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arxiv: 2606.02144 · v1 · pith:457KD4ASnew · submitted 2026-06-01 · 🧮 math.ST · stat.TH

Sharp Support Thresholds for Smeariness of Absolutely Continuous Measures on Spheres

Susovan Pal This is my paper

Pith reviewed 2026-06-28 12:17 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords smearinessFréchet meanspheresupport thresholdsHessian degeneracyabsolutely continuous measuresrotational symmetrydirectional smeariness
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The pith

For m≥4, support inside the geodesic ball of radius S_m around the Fréchet mean rules out smeariness of absolutely continuous measures on the sphere, while slightly larger supports allow explicit 2-smeary examples.

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

The paper determines sharp support thresholds that decide whether an absolutely continuous probability measure on the sphere can be smeary. Smeariness is produced by degeneracy of the Hessian of the Fréchet function and therefore prevents the classical central limit theorem and Wald-type inference from holding for the sample Fréchet mean. For rotationally symmetric densities the thresholds are tight: containment in the ball of radius S_m centered at the Fréchet mean forbids smeariness, yet for every ε>0 with S_m+ε<π there exist explicit rotationally symmetric 2-smeary densities supported inside the larger ball. Closed hemispherical support eliminates both full and directional smeariness for arbitrary densities.

Core claim

For rotationally symmetric densities full and directional smeariness coincide; the Hessian and fourth-order terms of the Fréchet function are controlled by two explicit geometry-dependent radii R_m<S_m. In dimensions m=2,3 such smeariness is impossible. For m≥4 support inside the geodesic ball of radius S_m centered at the Fréchet mean rules out smeariness, while for every ε>0 with S_m+ε<π explicit 2-smeary rotationally symmetric densities exist inside the ball of radius S_m+ε. Closed hemispherical support rules out both full and directional smeariness for general densities; support inside the closed ball of radius S_m already rules out full smeariness, and explicit directionally 2-smeary ex

What carries the argument

The pair of explicit geometry-dependent radii R_m<S_m that govern the sign of the Hessian and the fourth-order terms under rotational symmetry on the sphere.

If this is right

  • Rotationally symmetric smeariness cannot occur in dimensions m=2 or m=3.
  • Closed hemispherical support rules out both full and directional smeariness for general densities.
  • Support contained in the closed ball of radius S_m rules out full smeariness for general densities.
  • The explicit Hessian formulas supply a practical diagnostic for proximity to the non-classical regime.

Where Pith is reading between the lines

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

  • The same radii may mark the onset of smeariness for non-rotationally-symmetric densities once the rotational-symmetry assumption is dropped.
  • The diagnostic formulas could be used to decide, for a given sample, whether classical or non-classical asymptotics should be applied.
  • Directional smeariness may still permit certain one-dimensional projections of inference even when the full Hessian is degenerate.

Load-bearing premise

Smeariness occurs precisely when the Hessian of the Fréchet function degenerates.

What would settle it

A rotationally symmetric density supported strictly inside the ball of radius S_m whose Fréchet-function Hessian is nevertheless singular at the Fréchet mean.

Figures

Figures reproduced from arXiv: 2606.02144 by Susovan Pal.

Figure 1
Figure 1. Figure 1: Support-dependent behavior under rotational symmetry. If the radial support remains below the quartic threshold Sm, rotationally symmetric smeariness at N is ruled out; for every m ě 4, support extending past Sm allows the construction of rotationally symmetric 2-smeary densities. 5. Sharp support thresholds for smeariness under rotational symmetry We now combine the rotationally symmetric derivative formu… view at source ↗
Figure 2
Figure 2. Figure 2: below shows the functions bm and hm for m P t2, 3, 5, 10, 25, 50u [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
read the original abstract

We investigate support thresholds for fully smeary and directionally smeary absolutely continuous probability measures on the sphere \(\mathbb{S}^m\). The motivation is inferential: smeariness is caused by degeneracy of the Hessian of the Fr\'echet function, and such degeneracy can invalidate the classical central limit theorem (CLT) for Fr\'echet means and the corresponding Wald-type \(\chi^2\) inference. For rotationally symmetric densities, we show that full and directional smeariness are equivalent. The Hessian and fourth-order terms are governed by two explicit geometry-dependent radii \(R_m<S_m\). In dimensions \(m=2,3\), rotationally symmetric smeariness cannot occur. For \(m\ge4\), support contained in the geodesic ball of radius \(S_m\), centered at the Fr\'echet mean, rules out smeariness; conversely, for every \(\varepsilon>0\) with \(S_m+\varepsilon<\pi\), we construct examples of rotationally symmetric \(2\)-smeary densities supported in the ball of radius \(S_m+\varepsilon\). For general densities, closed hemispherical support rules out both full and directional smeariness. Support contained in the closed ball of radius \(S_m\) rules out full smeariness, while we construct explicit, directionally \(2\)-smeary examples supported in balls of radius \(\pi/2+\varepsilon\). As a byproduct, the explicit Hessian formulas in this paper also provide a practical diagnostic for detecting proximity to the Hessian-degenerate, non-classical regime.

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 / 3 minor

Summary. The paper establishes sharp geometric thresholds on the support of absolutely continuous probability measures on the sphere S^m that determine the presence or absence of full and directional smeariness (Hessian degeneracy of the Fréchet function). For rotationally symmetric densities it proves equivalence of the two notions of smeariness, identifies explicit radii R_m < S_m that control the Hessian and fourth-order terms, shows that smeariness is impossible for m=2,3, proves that support inside the geodesic ball of radius S_m rules out smeariness for m≥4, and supplies explicit 2-smeary constructions supported in any larger ball of radius S_m+ε<π. For general densities it shows that closed hemispherical support rules out both forms of smeariness, that support inside the closed ball of radius S_m rules out full smeariness, and constructs directionally 2-smeary examples supported in balls of radius π/2+ε. Explicit Hessian formulas are derived as a practical diagnostic.

Significance. If the derivations hold, the results supply the first geometrically explicit, dimension-dependent support conditions under which classical CLT and Wald-type inference for Fréchet means remain valid on spheres. The byproduct diagnostic formulas are immediately usable for applied work, and the explicit constructions demonstrate sharpness. These contributions strengthen the theoretical foundation for statistics on manifolds with constrained support.

minor comments (3)
  1. The introduction should state the explicit formulas for the radii R_m and S_m (currently referenced only by name) before the main theorems, so that the geometric meaning is clear from the outset.
  2. Notation for the Fréchet function and its Hessian should be introduced once in a preliminary section and then used consistently; occasional re-definition in later sections is unnecessary.
  3. A short remark on how the rotational-symmetry assumption can be relaxed or checked in practice would help readers apply the diagnostic formulas.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our results on support thresholds for smeariness, the assessment of their significance for inference on spheres, and the recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claims consist of explicit derivations of the geometry-dependent radii R_m < S_m from the sphere metric and rotational symmetry, together with direct constructions of 2-smeary densities and support-threshold rules that follow from the Hessian and fourth-order expansions of the Fréchet function. These steps are self-contained geometric calculations without fitted parameters renamed as predictions, without load-bearing self-citations, and without any reduction of the output to the input by definition. The stated equivalence of full and directional smeariness under symmetry is likewise obtained from the same explicit expansions rather than imported by ansatz or prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions from Riemannian geometry and Fréchet means on manifolds; no free parameters are fitted, no new entities are postulated, and the axioms invoked are background facts of differential geometry and probability on spheres.

axioms (2)
  • standard math The sphere S^m equipped with the geodesic distance is a complete Riemannian manifold on which the Fréchet function is twice differentiable away from the cut locus.
    Invoked to define the Hessian and to locate the critical radii R_m and S_m.
  • domain assumption Smeariness is defined precisely as degeneracy of the Hessian of the Fréchet function at the minimizer.
    Stated in the opening motivation paragraph as the inferential motivation.

pith-pipeline@v0.9.1-grok · 5805 in / 1643 out tokens · 31384 ms · 2026-06-28T12:17:19.064470+00:00 · methodology

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