Structural Memory, Free Energy, and Information in Recurrent Linear Systems

Delayed linear recurrence induces persistence through repeated reinjection of past states across fixed delay intervals. This paper introduces a recurrence-weighted structural free energy functional for discrete-time delayed systems of the form y[n] = u[n] + αy[n − N ]. The functional is defined as a geometrically weighted quadratic accumulation over delayed trajectory segments. Under the stability condition |α| < 1, convergence and boundedness of the functional are established. Explicit asymptotic scaling laws are derived for three regimes: homogeneous recurrence, bounded forcing, and white-noise excitation. In the homogeneous case, structural free energy decays exponentially at a rate determined by |α|. Under bounded input, accumulation exhibits cubic divergence as |α| → 1−. Under stochastic excitation, output variance diverges quadratically, consistent with recurrence amplification. A functional-analytic interpretation shows that delayed recurrence embeds scalar trajectories into a weighted Hilbert space induced by a geometrically scaled shift operator. The recurrence parameter simultaneously governs dynamical stability, contraction in delay space, energetic accumulation, spectral reshaping, and variance amplification. Numerical simulations confirm the analytic scaling laws. The results provide a quantitative framework for analyzing structural memory and energetic persistence in linear delayed systems, extending classical delay-system analysis through an explicit recurrence-weighted norm formulation.