claims about concrete polytopes, and none of them is defined in terms of the model's parameters. The Hirsch claim rests on the width-6 computation for the 24-vertex prismatoid listed in A.13; width is the facet-ridge distance between the base facets, a combinatorial quantity independent of how the object was generated. A.6 states that final evaluations use exact arithmetic ('Final evaluation of candidates always uses the precise mode'), so the reported property is checked after the search rather than being an output of the neural network. The fitness functions in A.7 and A.8 do directly encode width-related quantities, since the defect counts shortest paths and the objective is to minimize or otherwise improve those counts; this explains why the search succeeded, but it is goal-directed search, not a fitted parameter later renamed as a prediction. The 'new bounds' in §4.2 and §4.3 are measured values of the found objects, not out-of-sample statistical inferences. No load-bearing result depends on a self-citation: refs [7] and [30] by overlapping authors are cited only as examples of prior RL successes, and the prismatoid-to-Hirsch-counterexample reduction is credited to Santos [23] and Matschke–Santos–Weibel [20], both external. No uniqueness theorem from the authors' own prior work is invoked, no ansatz is smuggled in via citation, and no known result is merely renamed. The genuine weaknesses are reproducibility and correctness concerns, not circularity: A.13 prints rounded decimal coordinates rather than exact rationals, and the claim to 'bea[t] the state of the art of 25 vertices [20]' should be checked against the actual theorem content of [20], which may prove that 25 vertices is minimal for a width-6 five-prismatoid; if so, the 24-vertex example would indicate a computational error. Those issues affect verification, not whether the derivation is equivalent to its own inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

I count two explicit numeric heuristics as free parameters; many other architectural and search hyperparameters are not disclosed. The central mathematical facts rely on standard theorems plus the assumption that the unreleased exact-arithmetic computations are correct.

free parameters (2)
  • hop_margin = 0.8
    Algorithm 1, line 9: hyperplanes with distance less than 0.8 times the inscribed-ball radius are added to the hop region; this hand-chosen constant shapes all moves.
  • defect_cap = 127
    A.11.5: the auxiliary loss folds all defect values greater than or equal to 127 into one bucket, which is an arbitrary cutoff in the learned predictor.
assumptions (4)
  • standard math Santos's prismatoid reduction theorem
    Section 4.1: converts a d-dimensional prismatoid with n vertices and width w into a non-Hirsch polytope; the 19-dimensional record depends on this published theorem.
  • domain assumption Exact arithmetic verification via cddlib is bug-free
    A.6 and A.13: the width-6 claim rests on computation in 'precise mode'; no independent certificate is provided.
  • ad hoc to paper Neural network sampling produces feasible bounded regions often enough
    Algorithm 1 lines 5 to 11: if the sampled hyperplanes often fail geometric constraints, the search stalls; the paper gives no success-rate data.
  • domain assumption The MSW digraph criterion is the only structural explanation of human examples
    Section 4.1.1: the claim that AI examples are 'fundamentally different' assumes that failing the Proposition 2.3 digraph criterion indicates genuine novelty, checked on only 100 examples.

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Cite this review

Pith. "Pith review of Advancing Geometry with AI: Multi-agent Generation of Polytopes." pith.science (2026). https://pith.science/paper/SWLI7VOY

@misc{pith2026250205199,
  author       = {Pith},
  title        = {Pith review of: Advancing Geometry with AI: Multi-agent Generation of Polytopes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SWLI7VOY}},
  note         = {Machine review of arXiv:2502.05199}
}
read the original abstract

Polytopes are one of the most primitive concepts underlying geometry. Discovery and study of polytopes with complex structures provides a means of advancing scientific knowledge. Construction of polytopes with specific extremal structure is very difficult and time-consuming. Having an automated tool for the generation of such extremal examples is therefore of great value. We present an Artificial Intelligence system capable of generating novel polytopes with very high complexity, whose abilities we demonstrate in three different and challenging scenarios: the Hirsch Conjecture, the k-neighbourly problem and the longest monotone paths problem. For each of these three problems the system was able to generate novel examples, which match or surpass the best previously known bounds. Our main focus was the Hirsch Conjecture, which had remained an open problem for over 50 years. The highly parallel A.I. system presented in this paper was able to generate millions of examples, with many of them surpassing best known previous results and possessing properties not present in the earlier human-constructed examples. For comparison, it took leading human experts over 50 years to handcraft the first example of a polytope exceeding the bound conjectured by Hirsch, and in the decade since humans were able to construct only a scarce few families of such counterexample polytopes. With the adoption of computer-aided methods, the creation of new examples of mathematical objects stops being a domain reserved only for human expertise. Advances in A.I. provide mathematicians with yet another powerful tool in advancing mathematical knowledge. The results presented demonstrate that A.I. is capable of addressing problems in geometry recognized as extremely hard, and also to produce extremal examples different in nature from the ones constructed by humans.

Figures

Figures reproduced from arXiv: 2502.05199 by the authors.

Figure 1
Figure 1. An outline of the Hopper algorithm. (a) A polytope P and all the hyperplanes de￾termined by its vertices. (b) Of these regions, adding a point is only possible in the light orange shaded regions. In each of these regions, adding a point yields a combinatorially non-equivalent polytope. (c) After computation of all hyperplanes (red), the neural network determines a probabil￾ity distribution on these hyperplanes. (d) … view at source ↗
Figure 2
Figure 2. Multi-agent architecture. The agents (A) read and write polytopes from the shared polytope repository. Simultaneously pairs of polytopes and hyperplanes are written to the shared data repository. The pairs are labeled positively if the hyperplane was part of a boundary of a successful hop region and negatively otherwise. a floating point number representing a likelihood of hyperplane h being part of a boundary of a … view at source ↗
Figure 3
Figure 3. (a) The different scales of the constructions given by the Hopper algorithm, versus human examples. We normalize the bottom base facet and record the PCA eigenvalues of the top base facet, then plot the largest eigenvalue versus the smallest eigenvalue at log scale. The green, red, and purple dots correspond to Santos’s original construction, the 25 vertex constructions in20 , and the first member of the family of c… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Multi-objective training. Two fitness functions share the same global minimum corre￾sponding to a desired state. Alternating between the objectives can lead to bypassing a barrier of local minima. A.8 Heuristics In the Hirsch conjecture, the primary metric we want to o…
Figure 5
Figure 5. Figure 5: (a) The architecture of the Hopper brain. The input is a pair – a polytope and a hyperplane. The outputs are: the main prediction, of whether the input hyperplane was in a boundary of some successful hop region, and the auxiliary prediction – the defect (see A.8) of th…

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