Resolving Ratio Redundancy in Chemical Freeze-out Studies with Principal Component Analysis and Bayesian Calibration

We introduce a Principal Component Analysis (PCA)--Bayesian framework for extracting chemical freeze-out conditions in relativistic heavy-ion collisions that resolves long-standing ambiguities in hadron-ratio--based analyses. By constructing all possible hadron-yield ratios from a chosen set of species and transforming them into an orthogonal PCA basis, the method removes linear redundancies and eliminates the information loss and systematic uncertainties associated with ratio selection. Energy-wise Bayesian calibration of the Hadron Resonance Gas (HRG) model is then performed directly in this decorrelated space, with a Gaussian Process emulator enabling fast and accurate model evaluations. A detailed Sobol sensitivity analysis, together with the PCA loading structure, identifies the most informative ratio combinations and reveals a transition from chemical-potential--dominated to temperature-controlled freeze-out with increasing $\sqrt{s_{NN}}$. The calibrated model reproduces all measured ratios, and the extracted freeze-out parameters are consistent with previous HRG determinations.