Completion Necessity for Readout-Non-Equivalent Domains: Certified Active Slot Failure and the Obstruction to Full Public Closure (Shadow Theory, Paper 2)

This paper is the second 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. Paper 1 established the Readout Non-Equivalence Theorem, showing that an exact public readout or quotient presentation does not by itself imply realization-structure equivalence. Paper 2 builds directly on that foundation by identifying the precise conditions under which lost realization structure becomes a genuine obstruction to completing a public mathematical or physical target. The paper introduces the concepts of essential closure slots, certified active failure profiles, and completion necessity. It proves that when an essential realization role required for a target cannot be faithfully recovered or discharged, full certified public closure is formally blocked. Rather than treating missing structure as an informal limitation, the framework identifies it as a mathematically certifiable completion obligation. The work also develops a conditional bridge to first-order proof systems and establishes a formal handoff from certified obstruction to downstream completion theory, while carefully distinguishing completion failure from truth-refutation or global unprovability. Together with the first paper, this work advances the central premise of Shadow Theory: mathematics and physics are developed from within the public readout (shadow) rather than assumed to operate directly on source reality itself. Subsequent papers extend this foundation into categorical completion theory, the Tier-1 Shadow Compiler, the public Shadow Framework mathematics, and the complete synthesis architecture. Further information about the broader Shadow Theory research programme is available at https://www.everythingequation.com/.