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arxiv: 2605.08909 · v1 · submitted 2026-05-09 · 🧮 math.DG · math.CO

An Upper Bound for Discrete Isometric Filling of Cycles

Runtai He This is my paper

Pith reviewed 2026-05-12 01:46 UTC · model grok-4.3

classification 🧮 math.DG math.CO
keywords isometric fillingcycle graphdiscrete filling areaGromov filling problemannular triangulationgraph distance preservationasymptotic density
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The pith

A concentric annular construction gives isometric fillings of cycles with at most (1/6 + o(1))n² vertices.

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

The paper gives an explicit construction of abstract triangulations that fill the cycle graph C_n while preserving all boundary distances exactly. The resulting graphs use at most one-sixth the square of the cycle length in vertices, asymptotically, which improves the previous upper bound obtained by discretizing a hemisphere. Because the new density lies strictly below the continuous filling density, the discrete problem cannot be used to settle the original Gromov conjecture on the filling area of the circle.

Core claim

We construct isometric fillings K_n of C_n with |V(K_n)| ≤ (1/6 + o(1)) n², and hence D* ≤ 1/6 < 1/(π √3). This construction is a concentric annular filling.

What carries the argument

The concentric annular filling: a sequence of triangulated annular layers that connect the boundary cycle to an inner core while ensuring the graph distance between any two boundary vertices equals their distance along the cycle.

If this is right

  • The asymptotic density D* is at most 1/6.
  • The discrete isometric filling problem is a proper relaxation of Gromov's continuous filling area problem.
  • The discrete problem cannot settle Gromov's conjecture.
  • Concentric annular constructions achieve strictly better vertex efficiency than hemispherical discretizations.

Where Pith is reading between the lines

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

  • Similar layered constructions might be adapted to produce fillings of other graphs or surfaces with controlled distance distortion.
  • Closing the gap between the new upper bound of 1/6 and the known lower bound of 1/8 would require either denser packings inside the annuli or a different global geometry.
  • The o(1) term in the vertex count suggests that explicit error estimates could be derived by refining the triangulation rules within each layer.

Load-bearing premise

The annular layers can be triangulated and connected so that the graph distance between any two boundary vertices remains exactly the cycle distance, with no internal shortcuts.

What would settle it

An explicit computation, for a concrete large n, of the shortest-path distances between all pairs of boundary vertices in the constructed graph, to check whether any pair has distance strictly less than its cycle distance.

Figures

Figures reproduced from arXiv: 2605.08909 by Runtai He.

Figure 1
Figure 1. Figure 1: Equal-length annulus 3.2 Shrinking annuli We next give the corresponding construction when the inner cycle is shorter. Lemma 1 (Shrinking annulus triangulation). Let m ≥ M ≥ 3. Suppose the outer m-cycle U0, U1, . . . , Um−1 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Shrinking annulus 4 A drift estimate for edge paths Consider concentric cycles C0, C1, . . . , CJ , where |Cr| = mr, m0 = n, m0 ≥ m1 ≥ · · · ≥ mJ . The auxiliary coordinates on the cycles are chosen recursively as in the two elementary annuli above. Equivalently, each cycle Cr has a phase αr such that its cyclically ordered vertices have coordinates αr + ni/mr (mod n). Equal-length annuli update the phase … view at source ↗
Figure 3
Figure 3. Figure 3: Construction of the Discrete Isometric Filling [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
read the original abstract

We study the discrete graph-metric analogue of Gromov's filling area problem for the cycle graph \(C_n\). An abstract triangulation \(K\) is an isometric filling of \(C_n\) if \(\partial K=C_n\) and the graph distance between any two boundary vertices is not shortened inside the \(1\)-skeleton of \(K\). Let \(D(n;\epsilon)\) denote the minimum number of vertices in a \((1-\epsilon)\)-Lipschitz filling of \(C_n\), and set \[ D^*=\liminf_{\epsilon\to0^+}\liminf_{n\to\infty}\frac{D(n;\epsilon)}{n^2}. \] Previous work gives the general lower bound \(D^*\ge 1/8\), while discretizing the hemisphere gives the upper bound \[ D^*\le \frac{1}{\pi\sqrt3}. \] In this paper we give an explicit discrete construction which improves the hemispherical upper bound. More precisely, we construct isometric fillings \(K_n\) of \(C_n\) with \[ |V(K_n)|\le \left(\frac16+o(1)\right)n^2, \] and hence \[ D^*\le \frac16<\frac{1}{\pi\sqrt3}. \] This can directly illustrate the discrete filling area problem is a proper relaxation of Gromov's original filling area problem and cannot be used to settle Gromov's conjecture. The construction is a concentric annular filling.

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 presents an explicit combinatorial construction of isometric fillings K_n for the cycle graph C_n using concentric annular layers. This yields |V(K_n)| ≤ (1/6 + o(1)) n², implying an improved upper bound D* ≤ 1/6 for the asymptotic density of vertices in near-isometric fillings, which is better than the previous 1/(π √3) from hemispherical discretization.

Significance. This work strengthens the understanding of the discrete Gromov filling area problem by providing a tighter explicit upper bound through direct construction rather than continuous approximation. The result shows that the discrete version allows for more efficient fillings and cannot be used to prove the continuous conjecture. The explicit nature of the annular construction is a notable strength, as it permits potential computational verification.

major comments (1)
  1. The central claim that K_n is an isometric filling (i.e., the 1-skeleton preserves exact cycle distances between all boundary vertices) depends on the triangulation and connections within the concentric annular layers not introducing any shortcuts. The description asserts this property but lacks a detailed lemma or argument verifying that no inter-layer paths or chords reduce dist_K(u,v) below dist_{C_n}(u,v) for boundary vertices u,v; this verification is load-bearing for the stated bound on D*.
minor comments (2)
  1. The asymptotic count leading to the factor 1/6 should include an explicit breakdown of vertex contributions per annulus to make the o(1) term fully transparent.
  2. A small-n example (e.g., n=12 or n=24) with the explicit graph and distance matrix would help confirm the isometric property in the construction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for their thorough review and for highlighting the importance of rigorously establishing the isometric property of our construction. This feedback will help improve the clarity and completeness of the manuscript. We address the major comment below and plan to incorporate the necessary revisions.

read point-by-point responses

  1. Referee: The central claim that K_n is an isometric filling (i.e., the 1-skeleton preserves exact cycle distances between all boundary vertices) depends on the triangulation and connections within the concentric annular layers not introducing any shortcuts. The description asserts this property but lacks a detailed lemma or argument verifying that no inter-layer paths or chords reduce dist_K(u,v) below dist_{C_n}(u,v) for boundary vertices u,v; this verification is load-bearing for the stated bound on D*.

    Authors: We thank the referee for this important observation. The manuscript does provide a description of the concentric annular construction and states that it forms an isometric filling, but we acknowledge that the verification is not presented as a standalone lemma with full details. In the revised version, we will insert a new Lemma 3.1 (or similar) that proves no shortcuts are introduced. The proof will consist of showing that the triangulation within each annulus uses only edges that respect the angular ordering, and connections between annuli are strictly radial (connecting corresponding vertices in adjacent layers). Any potential path in the 1-skeleton that attempts to shortcut would have to cross layers in a way that its length is at least the boundary arc length, which we verify by comparing to the minimal path on the cycle. This detailed argument will be added to ensure the bound D* ≤ 1/6 is rigorously supported. revision: yes

Circularity Check

0 steps flagged

Explicit construction of isometric fillings provides independent upper bound

full rationale

The paper derives its improved upper bound on D* directly from an explicit construction of concentric annular triangulations K_n whose vertex count is bounded by (1/6 + o(1))n². This count is obtained by enumerating vertices in the described annuli and layers; the isometry condition (graph distances on the boundary preserved) is asserted to hold by the geometry of the construction rather than by any self-referential definition, fitted parameter, or load-bearing self-citation. No step reduces the claimed inequality to the input data or to a prior result by the same author; the lower bound D* ≥ 1/8 is imported from external previous work, and the hemispherical comparison is used only for context. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result relies on standard graph metric axioms and combinatorial construction rules with no fitted parameters or new postulated entities.

axioms (1)
  • standard math Graph distance is the shortest path length in the 1-skeleton
    Invoked in the definition of isometric filling and (1-ε)-Lipschitz condition.

pith-pipeline@v0.9.0 · 5548 in / 1265 out tokens · 63560 ms · 2026-05-12T01:46:03.985554+00:00 · methodology

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

  1. [1]

    M. Gromov. Filling Riemannian manifolds.Journal of Differential Geometry, 18(1):1–147, 1983

    work page 1983

  2. [2]

    Briggs and C

    J. Briggs and C. Wells. A discrete view of Gromov’s filling area conjecture. arXiv preprint arXiv:2602.17859, 2026. 15

    work page arXiv 2026