The Wisdom Closure Theorems: A Formal Theory of Failure-Learning Architectures

We present a formal theory of failure-learning architectures. An architecture is modeled as a measurable Markovian update system that carries state across failure episodes through a canonical antigen map into an equivalence-class space. We define wisdom as the normalized conditional mutual information that the architecture's state carries about future outcomes given task context.

The paper develops three closure results: a Failure Equivalence Principle, a Wisdom Horizon theorem for finitely resolving architectures, and a Wisdom--Compression Duality connecting wisdom to compression of an architecture's own failure stream. It further derives compression-generalization equivalence, architecture reflection necessity, and an optimal resolution tradeoff. The manuscript includes explicit technical caveats, falsification criteria, and an empirical protocol for testing the wisdom horizon on longitudinal failure-learning logs.

This record is a theory preprint. The included protocol self-test, if uploaded, is a synthetic software sanity check and should not be interpreted as external empirical validation. A post-hoc existing-log bridge over prior portfolio logs is included only as protocol feasibility and estimator-boundary evidence, not as preregistered theorem confirmation.