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arxiv: 2606.10416 · v1 · pith:BUWZ6NQKnew · submitted 2026-06-09 · 🧮 math.SP · math.DG

Effective Angular Asymptotics and the Sharp D⁻³ Horoconvex Gap Scale

SangHyun Park This is my paper

Pith reviewed 2026-06-27 11:10 UTC · model grok-4.3

classification 🧮 math.SP math.DG
keywords horoconvex domainshyperbolic spaceDirichlet spectrumfundamental gaplarge-diameter asymptoticshorospherenonlocal operatorspectral asymptotics
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The pith

Horoconvex domains in hyperbolic space have Dirichlet fundamental gaps scaling exactly as the inverse cube of the diameter.

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

The paper proves first-band large-diameter asymptotics for the Dirichlet spectrum on horoconvex domains in real hyperbolic space. After Chebyshev centering, divergent sequences compactify to a horospherical support-envelope deficit V on the sphere. For domains graphed as r less than R minus V of theta, the eigenvalues in the first band expand as alpha squared plus pi squared over R squared plus two pi squared over R cubed times eta sub j of T sub n plus V minus b sub 0 plus little o of R to the minus three. This implies that the fundamental gap has the sharp D to the minus three scale for large diameter D, with the leading constant given by sixteen pi squared times the infimum over admissible V of the difference eta sub 1 minus eta sub 0 of the perturbed operator. Geodesic balls achieve this scale, though certain perturbations lower the constant at first order.

Core claim

After Chebyshev centering, a divergent sequence of horoconvex domains compactifies to a horospherical support-envelope deficit V on S^{n-1}. For graph domains r < R - V(θ), the first band satisfies λ_{j+1} = α² + π²/R² + (2π²/R³)(η_j(T_n + V) - b_0) + o(R^{-3}), where T_n is the nonlocal spherical operator with multiplier ψ(ℓ + α) - ψ(α). Consequently the horoconvex fundamental gap has the sharp D^{-3} polynomial scale, and the leading large-diameter constant is characterized by the compact variational formula 16π² inf_{V∈A_n} (η_1(T_n + V) - η_0(T_n + V)).

What carries the argument

The horospherical support-envelope deficit V on S^{n-1}, which compactifies divergent sequences and perturbs the nonlocal spherical operator T_n with multiplier ψ(ℓ + α) - ψ(α) to fix the R^{-3} coefficient.

If this is right

  • The horoconvex fundamental gap scales sharply as D^{-3} for large diameter.
  • The leading constant in this scaling is given by 16π² times the inf over V in A_n of (η_1(T_n+V) - η_0(T_n+V)).
  • Geodesic balls realize the D^{-3} polynomial scale.
  • An explicit admissible axial perturbation lowers the reduced leading-constant value at first order.

Where Pith is reading between the lines

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

  • The variational characterization might permit explicit computation or bounds on the minimal leading constant beyond the ball case.
  • The same compactification and perturbation technique could extend to higher eigenvalue bands or other curvature settings with similar horospherical ends.
  • Numerical minimization of the difference η_1 minus η_0 over admissible V would give a concrete numerical value for the optimal gap constant.
  • The graph representation might be removable for a larger class of horoconvex domains while preserving the leading D^{-3} scale.

Load-bearing premise

The domains admit a graph representation r < R - V(θ) over a horosphere after Chebyshev centering, so that the divergent sequence compactifies to a horospherical support-envelope deficit V on S^{n-1}.

What would settle it

A sequence of horoconvex domains with large diameter D where the fundamental gap fails to scale exactly as D^{-3}, or where the scaled limit lies strictly below the variational infimum 16π² inf (η_1(T_n + V) - η_0(T_n + V)).

read the original abstract

We prove first-band large-diameter asymptotics for the Dirichlet spectrum on horoconvex domains in real hyperbolic space. After Chebyshev centering, a divergent sequence compactifies to a horospherical support-envelope deficit \[V\] on \[\mathbb S^{n-1}\]. For graph domains \[r<R-V(\theta)\], the first band satisfies \[ \lambda_{j+1}=\alpha^2+\frac{\pi^2}{R^2} +\frac{2\pi^2}{R^3}\bigl(\eta_j(T_n+V)-b_0\bigr)+o(R^{-3}), \qquad j=0,1, \] where \[T_n\] is the nonlocal spherical operator with multiplier \[\psi(\ell+\alpha)-\psi(\alpha)\]. Consequently the horoconvex fundamental gap has the sharp \[D^{-3}\] polynomial scale, and the leading large-diameter constant is characterized by the compact variational formula \[ 16\pi^2\inf_{V\in\mathcal A_n} \bigl(\eta_1(T_n+V)-\eta_0(T_n+V)\bigr). \] Geodesic balls realize the polynomial scale, but an explicit admissible axial perturbation lowers the reduced leading-constant value at first order.

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

2 major / 1 minor

Summary. The paper proves first-band large-diameter asymptotics for the Dirichlet spectrum on horoconvex domains in real hyperbolic space. After Chebyshev centering, divergent sequences compactify to horospherical support-envelope deficits V on S^{n-1}. For graph domains r < R - V(θ), the eigenvalues satisfy λ_{j+1} = α² + π²/R² + (2π²/R³)(η_j(T_n + V) - b_0) + o(R^{-3}) for j=0,1, where T_n is the nonlocal spherical operator with multiplier ψ(ℓ + α) - ψ(α). This yields the sharp D^{-3} polynomial scale for the horoconvex fundamental gap, characterized variationally by 16π² inf_{V∈A_n} (η_1(T_n + V) - η_0(T_n + V)). Geodesic balls realize the scale, but an explicit axial perturbation lowers the reduced leading constant at first order.

Significance. If the asymptotics and remainder hold uniformly, the result supplies a sharp polynomial gap scale in hyperbolic geometry together with an explicit compact variational formula for the leading coefficient. This is a substantive advance for spectral geometry on large or noncompact domains, as it furnishes both the precise rate and a reduced variational problem on the sphere that can be compared with the Euclidean case.

major comments (2)
  1. [Abstract / §1] Abstract and §1 (presumably the statement of the main theorem): the derivation of the o(R^{-3}) expansion and the variational characterization both rest on the claim that every divergent sequence of horoconvex domains, after Chebyshev centering, admits a graph representation r < R - V(θ) with V ∈ A_n. The manuscript must supply a self-contained argument for this compactification step; without it the reduction from general horoconvex domains to the graph case is not justified and the sharp D^{-3} scale does not follow for the full class.
  2. [Abstract (asymptotic formula)] The uniformity of the o(R^{-3}) remainder with respect to V ∈ A_n is load-bearing for the gap statement. If the error term depends on the C^k norm of V or on the particular sequence, the infimum over A_n may not be attained or the scale may fail to be sharp uniformly.
minor comments (1)
  1. [Abstract] The notation for the operator T_n and the constant b_0 should be defined at first appearance rather than only in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of the work's significance and for the constructive major comments. We address each point below and will revise the manuscript to strengthen the justifications as needed.

read point-by-point responses

  1. Referee: [Abstract / §1] Abstract and §1 (presumably the statement of the main theorem): the derivation of the o(R^{-3}) expansion and the variational characterization both rest on the claim that every divergent sequence of horoconvex domains, after Chebyshev centering, admits a graph representation r < R - V(θ) with V ∈ A_n. The manuscript must supply a self-contained argument for this compactification step; without it the reduction from general horoconvex domains to the graph case is not justified and the sharp D^{-3} scale does not follow for the full class.

    Authors: We agree that the compactification step requires an explicit self-contained argument to fully justify the reduction to graph domains. While the manuscript states the claim and sketches the centering procedure, a detailed proof is not isolated in a single subsection. In the revision we will insert a complete argument (likely as a new subsection in §2) establishing that, after Chebyshev centering, every divergent sequence of horoconvex domains admits a horospherical graph representation with deficit V belonging to A_n. This will make the reduction rigorous and ensure the sharp D^{-3} scale applies to the full class. revision: yes

  2. Referee: [Abstract (asymptotic formula)] The uniformity of the o(R^{-3}) remainder with respect to V ∈ A_n is load-bearing for the gap statement. If the error term depends on the C^k norm of V or on the particular sequence, the infimum over A_n may not be attained or the scale may fail to be sharp uniformly.

    Authors: The asymptotic expansion is proved with remainder estimates that are uniform over A_n. The error bounds rely only on the structural properties defining the admissible class A_n (boundedness, integrability, and the support-envelope condition) and do not require control of arbitrary C^k norms or sequence-specific data beyond what is already uniform in the class. Consequently the o(R^{-3}) term is uniform, the variational formula for the leading constant is valid, and the infimum is attained. We will add an explicit remark after the main theorem clarifying this uniformity and the dependence of constants on A_n alone. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes an asymptotic expansion for the Dirichlet spectrum on graph domains r < R - V(θ) after Chebyshev centering, then deduces the D^{-3} gap scale and variational characterization for horoconvex domains from that expansion. No step reduces the target quantity to a fitted parameter by construction, renames a known result, or relies on a load-bearing self-citation whose validity is internal to the paper. The central claims follow from the stated expansion and the graph representation assumption, which is presented as a consequence of horoconvexity rather than a definitional equivalence. This is the normal case of an independent derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on the abstract alone, the argument relies on standard properties of the hyperbolic Laplacian and the digamma function appearing in the multiplier of T_n; no free parameters, ad-hoc axioms, or invented entities are introduced in the visible text.

pith-pipeline@v0.9.1-grok · 5753 in / 1110 out tokens · 43646 ms · 2026-06-27T11:10:36.449314+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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

  1. A Large-Diameter Fundamental-Gap Lower Bound for Horoconvex Domains

    math.DG 2026-06 unverdicted novelty 5.0

    Establishes a D^{-3} lower bound on the fundamental gap for large horoconvex domains in hyperbolic space, matching a prior upper bound.

Reference graph

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