A Scale-Invariant Geometric Threshold for Deep Neural Networks: Deriving the Exact Semantic Coherence Limit via Topological Gradient Dynamics

The rapid scaling of Large Language Models (LLMs) and foundation architectures is fundamentally bottlenecked by representation degradation, manifesting empirically as attention saturation, mode collapse, and generative hallucinations. Current empirical models, such as the Kaplan and Chinchilla scaling laws, optimize parameter-to-token compute ratios but lack a deterministic, geometric boundary for multi-layer gradient coherence. This paper introduces a strict topo-dynamical framework for neural array scaling. By modeling the parameter matrix as a structural competition between the spatial propagation of semantic coherence and localized entropic noise, we derive a universal square-root geometric invariant (ℓcog). We explicitly contrast this limit with probabilistic scaling laws, provide rigorous mathematical proofs for how Sparse Mixture of Experts (MoE) circumvents this boundary, and validate universality across Multi-Task Learning and Neural Architecture Search (NAS). Finally, we present a hardware-aware blueprint and experimental validation paradigm for Active Topology Routing (ATR) in distributed training environments.

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