The rapid advancement of AI agents has exposed a critical bottleneck: memory management. Existing systems treat memory as passive storage, leading to unbounded growth, inefficient retrieval, and failure to consolidate knowledge across tasks. We introduce a fundamentally different perspective: \textbf{memory as a closed operator-driven dynamical system}. The system state evolves through a finite set of completely positive trace-preserving (CPTP) operators—fluctuation ($\fleq$), cyclic reset ($\cyceq$), phase nexter ($\nexteq$), phase reverser ($\reveq$), thresholding ($\liminfty$), pruning ($\termop$), and transformation ($\diveq$)—that together define a self-regulating mechanism. Capacity emerges not from external constraints but as a dynamical invariant from operator interactions. Memory efficiency is reinterpreted as \textbf{controlled entropy flow}, where stability arises from equilibrium between entropy injection (fluctuation) and reduction (pruning, compression). We evaluate SNT-MEM on long-context benchmarks with a Llama-3-8B model, comparing against vanilla RAG and Mem0-style baselines. SNT-MEM achieves $63.3\% \pm 2.1\%$ memory reduction versus RAG and $22.3\% \pm 1.8\%$ versus Mem0, with $3.8\times \pm 0.3\times$ retrieval speedup. On LongBench multi-document QA, it achieves $4.2\times$ token compression, $3.3\times$ faster time-to-first-token, and bounded memory at 8GB versus unbounded baseline growth, with minimal F1 degradation ($-2.2\%$). Ablation studies confirm that each operator contributes uniquely; removing any operator causes measurable performance loss, supporting functional minimality. A necessity theorem proves that any bounded memory system must implement pruning or compression. This work establishes memory as a self-organizing physical process governed by a small set of transformation rules, reframing memory from an engineering constraint to a principled dynamical system with theoretical guarantees and empirical validation on production-grade workloads.