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arxiv: 2606.28271 · v1 · pith:SJO57Z6Enew · submitted 2026-06-26 · 🧮 math-ph · gr-qc· hep-th· math.CA· math.MP

Which Saddles Contribute? The South-East Rule for Multidimensional Integrals

In\^es Aniceto , Job Feldbrugge , Christopher J. Howls This is my paper

Pith reviewed 2026-06-29 01:57 UTC · model grok-4.3

classification 🧮 math-ph gr-qchep-thmath.CAmath.MP
keywords asymptotic analysissaddle-point methodresurgencemultidimensional integralscritical pointsBorel planePicard-Lefschetz theorypath integrals
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The pith

A geometric South-East rule in the Borel plane selects which critical points contribute to the asymptotics of multidimensional integrals without computing complex gradient flows.

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

The paper shows that the relevant saddles for the large-k expansion of integrals over R^d of e^{i k f(x)} with analytic f can be identified by plotting the values of f at all critical points in the complex Borel plane, using adjacency relations from algebraic resurgence methods, and applying a new geometric South-East rule. This combination replaces the standard requirement to follow the flows of -Re(i ∇f) through complex space. The approach works for functions that stay bounded or become unbounded on the real domain. It is illustrated on both simple and advanced examples and is presented as a step toward handling instanton contributions in real-time path integrals.

Core claim

The combination of the values of f at all the critical points plotted in the complex Borel plane, the concept of adjacency between such points derived from algebraic resurgence/hyperasymptotic approaches and the new result here of a geometric South-East rule determines which critical points contribute to the asymptotic evaluation without computing the flows of -Re(i ∇f) in C^d.

What carries the argument

The South-East rule, a geometric test applied to the positions of critical values in the complex Borel plane that selects contributing saddles when combined with adjacency data.

If this is right

  • The algorithm removes the computational step of tracking gradient flows in complex space when evaluating multidimensional oscillatory integrals.
  • The method applies equally to functions that remain bounded and those that do not on the real integration domain.
  • It supplies a route to identifying which instantons contribute in real-time path integrals.
  • It addresses difficulties that arise when performing Wick rotations for such integrals.

Where Pith is reading between the lines

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

  • The rule may allow direct comparison between algebraic and geometric resurgence pictures in dimensions where flow tracking becomes impractical.
  • It suggests a way to test resurgence-based predictions against explicit contour deformations in low-dimensional cases.
  • If the rule holds, it could be checked by verifying that the selected saddles reproduce known asymptotic series for standard examples such as Airy or Pearcey integrals.

Load-bearing premise

The adjacency relations between critical points taken from algebraic resurgence methods remain valid when used together with the geometric South-East test for any analytic f.

What would settle it

A concrete analytic f in two or more variables where the saddles chosen by the South-East rule plus adjacency differ from those selected by explicit computation of the flows of -Re(i ∇f).

Figures

Figures reproduced from arXiv: 2606.28271 by Christopher J. Howls, In\^es Aniceto, Job Feldbrugge.

Figure 1
Figure 1. Figure 1: FIG. 1: Pearcey integral for [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The analytic continuation of the Pearcey integral Ψ [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The analytic continuation of the Pearcey integral Ψ [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Superposition of the singularity structure of the Borel planes for each of the 3 critical points of the Pearcey [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: A sketch of the steepest descent manifold [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: A sketch of the Borel plane for a constructed example, for illustrative purposes, for different angles [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: This is the Pearcey integral for [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The analytic continuation of Airy function at [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: The regulated Airy function for [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: The unfolding of the cusp caustic, with the fold caustic (the red curve) and the Stokes line (the green [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: The adjacency graph of the Pearcey integral associated with the unfolding of the cusp catastrophe in the [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: The Picard-Lefschetz deformation of the one-dimensional double Lorentzian lens model in the complex [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: The unfolding of the swallowtail caustic for [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: The regulated swallowtail diffraction integral ( [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: The unfolding of the hyperbolic umbilic caustic, with the fold caustic (the red curve) and the Stokes line [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: The adjacency graph (the red graph) in the complex [PITH_FULL_IMAGE:figures/full_fig_p024_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: Borel-Pad´e prediction of the adjacency graph in the complex [PITH_FULL_IMAGE:figures/full_fig_p024_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: The hyperbolic integral and the relevance of complex critical points. Left: The hyperbolic integral in [PITH_FULL_IMAGE:figures/full_fig_p027_18.png] view at source ↗
read the original abstract

In this paper, we introduce and demonstrate a simple geometric algorithm to determine which critical points, both complex as well as real, contribute to the asymptotic evaluation of multiple integrals with exponential integrands of the form $e^{ikf(\boldsymbol{x})}$ over $\mathbb R^d$, for finite $d\ge 1$ and $f$ is analytic. In so doing, the algorithm removes the need to compute the flows of $-\text{Re} (i\nabla f)$ in $\mathbb C^d$ that is required to identify such relevant critical points in Picard-Lefschetz approaches to the derivation of such asymptotic expansions. By contrast, our algorithm relies on the combination of three simple features: the values of $f$ at all the critical points plotted in the complex Borel plane, the concept of adjacency between such points derived from algebraic resurgence/hyperasymptotic approaches and the new result here of a geometric "South-East" rule. The algorithm incorporates functions $f$ that remain bounded or unbounded on $\mathbb R^d$. We illustrate this new approach with both pedagogical and advanced examples, and draw conclusions as to its importance for resolving issues associated with Wick rotations and its implications for path integrals. This is a significant step towards a systematic way of identifying instanton contributions in real-time path integrals.

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

2 major / 2 minor

Summary. The paper introduces a geometric 'South-East' rule, combined with values of f at critical points in the complex Borel plane and adjacency relations from algebraic resurgence/hyperasymptotics, to select which saddles contribute to the asymptotics of integrals ∫_{R^d} exp(ik f(x)) dx (f analytic) without computing flows of -Re(i ∇f) in C^d. The algorithm applies to bounded and unbounded f; it is illustrated on pedagogical and advanced examples and discussed for implications on Wick rotations and path integrals.

Significance. If the rule and its supporting assumptions hold, the approach offers a practical geometric shortcut for identifying contributing critical points in multi-dimensional oscillatory integrals, extending resurgence techniques and potentially aiding systematic treatment of instantons in real-time path integrals. The combination of Borel-plane plotting, pre-existing adjacency data, and the new rule is a concrete algorithmic contribution.

major comments (2)
  1. [Sections describing the algorithm and its justification (around the statement of the South-East rule and its combination] The load-bearing step is the assumption that adjacency relations derived from 1D algebraic resurgence remain exactly the relevant Stokes connections when lifted to the steepest-descent geometry of -Re(i ∇f) in C^d for arbitrary analytic f. The manuscript demonstrates agreement on specific low-dimensional examples but provides no general argument or counter-example search showing why additional multidimensional Stokes phenomena or non-adjacent but still connected saddles cannot appear; this directly affects the claim that the three-feature combination suffices without flow computations.
  2. [Examples section (pedagogical and advanced illustrations)] The examples section must include at least one higher-dimensional case (d ≥ 2) constructed so that the Picard-Lefschetz flow would reveal an extra connection not predicted by 1D adjacency; without such a test the generality of the rule remains unverified.
minor comments (2)
  1. Provide an explicit mathematical statement (theorem or proposition) of the South-East rule, including the precise geometric criterion in the Borel plane.
  2. Ensure all Borel-plane plots label the real and imaginary axes clearly and indicate which points are real versus complex critical values.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive suggestions. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: [Sections describing the algorithm and its justification] The load-bearing step is the assumption that adjacency relations derived from 1D algebraic resurgence remain exactly the relevant Stokes connections when lifted to the steepest-descent geometry of -Re(i ∇f) in C^d for arbitrary analytic f. The manuscript demonstrates agreement on specific low-dimensional examples but provides no general argument or counter-example search showing why additional multidimensional Stokes phenomena or non-adjacent but still connected saddles cannot appear.

    Authors: We agree this is the central assumption. The adjacency data are taken from the local Borel-plane singularity structure of f, which is determined by the analytic germ at each critical point and is therefore dimension-independent. The South-East rule then filters these adjacencies using only the ordering of the Borel values. While the manuscript verifies consistency on the examples provided, we do not claim a general theorem excluding all possible higher-dimensional Stokes phenomena. We will add an explicit paragraph in the discussion section stating the assumption, its scope, and the absence of a general proof. revision: partial

  2. Referee: [Examples section] The examples section must include at least one higher-dimensional case (d ≥ 2) constructed so that the Picard-Lefschetz flow would reveal an extra connection not predicted by 1D adjacency; without such a test the generality of the rule remains unverified.

    Authors: The manuscript already contains illustrations in d=2 and d=3. None of these were deliberately engineered to produce an extra multi-dimensional Stokes connection beyond 1D adjacency. We accept the referee’s point that a targeted test case would strengthen the claim. We will revise the examples section to include (or describe the construction of) such a d=2 example, or to explain why locating an analytic counter-example is non-trivial. revision: yes

standing simulated objections not resolved
  • A general theorem proving that 1D resurgence adjacencies lift without modification or additional multi-dimensional Stokes connections for arbitrary analytic f.

Circularity Check

0 steps flagged

No circularity: new geometric South-East rule is independent of inputs

full rationale

The paper presents a novel geometric South-East rule as its central contribution for selecting contributing critical points in multidimensional integrals, explicitly combining it with externally derived adjacency concepts from algebraic resurgence. No equations or steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the algorithm is described as removing the need for explicit flow computations via this independent geometric test. Demonstrations on examples do not indicate that the rule itself is forced by prior inputs. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on analyticity of f and on adjacency relations imported from resurgence theory; no free parameters or new invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption f is analytic
    Required for existence of critical points and validity of asymptotic expansions in the complex domain.
  • domain assumption Adjacency relations from algebraic resurgence correctly capture relevant connections between critical points
    Invoked to combine with the South-East rule; location is the description of the algorithm in the abstract.

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