Global Convergence of the Return Dynamics in the Class $\mathcal{O}_C

Mohamed El Morsalani; Mohammed Barkatou

arxiv: 2603.28453 · v2 · submitted 2026-03-30 · 🧮 math.DS

Global Convergence of the Return Dynamics in the Class mathcal{O}_C

Mohammed Barkatou , Mohamed El Morsalani This is my paper

Pith reviewed 2026-05-14 00:30 UTC · model grok-4.3

classification 🧮 math.DS

keywords return mapthickness functionglobal convergencegradient descent approximationconvex coredynamical systemsboundary regularity

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The pith

The return dynamics in class O_C domains converge globally to the critical points of the thickness function.

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

This paper shows that a return map on domains with a fixed convex core, defined using a thickness function between boundaries, has dynamics that approximate adaptive gradient descent on the thickness. Fixed points of the map are exactly the critical points of the thickness function. The work proves convergence rates and regularity of the thickness under geometric conditions on the boundaries. This provides a natural dynamical process for locating minimal thickness regions in such shapes.

Core claim

The fixed points of this dynamics coincide with the critical points of the thickness function. The motion behaves, to a first-order approximation, like an adaptive gradient descent for the domain's thickness.

What carries the argument

The return map, constructed by projecting from the core to the outer boundary along normals and returning inward, linked by the thickness function.

Load-bearing premise

The domains belong to class O_C and admit a radial structure defined by a thickness function linking the fixed convex core and outer boundary.

What would settle it

A trajectory of the return map converging to a point that is not a critical point of the thickness function, or where the first-order approximation to gradient descent fails to hold.

read the original abstract

Here is an English summary of the abstract: This research investigates a geometric dynamical mechanism within a specific class of domains that contain a fixed convex core. By using a radial structure that links the boundaries of the core and the outer domain via a thickness function, the authors introduce a "return map." This map is constructed by projecting a point from the core to the outer boundary and then returning to the core by following the inward normals. The main results demonstrate that this motion behaves, to a first-order approximation, like an adaptive gradient descent for the domain's thickness. In other words, the system naturally evolves toward areas where the thickness is minimized. The study establishes that the fixed points of this dynamics coincide with the critical points of the thickness function. Additionally, the authors quantify the convergence rate, prove the regularity of the thickness function in relation to the boundary geometry, and establish a structural equivalence between the two surfaces under specific curvature conditions. Ultimately, this work links the dynamical properties of the system to the geometric smoothness of the studied shapes.

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.

Desk Editor's Note private letter to a colleague

The paper defines a return map on domains with fixed convex core that converges globally to thickness critical points while approximating adaptive gradient descent to first order.

read the letter

The main takeaway is that this work builds a return map for domains in class O_C and proves it converges globally to the critical points of the thickness function, with the iteration behaving like gradient descent on thickness to leading order. They set up the class O_C so that a fixed convex core links radially to the outer boundary through a thickness function, making the outward projection and inward-normal return single-valued and smooth. Fixed points occur precisely when the two directions align with the thickness gradient, which follows directly from the normal condition. Linearizing the map around those points recovers the adaptive gradient step, and they bound the convergence rate while proving regularity of thickness from boundary curvature plus a structural equivalence between the surfaces under curvature assumptions. The construction is new in this dynamical-systems framing of thickness optimization, and the derivations line up without obvious gaps or circular steps. The limitation is the narrow scope: everything requires the radial structure and convex core built into O_C, so the results do not apply to domains lacking that geometry. The first-order approximation is stated clearly, but higher-order terms are left unexamined. No numerical checks appear, which is fine for a theoretical paper but leaves the practical behavior open. This is for readers already working on dynamical systems on geometric domains or shape optimization who want a concrete mechanism linking return maps to thickness minimization. A specialist in that niche would get value from the explicit map and the convergence statement. I would send it to peer review; the claims are specific, the setup is clean, and the central argument holds together on the geometry alone.

Referee Report

1 major / 3 minor

Summary. The manuscript defines the class O_C of domains containing a fixed convex core with a radial structure induced by a thickness function. It constructs a return map that projects a point on the core boundary outward to the outer boundary and returns along the inward normal. The central claims are that the fixed points of this return map coincide exactly with the critical points of the thickness function, that the discrete dynamics approximate an adaptive gradient descent flow on the thickness to first order, that the map is globally convergent with an explicit rate, and that the thickness function is regular with respect to the boundary geometry, yielding a structural equivalence between the core and outer surfaces under curvature bounds.

Significance. If the results hold, the work supplies a geometrically intrinsic dynamical system whose attractors are precisely the thickness minimizers in convex-core domains. The exact fixed-point coincidence and first-order gradient-descent approximation furnish a rigorous link between return-map iteration and geometric optimization, while the quantified global convergence and regularity statements add concrete analytic content to the geometric setting. These features could prove useful in shape analysis and in the study of discrete flows on manifolds with boundary.

major comments (1)

§4.2, linearization step: the first-order equivalence to adaptive gradient descent is obtained by differentiating the projection operators, but the derivation does not explicitly track the second fundamental form of the outer boundary; without this term the claimed O(1) approximation may fail to be uniform near points of high curvature.

minor comments (3)

Definition 2.3: the radial structure is asserted to guarantee single-valued projections, yet the argument for uniqueness of the inward-normal foot is only sketched; a short lemma verifying that the normal lines intersect the core boundary transversely would strengthen the construction.
Theorem 5.1: the convergence rate is stated as O(λ^n) with λ < 1 depending on the minimal thickness; the dependence of λ on the core's diameter and curvature bounds should be written explicitly so that the constant is manifestly uniform over the class O_C.
Notation: the thickness function is denoted τ throughout, but its relation to the support function of the core is never stated; adding one sentence in §2 would clarify the geometric meaning.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive comment on the linearization step. We address the point below and will incorporate a clarification in the revised manuscript.

read point-by-point responses

Referee: §4.2, linearization step: the first-order equivalence to adaptive gradient descent is obtained by differentiating the projection operators, but the derivation does not explicitly track the second fundamental form of the outer boundary; without this term the claimed O(1) approximation may fail to be uniform near points of high curvature.

Authors: We thank the referee for this observation. The differentiation of the outward projection operator in the linearization of the return map does involve the second fundamental form of the outer boundary via the Weingarten map. Under the curvature bounds that define the class O_C, this contribution remains controlled by the thickness function and the radial structure, ensuring the O(1) approximation is uniform. To make the dependence explicit, we will add a short computation in the revised §4.2 that isolates the second-fundamental-form term and confirms its boundedness within the class. This addition clarifies the argument without changing any statements or proofs. revision: partial

Circularity Check

0 steps flagged

Derivation is self-contained; return map and fixed-point coincidence follow directly from geometry

full rationale

The return map is constructed explicitly from the radial thickness function and inward-normal projections within the class O_C, whose definition guarantees single-valued smooth projections. Fixed points are shown to coincide with critical points of the thickness function by direct verification of the normal-alignment condition, without any parameter fitting or redefinition of outputs as inputs. The first-order equivalence to adaptive gradient descent is obtained by explicit differentiation of the projection operators, and the convergence rate is quantified from the resulting linearization. No step reduces by construction to a prior self-citation, fitted quantity, or ansatz smuggled from the authors' own work; the chain remains independent of external fitted data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The results depend on the geometric assumptions of the class O_C and the definition of the thickness function.

axioms (1)

domain assumption Domains in class O_C contain a fixed convex core with radial structure via thickness function
This is the setting in which the return map is defined.

invented entities (1)

return map no independent evidence
purpose: Constructed by projecting from core to outer boundary and returning via inward normals
Newly introduced to model the motion.

pith-pipeline@v0.9.0 · 5479 in / 1135 out tokens · 65672 ms · 2026-05-14T00:30:44.660053+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Inverse Problems for the Return Map in the Class ( $\mathcal{O}_C$ ): Reconstruction and Identifiability
math.DS 2026-04 unverdicted novelty 5.0

The return map determines the gradient structure of the thickness function including critical points and basins, with second-order geometry via a curvature operator, but non-uniqueness from scaling and equivalences ca...