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arxiv: 2606.25138 · v1 · pith:AGJYYK2Onew · submitted 2026-06-23 · 🧮 math.MG · math.DG

Stable recovery of a simple irreversible Finsler geometry from travel time data

Maarten V. de Hoop , Joonas Ilmavirta , Antti Kykk\"anen , Teemu Saksala This is my paper

Pith reviewed 2026-06-25 21:27 UTC · model grok-4.3

classification 🧮 math.MG math.DG
keywords Finsler geometrytravel time datainverse problemstabilityGromov-Hausdorff distanceirreversible metricmetric geometry
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The pith

A simple irreversible Finsler geometry can be recovered uniquely and with Lipschitz stability from its travel time data.

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

The paper shows that if an irreversible Finsler geometry is simple, then its structure can be uniquely recovered from travel time measurements between points, and this recovery is stable under small perturbations in a Lipschitz way. This is significant because it allows reconstructing the geometry from observable data in situations where the metric is not symmetric, which arises in some physical models. The authors introduce a specialized version of the Gromov-Hausdorff distance to handle irreversible cases and prove the result. Without the simplicity assumption, the notion of stability is not even well-defined, setting this apart from reversible geometries like Riemannian ones. This establishes a foundation for solving inverse problems in Finsler geometry using travel times.

Core claim

We show that a simple irreversible Finsler geometry can be recovered uniquely and Lipschitz-stably from its travel time data. We introduce and use a version of Gromov--Hausdorff distance adapted to irreversible metric spaces. In contrast to reversible (e.g. Riemannian) geometry, even the question of stability becomes ill-defined without simplicity.

What carries the argument

Adapted Gromov-Hausdorff distance for irreversible metric spaces, used to establish closeness and stability of the geometries.

If this is right

  • Unique recovery of the geometry from travel time data when the geometry is simple.
  • Lipschitz stability of the recovery map.
  • The simplicity condition is necessary for the stability question to be meaningful.
  • The result applies specifically to irreversible Finsler geometries, differing from reversible cases.

Where Pith is reading between the lines

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

  • Practical algorithms for geometry reconstruction could be developed based on this uniqueness and stability.
  • The method may extend to other inverse problems involving non-symmetric distances in physics or imaging.
  • Similar adapted distances might help in studying stability for other types of metric spaces.

Load-bearing premise

The geometry is assumed to be simple.

What would settle it

Finding two different simple irreversible Finsler geometries that have the same travel time data would show the claim is false.

read the original abstract

We show that a simple irreversible Finsler geometry can be recovered uniquely and Lipschitz-stably from its travel time data. We introduce and use a version of Gromov--Hausdorff distance adapted to irreversible metric spaces. In contrast to reversible (e.g. Riemannian) geometry, even the question of stability becomes ill-defined without simplicity.

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 proves that a simple irreversible Finsler geometry on a manifold with boundary can be recovered uniquely and with Lipschitz stability from its travel-time (boundary distance) data. The proof proceeds by introducing an adapted Gromov-Hausdorff distance on irreversible metric spaces and showing that this distance controls the Finsler structure under the simplicity hypothesis; the abstract explicitly notes that simplicity is required for the stability question to be well-posed.

Significance. If the result holds, it provides the first Lipschitz-stable recovery theorem for irreversible Finsler metrics, extending classical boundary rigidity and stability results from the reversible Riemannian setting. The adapted Gromov-Hausdorff construction is a technical contribution that may be useful for other directed or non-reversible geometries. The explicit necessity of simplicity clarifies the scope of the theorem.

minor comments (2)
  1. [Abstract] The abstract states the main theorem but does not indicate the length or key steps of the proof; a one-sentence outline of the strategy (e.g., reduction to the reversible case or use of the adapted GH distance) would improve readability.
  2. Notation for the irreversible distance function and the adapted GH distance should be introduced with a short comparison table to the classical reversible versions to aid readers familiar with the Riemannian literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; theorem is self-contained under explicit simplicity hypothesis

full rationale

The paper states a uniqueness-plus-Lipschitz-stability theorem for recovering a simple irreversible Finsler structure from boundary travel-time data via an adapted Gromov-Hausdorff distance on irreversible spaces. The abstract explicitly flags that simplicity is required for the stability question to be well-posed, and the result is conditioned on that hypothesis. No self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the provided abstract or description. The derivation chain is therefore independent of its own outputs and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no specific free parameters, axioms, or invented entities are identifiable from the provided information.

pith-pipeline@v0.9.1-grok · 5589 in / 926 out tokens · 26283 ms · 2026-06-25T21:27:02.278965+00:00 · methodology

discussion (0)

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Reference graph

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