We propose a minimal thermodynamic model for open energized systems in which recurrent functional structure arises as the capacity to convert part of the thermodynamically available power into organized dissipative channels. The model distinguishes total input power from the entropic cost of irreversibility and introduces an effective useful power, defined as the state-dependent fraction of the remaining available power that is actually captured by recurrent structural routes. The resulting dynamics is governed by a single nonnegative structural variable and an associated organized dissipation capacity. Under natural regularity assumptions, we prove global existence, positivity, and boundedness of trajectories and, under constant forcing together with a sufficient small-gain condition, existence, uniqueness, and global asymptotic stability of a nontrivial stationary regime. We then show that the abstract efficiency law of the model is explicitly realized by two physically distinct mechanisms: saturating autocatalytic chemistry and filamentary resistive switching in memristive devices. Worked examples are given in a Formose-like sugar-production setting and in a SET-transition memristor regime. The framework does not claim a universal derivation of function from first principles; rather, it provides a compact and mathematically explicit minimal model showing how self-reinforcing yet saturating functional organization can be consistently represented in open nonequilibrium systems under energetic drive and entropic constraints.