Sealed or Leaky: The Source Tetralemma, Bell-Certified Hidden Information, and Finite-Resource Witnesses

When does an observable readout contain all the information needed to determine its underlying source, and when does source information remain inaccessible? This paper extends Shadow Theory’s source–readout framework by distinguishing information preserved in the underlying dynamics from information accessible through a specified class of observations. It develops a conditional source tetralemma, quantitative bounds on hidden information from Bell violations, and operational criteria for distinguishing sealed information from information that can become observable. Under explicitly stated assumptions concerning definite outcomes, measurement independence, and conditional no-signalling, the paper derives lower bounds on source information missing from an initial classical readout. It also examines operationally indistinguishable surrogate models, reversible dynamics that preserve information while restricting access, and finite-resource witnesses of incomplete readouts. The results distinguish mathematical conclusions from physical assumptions and model-specific applications. They do not establish a unique underlying ontology or claim that Bell violations alone prove Shadow Theory. The accompanying Python verification script checks specified numerical bounds, finite constructions, and manuscript consistency conditions. It uses the Python standard library and includes exact rational arithmetic. Version 3.1. Author: Jeremy Rodgers, independent researcher.Project website: https://www.everythingequation.comDOI: 10.5281/zenodo.23077474