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arxiv: 2605.21705 · v1 · pith:4AXLJR3Snew · submitted 2026-05-20 · 🧮 math.AP · math.DG

A Sharp Regularity Threshold for Uniqueness in Riemannian Calder\'on-type Problems

Thierry Daud\'e , Alberto Enciso , Bernard Helffer , Niky Kamran , Fran\c{c}ois Nicoleau This is my paper

Pith reviewed 2026-05-22 08:38 UTC · model grok-4.3

classification 🧮 math.AP math.DG
keywords Calderón probleminverse problemsRiemannian metricGevrey classDirichlet-to-Neumann mapanalytic regularityuniqueness threshold
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The pith

Analyticity is the exact threshold for uniqueness in Riemannian Calderón-type problems.

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

The paper shows that analytic metrics are uniquely determined by the Dirichlet-to-Neumann map in the Riemannian Schrödinger problem with a fixed nonconstant analytic potential and in the anisotropic Calderón problem at fixed nonzero frequency. Uniqueness holds modulo the gauge of boundary-fixing diffeomorphisms that preserve the potential or frequency. In every Gevrey class with index greater than one, uniqueness fails densely, and the counterexamples consist of metrics that cannot be connected by any such diffeomorphism. This establishes analyticity as the precise regularity dividing line between unique determination and the existence of distinct metrics producing identical data.

Core claim

Analytic metrics are uniquely determined modulo the gauge by a minor adaptation of the Lassas-Uhlmann theorem, while uniqueness fails densely in every non-analytic Gevrey class G^σ for σ>1, with counterexamples not connected by any boundary-fixing diffeomorphism preserving V. The analogous sharp threshold holds for the anisotropic Calderón problem at fixed nonzero frequency. The two constructions use different scalar mechanisms: the nonconstant potential itself provides a local coordinate in the fixed-potential case, while at nonzero frequency one uses a compactly supported prescribed-Jacobian lemma in Gevrey spaces.

What carries the argument

The nonconstant analytic potential serving as a local coordinate for fixed-potential non-uniqueness, together with a compactly supported prescribed-Jacobian lemma in Gevrey spaces for the fixed-frequency case, to produce non-isometric counterexamples.

If this is right

  • Analytic metrics are recoverable from the Dirichlet-to-Neumann map up to the stated gauge in both the fixed-potential and fixed-frequency settings.
  • In every Gevrey class smoother than analytic, there exist dense families of non-unique metrics that are not gauge-equivalent.
  • The same analytic threshold separates uniqueness from non-uniqueness for the two distinct inverse problems considered.
  • Counterexamples can be constructed without the metrics being isometric via any diffeomorphism that fixes the boundary and the auxiliary data.

Where Pith is reading between the lines

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

  • The result indicates that numerical reconstruction algorithms for these inverse problems may need to enforce or detect analytic regularity to avoid ambiguity.
  • Similar sharp thresholds between analytic and Gevrey regularity could be examined in related geometric inverse problems such as boundary rigidity.
  • One could seek concrete, low-dimensional examples of the Gevrey counterexamples to test stability under discretization.

Load-bearing premise

The constructions of the counterexamples in Gevrey classes rely on the existence of a compactly supported prescribed-Jacobian lemma in Gevrey spaces and on the nonconstant analytic potential serving as a local coordinate.

What would settle it

An explicit pair of distinct metrics belonging to some Gevrey class G^σ with σ>1 that produce identical Dirichlet-to-Neumann data yet are not related by any boundary-fixing diffeomorphism preserving the potential would confirm the non-uniqueness claim.

read the original abstract

We prove a sharp regularity threshold for uniqueness in two anisotropic Calder\'on-type inverse problems in dimension $n\ge 3$. The main setting is the Riemannian Schr\"odinger problem with fixed scalar potential: for a prescribed nonconstant analytic function $V$, we study whether the Dirichlet-to-Neumann map of $-\Delta_g+V$ on a domain $\Omega\subset\mathbb{R}^n$ determines the unknown metric $g$. The natural gauge is the group of boundary-fixing diffeomorphisms preserving $V$. We show that, while analytic metrics are uniquely determined modulo this gauge by a minor adaptation of the Lassas--Uhlmann reconstruction theorem, uniqueness fails densely in every non-analytic Gevrey class $G^\sigma$, $\sigma>1$. In fact, our counterexamples are not isometric in the sense that they are not connected by the pushforward of any diffeomorphism of $\overline\Omega$. We also prove the analogous sharp threshold for the anisotropic Calder\'on problem at fixed nonzero frequency, thereby upgrading the previously known finite-regularity counterexamples to Gevrey and $C^\infty$ regularity. The two constructions use different scalar mechanisms: for fixed potentials, the nonconstant potential itself provides a local coordinate, while at nonzero frequency one uses a compactly supported prescribed-Jacobian lemma in Gevrey spaces. Thus analyticity is the exact threshold for uniqueness in both problems.

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

1 major / 2 minor

Summary. The manuscript establishes a sharp regularity threshold for uniqueness in two anisotropic Calderón-type inverse problems in dimensions n ≥ 3. For the Riemannian Schrödinger problem with fixed nonconstant analytic potential V, analytic metrics are uniquely determined by the Dirichlet-to-Neumann map modulo the gauge of boundary-fixing diffeomorphisms preserving V, via a minor adaptation of the Lassas-Uhlmann reconstruction theorem. Uniqueness fails densely in every Gevrey class G^σ for σ > 1, with counterexamples not related by any such gauge transformation; the construction uses the potential V itself as a local coordinate. An analogous sharp threshold is proved for the anisotropic Calderón problem at fixed nonzero frequency by upgrading prior finite-regularity counterexamples, using a compactly supported prescribed-Jacobian lemma in Gevrey spaces. Analyticity is therefore the exact threshold for uniqueness in both problems.

Significance. If the stated constructions hold, the result is significant: it identifies analyticity as both sufficient and necessary for uniqueness, sharpening the known landscape of inverse problems by showing that C^∞ and Gevrey regularity are insufficient while analytic regularity suffices. The work upgrades earlier finite-regularity non-uniqueness results to the Gevrey and smooth categories and supplies explicit, mechanism-specific constructions (V-as-coordinate and prescribed-Jacobian) that are falsifiable. These features, together with the adaptation of an existing reconstruction theorem, strengthen the contribution to the field of geometric inverse problems.

major comments (1)
  1. [Sections 4 and 5] The non-uniqueness statements in both settings rest on the technical lemmas cited in the abstract (compactly supported prescribed-Jacobian lemma in Gevrey spaces for the frequency case; V serving as local coordinate while preserving Gevrey class for the potential case). Section 4 and Section 5 should contain explicit verification that the resulting metrics lie outside the boundary-fixing, V-preserving gauge and that the pulled-back coefficients remain in the claimed Gevrey class; without these verifications the dense failure claim would not be established.
minor comments (2)
  1. [Introduction] The introduction would benefit from a brief recall of the precise definition of the Gevrey class G^σ used throughout (including the radius of analyticity parameter) to aid readers unfamiliar with the variant employed.
  2. [Section 2] Notation for the gauge group (boundary-fixing diffeomorphisms preserving V) is introduced late; an early definition or reference to the precise group action would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their positive overall assessment, and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Sections 4 and 5] The non-uniqueness statements in both settings rest on the technical lemmas cited in the abstract (compactly supported prescribed-Jacobian lemma in Gevrey spaces for the frequency case; V serving as local coordinate while preserving Gevrey class for the potential case). Section 4 and Section 5 should contain explicit verification that the resulting metrics lie outside the boundary-fixing, V-preserving gauge and that the pulled-back coefficients remain in the claimed Gevrey class; without these verifications the dense failure claim would not be established.

    Authors: We thank the referee for highlighting this point. The constructions are built so that the resulting metrics are not related by any boundary-fixing diffeomorphism preserving V (in the Schrödinger case) or satisfying the natural gauge condition at fixed frequency. This follows directly from the choice of the diffeomorphism (using V itself as a local coordinate in Section 4, and the compactly supported prescribed-Jacobian map in Section 5) together with the fact that the metrics differ by a non-gauge term that cannot be absorbed by such a diffeomorphism. The lemmas already ensure that the pulled-back coefficients remain in the stated Gevrey class G^σ. Nevertheless, we agree that an explicit, self-contained verification of both the non-gauge-equivalence and the regularity preservation will strengthen the exposition. In the revised manuscript we will insert short, direct arguments (approximately one paragraph each) at the end of Sections 4 and 5 that carry out these checks explicitly, without altering any of the stated results or proofs. revision: yes

Circularity Check

0 steps flagged

No circularity: analytic uniqueness via external theorem adaptation; Gevrey non-uniqueness via explicit constructions

full rationale

The derivation chain relies on an adaptation of the external Lassas-Uhlmann reconstruction theorem for the analytic uniqueness direction and on direct constructions of counterexamples in Gevrey classes using a prescribed-Jacobian lemma and the analytic potential as a local coordinate. These steps do not reduce to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations that render the threshold claim tautological by construction. The paper upgrades prior finite-regularity results through new mechanisms stated within the work itself, keeping the central claim self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard properties of Gevrey classes, the Lassas-Uhlmann reconstruction theorem, and two technical lemmas whose validity is assumed for the constructions; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Properties of Gevrey classes G^σ for σ>1 and existence of compactly supported functions with prescribed Jacobian in those classes
    Invoked to construct the non-uniqueness examples in the frequency case
  • standard math Lassas-Uhlmann reconstruction theorem for analytic metrics
    Used as the base for the uniqueness direction after minor adaptation

pith-pipeline@v0.9.0 · 5797 in / 1401 out tokens · 44091 ms · 2026-05-22T08:38:57.064590+00:00 · methodology

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