Equifinality—where multiple parameter combinations produce indistinguishable climate states—is a fundamental obstacle to calibrating Earth system models (ESMs). Parameter sets that agree well with the observed mean state can produce dramatically different responses to external perturbations, reflecting a mathematical non-uniqueness that standard calibration approaches cannot resolve. This poses a critical challenge for constraining the effective radiative forcing due to aerosol-cloud interactions (ERF_ACI), which depends sensitively on cloud responses to aerosol perturbations. We use a perturbed parameter ensemble that, as a minimal working example, varies only two parameters related to warm-rain formation (autoconversion and accretion) and constrain on synthetic observations in a perfect-model setup. Even under ideal conditions, radiation-only constraints produce a fundamentally degenerate, bimodal ERF_ACI posterior. Incorporating the observable cloud state (liquid water path, 𝔏) for warm marine stratocumulus clouds can resolve this bimodality, but its utility diminishes with large observational uncertainty. A far more robust constraint is achieved by directly targeting the underlying physical process rates that modulate 𝔏. Process-rate constraints effectively eliminate bimodality and dramatically reduce posterior uncertainty even under large assumed observational uncertainties. As process rates are not directly observable, we test their observable proxies as a tractable alternative, finding they offer only limited additional value over cloud state due to their emergent behavior. These results establish a critical principle for ESM calibration: to achieve robust constraints on ERF_ACI and beyond, it is necessary to constrain both the emergent climate state and the physical pathways that govern its formation.