A Pre-Registered Test of Entropy–Structure Coupling in Simulated and Observed Galaxy Clusters

We present a pre-registered, definition-matched test of the relationship between entropy profile shape and cool-core thermodynamic state in galaxy clusters. Observational analyses of X-ray clusters (ACCEPT; Cavagnolo et al. 2009) report a positive rank correlation between the entropy profile slope parameter α and central cooling time, indicating that clusters with steeper entropy gradients tend to have higher central cooling times. We test whether this coupling is reproduced in the TNG-Cluster cosmological simulation suite.As a computationally efficient pre-test, we construct an entropy-gradient proxy from the catalogued electron density slope, α_proxy = (2/3)·ne_slope, and evaluate its rank correlation with central entropy K₀ and cooling time t_cool across 352 simulated clusters at z = 0. The proxy analysis yields a strong negative correlation (ρ ≈ −0.87) that is robust to mass control and consistent across mass bins, opposite in sign to the observational result.To eliminate remaining systematic ambiguities, we then compute full entropy profiles directly from gas particle data via the TNG API and fit the Cavagnolo-model K(r) = K₀ + K₁₀₀(r/100 kpc)^α over a fixed physical interval [20, 400] kpc. This definition-matched α_fit removes potential biases from temperature-slope variation, local-vs-global slope definitions, and radius mismatch. The resulting α_fit values are subjected to the same pre-registered statistical tests.The outcome of this decisive test establishes whether the sign difference between simulation and observation persists under matched observable definitions. Regardless of the final sign, the analysis provides a transparent, rank-based diagnostic of entropy–structure coupling in cluster cores and a reproducible framework for comparing simulations and X-ray observations.