An Operator Geometric Framework for Cognitive Engineering and Computational Meanings: BLOC Theory v2.0

This paper presents BLOC Theory v2.0, an operator–geometric extension of the original Binary Limitation of Organic Cognition (BLOC) framework. The extended theory introduces a unified mathematical architecture for modeling Human–AI cognitive systems through cognitive operators, geometric state spaces, manifold structures, metric formulations, and algebraic transformations. Semantic transformation is formalized through the Meaning Operator M, alignment dynamics through the Alignment Operator Algn, and organic binary divergence through the continuum discrete functor F together with the recovery functor G. These operators provide a computable representation of semantic change, alignment stability, information loss, and partial recoverability across coupled Human–AI systems. New geometric constructions including temporal manifolds, capability manifolds, loss curvature, and phenomenological state spaces establish a structured basis for analyzing cognitive asymmetry, constrained reasoning, interpretability dynamics, and state transition behavior. These structures enable formal analysis of meaning propagation, alignment deformation, and cognitive state evolution within adaptive cognitive architectures.BLOC Theory v2.0 advances the original framework from a conceptual limitation theory to a mathematically structured cognitive systems model capable of supporting computable reasoning, formal verification pathways, and generalized cognitive engineering analysis..