What question did this study set out to answer?

The aim is to create a classification theory for Newton polygons using the WKB geometry approach.

May 27, 2026Open Access

The WKB Geometry of Newton Polygons: A Functorial Classification Theory

Key Points

The aim is to create a classification theory for Newton polygons using the WKB geometry approach.
Development of a computational atlas through a phase analysis of Newton polygons.
Application of the WKB ansatz to identify characteristic polynomials and their properties.
Utilization of algebraic identities to classify polynomial behaviors across multiple edge slopes.
Identified 23 single-edge classes and expanded to 39 with further analysis on multi-edge and nested-ramification cases.
Showed per-edge factorisation along slope filtration with implications for algebraic independence and cyclotomic structure.
Established connections between anti-Stokes directions and the wave patterns in the context of single-edge operators.

Abstract

This release publishes the camera-ready manuscript The WKB Geometry of Newton Polygons: A Functorial Classification Theory together with its full Phase I–V computational atlas, refereed Round-1 and Round-2 revision packages, and submission-ready arXiv and journal bundles. For a scalar local linear operator L (differential, difference, or q-difference) with one irregular singularity at x = ∞ whose Newton polygon has a unique dominant edge of slope −p/q with r + 1 lattice points, the WKB ansatz on that edge produces a characteristic polynomial χ ∈ Cc. Our central algebraic identity is χ (c) = χw (cq), where χw ∈ Cw is the classical edge-indicial polynomial of degree r. The exponent constellation C (L) ⊂ C is therefore a μq-equivariant union of r fibres of c → cq over Roots (χw). This factorisation organises a finite stratified atlas: a computational sweep yields 23 single-edge classes for −p/q ∈ 1/2, 1/3, 1/4, 2/3, 3/2 and r ≤ 3, refining to 39 classes once multi-edge, nested-ramification, and prime-q arithmetic strata are tracked. We prove a per-edge factorisation along the slope filtration, an iterated μq′-pullback law for nested inner polynomials, a block-diagonal pullback for systems, and a prime-q Kummer dichotomy separating the algebraically independent stratum from the cyclotomic stratum. The analytic input attaching anti-Stokes directions is classical (Levelt–Turrittin, Sibuya, Wasow) ; on the q-fold cover the rays of a generic single-edge operator satisfy arg Δij + pφ ≡ π/2 (mod π). One explicit consequence: the slope numerator p, invisible to C (L), is visible to the cover-side ray pattern. The analytic statements (Theorems AT1, AT3, AT5, AT7) are μq/μq′-equivariant repackagings of classical formal-classification theory. Release contents. Camera-ready LaTeX manuscript (26 pp, 898 KB PDF, byte-identical compile across all bundles), the full Phase I–V computational atlas (19 JSON artifacts, ~2. 6 MB), the arXiv-ready and journal-ready submission bundles, the eight in-manuscript theorem proofs (M1, N1, D2, S1, AT1, AT3, AT5, AT7), the agent-pipeline scripts (fleet. py … fleet5. py), the phase reports, and the Round-1 + Round-2 refereed revision packages. Repository. A stable, versioned public GitHub mirror of this Zenodo release is at https: //github. com/papanokechi/wkb-newton-polygons; the manuscript points to it from the Reproducibility paragraph of the Computational and AI Disclosure section. Dual licensing. All text, figures, and the JSON atlas are released under CC-BY-4. 0; all code and fleet scripts are released under MIT. See LICENSE-TEXT and LICENSE-CODE in the record.

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