Fibonacci-Type Dynamics as Minimal Compensated Growth: Spectral Emergence in Hierarchical Positive Systems

This paper develops a minimal compensated-growth dynamical closure that induces Fibonacci-type recurrences under explicit structural hypotheses: positivity, irreducible interdependence, and a pure one-step delayed channel. We formalize the admissible transformation class preserving component-wise meaning and show that the compensated closure reduces the dynamics to a one-parameter family of positive linear operators. A hierarchical embedding theorem establishes that, for structured high-dimensional positive systems with spectral dominance and near-pure delay, a macroscopic two-dimensional reduction emerges whose leading coordinate follows an effective Fibonacci-type recurrence up to controlled error. The asymptotic ratio convergence is governed by the Perron--Frobenius eigenvalue of the effective operator; the golden ratio appears only in a neutral calibrated case and is not asserted to be universal. The analysis is purely spectral and dynamical and is compatible with ratio-based structural frameworks such as EROI without requiring them. Version v0. February 2026.