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Curvy points, the perimeter, and the complexity of convex toric domains

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Pith's one-line read The affine perimeter of a convex toric domain equals the liminf of its subleading ECH capacities, and that perimeter is an additive obstruction to full symplectic packings.

desk verdict Important paper with real results and two genuine but repairable gaps in the written proofs; the main theorems likely stand, but the details need a careful fix before I'd certify them. read the letter →

arxiv 2506.23498 v2 pith:SBYDDJKC submitted 2025-06-30 math.SG

Dan Cristofaro-Gardiner , Nicki Magill , Dusa McDuff This is my paper · ORCID
classification math.SG MSC 53D05
keywords symplecticembeddingsconvextoricdomainsECHcapacitieselementaryaffineperimeterfullfillingsinfinitestaircasespackingstability
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

This paper studies four-dimensional symplectic domains built from a compact convex plane region $\Omega$ via the moment map of $\mathbb{C}^2$, dropping the usual requirements that $\Omega$ touch the axes or be rational. The central discovery is a refined law for the ECH capacities (a sequence of symplectic embedding obstructions): for every such convex toric domain $X_\Omega$, the liminf of the gap $c_k(X_\Omega)-\sqrt{2k\,\mathrm{Vol}(X_\Omega)}$ is exactly $-\mathrm{Per}(\Omega)/2$, where $\mathrm{Per}(\Omega)$ is the affine perimeter of the bounding curve. Because ECH capacities are monotone under symplectic embeddings, this makes the perimeter an additive obstruction: a disjoint union of toric domains can fully fill a target only if the sum of their perimeters is at least the target's perimeter. Concrete consequences follow, including that zero-perimeter domains cannot fully fill a ball or $\mathbb{CP}^2$, producing the first examples of failure of packing stability by open subsets of compact manifolds with smooth or empty boundary, and hence long-term super-recurrence. A second thread shows that one positively curved point on the toric boundary rules out an infinite staircase, leading to a complete classification of smooth convex toric domains with infinite staircases.

What carries the argument

The load-bearing objects are the generalized convex toric domains $X_\Omega=\Phi^{-1}(\Omega)$, with $\Omega\subset\mathbb{R}^2_{\geq 0}$ compact and convex and $\Phi$ the moment map of $\mathbb{C}^2$; the affine perimeter $\mathrm{Per}(\Omega)$, the sum of the affine lengths of the rational-slope segments of $\partial\Omega$, which is $SL(2,\mathbb{Z})$-invariant; the ECH capacities $c_k$ and the elementary ECH capacities, which satisfy monotonicity, scaling, disjoint-union, and volume-limit axioms; the weight expansion $(b;b_1,b_2,\dots)$ produced by the cutting algorithm that decomposes $\Omega$ into standard triangles; the Cremona action, a reordering operation on weight tuples that preserves ECH capacities; and the accumulation point theorem, which says that the nonsmooth points of the ellipsoid embedding function converge to the unique root $a_0\ge 1$ of $z^2-(\mathrm{Per}^2/\mathrm{Vol}-2)z+1=0$, with $a_0$ unobstructed when there are infinitely many such points. The cutting algorithm and its Cremona action provide the combinatorial backbone for staircases, while the subleading capacity asymptotics convert the perimeter into an embedding obstruction; the proof that elementary and standard ECH capacities agree on generalized convex toric domains is what opens the door to $\mathbb{CP}^2$ as a target.

What would settle it

Exhibit a smooth convex toric domain whose moment polygon contains both a rational line segment and a strictly convex arc, and show it has an infinite staircase; that would refute Theorem 1.1.5. Alternatively, compute the liminf of $c_k(X_\Omega)-\sqrt{2k\,\mathrm{Vol}(X_\Omega)}$ for such a mixed-boundary domain and check whether it equals $-\mathrm{Per}(\Omega)/2$, directly testing Theorem 1.2.5.

Watch

Extended reading notes

Core claim

The paper's main theorem (Theorem 1.2.5) states that for any convex toric domain $X_\Omega$, if $c_k$ denotes either the ECH or the elementary ECH capacities, then $\liminf_{k\to\infty}\bigl(c_k(X_\Omega)-\sqrt{2k\,\mathrm{Vol}(X_\Omega)}\bigr)=-\mathrm{Per}(\Omega)/2$, where $\mathrm{Per}(\Omega)$ is the affine $SL(2,\mathbb{Z})$-invariant length of the boundary of the moment polygon. The novelty is that no genericity, rationality, or axis-intersection hypothesis is imposed on $\Omega$; for domains such as the $4$-ball the error terms do not have a limit, only a liminf, and the paper shows the liminf still carries the geometric information of the perimeter. From this subleading asymptotics the paper derives Theorem 1.1.1: a full filling $X_{\Omega_1}\sqcup\cdots\sqcup X_{\Omega_n}\hookrightarrow X$ forces $\sum_i \mathrm{Per}(\Omega_i)\ge \mathrm{Per}(X)$, for $X$ a generalized convex toric domain or $\mathbb{CP}^2$. The statement for elementary ECH capacities is essential because the ECH capacities of $\mathbb{CP}^2$ are not yet known, and it yields Corollary 1.1.2: finitely many zero-perimeter domains cannot fully fill a ball or $\mathbb{CP}^2$.

Load-bearing premise

The classification proof assumes that a smooth convex toric domain containing the origin is either an ellipsoid or has zero affine perimeter, without separately proving that a smooth boundary cannot contain both a rational line segment and a positive-curvature arc that would give positive perimeter.

Editorial extensions

If this is right

  • Zero-perimeter convex toric domains can never fully fill a ball or $\mathbb{CP}^2$, and long-term super-recurrence occurs for any open zero-perimeter domain in those targets.
  • Smooth convex toric domains with an infinite staircase are exactly the ball, scalings of $E(1,2)$, and scalings of $E(1,3/2)$; every other smooth convex toric domain has only finitely many obstructive classes.
  • For any convex toric domain, all irrational $a<a_0$ satisfy $c_\Omega(a)>V_\Omega(a)$, so the obstructed set in $[1,a_0]$ has full measure, while all sufficiently large $a$ are unobstructed.
  • There are rational convex toric domains of arbitrarily large cut-length that support infinite staircases, so high combinatorial complexity does not preclude a staircase.
  • Rational symplectic $4$-manifolds with $c_1(\omega)\cdot[\omega]\le 0$ admit no infinite staircase.

Reading between the lines

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

  • If the liminf formula for elementary ECH capacities extends to other closed symplectic $4$-manifolds whose elementary capacities are known, then the perimeter would give a widely computable full-packing obstruction in dimension $4$; the paper states it expects Theorem 1.1.1 to hold for many other closed $4$-manifolds but works out only $\mathbb{CP}^2$.
  • The classification result suggests that every infinite staircase in a smooth convex toric domain is generated by one of the three 'perfect seed' recursions behind the ball and the two rational ellipsoids; a testable extension is to check whether any rational convex toric domain with an infinite staircase must Cremona-reduce to one of these seeds.
  • The 'ghost stairs' in irrational ellipsoids—infinitely many obstructive classes that are invisible because another class overshadows them—show that counting obstructive classes cannot decide staircase existence; the paper's Open Question 1.3.1, whether an infinite weight expansion can ever have an infinite staircase, is the precise test of whether curvy complexity is compatible with staircases.
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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 / 5 minor

Summary. The paper studies convex toric domains in dimension four without requiring the moment polygon to contain the origin or be rational. It introduces affine perimeter and curvy points as symplectic invariants, and proves several theorems: Theorem 1.2.5, that the liminf of the subleading ECH and elementary ECH capacities equals -Per(Omega)/2; Theorem 1.1.1, that the perimeter is an obstruction to full fillings; Corollary 1.1.2, that zero-perimeter domains cannot fully fill balls or CP^2, implying long-term super-recurrence; Theorem 1.1.4, that a curvy point prevents infinite staircases; Theorem 1.1.5, a classification of smooth convex toric domains with infinite staircases; and Theorem 1.3.2, a family of rational domains with increasing cut-length that do support staircases. It also extends the concave-to-convex embedding theorem and the accumulation point theorem to generalized convex toric domains.

Significance. If the results are correct, this is a significant advance in the symplectic embedding theory of toric domains. The perimeter as a full-filling obstruction is a clean and powerful statement, and the removal of genericity assumptions in Theorem 1.2.5 is a genuine improvement over prior work. The classification result for smooth domains with infinite staircases addresses an open question about irrational ellipsoids, and the new examples of staircases with arbitrarily high cut-length are interesting. The paper draws on established ECH capacity technology and prior work on staircases; there is no circularity in the main arguments. However, two load-bearing technical gaps need repair: an invalid inequality in the proof of Lemma 5.1.4, and an omitted case in the proof of Theorem 1.1.5. Both appear repairable within the scope of the paper.

major comments (3)
  1. [§5.1, Lemma 5.1.4] The proof of the upper bound in Theorem 1.2.5 contains a reversed inequality. After deriving 2kVol >= m^2 Vol^2 (1 + (Per - eps/2)/(mVol)), the text concludes sqrt(2kVol) >= mVol sqrt(1 + Per/(mVol) - eps/(2mVol)) >= mVol (1 + Per/(2mVol) - eps/(2mVol)). But sqrt(1+x) < 1 + x/2 for every nonzero x, so the second inequality is false in every nontrivial case, including the zero-perimeter domains that drive Corollary 1.1.2. This invalidates the proof of liminf_k e_k <= -Per/2. The gap is repairable by retaining the second-order term -x^2/8 and slightly enlarging eps, but as written the proof of Theorem 1.2.5, and hence Theorem 1.1.1, is incomplete.
  2. [§6.1, proof of Theorem 1.1.5] The proof asserts that if a smooth convex toric domain contains a neighborhood of the origin, then either it is an ellipsoid or the perimeter of the boundary curve is zero. This dichotomy is not proved and is false as stated. A smooth convex curve can contain a rational line segment of positive affine length together with a positive-curvature arc; the corresponding toric domain is neither an ellipsoid nor of zero perimeter, yet it has a curvy point. Such domains are not covered by the two alternatives in the proof. The missing case is repairable by appealing to Theorem 1.1.4, but the classification proof as written omits it.
  3. [§6.1, proof of Theorem 1.1.5] The proof also states without justification that zero perimeter forces the boundary to have a curvy point. While this is plausible for a smooth compact convex curve, it requires an argument: one must rule out the possibility that the boundary is a union of irrational straight segments with no positive-curvature points, or explain why such a boundary cannot be smooth. As written, the application of Proposition 6.1.1 to the zero-perimeter case is not fully justified.
minor comments (5)
  1. [§1, paragraph 2] The text reads "disjoint nnion" and should be "disjoint union".
  2. [§1.5, organization paragraph] The phrase "Corollay 1.1.2" is a typo and should be "Corollary 1.1.2".
  3. [§2.1, cutting algorithm] The introduction states "We make neither or these assumptions"; this should be "neither of these assumptions".
  4. [§6.3, proof of Lemma 6.3.2] The word "perfact" is a typo and should be "perfect".
  5. [§4.2, Equation (4.2.5)] The sentence "the the third equality uses" contains a duplicated article and should read "the third equality uses".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are derived from independent ECH results and prior published classification work, with no fitted parameter renamed as a prediction.

full rationale

The paper's derivation chain is self-contained with respect to its announced conclusions. Theorem 1.2.5 is proved by combining the ECH subtraction formula (Lemma 3.3.1, proved here for generalized convex toric domains from lattice-path length identities) with estimates on ball ECH capacities (Lemma 5.1.1) and Diophantine approximation of the weight sequence (Lemmas 5.1.2-5.1.3); the perimeter Per(Omega) enters as the independently defined affine length 3b - sum b_j, not as a fitted parameter, and no c_k value is assigned so as to force the identity. Theorem 1.1.1 follows from Theorem 1.2.5 plus the disjoint-union liminf inequality (Lemma 5.2.1), not from an assumption equivalent to the theorem. Theorem 1.2.2 is proved in Section 4 from weight-expansion obstructions, extending rather than assuming [CGHMP]; the equation defining the accumulation point is derived, and the use of [CG1] for the concave-to-convex embedding theorem is an independent, published result that this paper explicitly generalizes. The self-citations that play a load-bearing role are [CG2] for the ellipsoid staircase classification in Theorem 1.1.5 and the textbook uniqueness results [McSal1, McSal2] in Proposition 3.1.1; these are prior independent theorems with proofs, not restatements of this paper's hypotheses, and neither is fitted to the data here. I found no equation in which a predicted quantity reduces by construction to an input, no parameter fitted to a subset and then called a prediction, and no uniqueness theorem imported solely from the authors to forbid alternatives. The reversed square-root inequality noted in Lemma 5.1.4 is a possible correctness gap in the written proof, but it is not a circularity: the liminf bound is not obtained by assuming the theorem.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The ledger is small: the paper introduces no free fitting parameters and no new theoretical entities. Its new results are assembled from standard ECH capacity axioms, the ECH index package, Gromov and McDuff-Salamon uniqueness results, and prior classification and exceptionality theorems from [Hut1, Hut2, Hut3, CG2, Sal, BHM]. The only unproved geometric assertion that functions as an input is the ellipsoid-versus-zero-perimeter dichotomy in the proof of Theorem 1.1.5, which is recorded in red_flags.

assumptions (6)
  • domain assumption ECH capacities of convex and concave toric domains satisfy monotonicity, scaling, disjoint union, and the volume limit.
    Invoked in Section 3.1 as background from [Hut1, Hut2]; for generalized domains the paper defines capacities via Liouville subdomains, but the axioms are assumed at the start.
  • domain assumption The ECH index inequality, spectrality, and compactness statements needed for elementary ECH capacities hold for the nondegenerate perturbations used in Section 3.4.
    Used in Proposition 3.4.1 to prove that elementary ECH capacities agree with ECH capacities for generalized convex toric domains, citing [Hu] and [Hut3].
  • standard math Gromov's theorem on the symplectomorphism group of S^2 x S^2, and the [McSal1] uniqueness results for symplectic forms on S^2 x S^2.
    Used in Lemma 3.1.3 to prove Proposition 3.1.1, which supports Theorem 1.2.1.
  • domain assumption Hutchings' limsup bound for the ECH capacities of unions of balls, [Hut2, Lem.3.8], and its elementary analogue.
    Used in Lemma 5.1.5 for the lower bound on liminf_k e_k(X).
  • domain assumption The ellipsoid staircase classification of [CG2] and [Sal]: among E(1,b), only balls, E(1,2), and E(1,3/2) have infinite staircases.
    Used in Theorem 1.1.5; [Sal] is an unpublished thesis, so this external input is not fully checkable from the preprint.
  • domain assumption Exceptionality of the classes B_k(n) from [BHM, Prop. 79].
    Used in Lemma 7.3.6 to prove that the recursively built classes E_k(n) are perfect, a necessary condition for staircases in Proposition 7.1.6.

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Pith. "Pith review of Curvy points, the perimeter, and the complexity of convex toric domains." pith.science (2026). https://pith.science/paper/SBYDDJKC

@misc{pith2026250623498,
  author       = {Pith},
  title        = {Pith review of: Curvy points, the perimeter, and the complexity of convex toric domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SBYDDJKC}},
  note         = {Machine review of arXiv:2506.23498}
}
read the original abstract

We study the related notions of curvature and perimeter for toric boundaries and their implications for symplectic packing problems in dimension 4; a natural setting for this is a generalized version of convex toric domain which we also study, where there are no conditions on the moment polytope at all aside from convexity. We show that the subleading asymptotics of the ECH and elementary ECH capacities recover the perimeter of such domains in their liminf, without any genericity required, and hence the perimeter is an obstruction to a full filling. As an application, we give the first examples of the failure of packing stability by open subsets of compact manifolds with smooth boundary or with no boundary at all; this has implications for long-term super-recurrence. We also show that a single smooth point of positive curvature on the toric boundary obstructs the existence of an infinite staircase, and we build on this to completely classify smooth (generalized) convex toric domains which have an infinite staircase. We also extend a number of theorems to generalized convex toric domains, in particular the "concave to convex", embedding theorem and the "accumulation point theorem". A curvy point forces "infinite complexity"; we raise the question of whether an infinitely complex domain can ever have an infinite staircase and we give examples with infinite staircases and arbitrarily high finite complexity.

Figures

Figures reproduced from arXiv: 2506.23498 by the authors.

Figure 2.1
Figure 2.1. This illustrates the cutting algorithm. The first triangle has size a; the second cuts have sizes a1, a2 where a1 + a2 ≤ a, the third set of cuts have normal vectors (1, 3),(2, 3),(3, 2),(3, 1) and sizes a11, a12, a21, a22, where a11 + a12 ≤ a1, a21 + a22 ≤ a2, and a12 + a21 ≤ a − (a1 + a2). After three cuts, there are four concave regions Ri1i2 given by the closures of the components of Ω ′∖ [PITH_FULL_IMAGE:figur… view at source ↗
Figure 6.1
Figure 6.1. This figure illustrates the cutting algorithm in the proof of Proposition 6.1.1. for some constants c, d so that f ′ (xk) = cxk + dx2 = 1 k . Let us also assume for convenience that d > 0. Then this has positive solution xk = −c + q c 2 + 4 d k 2d = c 2d [PITH_FULL_IMAGE:figures/full_fig_p049_6_1.png] view at source ↗
Figure 7.1
Figure 7.1. Here is one way to construct a domain Ωn with weights (1; b1, b2, b×2 3 , b×(5+2n) 4 ) and 7 sides. This domain is geometrically Cremona equivalent to Ω ′ n with weight sequence (1 − b3; b1 − b3, b2 − b3, b3, b×(5+2n) 4 ). The domain Ωn (in green) can be seen from cutting the black triangle of size 1 or cutting the blue triangle of size 1−b3 = b1+b2−b3. The unlabeled scalene triangle is made from 5 + 2n cuts of size… view at source ↗

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