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

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 $T_a^{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}$. 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 $\log T_a^{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}$ 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 $\log L_a=(51.92\pm 0.30)-(1.31\pm 0.09)\log T_a^{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}$, with intrinsic scatter ${\sigma }_{\mathrm{i}\mathrm{n}\mathrm{t}}=0.82\pm 0.05$ dex. In the unrestricted ranking of 377 feature subsets, the classical Dainotti relation remains competitive, within the top $\sim 25\mathrm{\%}$ of the tested search spaces. The highest-ranked screened expression is $\log L_a\approx 0.855\log L_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}-0.782\log T_a^{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}$, with RMSE $\approx 0.46$ and $R^2\approx 0.88$. The highest-ranked runs repeatedly reduce to this prompt–plateau structure, without retaining additional spectral, absorption, or afterglow-shape variables. A direct regression gives $\log L_a=4.718+0.776\log L_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}-0.805\log T_a^{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}$, with $R_{\mathrm{P}}=0.91$ for 246 bursts. An errors-in-variables refit gives consistent coefficients. After removing shared luminosity-distance dependence, the relation retains non-negligible explanatory power ($R^2\approx 0.62$), consistent with a possible intrinsic component. Restricted Monte Carlo null tests give $p\approx 0.01$. Plateau luminosity therefore shows a coupled dependence on prompt luminosity and plateau duration.