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arxiv: 2604.04449 · v1 · submitted 2026-04-06 · 🧮 math.AG · math.CA· math.CV

Stokes structure of wild difference modules

Yota Shamoto This is my paper

Pith reviewed 2026-05-10 19:55 UTC · model grok-4.3

classification 🧮 math.AG math.CAmath.CV
keywords Riemann-Hilbert correspondencewild difference modulesStokes filtrationA_per-modulesmeromorphic connectionsirregular singularitiesdifference equations
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The pith

A Riemann-Hilbert correspondence equates wild difference modules to wild Stokes-filtered A_per-modules.

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

The paper formulates and proves an equivalence of categories between wild difference modules and wild Stokes-filtered A_per-modules. This extends the Deligne-Malgrange correspondence from meromorphic connections to the setting of difference equations and builds on earlier results limited to mild difference modules. If the equivalence holds, questions about asymptotic behavior and Stokes phenomena in wild difference equations become equivalent to questions about filtered modules equipped with periodic structures. A sympathetic reader would care because the link supplies a dictionary for translating existence, uniqueness, and classification problems across the two categories.

Core claim

We formulate and prove a Riemann-Hilbert correspondence between two categories: wild difference modules and wild Stokes-filtered A_per-modules. This correspondence is motivated by the Riemann-Hilbert correspondence for germs of meromorphic connections in one variable due to Deligne-Malgrange. It also generalizes the Riemann-Hilbert correspondence for mild difference modules.

What carries the argument

The wild Stokes filtration on A_per-modules, which encodes the irregular asymptotic data of the difference modules in a manner compatible with periodic structures.

Load-bearing premise

The categories of wild difference modules and wild Stokes-filtered A_per-modules are defined compatibly with the Deligne-Malgrange framework so that the filtrations and periodic structures behave as assumed.

What would settle it

A concrete wild difference module for which the associated Stokes data cannot be realized by any wild Stokes-filtered A_per-module, or for which the functor fails to be an equivalence of categories.

read the original abstract

We formulate and prove a Riemann--Hilbert correspondence between two categories: wild difference modules and wild Stokes-filtered $\mathscr{A}_{\rm{per}}$-modules. This correspondence is motivated by the Riemann--Hilbert correspondence for germs of meromorphic connections in one variable due to Deligne--Malgrange. It also generalizes the Riemann--Hilbert correspondence for mild difference modules.

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

0 major / 2 minor

Summary. The paper formulates and proves a Riemann-Hilbert correspondence between the categories of wild difference modules and wild Stokes-filtered A_per-modules. This generalizes the Deligne-Malgrange correspondence for germs of meromorphic connections in one variable and extends the known correspondence for mild difference modules, using periodic structures on A_per to handle wild filtrations and Stokes data.

Significance. If the result holds, the work provides a meaningful extension of the Riemann-Hilbert correspondence into the wild irregular setting for difference modules. It unifies Stokes structures with difference operators via explicit functors that preserve filtrations and periodic data, building directly on Deligne-Malgrange techniques and standard methods from the mild case. This could facilitate further study of irregular singularities in difference equations and related algebraic geometry contexts.

minor comments (2)
  1. [Introduction / §2] The definition of the periodic module A_per and its wild filtration could be recalled or cross-referenced more explicitly in the statement of the main theorem (likely around the formulation in the introduction or §2) to improve self-contained readability for readers familiar with the mild case.
  2. [§4 or §5] An illustrative example of a simple wild difference module, its Stokes filtration, and the corresponding A_per-module would help verify the functorial construction and growth control arguments used in the proof.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report accurately captures the main result: a Riemann-Hilbert correspondence between wild difference modules and wild Stokes-filtered A_per-modules that extends the Deligne-Malgrange correspondence and the known mild case.

Circularity Check

0 steps flagged

No significant circularity; explicit functorial construction of the correspondence

full rationale

The paper defines wild difference modules and wild Stokes-filtered A_per-modules compatibly with the Deligne-Malgrange framework and the mild case, then constructs the Riemann-Hilbert correspondence via explicit functors that recover Stokes data and difference structures in both directions. Proofs extend standard mild-case techniques by controlling growth of wild irregular parts. No load-bearing step reduces by definition, by fitting, or by self-citation chain to its own inputs; the equivalence is verified directly rather than assumed or renamed. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard category-theoretic and algebraic geometry axioms for defining modules, filtrations, and equivalences, plus domain assumptions from prior Riemann-Hilbert literature; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • standard math Standard axioms of abelian categories, filtrations, and equivalences of categories in algebraic geometry.
    The Riemann-Hilbert correspondence is formulated using these established structures for modules and Stokes filtrations.
  • domain assumption Domain assumptions from Deligne-Malgrange correspondence for meromorphic connections and mild difference modules.
    The wild case is presented as a direct generalization, relying on the validity and setup of those prior correspondences.

pith-pipeline@v0.9.0 · 5340 in / 1462 out tokens · 39161 ms · 2026-05-10T19:55:50.051788+00:00 · methodology

discussion (0)

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Reference graph

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12 extracted references · 12 canonical work pages

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