Structural Limits and Sensitivity Hierarchy of Energetic Continuous Computation

This twelve-part series develops a resource-explicit structural theory of energetic continuous computation. The framework models computation as smooth dynamical evolution on finite-dimensional manifolds, parameterized by state dimension, descriptive complexity, evolution time, and accumulated dynamical sensitivity Σ(n)=∫0τ(n)∥DFφ(x(t))∥dt.\Sigma(n) = \int_0^{\tau(n)} \|DF_\varphi(x(t))\| dt .Σ(n)=∫0τ(n)∥DFφ(x(t))∥dt. The central result establishes the structural equivalence Epoly=P,E_{\text{poly}} = P,Epoly=P, showing that smooth energetic systems with polynomially bounded physical resources and logarithmic accumulated sensitivity are polynomial-time simulable by deterministic Turing machines, and conversely that every polynomial-time computation admits such an energetic realization. The series further introduces a sensitivity-based hierarchy in which computational amplification scales as eΣ(n)e^{\Sigma(n)}eΣ(n), yielding a unified resource conservation principle: super-polynomial computational power requires super-polynomial growth of accumulated dynamical sensitivity or other physical resources. The work does not resolve classical separation questions such as P≠NPP \neq NPP=NP; rather, it provides a structural dynamical characterization of polynomial-time computation and identifies accumulated sensitivity as the fundamental bridge between continuous energetic evolution and discrete complexity theory.