Addressing Mathematical Rigor in the Open General Intelligence Framework: A Critical Analysis and Formal Implementation

The Open General Intelligence (OGI) framework (Dollinger & Singleton, arXiv:2411.15832) proposed a modular cognitive architecture for artificial general intelligence. While architecturally compelling, the original framework lacked formal mathematical specification. This paper provides those foundations. Contributions include: (1) a formal attention-based definition of OGI's dynamic weighting function with Lipschitz stability proof; (2) a Global Coherence Objective linking computational efficiency with cognitive consistency; (3) complexity analysis of the fabric interconnect protocol, establishing O(n²dₘ) as a fundamental scalability barrier; (4) Top-K Gating (Theorem 3), a biologically-motivated resolution reducing complexity to O(k²dₘ + n(d_c d_a)); (5) Theorem 4 with formal sufficient conditions and two supporting lemmas establishing semantic preservation in GRU-based multi-modal fusion; and (6) simulation results validating temporal stability across 200 processing cycles under Gaussian noise levels σ ∈ {0.0, 0.25, 0.50, 0.75, 1.0}. Simulation results show cosine similarity of 0.9940 at σ = 0.5 and 0.9724 at σ = 1.0, with 181/200 and 173/200 cycles stable above threshold respectively. The measured Coherence Tax (computational overhead of the MINE-based coherence objective) is +73% per trial versus a task-loss-only baseline on CPU at dim = 128. The absence of accuracy gain in single-modality conditions is theoretically explained: the coherence objective's disambiguation benefit requires genuine cross-modal ambiguity, which constitutes the primary future experimental direction. This paper is a companion to arXiv:2411.15832 and represents step two of a three-part research sequence: architectural vision (OGI, 2024), formal mathematical foundations (this paper, 2025), and implementation with multi-modal validation (future work). Benchmark code is available at: https://github.com/singleton2787/ogi-math-foundations