Pulsar timing array (PTA) experiments have reported compelling evidence for a stochastic gravitational-wave background in the nanohertz frequency band. While these observations represent a significant advance, their physical interpretation remains uncertain, as multiple astrophysical and cosmological scenarios can reproduce the gross features of the inferred signal. Moreover, the structure of PTA datasets—characterized by long observational baselines, irregular sampling, and correlated noise—complicates direct model discrimination. We present the Silvernail Paradigm, a model-agnostic framework for the analysis and interpretation of PTA data that is designed to minimize theoretical assumptions. The paradigm formulates an operationally closed representation of PTA observables in terms of a small set of empirically defined descriptors, capturing the amplitude, spectral properties, correlation structure, and internal consistency of the inferred background. All quantities entering the framework are defined at the level of measurable observables or statistically inferable constructs. The framework is organized into three layers: an empirical signal representation, a statistical inference layer specifying likelihood and stability structure, and an interpretational mapping that enables systematic comparison with a broad class of physical models. By separating data-driven structure from model-dependent interpretation, the Silvernail Paradigm provides a transparent and reproducible methodology for organizing PTA observations. This approach facilitates consistent comparison across competing theoretical explanations and establishes a flexible foundation for the analysis of present and future nanohertz gravitational-wave data.