Stationary Organization and Functional Persistence in Collective Kinetic Systems

This fourth paper in the Kinetic Organization and Biological Function series establishes stationary organization, full kinetic local stability, and persistent physical function in a nonlinear collective kinetic–chemical–electrical system derived from interacting mechanochemical transporters. The physical model retains periodic mechanical position, unbounded momentum, four conformational phases, finite shared ionic and chemical resources, and an inventory-derived membrane capacitor. The investigation determines how continuing microscopic transport and energetic transformation can sustain a time-independent macroscopic organization with nonzero material currents. An exact stationary reduction, incorporating the full frozen kinetic invariant law, reduces maintained self-consistency to three independent physical coordinates. Complete physical-domain exclusion, continuum Poisson-response estimates, and rigorous interval inclusion establish a unique maintained stationary organization. Its normalized membrane charge is approximately 1.13280067, while its signed material circulation is approximately 0.10303471. The four conformational currents, passive return, and chemical regeneration satisfy exact stationary balances with strictly positive circulation and zero net charge accumulation. The stationary mechanical state exhibits nontrivial phase–position correlations and non-Gaussian momentum organization. Its irreversible entropy production and caloric exchange obey the exact physical identities ( D(X_)=6J_) and ( Qout,=(15/2)J_). The full infinite-dimensional kinetic–resource linearization admits an exponential semigroup estimate, yielding a quantitatively explicit nonlinear locally attracting neighborhood. Beyond this local organization, physical resource absorption, kinetic regularization, and weighted momentum-tail compactness establish a compact attracting set for the maintained collective evolution. A source-specific Hellinger quadratic establishes a strictly positive dissipation floor exceeding (1/5) throughout this set and invariant-mean material circulation exceeding (1/30). An augmented free-energy identity and restricted-charge inequalities further establish quantitative finite-window recurrence and a long-time occupation fraction of at least (77/16077) for the selected normalized charge band ([1,3]). The investigation also identifies the constrained information geometry of the physical state. The second variation of the Helmholtz functional contains the Fisher–Rao metric of the phase-resolved kinetic probability distribution, together with positive ionic, chemical, and capacitor contributions. Its relationship to Hellinger reaction dissipation connects admissible state variation, irreversible turnover, and persistent collective function. Mechanism-specific controls distinguish active transport, passive return, chemical regeneration, and mechanical coupling. An exact finite-population exit theorem additionally establishes that uninterrupted indefinite charge-band survival vanishes at each fixed finite population, whereas the corresponding stationary collective preparation remains within the band, producing noncommuting population and observation-time limits. Continuing the series' investigation of 氣機 (qìjī) and 氣化 (qìhuà), the work develops a mathematical account of organized movement and continuing material transformation as mutually conditioning physical processes. The historical correspondence is structural and interpretive; the mathematical conclusions concern the specified mechanochemical source. Together, the results establish a rigorous framework for stationary kinetic organization and persistent biological function, preparing the subsequent investigation of global dynamical selection.