Partial Observation and the Limits of Finite Recovery

This paper develops a general theory of partial observation on finite partition lattices, addressing where exact recovery succeeds and where it fails for structural reasons in finite deterministic systems. We introduce a recovery-problem framework specifying structural classes, admissible reconstructions, output targets, and the operational cost of exact recovery from partial data. Within this framework, the paper establishes four principal results: 1. Theorem 5 (Chapter 4): a complete characterization of exact recovery in the two-observer case, delineating the exceptional regime where completeness can be classified exhaustively. 2. Theorem A' (Chapter 6) and Theorem A'' (Chapter 7): reflexive incompleteness for single and finite-family self-observing systems, showing that exact self-determination fails structurally under self-inclusion. 3. The Unbounded Depth Theorem (Chapter 10): beyond the two-observer regime, no uniform finite bound exists on the operational depth required for exact recovery, establishing unbounded constructive obstruction. The proof uses residual finiteness of the free lattice FL(3) and the Pudlák–Tůma embedding theorem. 4. The Incompleteness Principle of Partial Observation (Chapter 11): synthesis of three distinct mechanisms (boundary completeness, constructive unboundedness, reflexive obstruction) into a general incompleteness principle, characterizing system-theoretic limits of exact recovery, distinct from formal incompleteness in mathematical logic. Experimental validation on 2,635 three-generator systems confirms the theoretical predictions (Appendix A). This work continues the Ma Theory program initiated in an earlier paper on optimization-induced dynamics and the inverted-U (Zenodo record 10.5281/zenodo.19439785, 2026). This work was developed through a collaborative cognitive framework between a human researcher and multiple large language model agents (Claude, GPT, Gemini), with mathematical correctness as the sole evaluation criterion. The paper passed three-party internal review before release.