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Ranking Prompt--Plateau Correlations in Gamma-Ray Bursts with Symbolic Regression

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Fatemeh Rastegarnia, Rahim Moradi, Brwa Shukur Ismael, Mohsen Asaadi, Yu Wang

High Energy Astrophysical Phenomena · Submitted 16 Oct 2026 · 295 references

Pith's verdict: Competent symbolic-regression study that systematically ranks GRB plateau correlations and recovers a prompt-plateau plane; limitations on the intrinsic claim are clearly stated and honestly scoped.

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arXiv ID2610.23155 (version 1)
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CitationBrwa Shukur Ismael, Fatemeh Rastegarnia, Mohsen Asaadi, Rahim Moradi, Yu Wang. "Ranking Prompt--Plateau Correlations in Gamma-Ray Bursts with Symbolic Regression." arXiv:2610.23155 (2026). https://pith.science/paper/PGX4YJOO
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The paper

The paper claims

Symbolic regression reveals a tighter GRB plateau relation linking prompt and afterglow emission.

The reading

This paper uses symbolic regression to search over 377 variable combinations for the best low-dimensional description of gamma-ray burst (GRB) plateau luminosity. It finds that the classical Dainotti relation—an anticorrelation between plateau luminosity and duration—is not the tightest: a plane that also includes the prompt peak luminosity performs significantly better. The discovery is robust across searches, uncertainty-aware refits, and distance-controlled tests, though the authors caution that selection effects and luminosity evolution have not been fully corrected. If the result holds, it means the plateau phase carries information about the prompt energy release, and the classical Dainotti relation is a projection of a more fundamental prompt–plateau connection.

What carries the argument

Symbolic regression, implemented via the PySR framework (Cranmer 2023), searches over mathematical expressions built from $+, -, \times, /, \exp, \log, \sqrt{}$ without assuming a functional form. It evaluates each candidate on a held-out validation split, ranking feature subsets by validation mean absolute error. This allows the authors to ask not just what the best fit is, but how the classical Dainotti relation ranks among all low-dimensional alternatives. The method is used in a Dainotti-anchored search (where $\log T_a^{\rm rest}$ is always present) and an unrestricted search over all non-empty subsets of at most three of the 13 candidate variables.

Extended reading notes

Core claim

The central claim is that the X-ray plateau luminosity $L_a$ depends on both the prompt peak luminosity $L_{\rm peak}$ and the rest-frame plateau duration $T_a^{\rm rest}$, not on $T_a^{\rm rest}$ alone. Symbolic regression ranks the two-variable relation $\log L_a \approx 0.855 \log L_{\rm peak} - 0.782 \log T_a^{\rm rest}$ as the best among 377 candidate spaces, with validation $R^2 \approx 0.88$. Direct regression yields $\log L_a = 4.718 + 0.776 \log L_{\rm peak} - 0.805 \log T_a^{\rm rest}$, with $R_P = 0.91$ for 246 bursts. After removing the shared luminosity-distance dependence (which both $L_a$ and $L_{\rm peak}$ inherit from $D_L^2$), the relation retains $R^2 \approx 0.62$, consistent with an intrinsic physical component. The classical Dainotti relation ($\log L_a$ vs. $\log T_a^{\rm rest}$ alone) ranks within the top $\sim25\%$ of the search spaces, confirming it is competitive but not the tightest.

Load-bearing premise

The apparent intrinsic component of the prompt–plateau relation is estimated by removing only the linear dependence on luminosity distance $D_L$, not by a full Efron–Petrosian treatment of sample truncation and luminosity evolution; if those biases are significant, the residual correlation could be an artifact rather than a true physical relation.

Editorial extensions

If this is right

  • The Dainotti relation is shown to be a two-dimensional projection of a three-dimensional prompt–plateau plane, placing earlier one-dimensional fits in a broader context.
  • Plateau luminosity carries information about the prompt energy release, meaning that any model of the plateau phase must couple to the central engine activity that powers the prompt emission.
  • Additional spectral, absorption, and afterglow-shape variables (photon index, column density, decay indices) do not improve the plateau luminosity prediction beyond the prompt–plateau pair, suggesting the dominant low-dimensional physics is captured by these two observables.
  • The prompt–plateau plane may help standardize GRBs for cosmological distance measurements, analogous to how the Dainotti relation has been used, potentially with lower scatter.
  • The residual trend after removing shared distance dependence ($R^2 \approx 0.62$) supports an intrinsic component, but a full Efron–Petrosian truncation analysis is needed to separate it from evolution and selection effects.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Symbolic regression could be applied to other GRB correlations (e.g., the Amati relation, the Yonetoku relation) to test whether they are projections of higher-dimensional planes, or whether the apparent correlation changes when additional variables are considered.
  • The fact that the ratio $L_a T_a / L_{\rm peak}$ is not constant (it increases with $T_a$) argues against simple constant-efficiency energy injection models; future theoretical work should explore prompt–plateau coupling with variable conversion efficiency.
  • The class-dependent fits (XRR, LGRB, SNe-GRB, short/SEE) show the same sign pattern, hinting that the prompt–plateau link may be universal across GRB subclasses, though small-number statistics prevent a definitive test here.
Original abstract

Gamma-ray burst (GRB) correlations probe central engines, relativistic outflows, and radiation processes. The Dainotti relation is an anticorrelation between the X-ray plateau luminosity $L_a$ and rest-frame plateau duration $T_a^{\rm rest}$. We use symbolic regression to compare it with other low-dimensional correlations in 251 GRBs with Swift-XRT plateau properties and Swift-BAT prompt-emission parameters. We perform a Dainotti-guided search, in which $\log T_a^{\rm rest}$ is always present, and an unrestricted search over subsets of at most three variables. The guided search recovers the Dainotti dependence; a D'Agostini fit gives $\log L_a=(51.92\pm0.30)-(1.31\pm0.09)\log T_a^{\rm rest}$, with intrinsic scatter $\sigma_{\rm int}=0.82\pm0.05$ dex. In the unrestricted ranking of 377 feature subsets, the classical Dainotti relation remains competitive, within the top $\sim25\%$ of the tested search spaces. The highest-ranked screened expression is $\log L_a\approx0.855\log L_{\rm peak}-0.782\log T_a^{\rm rest}$, with RMSE $\approx0.46$ and $R^2\approx0.88$. The highest-ranked runs repeatedly reduce to this prompt--plateau structure, without retaining additional spectral, absorption, or afterglow-shape variables. A direct regression gives $\log L_a=4.718+0.776\log L_{\rm peak}-0.805\log T_a^{\rm rest}$, with $R_{\rm P}=0.91$ for 246 bursts. An errors-in-variables refit gives consistent coefficients. After removing shared luminosity-distance dependence, the relation retains non-negligible explanatory power ($R^2\approx0.62$), consistent with a possible intrinsic component. Restricted Monte Carlo null tests give $p\approx0.01$. Plateau luminosity therefore shows a coupled dependence on prompt luminosity and plateau duration.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-20: 1 request, 9,878 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.54 · human participants 0.01 · controlled comparison 0.19 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.91 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Gamma-ray burst (GRB) correlations probe central engines, relativistic outflows, and radiation processes. The Dainotti relation is an anticorrelation between the X-ray plateau luminosity $L_a$ and rest-frame plateau duration $T_a^{\rm rest}$. We use symbolic regression to compare it with other low-dimensional correlations in 251 GRBs with Swift-XRT plateau properties and Swift-BAT prompt-emission parameters. We perform a Dainotti-guided search, in which $\log T_a^{\rm rest}$ is always present, and an unrestricted search over subsets of at most three variables. The guided search recovers the Dainotti dependence; a D'Agostini fit gives $\log L_a=(51.92\pm0.30)-(1.31\pm0.09)\log T_a^{\rm rest}$, with intrinsic scatter $\sigma_{\rm int}=0.82\pm0.05$ dex. In the unrestricted ranking of 377 feature subsets, the classical Dainotti relation remains competitive, within the top $\sim25\%$ of the tested search spaces. The highest-ranked screened expression is $\log L_a\approx0.855\log L_{\rm peak}-0.782\log T_a^{\rm rest}$, with RMSE $\approx0.46$ and $R^2\approx0.88$. The highest-ranked runs repeatedly reduce to this prompt--plateau structure, without retaining additional spectral, absorption, or afterglow-shape variables. A direct regression gives $\log L_a=4.718+0.776\log L_{\rm peak}-0.805\log T_a^{\rm rest}$, with $R_{\rm P}=0.91$ for 246 bursts. An errors-in-variables refit gives consistent coefficients. After removing shared luminosity-distance dependence, the relation retains non-negligible explanatory power ($R^2\approx0.62$), consistent with a possible intrinsic component. Restricted Monte Carlo null tests give $p\approx0.01$. Plateau luminosity therefore shows a coupled dependence on prompt luminosity and plateau duration.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Gamma-ray burst (GRB) correlations probe central engines, relativistic outflows, and radiation processes. background 1.00 0.05 0.02 asserted 1.00
s1 The Dainotti relation is an anticorrelation between the X-ray plateau luminosity $L_a$ and rest-frame plateau duration $T_a^{\rm rest}$. background 1.00 0.05 0.03 asserted 1.00
s2 We use symbolic regression to compare it with other low-dimensional correlations in 251 GRBs with Swift-XRT plateau properties and Swift-BAT prompt-emission parameters. method 0.77 0.94 0.04 asserted 0.97
s3 We perform a Dainotti-guided search, in which $\log T_a^{\rm rest}$ is always present, and an unrestricted search over subsets of at most three variables. method 0.99 0.94 0.03 asserted 0.94
s4 The guided search recovers the Dainotti dependence; a D'Agostini fit gives $\log L_a=(51.92\pm0.30)-(1.31\pm0.09)\log T_a^{\rm rest}$, with intrinsic scatter $\sigma_{\rm int}=0.82\pm0.05$ dex. result 1.00 0.95 0.11 asserted 1.00
s5 In the unrestricted ranking of 377 feature subsets, the classical Dainotti relation remains competitive, within the top $\sim25\%$ of the tested search spaces. result 1.00 0.95 0.09 asserted 1.00
s6 The highest-ranked screened expression is $\log L_a\approx0.855\log L_{\rm peak}-0.782\log T_a^{\rm rest}$, with RMSE $\approx0.46$ and $R^2\approx0.88$. result 1.00 0.95 0.17 asserted 1.00
s7 The highest-ranked runs repeatedly reduce to this prompt--plateau structure, without retaining additional spectral, absorption, or afterglow-shape variables. result 0.99 0.93 0.12 asserted 1.00
s8 A direct regression gives $\log L_a=4.718+0.776\log L_{\rm peak}-0.805\log T_a^{\rm rest}$, with $R_{\rm P}=0.91$ for 246 bursts. result 0.99 0.94 0.12 asserted 1.00
s9 An errors-in-variables refit gives consistent coefficients. result 0.77 0.89 0.03 asserted 1.00
s10 After removing shared luminosity-distance dependence, the relation retains non-negligible explanatory power ($R^2\approx0.62$), consistent with a possible intrinsic component. result 0.84 0.93 0.10 hedged possibility 0.95
s11 Restricted Monte Carlo null tests give $p\approx0.01$. result 0.93 0.92 0.04 asserted 0.99
s12 Plateau luminosity therefore shows a coupled dependence on prompt luminosity and plateau duration. interpretation 0.89 0.87 0.84 asserted 1.00

What kind of paper the classifier reports this is: empirical study 0.91 · evidence from analysis of existing data 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Ranking of all 377 unrestricted feature-subset search spaces explored in the search, sorted by validation MAE (lower is better). The pure symbolic Dainotti model and the classical linear Dainotti baseline are highlighted relative to the best-performing symbolic relation. The plot also includes a literature fundamental-plane baseline using the slopes reported by Dainotti et al. (2023); only the intercept is re-normalized on the present training split because the prompt-luminosity calibration differs from that work. The Dainotti relation falls within the top ∼ 25% of the ranking,…
Figure 2. Feature usage among the top 20 MAE-ranked models from the unrestricted symbolic search. This statistic refers only to the unrestricted search, where log T rest a was not forced into the candidate models, and should be read as a descriptive top-20 diagnostic rather than as a formal threshold test. Prompt peak luminosity appears in all of these top-20 relations, suggesting that it is the variable most closely associated with the tightest plateau-luminosity correlations. By contrast, log T rest a appears less often, consistent with the idea that the preferred relations involve both…
Figure 3. Raw and distance-controlled prompt–plateau planes. Left: the plane in the space of log Lpeak, log T rest a , and log La, with the best-fit surface log La = 4.718 + 0.776 log Lpeak − 0.805 log T rest a . Right: the same relation after removing the shared luminosity-distance dependence. The plane remains visible in residual space, leaving room for an intrinsic component in addition to distance-driven covariance. 42 44 46 48 50 logLa observed 42 44 46 48 50 logLa predicted -2 -1 0 1 2 logLa residual -2 -1 0 1 2 logLa residual predicted
Figure 4. Predicted versus observed plateau luminosity for the explicit prompt–plateau plane fit. Left: raw plane prediction with RMSE ≈ 0.535 and R 2 ≈ 0.824. Right: distance-controlled prediction with RMSE ≈ 0.569 and R 2 ≈ 0.622. The three￾parameter plane remains visible after removing shared luminosity-distance scaling, although with more scatter. Horizontal bars show the catalog uncertainty in log La, and vertical bars show the propagated uncertainty from log Lpeak and log T rest a through the fitted plane
Figure 5. Plateau energy proxy, Eplateau = LaT rest a , versus prompt peak luminosity. Left: raw relation showing a positive OLS correlation with slope 0.704 ± 0.036, RP = 0.778, and R 2 ≈ 0.605. Right: distance-controlled relation after removing the shared luminosity-distance dependence. The correlation weakens substantially (RP = 0.499, R 2 ≈ 0.249), which indicates that part of the raw trend comes from cosmological scaling. The remaining trend is consistent with a weaker prompt–plateau energetics link. 2 3 4 5 logT rest a -9 -8 -7 -6 -5 log(LaT rest a /Lpeak) Mean = -7.784 Scatter = 0.620…
Figure 6. Diagnostic test of the symbolic relation using the ratio log(LaT rest a /Lpeak) as a function of log T rest a . A constant-ratio model (horizontal dashed line) would corre￾spond to La ∝ Lpeak/Ta. The data instead show substantial scatter (σ ≈ 0.62) and an OLS residual trend with slope 0.337 ± 0.049 (RP ≈ 0.40, R 2 ≈ 0.16). The ratio is there￾fore not constant, supporting a prompt–plateau description beyond a one-dimensional constant-ratio scaling. sion with B = −0.85 ± 0.12 and C = 0.49 ± 0.13. In the same notation, our explicit regression plane in Eq. 8 has B = −0.805 for log T
Figure 7. Monte Carlo null-test distributions for validation performance. Left: best validation MAE from shuffled datasets. Right: best validation RMSE. In both panels, the vertical dashed line marks the refined real-data relation used for this comparison (MAE ≈ 0.367, RMSE ≈ 0.475), which lies far outside the restricted null distributions. The null models cluster at much worse values (MAE ∼ 0.9, RMSE ∼ 1.2), indicating that the recovered relation is not reproduced by these random permutations of the data. 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Best distance-controlled MAE in null realizations 0 2 4…
Figure 8. Monte Carlo null-test distributions for distance-controlled performance. Left: best distance-controlled MAE. Right: best distance-controlled RMSE. The refined real-data relation used for this comparison (MAE ≈ 0.368, RMSE ≈ 0.469) again lies well away from the restricted null distributions, showing that the real-data performance remains separated from the null results in this diagnostic. In this diagnostic, plateau energy is not constant and increases with prompt luminosity. A luminosity–time re￾lation alone does not include this dependence on prompt emission. Possible…

Figures reproduced from arXiv:2610.23155 by the authors.

References

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