This paper is the fifth installment in the six-paper Shadow Theory architecture, a proposed Theory of Everything framework that develops a formal mathematics of readout (shadow) domains and their relationship to underlying source structure. The first four papers establish the foundational architecture of Shadow Theory: the Readout Non-Equivalence Theorem, Completion Necessity, the Canonical Completion Object Theorem, and the Tier-1 Shadow Compiler. Paper 5 develops the complete public runtime mathematics governing how the framework operates after compilation. Rather than introducing a single new theorem, this paper presents the operational calculus of Shadow Theory. It formalizes the runtime objects, status systems, residue handling, equation artifact hierarchy, audit rules, score vectors, certification procedures, and framework-level execution semantics that govern the production, classification, and evaluation of public mathematical and physical outputs. The framework treats public mathematics as a mathematically rigorous shadow-side calculus rather than a direct representation of source reality. Runtime execution therefore preserves completion status, compiler certification, admissibility conditions, residue accounting, and auditability throughout the production of Tier-1 artifacts. The paper also develops the framework's testing protocols, failure classifications, promotion rules, and certification boundaries, ensuring that incomplete or unsupported objects cannot be promoted beyond their formally established status. Together with the preceding papers, this work completes the operational mathematical machinery of Shadow Theory. The sixth and final paper synthesizes these components into the complete Shadow Theory architecture and presents the overall framework as an integrated research programme. Further information about the broader Shadow Theory research programme is available at https://www.everythingequation.com/.
Jeremy Rodgers (Sat,) studied this question.