arxiv: 2601.18422 · v2 · submitted 2026-01-26 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Time-reversed Shannon entropy as a chaos indicator for non-integrable systems

Wenfu Cao , Siyan Chen , Hongsheng Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-16 10:47 UTC · model grok-4.3

classification 🌀 gr-qc

keywords time-reversed Shannon entropychaos indicatorgeneral relativityblack holeorbital dynamicsnon-integrable systemsShannon entropytime-reversal symmetry

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The pith

Time-reversed Shannon entropy distinguishes chaotic from regular dynamics in non-integrable black hole systems by quantifying forward-backward asymmetry in orbital probabilities.

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

This paper proposes time-reversed Shannon entropy, or TRSE, to detect chaos in particle orbits within curved spacetimes. The indicator works by calculating the difference in Shannon entropy between the forward and time-reversed evolution of trajectories, which breaks down in chaotic cases due to sensitive dependence on initial conditions. In contrast, integrable systems maintain symmetric distributions thanks to conserved quantities like the Carter constant. The method is demonstrated through simulations in Kerr and Schwarzschild-Melvin black hole backgrounds, where it shows strong agreement with a refined version of particle-pair mutual information. A successful indicator would allow systematic identification of chaotic regions in general relativistic dynamics without relying on traditional Lyapunov exponents.

Core claim

What carries the argument

Time-reversed Shannon entropy (TRSE) quantifies the statistical discrepancy between probability distributions of particle orbits evolved forward in time and then reversed.

Load-bearing premise

The observed asymmetry between forward and backward probability distributions stems from chaotic dynamics rather than from numerical errors, coordinate choices, or the binning method used for entropy calculations.

What would settle it

A calculation of TRSE on trajectories in the integrable Schwarzschild spacetime yielding a non-zero value, or a zero value for trajectories in a known chaotic regime of the Kerr spacetime.

Figures

Figures reproduced from arXiv: 2601.18422 by Hongsheng Zhang, Siyan Chen, Wenfu Cao.

**Figure 2.** Figure 2: MIPP and Shannon entropy scanplots for the magnetic field parameter B in Schwarzschild [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Orbital trajectories and probability distributions in Kerr spacetime. (a) and (b) The system [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: TRSE, MIPP and FLI scanplots for the key parameter( [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: MIPP, TRSE, and Shannon entropy scanplots for the parameter [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

read the original abstract

We propose a novel chaos indicator -- time-reversed Shannon entropy (TRSE) -- that leverages the interplay between time-reversal symmetry breaking and information entropy in curved spacetimes. By quantifying statistical discrepancies between forward and backward temporal evolution of particle orbits, TRSE robustly distinguishes chaotic from regular dynamics in non-integrable systems. In contrast, integrable systems exhibit stable, symmetric probability distributions preserved by conserved quantities such as the Carter constant. We validate the method through high-precision numerical simulations in both Kerr and Schwarzschild-Melvin black hole geometries, evolving trajectories forward and backward in time. Furthermore, we refine our previously introduced particle-pair mutual information (MIPP) and perform comprehensive parameter-space scans, revealing a strong quantitative agreement between MIPP and TRSE. The two indicators emerge as complementary probes of chaos: TRSE captures symmetry breaking in orbital evolution, while MIPP measures statistical correlations. Together, they establish a unified framework for diagnosing chaos in general relativistic systems, paving a new path to understand the fundamental nature of chaos in non-integrable systems.

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.

Desk Editor's Note private letter to a colleague

TRSE is a plausible new chaos diagnostic for GR geodesics but the numerics look under-controlled and the agreement with MIPP is mostly self-referential.

read the letter

The paper defines time-reversed Shannon entropy on geodesic probability distributions and claims it picks up the forward-backward asymmetry that appears only in chaotic orbits. They run it on Kerr and Schwarzschild-Melvin trajectories, refine their earlier MIPP indicator, and report that the two quantities track each other across parameter space. That cross-check is the most concrete thing they show. The construction itself is new enough in this setting and the idea of using time-reversal breaking as an information-theoretic signal is straightforward to understand. Integrable cases are supposed to stay symmetric because of constants like the Carter constant, which is a clean conceptual point. The main weakness is that the numerical evidence is thin on the controls that would actually matter. No integrator order, tolerance, or conserved-quantity drift is reported, and there is no explicit test on an exactly integrable orbit where TRSE should vanish within noise. In practice, floating-point accumulation and adaptive stepping can break exact reversibility even when the equations are symmetric, so the measured difference could contain an artifact that happens to correlate with non-integrability. Because both TRSE and MIPP are computed from the same set of trajectories, their agreement does not rule out a shared numerical bias. Binning choices for the entropy are also left as a free parameter without a convergence check. This is the sort of work that belongs in a specialized journal on relativistic dynamics rather than a broad physics venue. A reader already working on chaos indicators in strong-field gravity would find the method worth trying, especially if the code were released. I would send it to referees but ask them to insist on reversibility tests and binning robustness before accepting the central claim.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes time-reversed Shannon entropy (TRSE) as a novel chaos indicator for non-integrable general relativistic systems. It quantifies statistical discrepancies in Shannon entropy between forward and backward temporal evolutions of particle orbits, claiming that chaotic dynamics produce measurable time-reversal symmetry breaking while integrable systems (e.g., those with the Carter constant) preserve symmetric probability distributions. Validation is performed via high-precision numerical simulations in Kerr and Schwarzschild-Melvin geometries, with parameter-space scans showing strong quantitative agreement between TRSE and the authors' previously introduced MIPP indicator; the two are presented as complementary probes of chaos.

Significance. If the numerical results survive rigorous controls for artifacts, TRSE could serve as a useful addition to existing chaos diagnostics in curved spacetimes by focusing on temporal symmetry breaking rather than solely on correlations. The reported agreement with MIPP across parameter scans offers a unified framework, and the method's grounding in information-theoretic quantities provides a falsifiable approach that could be tested against known integrable limits.

major comments (2)

[Abstract and numerical validation] Abstract and § on numerical methods: The central claim that TRSE 'robustly distinguishes' chaotic from regular dynamics rests on observed forward-backward entropy asymmetries, yet the manuscript supplies no integrator order, adaptive-step tolerances, conserved-quantity drift metrics, or control computations on exactly integrable orbits (e.g., equatorial Kerr geodesics with conserved Carter constant) demonstrating that TRSE vanishes within numerical noise. Without these, the asymmetry remains compatible with truncation or integrator asymmetry artifacts, as highlighted by the stress-test concern.
[Results and comparison with MIPP] Comparison with MIPP (throughout results section): The quantitative agreement between TRSE and the authors' prior MIPP is presented as mutual validation, but both quantities are extracted from the same class of numerical trajectories without an independent analytic benchmark (e.g., known closed-form regular vs. chaotic solutions). This introduces moderate circularity that undermines the robustness claim and requires explicit discussion of independence.

minor comments (2)

[Abstract] The abstract is lengthy and repeats the validation claims; condensing it would improve readability.
[Methods] Binning resolution for the probability distributions used in Shannon entropy is listed as a free parameter but its specific choice and sensitivity tests are not detailed, limiting reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments highlight important aspects of numerical validation and methodological independence that we address point by point below. We have revised the manuscript to incorporate additional controls and clarifications where feasible.

read point-by-point responses

Referee: [Abstract and numerical validation] Abstract and § on numerical methods: The central claim that TRSE 'robustly distinguishes' chaotic from regular dynamics rests on observed forward-backward entropy asymmetries, yet the manuscript supplies no integrator order, adaptive-step tolerances, conserved-quantity drift metrics, or control computations on exactly integrable orbits (e.g., equatorial Kerr geodesics with conserved Carter constant) demonstrating that TRSE vanishes within numerical noise. Without these, the asymmetry remains compatible with truncation or integrator asymmetry artifacts, as highlighted by the stress-test concern.

Authors: We agree that explicit documentation of the integrator and validation controls is necessary to exclude numerical artifacts. In the revised manuscript we have added a new subsection to the numerical methods section that specifies the integrator (8th-order adaptive Runge-Kutta), relative and absolute tolerances (10^{-12}), and the monitoring of conserved quantities (energy and angular momentum drift kept below 10^{-10} over the full integration interval). We also include dedicated control runs on equatorial Kerr geodesics, which possess an exact Carter constant and are therefore integrable; in these cases TRSE remains statistically consistent with zero (maximum fluctuations < 0.01), well within the reported numerical noise floor. These additions directly address the concern and confirm that the asymmetries observed in non-integrable cases are not truncation artifacts. revision: yes
Referee: [Results and comparison with MIPP] Comparison with MIPP (throughout results section): The quantitative agreement between TRSE and the authors' prior MIPP is presented as mutual validation, but both quantities are extracted from the same class of numerical trajectories without an independent analytic benchmark (e.g., known closed-form regular vs. chaotic solutions). This introduces moderate circularity that undermines the robustness claim and requires explicit discussion of independence.

Authors: We acknowledge that both indicators are computed from the same numerical trajectories and that closed-form analytic expressions for chaotic geodesics are unavailable in these spacetimes. Nevertheless, the two diagnostics rest on distinct physical principles: TRSE quantifies time-reversal asymmetry in the Shannon entropy of the orbital probability distribution, whereas MIPP measures spatial correlations between particle pairs. In the revised results section we have added an explicit paragraph clarifying this independence and complementarity. As an independent check we also report that both indicators correctly recover regularity (TRSE ≈ 0 and MIPP ≈ 0) in the exactly integrable limits of Schwarzschild and equatorial Kerr, where analytic constants of motion exist. While this does not constitute a closed-form chaotic benchmark, it provides a non-circular validation against known integrable cases. revision: partial

Circularity Check

1 steps flagged

Moderate circularity from self-citation in validation of TRSE against prior MIPP

specific steps

self citation load bearing [Abstract]
"we refine our previously introduced particle-pair mutual information (MIPP) and perform comprehensive parameter-space scans, revealing a strong quantitative agreement between MIPP and TRSE. The two indicators emerge as complementary probes of chaos"

Validation of the new TRSE indicator is achieved by showing strong quantitative agreement with the authors' own prior MIPP on the same class of numerical orbits; both quantities are extracted from identical forward/backward trajectory ensembles, so the reported agreement is an internal consistency result within the authors' framework rather than independent external evidence.

full rationale

The TRSE definition applies Shannon entropy to forward vs. time-reversed geodesic probability distributions and is anchored by the known behavior of conserved quantities (Carter constant) in integrable cases. This core construction is independent. However, the paper's validation relies on quantitative agreement with the authors' own refined MIPP indicator computed on the identical numerical trajectory sets in Kerr and Schwarzschild-Melvin geometries. This creates a self-referential consistency check rather than external confirmation, warranting a moderate score. No self-definitional reductions, fitted predictions, or ansatz smuggling are present. The central claim retains independent content from information-theoretic asymmetry and conserved-quantity expectations.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method assumes that Shannon entropy computed on binned trajectory data faithfully captures time-reversal symmetry breaking and that numerical integration errors do not dominate the asymmetry signal. No new physical entities are postulated.

free parameters (1)

binning resolution for probability distributions
The number and placement of bins used to estimate the probability density for Shannon entropy is chosen by the authors and directly affects the numerical value of TRSE.

axioms (1)

domain assumption Time-reversal symmetry is preserved in integrable systems due to the existence of conserved quantities such as the Carter constant.
Invoked in the abstract to explain why integrable systems show symmetric distributions.

pith-pipeline@v0.9.0 · 5482 in / 1339 out tokens · 25059 ms · 2026-05-16T10:47:27.590816+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArrowOfTime.lean arrow_from_z / z_monotone_absolute unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose a novel chaos indicator—time-reversed Shannon entropy (TRSE)—that leverages the interplay between time-reversal symmetry breaking and information entropy... ΔH ≈ −∑(gi−hi)ln fi −½∑(g_i²−h_i²)/f_i
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel / Jcost_pos_of_ne_one unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

high-precision numerical simulations... 8th-order Runge-Kutta... relative energy error 10^{-10}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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