Coherence-Theoretic Foundations for Cross-Domain Structural Transfer in Autonomous Reasoning Systems

This position paper identifies four architectural gaps in Google DeepMind's Aletheia autonomous reasoning agent (arXiv:2602.10177, arXiv:2602.03837) where identity divergence theory could improve performance. Aletheia achieves 95.1% on Olympiad-level benchmarks but only 6.5% accuracy on 700 open Erdős problems, with 68.5% of responses fundamentally wrong and a substantial proportion "mathematically empty. " We argue that the Law of Identity Divergence (Elghandour, 2026; DOI: 10.5281/zenodo.18616149) — which proves that persistent behavioral equivalence between distinct system instantiations has probability zero — provides a missing theoretical foundation for evaluating cross-domain structural transfer. The framework's six-component identity signature, Lean 4 mechanization, and empirical validation offer tools applicable to: (1) structural pre-filtering before transfer attempts, (2) semantic validity verification beyond logical correctness, (3) predictive theory for cross-domain transfer success, and (4) safety certification for autonomous reasoning at scale.