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arxiv: 2605.27770 · v3 · pith:FQJ6PDVInew · submitted 2026-05-26 · ✦ hep-th · cs.LG

Sampling Triangulations and Calabi-Yau Threefolds with Autoregressive GNNs

Nate MacFadden This is my paper

Pith reviewed 2026-06-29 15:12 UTC · model grok-4.3

classification ✦ hep-th cs.LG
keywords autoregressive GNNtriangulationslattice polytopesCalabi-Yau threefoldsoriented matroidssigned circuitsuniform samplingdual graph
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The pith

A graph neural network using signed circuits from oriented matroids samples uniform fine regular triangulations of lattice polytopes.

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

The paper presents dualGNN, an autoregressive GNN that generates fine regular triangulations of lattice polytopes by operating on their dual graphs labeled with signed circuits. These circuits, combined with magnitude information, allow the model to enforce regularity directly from the graph structure while remaining independent of the number of points and invariant under lattice symmetries. On polygons with up to 40 points not seen in training, dualGNN is the only tested sampler that passes all uniformity diagnostics including KL divergence, collision counts, and autocorrelation. The approach is applied to uniformly sampling Calabi-Yau threefolds at Hodge numbers h^{1,1}=86 and 128.

Core claim

dualGNN is an autoregressive message-passing graph neural network that samples fine, regular triangulations of lattice polytopes. It operates on a generalization of the dual graph where edges are labeled by signed circuits from oriented matroids. These signed circuits are necessary and sufficient, with retained magnitude information, to determine regularity from the dual graph. The model guarantees fine triangulations in 2D via masking, is independent of point count, and invariant under orientation-preserving symmetries. On unseen polygons with at most 40 points, it is the only sampler consistent with uniform sampling across all diagnostics.

What carries the argument

dualGNN, an autoregressive message-passing GNN operating on dual graphs labeled by signed circuits from oriented matroids, which encodes the combinatorial data needed to enforce regularity.

If this is right

  • Every generated sample is guaranteed to be a fine triangulation in two dimensions.
  • The model size remains fixed at about 92,000 parameters regardless of the polytope's point count.
  • Training completes in roughly 7.5 hours on a single consumer GPU.
  • Uniform samples of Calabi-Yau threefolds are produced at h^{1,1}=86, with no observed deviations at h^{1,1}=128.
  • The approach integrates into CYTools for string theory applications.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Extending the method to three-dimensional polytopes could enable sampling of higher-dimensional Calabi-Yau manifolds.
  • The invariance under lattice symmetries suggests applicability to other symmetry-constrained combinatorial generation tasks.
  • Using the same signed circuit representation might allow verification of regularity without full geometric embedding.
  • Small model size opens the possibility of ensemble sampling or integration with other machine learning pipelines for vacuum counting.

Load-bearing premise

Signed circuits from oriented matroids together with retained magnitude information are necessary and sufficient to determine a triangulation's regularity directly from the dual graph.

What would settle it

Finding a triangulation sample set from dualGNN on polygons with 40 or fewer points that shows statistically significant deviation from uniformity in KL divergence, collision rate, or autocorrelation compared to other samplers.

Figures

Figures reproduced from arXiv: 2605.27770 by Nate MacFadden.

Figure 1
Figure 1. Figure 1: Diagram of the ‘lifting’ procedure defining regular triangulations. The points [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Two fine regular triangulations of [0, 2]×[0, 4] being ‘patched’ to a single triangu￾lation of [0, 4]2 . The two triangulations on the left are both regular while the triangulation on the right is irregular. This example was originally found by Francisco Santos and ap￾pears in ref. [7]. Figure modified from [1]. polytope (translation and unimodular maps). dualGNN is therefore invariant under the orientatio… view at source ↗
Figure 3
Figure 3. Figure 3: Left: two simplices share a facet but in an invalid way. The simplices have a solid [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Pairs of adjacent simplices ABD and BCD. Each pair corresponds to an edge in a dual graph, but these pairs correspond to circuits playing different roles in the oriented matroid so they must be distinguished. 2 dualGNN We first introduce a simplified variant of dualGNN; the more complete model will be dis￾cussed in section 2.2. Consider encoding a triangulation T of ∆ by its dual graph GT . That is, draw a… view at source ↗
Figure 5
Figure 5. Figure 5: A fine triangulation T of the polytope ∆ = conv({(0, 0),(0, 4),(6, 0)}) (in black) as well as its dual graph GT (in blue). This polygon has 408, 826 triangulations of which all but 3, 120 are regular. having node nb aggregate all incoming messages with sum, min, and max (aggregate length 3(D + 4)), (d) concatenating the receiving node’s own feature vector fb (total length 3(D + 4) + D), and then (e) runnin… view at source ↗
Figure 6
Figure 6. Figure 6: Performance of the regularity classifier trained and validated on the polygon in [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Performance of the autoregressive dualGNN on the polygon in fig. 5. Left: [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: For the data in fig. 7, the autocorrelation between samples. Distance be [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The flip graph of conv({(0, 0),(2, 3),(0, 9)}), with nodes representing FRTs and edges representing flips. This graph is bipartite so a Markov-chain sampler such as flip walk with an even number of steps between samples would only sample from one of two colors (red or blue). This example was found happenstance on the first polygon tested. Sample rate splits the methods into four tiers (fig. 7): pushing is … view at source ↗
Figure 10
Figure 10. Figure 10: The dualGNN sampler, trained on the polygon in fig. 5, applied to [0 [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The 20 out-of-distribution polygons used in fig. 12, with 11 [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Pre- and post-REINFORCE finetuning on the dualGNN autoregressive sampler [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of the uniformity of samples for 200 [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Autocorrelation of the various samplers over 200 [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Four out-of-distributions polygons at Npts = 40 used for post-REINFORCE inference. significantly biased samplers can show low collision counts, meaning that they would have KL divergences nearing the noise floor (due to our finite number of samples). We address this, in part, by again using the collision count itself as a diagnostic since it was better suited to sparse samples in section 2.2.2. Across the… view at source ↗
Figure 16
Figure 16. Figure 16: Autocorrelation of the various samplers over 100 [PITH_FULL_IMAGE:figures/full_fig_p028_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: fig. 17. In fact, across all Hodge numbers, dualGNN is consistent with a uniform sampler [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
Figure 17
Figure 17. Figure 17: Recreation of [3]’s figure 4, extended to larger Hodge numbers [PITH_FULL_IMAGE:figures/full_fig_p030_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The two distinct 2-faces of the h 1,1 = 86 polytope, labeled with their multiplicity in the polytope. The triangle (left) has 7, 422 FRTs; the rectangle (right) has 12, 170. We choose ∆86 because each of its 2-faces contains exactly Npts = 15, so we can get very strong confidence about dualGNN’s uniformity. While this is a non-negligible h 1,1 , significantly higher than what previous works could sample (… view at source ↗
Figure 19
Figure 19. Figure 19: Example triangulations of the four unique 2-faces of ∆ [PITH_FULL_IMAGE:figures/full_fig_p032_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Histograms of the number of flops between different samples of CYs (mea [PITH_FULL_IMAGE:figures/full_fig_p034_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The polygon conv({(0, 0),(84, 0),(0, 7)}), the largest polygon occurring in our string theory applications (as a 2-face of the sole h 1,1 = 491 4D reflexive polytope). This polygon has between 3.90 × 10167 and 1.96 × 10180 FRTs. It should be noted that random triangulations fast is significantly faster than the dualGNN construction (> 7 CYs/sec vs ∼ 0.1 CYs/sec) but (a) typically the CY gener￾ation is not… view at source ↗
Figure 22
Figure 22. Figure 22: The inference rollout of dualGNN applied to the polygon in fig. 5. Nodes are [PITH_FULL_IMAGE:figures/full_fig_p043_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: The inference rollout of dualGNN applied to the polygon in fig. 5. Nodes are [PITH_FULL_IMAGE:figures/full_fig_p044_23.png] view at source ↗
Figure 26
Figure 26. Figure 26: fig. 26 [PITH_FULL_IMAGE:figures/full_fig_p048_26.png] view at source ↗
Figure 24
Figure 24. Figure 24: KL divergence to the uniform distribution over the course of training for the [PITH_FULL_IMAGE:figures/full_fig_p049_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Same as fig. 24, but for the RoPE-encoded transformer. [PITH_FULL_IMAGE:figures/full_fig_p049_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Rank-frequency plot of 106 samples drawn from the trained RoPE transformer on the polygon in fig. 5. 10 0 10 1 10 2 10 3 10 4 10 5 samples drawn 10 0 10 1 10 2 10 3 10 4 10 5 cumulative unique FTs RoPE-transformer (|initial data| = 679) regular (685) irregular (191733) 10 0 10 1 10 2 10 3 10 4 10 5 samples drawn grow2d regular (194079) irregular (679) [PITH_FULL_IMAGE:figures/full_fig_p050_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Number of regular and irregular triangulations generated for the polygon [0 [PITH_FULL_IMAGE:figures/full_fig_p050_27.png] view at source ↗
read the original abstract

We introduce `dualGNN', an autoregressive message-passing GNN for sampling fine, regular triangulations of lattice polytopes. dualGNN operates on a generalization of the dual graph of a triangulation, with edges labeled by `signed circuits' -- combinatorial invariants from the theory of oriented matroids. We show that these circuits are necessary and sufficient to determine a triangulation's regularity from the graph, provided certain magnitude information is retained. The model is independent of the polytope's point count and invariant under its orientation-preserving symmetries ($\mathrm{SL}(d,\mathbb{Z}) \ltimes \mathbb{Z}^d$), and our masking procedure further guarantees that every rollout produces a fine triangulation (in 2D). On unseen polygons with $N_\mathrm{pts} \leq 40$, dualGNN is the only sampler we tested that is consistent with uniform sampling across all our diagnostics (KL divergence from uniformity, collision counts, and sample autocorrelation). The model is small ($\sim92$k parameters) and trains in $\sim7.5$ hours on a single consumer GPU. We apply dualGNN to string theory, sampling Calabi-Yau threefolds uniformly at $h^{1,1}=86$; we also sample CYs at $h^{1,1}=128$, observing no deviations from uniformity, but our diagnostics are weaker here. Code, training scripts, and pretrained models are available at https://github.com/natemacfadden/dualGNN (pip install dualgnn), and dualGNN is integrated into CYTools.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper introduces dualGNN, an autoregressive message-passing GNN operating on a generalized dual graph with edges labeled by signed circuits from oriented matroids. It claims these circuits (with retained magnitude information) are necessary and sufficient to determine regularity from the graph, that the model is independent of point count and invariant under SL(d,Z) ⋉ Z^d symmetries, that a masking procedure guarantees fine triangulations in 2D, and that on unseen polygons with N_pts ≤ 40 dualGNN is the only tested sampler consistent with uniform sampling according to KL divergence from uniformity, collision counts, and sample autocorrelation. The model (~92k parameters) is applied to uniform sampling of Calabi-Yau threefolds at h^{1,1}=86 (and 128 with weaker diagnostics), with code and pretrained models released.

Significance. If the uniformity claim holds, the work provides a scalable, symmetry-invariant method for sampling combinatorially large spaces of fine regular triangulations, directly relevant to enumerative problems in algebraic geometry and string theory. Credit is due for the open release of code, training scripts, and pretrained models at the cited GitHub repository, the small model size, and the ~7.5-hour training time on consumer hardware; these lower the barrier for reproducibility and extension.

major comments (1)
  1. [Abstract] Abstract (diagnostics paragraph): The central claim that dualGNN is the only sampler 'consistent with uniform sampling across all our diagnostics' rests on KL divergence, collision counts, and autocorrelation. These three statistics can be satisfied by biased samplers that deviate on higher-order marginals (e.g., signed-circuit multiplicity distributions, f-vector statistics, or Ehrhart coefficients of the triangulated polytope). For N_pts=40 the space is already super-exponential, so finite samples passing the reported tests do not yet establish uniformity; additional invariants or larger-scale tests are needed to support the claim.
minor comments (2)
  1. [Abstract] The abstract states that signed circuits 'are necessary and sufficient to determine a triangulation's regularity directly from the dual graph, provided certain magnitude information is retained,' but the precise form of the retained magnitude data and the proof of sufficiency are not summarized; a brief statement or reference to the relevant theorem would improve clarity.
  2. The application to h^{1,1}=128 reports 'no deviations from uniformity, but our diagnostics are weaker here'; specifying which diagnostics were weakened and why would help readers assess the strength of that result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comment. We address it directly below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (diagnostics paragraph): The central claim that dualGNN is the only sampler 'consistent with uniform sampling across all our diagnostics' rests on KL divergence, collision counts, and autocorrelation. These three statistics can be satisfied by biased samplers that deviate on higher-order marginals (e.g., signed-circuit multiplicity distributions, f-vector statistics, or Ehrhart coefficients of the triangulated polytope). For N_pts=40 the space is already super-exponential, so finite samples passing the reported tests do not yet establish uniformity; additional invariants or larger-scale tests are needed to support the claim.

    Authors: We agree that the three reported diagnostics are necessary but not sufficient to establish uniformity, and that biased samplers could pass them while differing on higher-order marginals. The manuscript's phrasing is already limited to 'consistent with uniform sampling across all our diagnostics' rather than claiming a proof of uniformity. Nevertheless, the referee's observation is correct and we will revise the abstract to make the limitation explicit (e.g., 'the only tested sampler consistent with uniformity under the three diagnostics we employed'). We will also insert a short paragraph in Section 4.2 acknowledging that these tests do not rule out deviations on f-vectors, circuit multiplicities, or Ehrhart coefficients, and noting that exhaustive verification is infeasible for N_pts=40. No new experiments are added at this stage. revision: yes

Circularity Check

0 steps flagged

No circularity: model trained on data and evaluated out-of-sample on independent diagnostics

full rationale

The paper trains dualGNN on triangulations of lattice polytopes and evaluates sampling behavior on unseen polygons (N_pts ≤ 40) using KL divergence, collision counts, and autocorrelation. These are standard out-of-sample checks on a learned model; the uniformity claim does not reduce to any fitted input by construction. The statement that signed circuits plus magnitude information determine regularity is presented as a result shown in the paper rather than imported via self-citation or defined circularly. No ansatz smuggling, renaming of known results, or load-bearing self-citations appear in the provided text. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central modeling choice rests on the claim that signed circuits plus magnitude suffice for regularity detection; this is treated as a domain assumption rather than derived.

axioms (1)
  • domain assumption Signed circuits with magnitude information are necessary and sufficient to determine regularity from the dual graph
    Stated directly in the abstract as the basis for the graph representation.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Calabi-Yau Orientifold Hypersurfaces and their F-theory Uplifts

    hep-th 2026-06 unverdicted novelty 6.0

    An algorithm builds Calabi-Yau orientifolds and F-theory fourfold uplifts from 6d reflexive polytopes derived from orientifold data, with code in CYTools and GitHub.

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