A Computability-Theoretic Proof of Self-Referential Ignorance Formal Supplement to the Unified Theory of Self-Referential Ignorance (UT-SRI)

Abstract The original informal argument for Self-Referential Ignorance (SRI) relied on intuitive analogies to Gödelian incompleteness and self-modeling regress. This supplement replaces those analogies with a rigorous, literal proof grounded in computability theory. Using the uncomputability of Kolmogorov complexity, Chaitin's incompleteness theorem, Kleene's recursion theorem, and Rice's theorem, we establish that every finite computable observer possesses a provably nonempty, infinite set of true propositions about its own complexity, its exact self-referential ratio Ψ = M/K, and its future behavioral trajectory that it cannot establish from within its own formal framework. This establishes SRI(O) > 0 as a strict mathematical theorem rather than an empirical conjecture. To prevent structural over-claiming, we also extend this framework using Probably Approximately Correct (PAC) learning theory and statistical mechanics to formalize the bounds governing external-world modeling and complex-system prediction, successfully completing the mathematical foundations for all five pillars of the UT-SRI program.