Dynamic Nested Hierarchies: Pioneering Self-Evolution in Machine Learning Architectures for Lifelong Intelligence

Contemporary machine learning models, including large language models, exhibit remarkable capabilities in static tasks yet falter in non-stationary environments due to rigid architectures that hinder continual adaptation and lifelong learning. Building upon the nested learning paradigm, which decomposes models into multi-level optimization problems with fixed update frequencies, this work proposes dynamic nested hierarchies as the next evolutionary step in advancing artificial intelligence and machine learning. Dynamic nested hierarchies empower models to autonomously adjust the number of optimization levels, their nesting structures, and update frequencies during training or inference, inspired by neuroplasticity to enable self-evolution without predefined constraints. This innovation addresses the anterograde amnesia in existing models, facilitating true lifelong learning by dynamically compressing context flows and adapting to distribution shifts. Through rigorous mathematical formulations, theoretical proofs of convergence, expressivity bounds, and sublinear regret in varying regimes, alongside empirical demonstrations of superior performance in language modeling, continual learning, and long-context reasoning, dynamic nested hierarchies establish a foundational advancement toward adaptive, general-purpose intelligence.

A Mathematical Framework for AI Singularity: Conditions, Bounds, and Control of Recursive Improvement

AI systems improve by drawing on more compute, data, energy, and better training methods. This paper asks a precise, testable version of the "runaway growth" question: under what measurable conditions could capability escalate without bound in finite time, and under what conditions can that be ruled out? We develop an analytic framework for recursive self-improvement that links capability growth to resource build-out and deployment policies. Physical and information-theoretic limits from power, bandwidth, and memory define a service envelope that caps instantaneous improvement. An endogenous growth model couples capital to compute, data, and energy and defines a critical boundary separating superlinear from subcritical regimes. We derive decision rules that map observable series (facility power, IO bandwidth, training throughput, benchmark losses, and spending) into yes/no certificates for runaway versus nonsingular behavior. The framework yields falsifiable tests based on how fast improvement accelerates relative to its current level, and it provides safety controls that are directly implementable in practice, such as power caps, throughput throttling, and evaluation gates. Analytical case studies cover capped-power, saturating-data, and investment-amplified settings, illustrating when the envelope binds and when it does not. The approach is simulation-free and grounded in measurements engineers already collect. Limitations include dependence on the chosen capability metric and on regularity diagnostics; future work will address stochastic dynamics, multi-agent competition, and abrupt architectural shifts. Overall, the results replace speculation with testable conditions and deployable controls for certifying or precluding an AI singularity.