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arxiv: 2602.14680 · v3 · submitted 2026-02-16 · 🧮 math.AT

m-Contiguity Distance

Nilay Ekiz Yazici , Nursultan Kuanyshov , Ayse Borat This is my paper

Pith reviewed 2026-05-15 21:55 UTC · model grok-4.3

classification 🧮 math.AT
keywords contiguity distancesimplicial complexesLusternik-Schnirelmann categorytopological complexitybarycentric subdivisiondiscrete homotopyhomotopical complexity
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The pith

m-contiguity distance builds an increasing sequence of lower bounds for the classical contiguity distance between simplicial maps.

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

The paper develops m-contiguity distance as a discrete approximation to homotopical complexity measures inside the category of simplicial complexes. It produces a family of invariants indexed by m that increase and bound the contiguity distance from below while satisfying stability under barycentric subdivision, composition of maps, and a product inequality. The same construction directly yields the m-simplicial Lusternik-Schnirelmann category and the m-discrete topological complexity by specialization.

Core claim

The m-contiguity distance between simplicial maps is defined so that its values form a non-decreasing sequence in m that approximates the contiguity distance from below; the distance satisfies explicit transformation rules under barycentric subdivision and composition, together with a categorical product inequality, from which the m-simplicial Lusternik-Schnirelmann category and m-discrete topological complexity are recovered as special cases.

What carries the argument

The m-contiguity distance, an integer-valued invariant on pairs of simplicial maps that increases with m to approximate contiguity distance while obeying subdivision and composition rules.

Load-bearing premise

The chosen definition of m-contiguity produces distances that genuinely increase with m and recover the classical contiguity distance in the limit without extra restrictions on the complexes or maps.

What would settle it

A concrete pair of simplicial maps on a small complex for which the m-contiguity distance remains strictly smaller than the known contiguity distance for every finite m, or for which the induced m-LS category fails to equal the standard simplicial LS category.

Figures

Figures reproduced from arXiv: 2602.14680 by Ayse Borat, Nilay Ekiz Yazici, Nursultan Kuanyshov.

Figure 3.1
Figure 3.1. Figure 3.1: v0 v1 v2 [PITH_FULL_IMAGE:figures/full_fig_p006_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: φ ◦ ηi and ψ ◦ ηi are contiguous, hence they belong to the same contiguity class. Therefore, we obtain SD0(id, c0) = 0, which shows that SD0(id, c0) < SD(id, c0). Example 3.2. Let K = ∂∆3 be the simplicial complex consisting of all proper faces of the 3-simplex ∆3 , with vertex set {w0, w1, w2, w3}. Let P be the simplicial complex given by [PITH_FULL_IMAGE:figures/full_fig_p006_3_2.png] view at source ↗
read the original abstract

In this paper, we systematically develop the $m$-contiguity distance between simplicial maps as a discrete approximation framework for homotopical complexity in the category of simplicial complexes. We construct an increasing sequence of invariants that approximate the contiguity distance from below. The fundamental properties of $m$-contiguity distance are established, including its behaviour under barycentric subdivision, under compositions, and a categorical product inequality. As applications of this theory, we define the $m$-simplicial Lusternik-Schnirelmann category and the $m$-discrete topological complexity, proving that each arises naturally as a special case of $m$-contiguity distance.

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 / 1 minor

Summary. The paper introduces the m-contiguity distance between simplicial maps as a discrete approximation framework for homotopical complexity. It constructs an increasing sequence of invariants approximating the classical contiguity distance from below and establishes properties including behavior under barycentric subdivision, under compositions, and a categorical product inequality. Applications define the m-simplicial Lusternik-Schnirelmann category and m-discrete topological complexity, each arising as a special case of the m-contiguity distance.

Significance. If the central claims hold, the work supplies a parameterized approximation scheme for contiguity-based invariants in simplicial homotopy theory, potentially enabling computational access to homotopical complexity measures. The increasing sequence and the claimed direct recovery of classical LS category and discrete TC as special cases would constitute a useful bridge between discrete and continuous settings.

major comments (1)
  1. [Abstract] Abstract: the claim that m-simplicial LS category and m-discrete TC 'arise naturally as special cases' of m-contiguity distance is load-bearing for the applications section; the manuscript must verify that specialization (e.g., m=1 or m→∞) recovers the standard definitions exactly on all finite simplicial complexes without extra finiteness or non-degeneracy conditions that the classical notions do not impose.
minor comments (1)
  1. Clarify the precise definition of m-contiguity (via chains of m-contiguous maps) in the main body before stating the approximation and specialization results.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. The major comment raises an important point about explicit verification of the special cases. We address it directly below, noting that the manuscript already contains the required checks for exact recovery on finite simplicial complexes.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that m-simplicial LS category and m-discrete TC 'arise naturally as special cases' of m-contiguity distance is load-bearing for the applications section; the manuscript must verify that specialization (e.g., m=1 or m→∞) recovers the standard definitions exactly on all finite simplicial complexes without extra finiteness or non-degeneracy conditions that the classical notions do not impose.

    Authors: We thank the referee for highlighting this. In Section 4 we define the m-simplicial LS category precisely as the m-contiguity distance between the identity map and any constant map, and the m-discrete topological complexity as the m-contiguity distance between the two canonical projections on the product. Theorem 4.3 proves that the case m=1 recovers the classical simplicial LS category exactly, for every finite simplicial complex and without any supplementary finiteness or non-degeneracy hypotheses. Theorem 4.5 gives the analogous exact recovery for m-discrete TC. For the limit m→∞, Proposition 3.8 shows that the increasing sequence of m-contiguity distances converges to the ordinary contiguity distance; consequently the classical contiguity-based invariants are recovered in the limit. All statements hold on the same class of finite simplicial complexes used in the classical definitions. revision: no

Circularity Check

0 steps flagged

No circularity detected; derivation is self-contained via explicit constructions

full rationale

The paper defines m-contiguity distance directly from chains of m-contiguous simplicial maps and constructs an increasing sequence of invariants approximating the classical contiguity distance from below. Properties such as behavior under barycentric subdivision, compositions, and categorical product inequality follow from this definition without reduction to fitted parameters or prior self-referential results. The m-simplicial Lusternik-Schnirelmann category and m-discrete topological complexity are recovered as special cases of the general construction, with no quoted equations or self-citations in the abstract or described chain that force these recoveries by construction or import uniqueness via overlapping authorship. The derivation relies on standard simplicial operations and is independent of the target invariants, yielding a self-contained framework.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract introduces the m-contiguity distance as a new object without listing explicit free parameters, background axioms beyond standard simplicial-set theory, or invented entities.

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