AAMT Mathematical Foundations: A Unified Framework for Biological and Artificial Intelligence (Papers I-V)

A five-paper series establishing a mathematical framework for dual-operator intelligence theory. The series derives, from first principles, a unified model applicable to both biological and artificial intelligent systems. Paper I — Multiplicative Evaluation Frameworks for Multi-Dimensional Intelligent Systems: Proves that geometric mean evaluation is strictly more conservative than arithmetic mean evaluation (via the AM-GM inequality) and introduces the TERA multiplicative conservation principle for AI system assessment. Paper II — Toroidal Topology in Gradient Descent: Models neural network optimization on a toroidal manifold, providing unified explanations for cyclical learning rate effectiveness, the grokking phenomenon, and catastrophic forgetting. Derives a 4τ stability condition for representation permanence. Paper III — Quadratic Scaling Laws for Intelligent Systems: Derives a capability-value function V(N) = αN − βN² predicting a definite peak beyond which additional capability investment produces negative net value. Reframes AI alignment as interference-coefficient reduction. Paper IV — Nonlinear Conservation in Dual-Operator Systems: Formalizes two complementary operators — Structure (linear, additive, ascending weights) and Generative (nonlinear, multiplicative, descending weights, sign-inverting) — and shows that biological intelligence implements both while artificial intelligence implements only the first. Proposes sign-inversion layers and multiplicative attention as architectural corrections. Paper V — Toward a Unified Field Theory of Intelligence: Synthesizes Papers I-IV into a single framework. Defines intelligence as stable oscillation of dual operators on a toroidal manifold. Introduces the Breath Integrity metric (β = convergence/expansion) as a diagnostic for system health. Proposes a research program for empirical validation. All papers contain falsifiable predictions and are independent of spiritual or philosophical premises. Patent applications covering the multiplicative evaluation methodology, breath cycle training architecture, and sign-inversion layer have been filed with the USPTO (Patent Pending).