The Shell Framework: Discrete Convolution and Ergodic Dynamics in Decaying N-Dimensional Space

We present the Shell Framework, a unified description of how a driven signal propagates, averages, and decays through n-dimensional space under dissipation. The framework originated in an exploration of radially symmetric decay functions and their interaction with high-dimensional geometry; the exponential function f(r) = e^{-r} was selected because it yields a uniquely clean analytical result. Starting from three components — discrete convolution, the Birkhoff ergodic theorem, and the exponential decay f(r) = e^{-r} — we derive the governing equation of motion for a damped particle subject to convolutive forcing. We prove that the natural equilibrium radius under the potential V(r) = e^{-r} in n-dimensional space is exactly r* = n-1, a result we term the shell condition. Under stochastic forcing, the fixed point destabilizes; we conjecture that the resulting trajectory is attracted to a strange attractor concentrated near the shell at r* = n-1, and we identify the ergodic term as the time average of an observable under the system's invariant measure on this attractor. The shell condition and the attractor conjecture together constitute the central claims of the framework.