THEIA: Learning Complete Kleene Three-Valued Logic in a Pure-Neural Modular Architecture

Compositional learning under missing information requires systems to preserve uncertainty while delaying commitments until the relevant composition step. We study this setting in Kleene's strong three-valued logic (K3) using THEIA, a pure-neural modular architecture with separate arithmetic, order, set, and logic engines. THEIA learns the full 39-rule K3 diagnostic from task data without runtime symbolic inference or hand-coded K3 gate primitives. Its internal computation exhibits an uncertainty-verdict asymmetry: the Has-Unknown signal is preserved at upstream engine boundaries, whereas final-verdict decodability remains at or below a U-vs-non-U oracle reference until the Logic boundary; activation patching rules out residual shortcuts on matched non-absorbent cases. In an absorbing-state-free mod-3 composition task, THEIA generalizes from 5-step training to 500-step evaluation with 99.96% $\pm$ 0.04% accuracy across 5 seeds. Flat MLPs collapse under the same discretized protocol, a larger ResMLP grid reaches $\geq$99% on only 3/20 trials, and a larger Transformer reaches 99.24% $\pm$ 0.34% but follows a qualitatively different representational trajectory. These results position THEIA as a controlled testbed for interpretable compositional learning under uncertainty.