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arxiv: 2605.23817 · v1 · pith:ZSR6NHGPnew · submitted 2026-05-22 · 🧮 math.PR · math.DG

Visibility in the Boolean Model on Harmonic Manifolds

Enkelejd Hashorva , Christoph Th\"ale This is my paper

Pith reviewed 2026-05-25 03:13 UTC · model grok-4.3

classification 🧮 math.PR math.DG
keywords Poisson Boolean modelharmonic manifoldsvisible rangeexponential distributiontube volumesRiemannian manifoldsstochastic geometry
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The pith

The directional visible range from an uncovered point in Boolean models is exponentially distributed on all simply connected non-compact homogeneous harmonic manifolds.

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

The paper shows that a known property of Poisson Boolean models with ball-shaped grains extends from Euclidean and hyperbolic spaces to a wider class of spaces. In these models, the distance one can see in a given direction from a random uncovered point follows an exponential distribution. This holds on every simply connected non-compact homogeneous harmonic manifold because tube volumes around geodesic segments grow in an affine linear way. The result allows identification of when the expected volume of the visible region is finite and gives a geometric meaning to the critical intensity in cases with positive entropy. Constructions of other manifolds reveal that the exact exponential form depends on having precisely linear tube growth.

Core claim

In Poisson Boolean models with deterministic ball grains, the directional visible range from an uncovered point is exponentially distributed on every simply connected non-compact homogeneous harmonic manifold. The geometric mechanism is the affine-linear growth of tube volumes around geodesic segments. As a consequence, the finiteness regime for the expected volume of the visible region is identified, including a geometric interpretation of the critical threshold in the positive-entropy case. Explicit complete non-homogeneous Riemannian manifolds are constructed to show that exact exponentiality is tied to exact tube linearity, with superlinear tube growth leading to Weibull-type tails and 2

What carries the argument

affine-linear growth of tube volumes around geodesic segments, which produces the exponential distribution of visible ranges

If this is right

  • The expected volume of the visible region stays finite above a critical intensity threshold set by the manifold geometry.
  • In the positive-entropy case the critical threshold receives a direct geometric interpretation.
  • Exact exponential tails require precise affine-linear tube growth rather than merely asymptotic linearity.
  • Superlinear tube growth on non-homogeneous manifolds produces Weibull-type tails for the visible range.

Where Pith is reading between the lines

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

  • The same exponential visibility might hold on other manifolds whose tube volumes grow linearly even if they lack homogeneity.
  • Numerical checks of the visible-range distribution on concrete examples such as hyperbolic space could confirm the exponential form at the predicted rate.
  • The counterexamples suggest exploring whether asymptotic linearity alone suffices for exponential decay in related coverage models.

Load-bearing premise

The manifolds are simply connected non-compact homogeneous harmonic manifolds on which tube volumes around geodesic segments grow affinely linearly.

What would settle it

A simulation or explicit calculation of the visible range distribution on a specific simply connected non-compact homogeneous harmonic manifold that shows a non-exponential tail would disprove the claim.

read the original abstract

In Poisson Boolean models with deterministic ball grains, the directional visible range from an uncovered point is known to be exponentially distributed in Euclidean and real hyperbolic space. We show that the same phenomenon holds on every simply connected non-compact homogeneous harmonic manifold. The geometric mechanism behind this fact is the affine-linear growth of tube volumes around geodesic segments. As a consequence, we identify the finiteness regime for the expected volume of the visible region, including a geometric interpretation of the critical threshold in the positive-entropy case. We also construct explicit complete non-homogeneous Riemannian manifolds showing that exact exponentiality is tied to exact tube linearity: superlinear tube growth leads to Weibull-type tails, while asymptotic tube linearity still yields an exponential decay rate.

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 the directional visible range from an uncovered point in a Poisson Boolean model with deterministic ball grains is exponentially distributed on every simply connected non-compact homogeneous harmonic manifold. The proof relies on the affine-linear growth of tube volumes around geodesic segments, which yields exponential tails via the standard void-probability mechanism. As consequences, the authors identify the finiteness regime for the expected visible volume (with a geometric reading of the critical threshold in the positive-entropy case) and construct explicit complete non-homogeneous Riemannian manifolds showing that exact exponentiality requires exact tube linearity while superlinear growth produces Weibull tails and asymptotic linearity still produces an exponential decay rate.

Significance. If the geometric reduction holds, the result cleanly extends the Euclidean and real-hyperbolic cases to the full class of simply connected non-compact homogeneous harmonic manifolds by isolating the precise tube-volume condition that drives exponentiality. Credit is due for the explicit non-homogeneous counter-examples that separate exact from asymptotic linearity and for the identification of the critical threshold for expected visible volume; these sharpen the understanding of the geometric-probabilistic link beyond what was previously known.

minor comments (2)
  1. The introduction could include a one-paragraph derivation sketch of how affine-linear tube volume V(t) = a + bt implies the exponential tail for the visible range after conditioning on the origin being uncovered; this would make the central mechanism self-contained for readers outside geometric probability.
  2. Notation for the directional visible range and the visible region should be introduced with a single displayed equation early in §1 or §2 to avoid later ambiguity when the expected-volume finiteness threshold is discussed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment, accurate summary of the main theorem, and recognition of the geometric mechanism (affine-linear tube growth) together with the value of the non-homogeneous counter-examples and the expected-volume threshold. The recommendation of minor revision is noted. No specific major comments appear under the MAJOR COMMENTS heading, so we provide no point-by-point rebuttals.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation rests on the established geometric fact that tube volumes around geodesic segments grow affinely linearly on simply connected non-compact homogeneous harmonic manifolds; the exponential tail for the visible range then follows directly from the void probability formula exp(−λ V(t)) with linear V(t). The paper treats this volume growth as an input property of the manifold class (not derived from the Boolean model) and separately constructs explicit non-homogeneous counterexamples to show that exact linearity is necessary for exact exponentiality. No equation reduces a claimed prediction to a fitted parameter by construction, no load-bearing uniqueness theorem is imported via self-citation, and the central claim remains independent of the probabilistic conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definition of harmonic manifolds and the Poisson Boolean model; the key unverified element in the abstract is the affine-linear tube volume growth property that is invoked as the mechanism.

axioms (1)
  • domain assumption Simply connected non-compact homogeneous harmonic manifolds exhibit affine-linear growth of tube volumes around geodesic segments.
    This property is presented in the abstract as the geometric mechanism enabling the exponential distribution.

pith-pipeline@v0.9.0 · 5642 in / 1141 out tokens · 29617 ms · 2026-05-25T03:13:27.131413+00:00 · methodology

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Reference graph

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12 extracted references · 2 canonical work pages · 1 internal anchor

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