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The Fisher Information Sector of Perelman's Functionals on Black Hole Spatial Slices

Information Geometry Meets Black Hole Physics

Author: Lark Laflamme

Date: April 2026

Status: Version 3.2 Final - Submission-ready

Abstract

We study the Fisher information component F_∇ of Perelman's F-functional on black hole spatial slices, using the heat kernel to determine the Boltzmann measure from first principles.

Mathematical results (no physical assumptions): On constant-curvature slices, the curvature component F_R is a constant independent of the black hole, and F_∇ is the unique carrier of state-dependent gravitational information (Theorem 1). On the BTZ spatial slice, exact computation gives F_∇ = r₊/ℓ².

Near-horizon structural result: F_∇ = 1/(2τ), where τ is the heat kernel diffusion parameter. The horizon area cancels; the result is dimension-independent.

Conditional geometric interpretation: If τ = 1/(2κ), then F_∇ = κ (surface gravity), and the Hawking temperature T = κ/(2π) follows without thermodynamic input.

1. Introduction

Perelman's Ricci flow functionals (2002) have a natural statistical mechanics structure: the F-functional decomposes into a curvature component F_R and a gradient component F_∇, where the latter is a Fisher information in the sense of Amari (1985). Despite significant interest in connecting Perelman's mathematics to gravitational physics, no explicit computation of these functionals on a black hole geometry has appeared in the literature.

This paper fills that gap. We compute Perelman's F and W on the BTZ spatial slice and discover a structural result: on constant-curvature backgrounds, the curvature component carries no state-dependent information, and the Fisher information component carries all of it.

2. Summary of Results

Tier 1: Mathematical Results (No Physical Assumptions)

Theorem 1 (Fisher Information Theorem): On a constant-curvature surface, F_R = R₀ (constant, independent of the Boltzmann measure and black hole parameters). All state dependence resides in F_∇.

The BTZ Computation: With the exact heat kernel: F_∇ = r₊/ℓ² and W = ½ln(r₊) + const.

Tier 2: Conditional on the Diffusion-Parameter Identification

Theorem 2 (Near-Horizon Formula): For the heat kernel centered at the horizon: F_∇ = 1/(2τ). This is dimension-independent; the horizon area cancels between numerator and normalization.

Proposition 1 (Surface Gravity): If τ = 1/(2κ) (motivated by Euclidean regularity, not derived from Perelman's framework), then F_∇ = κ. Temperature follows: T = κ/(2π).

Proposition 2 (Structural Cancellation): In the W-functional, τ · F_∇ = 1/2 (constant). This is a non-trivial structural constraint.

Tier 3: Interpretive Proposals

F_∇ provides an information-geometric characterization of surface gravity.
On constant-curvature backgrounds, state-dependent physics is encoded in the measure geometry, not the metric curvature.

3. Physical Significance

The central discovery is a division of labor within Perelman's functional: curvature sets the stage, but the measure gradient carries the state-dependent physics. On constant-curvature backgrounds — which include the BTZ black hole and the near-horizon region of any black hole — the curvature component F_R is a fixed constant. All information about which black hole we are looking at resides in the Fisher information F_∇.

The near-horizon formula F_∇ = 1/(2τ) is striking in its simplicity: the horizon area cancels, the result is dimension-independent, and F_∇ measures a local quantity (diffusion sharpness of the thermal measure) rather than a global one like entropy.

Significance: This paper provides the first explicit computation of Perelman's Ricci flow functionals on black hole geometry. The information-geometric content resides in the gradient sector of the Boltzmann measure, suggesting a deep connection between Fisher information geometry and black hole thermodynamics that operates through the measure, not through the curvature.