More counterexamples to Lagrangian Poincar\'e recurrence in dimension four

Joel Schmitz

REVIEW 2 major objections 5 minor 3 references

More counterexamples to Lagrangian Poincar\'e recurrence in dimension four

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper constructs, in every non-monotone toric symplectic four-manifold of dimension four, a Hamiltonian diffeomorphism and a Lagrangian torus whose forward orbit never intersects the torus.

desk verdict Extends Lagrangian recurrence counterexamples to all non-monotone toric four-manifolds, but the proof depends on an unproved width condition and on the author's unpublished nodal-tangle preprint. read the letter →

arxiv 2507.13924 v2 pith:KCRCEKKS submitted 2025-07-18 math.SG math.DS

Joel Schmitz This is my paper · ORCID

classification math.SGmath.DS

keywords LagrangianPoincarérecurrencetoricsymplecticfour-manifoldHamiltoniandiffeomorphismsubmanifoldnodalintegralaffinebasetanglealmostfibrationmappingclassgroup

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The pith

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

The reading

This note aims to prove that Lagrangian Poincar\'e recurrence fails for every non-monotone toric symplectic four-manifold: for each such manifold there is a Lagrangian torus $L$ and a Hamiltonian diffeomorphism $\psi$ with $\psi^n(L) \cap L = \emptyset$ for every $n \in \mathbb{N}$. The construction is the previous one from the author's earlier paper, now extended to close the exceptional gaps, which were the non-monotone products $S^2 \times S^2$ with unequal factor areas and the one- or two-fold blow-ups of such products at critical size. The proof encodes the manifold as an almost toric fibration over a nodal integral affine base, performs nodal slides to produce an integral affine isomorphism of the base, and lifts that isomorphism to a symplectomorphism of the total space. Because the induced map translates Lagrangian fibers along closed level sets by an amount whose ratio to the level-set length can be made irrational, the orbit of the fiber never returns. If the construction is correct, the known counterexamples to Lagrangian recurrence now cover all non-monotone toric four-manifolds.

What carries the argument

The central object is the nodal integral affine base $(B_0, \mathfrak{N}_0)$ of an almost toric fibration: the usual Delzant polygon data plus a choice of hat and parked nodes, with a height function $\mathcal{F}$ measuring integral affine distance to the boundary. The load-bearing identity is the level-set length formula $g(h) = 2w - k(M-h) + \sum_i \min\{h - \alpha_i, 0\}$, where $w$ is the width of the hat, $M$ is the maximum height, $k$ is a constant determined by the hat class, and $\alpha_i$ are the heights of the parked nodes. A nodal tangle that slides nodes and modifies the hat produces a second base $(B_1, \mathfrak{N}_1)$ that is integral affine isomorphic to the first through an explicit shear-and-reglue map, and the supporting theorem lifts this isomorphism to a genuine symplectomorphism of the total space. The induced map translates fibers along the closed level sets, and the irrationality of $g(h)/(2(M-h))$ is what forces the Lagrangian orbit never to close up.

What would settle it

Pick a non-monotone toric symplectic four-manifold whose Delzant polygon is small enough to check by hand and test the two load-bearing inputs: verify whether the shear-plus-reglue map is a true integral affine isomorphism at the ridge $\Delta_M$, and compute the hat width $w$; a single manifold where the map has a corner singularity or $w=0$ would show that the construction does not cover all non-monotone cases.

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Extended reading notes

Core claim

On the author's own terms, the central claim is: for every non-monotone toric symplectic four-manifold $X$ there exists a Lagrangian torus $L \subset X$ and a Hamiltonian diffeomorphism $\psi$ of $X$ such that $\psi^n(L) \cap L = \emptyset$ for all $n \in \mathbb{N}$. The manifold is described by a nodal integral affine base $(B_0, \mathfrak{N}_0)$ with height function $\mathcal{F}$; after a nodal tangle that slides parked nodes and modifies the hat, the new base $(B_1, \mathfrak{N}_1)$ is integral affine isomorphic to the original via a map composed of a shear and a triangle cut-and-reglue. Using a lift theorem from the author's companion preprint, this base isomorphism becomes a symplectomorphism $\psi$, which acts on the fibers above a level set as translation by integral affine distance $2(M-h)$. Since the level set has integral affine length $g(h) = 2w - k(M-h) + \sum_i \min\{h - \alpha_i, 0\}$, choosing $h$ with $g(h)/(2(M-h))$ irrational makes the iterates $\psi^n(\pi_0^{-1}(x))$ pairwise disjoint from the starting fiber; finiteness of the symplectic mapping class group then gives a Hamiltonian iterate, namely $\psi^m$ for some $m$, which still satisfies $(\psi^m)^n(L) \cap L = \emptyset$ for all $n$. This directly covers the previously missing non-monotone $S^2 \times S^2$ and critical blow-up cases.

Load-bearing premise

The construction leans on the unpublished theorem that an integral affine isomorphism between nodal bases always lifts to a symplectomorphism, together with the unverified assumption that every non-monotone toric four-manifold has a hat of positive width, so if either gives way the claimed all-non-monotone counterexamples are not established.

Editorial extensions

If this is right

The previous exceptional cases—non-monotone $S^2 \times S^2$ with unequal factor areas, and the one- or two-fold blow-ups of such products at size $c = \min\{a,b\}/2$—now also admit Hamiltonian counterexamples to Lagrangian Poincar\'e recurrence.
In every non-monotone toric symplectic four-manifold, the constructed Lagrangian torus is a fibre of an almost toric fibration, so the non-recurrence occurs on a geometrically natural Lagrangian rather than an exotic one.
Because the symplectic mapping class group is finite in these manifolds, the power $\psi^m$ actually lies in $\mathrm{Ham}(X)$, so the counterexample can be taken Hamiltonian even though the raw map $\psi$ is only a symplectomorphism.
The arithmetic condition $g(h)/(2(M-h))$ irrational is open, so each manifold covered by the construction yields many non-returning Lagrangian fibres, one for each admissible irrational ratio.

Reading between the lines

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

A natural extension would be to formulate the same height-function and irrational-rotation mechanism in higher-dimensional toric manifolds, provided corresponding integral affine bases with closed level sets and integral affine automorphisms exist; the paper does not address this.
The completeness of the all-non-monotone statement currently depends on the unpublished companion theorem [3, Theorem 2.26]; a self-contained proof of that lift theorem would remove the main external dependency.
A direct test of the construction is to compute the hat width $w$ for every Delzant polygon of the claimed class; if some non-monotone polygon forces $w=0$, the note's general assumption \'suppose $w>0$\' would miss that case.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Joel, here's my take. The note does what it says: it extends the counterexample construction from [2] to the two remaining families of non-monotone toric four-manifolds. The new ingredient is a slightly different tangle that works when the hat has positive width, and the S^2×S^2 case is worked out carefully. If the machinery in [3] is sound, the result is almost certainly correct, and completing the toric picture is a legitimate advance.

What I like: the construction is concrete, the figures help, and the use of the shear matrix to get the integral affine isomorphism is neat. The paper is short but not sloppy in its main argument.

The main problem is the hat-width assumption. Section 2 begins with 'Suppose that the hat H has width w > 0' and then proceeds. The note never proves that every non-monotone toric four-manifold can be represented by a canonical nodal integral affine surface with positive width. Figure 3 shows that w=0 can occur for every type, so this is not vacuous. If some non-monotone manifold only admits w=0 representatives, the construction doesn't apply, and the abstract's 'all non-monotone' claim is not established. This is a real gap, not a minor omission. The author should either prove the width condition or point to a theorem in [3] that does it.

The other issue is the heavy reliance on [3]: Theorem 2.26 and Definition 2.30 are central, and they come from an unpublished preprint. That makes the note hard to evaluate independently. A referee would need to verify those parts. Not a fatal flaw, but a reason to be cautious.

Minor slips: 'n ≥ 0' should be 'n ≥ 1' in the intersection statements, and at the end of Section 2 the power m should be positive. These are easily fixed.

My bottom line: this deserves a serious referee. The construction is plausible, the extension is new, and the gaps are checkable. I would recommend sending it to review with the request that the author clarify the positive-width condition and make the dependence on [3] precise. I wouldn't cite the full 'all non-monotone' claim until that's settled, but the specific new cases are worth knowing about.

Referee Report

2 major / 5 minor

Summary. This note claims to extend counterexamples to Lagrangian Poincaré recurrence in dimension four to all non-monotone toric symplectic four-manifolds. It constructs a Hamiltonian diffeomorphism ψ and a Lagrangian torus L such that ψ^n(L)∩L=∅ for all n≥0. The construction uses nodal integral affine bases and nodal tangles. Section 1 treats non-monotone S^2×S^2 explicitly: starting from the base B0 of the almost toric fibration, the author performs nodal slides to obtain B1, defines an integral affine isomorphism τ:B0→B1, and invokes Theorem 2.26 of [3] to lift τ to a symplectomorphism ψ. Section 2 generalizes this to a nodal integral affine surface (B0,𝔑0) of canonical type (H,M,α_1,…,α_n), assuming the hat H has positive width w>0. The author again constructs a nodal tangle and an integral affine isomorphism τ, lifts it via [3, Theorem 2.26], and selects Lagrangian tori over levels where the ratio g(h)/(2(M-h)) is irrational. The paper concludes by using finiteness of the symplectic mapping class group to pass from a symplectomorphism to a Hamiltonian diffeomorphism.

Significance. If correct, the result would close the remaining cases 2 and 3 from [2, Remark 3.2] and establish that Lagrangian Poincaré recurrence fails for every non-monotone toric symplectic four-manifold, a substantial strengthening of the earlier construction. The note is concise and well illustrated, with explicit formulas for level-set lengths and a clear geometric mechanism. However, two load-bearing points prevent the result from being fully established as written: the construction depends on an unproved theorem from the author's unpublished preprint [3], and the positive-width hat hypothesis is not shown to hold for all non-monotone manifolds. With those gaps filled, the result would be a significant contribution to the study of Lagrangian dynamics on rational toric manifolds.

major comments (2)

[Section 2, 'Suppose that the hat H has width w>0' (page 3)] The abstract promises counterexamples for all non-monotone toric symplectic four-manifolds, but the construction in Section 2 is carried out only under the hypothesis that the canonical hat H has positive width w>0. Figure 3 explicitly displays zero-width hats for each type A–E, and the text does not explain why a non-monotone manifold must have a canonical representative with w>0, or how to modify a zero-width representative without changing the symplectic manifold. If some non-monotone manifolds admit only zero-width canonical hats, the supplied construction does not apply to them, and the earlier construction in [2] was already stated to exclude cases 2 and 3. Thus the 'all non-monotone' claim is not established as written. The author should either prove that every non-monotone toric four-manifold admits a base with w>0, or restrict the stated theorem and explain how the remaining zero-width cases are handled.
[Section 1 and Section 2, use of [3, Theorem 2.26]] The map that produces the desired symplectomorphism ψ is obtained entirely from [3, Theorem 2.26], which asserts that an integral affine isomorphism between nodal integral affine bases lifts to a symplectomorphism fitting the stated diagram. This theorem is not stated or proved in the note, and [3] is an unpublished preprint, so the note is conditional on an external result whose proof the reader cannot check. Since Theorem 2.26 is load-bearing for both the S^2×S^2 case (Section 1) and the general case (Section 2), the author should include the precise statement, a proof or sketch, or clearly restrict the note to depend on [3] only as a published reference; as it stands, the main theorem cannot be independently verified from this note alone.

minor comments (5)

[Page 1, paragraph after abstract] The text says 'than repeat the construction' and should say 'then repeat the construction'.
[Section 1, paragraph after Figure 1] The notation t· for the ℝ-action on B0∖ℱ^{-1}(M) is used without definition; please specify that it denotes the flow along the level-set circle in the chosen clockwise direction.
[Section 1, level-set length formula] The formula 'integral affine length 2w+8(M−h)' uses w, but w is introduced in this section only as a parameter in the symplectic form; please state explicitly that this w is also the hat width appearing in the length formula.
[Section 2, definition of f] The sentence 'Let f:B0∖(ℓ∪Δ_M)→R^2 be the length of the straight line segment...' is linguistically garbled; it should say 'Let f(x) be the length of the straight line segment...' and specify that f takes values in R.
[Figure 3 caption] The caption labels hats with 'A , w=0' etc.; the spacing and comma placement should be cleaned up for readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found; the construction depends heavily on the author's own unpublished [3] for the lifting theorem, and the 'all non-monotone' claim has an unproved positive-width assumption.

full rationale

I traced the derivation chain: the paper constructs nodal integral affine bases B0 and B1, an integral affine isomorphism tau identifying them, and then invokes [3, Theorem 2.26] to lift tau to a symplectomorphism psi. The non-recurrence conclusion follows from the irrationality of g(h)/(2(M-h)) along level sets, which is a generic choice, not a fitted parameter. The construction does not define the Lagrangian L or the diffeomorphism psi in terms of the desired conclusion, and no equation in the paper reduces a predicted quantity to an input by definition. The main structural weaknesses are not circularity: (1) the central lifting theorem and the nodal base classification are imported from the author's own unpublished preprint [3] rather than proved in the note, which is heavy self-citation and a support risk, but [3] is cited as an independent tool with stated assumptions rather than as the target result; (2) Section 2 assumes 'Suppose that the hat H has width w > 0' and Figure 3 explicitly lists hats with w=0, while the note never proves that every non-monotone toric four-manifold admits a canonical representative with positive width, so the advertised extension to 'all non-monotone' toric four-manifolds is not fully established. These are correctness/completeness gaps, not circular reductions, and the final score of 2 reflects the heavy self-citation dependence rather than an equation-level circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters or new physical entities. It relies on a collection of theorems from the author's earlier preprint [3] (nodal tangle theory) and on [1, Corollary 4.6]. The main unproved input is [3, Theorem 2.26], which provides the symplectomorphism from an integral affine isomorphism. The note also assumes the hat width w > 0 without proof. These are domain assumptions from the author's own prior work.

assumptions (5)

domain assumption [3, Theorem 2.26]: integral affine isomorphism of nodal bases lifts to a symplectomorphism of total spaces
This is the key tool that produces the symplectomorphism ψ. It is proved in the author's preprint [3], not in this note.
domain assumption [3, Proposition 3.17]: every toric symplectic four-manifold admits a nodal base of canonical type
The construction assumes the existence of such a nodal base with hat H and parked nodes.
domain assumption [3, Definition 2.30]: existence of a lift of the nodal tangle supported on a given region
The Hamiltonian lift π is taken from this definition/theorem.
domain assumption [1, Corollary 4.6]: the symplectic mapping class group of X is finite
Used to pass from symplectomorphism to Hamiltonian by taking a power.
ad hoc to paper The hat H has width w > 0
The construction in Section 2 requires positive width w > 0; the paper does not show that every non-monotone toric four-manifold satisfies this.

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Cite this review

Pith. "Pith review of More counterexamples to Lagrangian Poincar\'e recurrence in dimension four." pith.science (2026). https://pith.science/paper/KCRCEKKS

@misc{pith2026250713924,
  author       = {Pith},
  title        = {Pith review of: More counterexamples to Lagrangian Poincar\'e recurrence in dimension four},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KCRCEKKS}},
  note         = {Machine review of arXiv:2507.13924}
}

read the original abstract

In earlier work, we constructed counterexamples to Lagrangian Poincar\'e recurrence for many toric symplectic four manifolds. Here we provide a few more examples extending the family of counterexamples to include all non-monotone toric symplectic four manifolds. This work has been incorporated into arXiv:2506.23754v2 Section 3.3

Figures

Figures reproduced from arXiv: 2507.13924 by the authors.

**Figure 1.** The construction for non-monotone 𝑆 2 × 𝑆2 . The nodal tangle. Modify 𝐵0 by nodal slides to obtain the nodal integral affine surface 𝐵1 pictured in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Two nodal charts for the same canonical type. of 𝐵0 . The integral affine length of Δ𝑀 is the width 𝑤 of the hat 𝐻 ([3, Definition 3.20]). The integral affine length of ℱ −1(ℎ) is given by the function 𝑔 ∶ ℝ → ℝ ℎ ↦ 2𝑤 − 𝑘(𝑀 − ℎ) + 𝑛 ∑ 𝑖=1 min{ℎ − 𝛼𝑖 , 0}, where 𝑘 is a constant depending on the hat class of 𝐻 as follows: B C D E 𝑘 = 8 𝑘 = 8 𝑘 = 7 𝑘 = 6 where we used the same notation as in [PITH_FULL_IMAGE:figures/… view at source ↗

**Figure 3.** All possible 𝜀-hats up to nodal tangle. to the end or two. Depending on the types of ends of Δ𝑀 , modify 𝐻 by a nodal tangle as in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Nodal tangle transforming part of an 𝜀-hat of positive width. On the left the nodal tangle for an end of Δ𝑀 with two incoming nodes, on the right the one for an end with three incoming nodes. The top row shows the nodal tangle in a nodal chart as in [PITH_FULL_IMAGE:f…

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Works this paper leans on

3 extracted references · 2 canonical work pages

[2]

A counterexample to Lagrangian Poincaré recurrence in dimension four

Joel Schmitz. A counterexample to Lagrangian Poincaré recurrence in dimension four . Preprint, arXiv:2410.24102. To appear in Compositio Mathematica. 2024

work page arXiv 2024
[3]

Nodal Tangles

Joel Schmitz. Nodal tangles. Preprint, arXiv:2506.23754. 2025. 5

work page Pith review arXiv 2025
[1]

Symplectic Torelli groups of rational surfaces

Jun Li, Tian-Jun Li, and Weiwei Wu. Symplectic Torelli groups of rational surfaces . Preprint, arXiv:2212.01873. 2022

arXiv 2022

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [2]

A counterexample to Lagrangian Poincaré recurrence in dimension four

Joel Schmitz. A counterexample to Lagrangian Poincaré recurrence in dimension four . Preprint, arXiv:2410.24102. To appear in Compositio Mathematica. 2024

work page arXiv 2024

[2] [3]

Nodal Tangles

Joel Schmitz. Nodal tangles. Preprint, arXiv:2506.23754. 2025. 5

work page Pith review arXiv 2025

[3] [1]

Symplectic Torelli groups of rational surfaces

Jun Li, Tian-Jun Li, and Weiwei Wu. Symplectic Torelli groups of rational surfaces . Preprint, arXiv:2212.01873. 2022

arXiv 2022