From Individual Transport to Collective Kinetic Function

This third paper in the Kinetic Organization and Biological Function series develops a mathematical-physics theory of collective kinetic organization and the information required to predict its physical functions. Extending the finite-resource mechanochemical system established in Paper II, the investigation constructs an interacting population of momentum-bearing transporters coupled through shared ionic and chemical reservoirs and an inventory-derived membrane capacitor. The exact finite diffusion–jump generator preserves mechanical position, momentum, conformational state, physical stoichiometry, electrical work, and energetic accounting. The population limit yields a nonlinear kinetic–chemical–electrical evolution in which phase-conditioned mechanical distributions determine transport currents, while those currents transform the shared environment governing subsequent kinetic activity. Positive-domain well-posedness, empirical martingale convergence, and kinetic compactness establish finite-time convergence to this collective law. Quantitative population estimates, conditional propagation of chaos, and selected Gaussian fluctuation theorems characterize the resulting statistical organization, retaining the physical cross-covariations generated by shared transport and chemical events. A central contribution is a quantitative theory of mathematical compression and predictive information. Explicit equal-energy preparations demonstrate that identical resource concentrations, conformational occupations, membrane voltage, and mechanical energy can produce different transport currents. Observation-fiber minimax theorems establish sharp lower bounds on the resulting predictive error, while an infinite-dimensional kinetic-generator span theorem obstructs exact closure within specified finite linear moment representations. These results identify physical information lost under particular macroscopic observations and quantify its consequences for collective functional prediction. The complementary constructive result develops a finite Boltzmann-template kinetic approximation with an explicitly retained physical residual. A positive, resource-sensitive error comparison propagates the discarded kinetic information into rigorous bounds on selected observables. For a specified chemically maintained collective preparation, an independently checked 100-slab rational calculation establishes [1.8932060449<χ(0.05)<1.9954528308<χ(0)=2,] certifying a finite-time decrease in the normalized membrane charge of the complete nonlinear kinetic system. The analysis additionally establishes finite-population charge-band persistence bounds, collective energy and free-energy balances, physical caloric observability, and positive-time kinetic regularity. A conditional full-joint entropy result identifies additional preparation and regularity requirements for collective path-entropy convergence. Continuing the series' investigation of 氣機 (qìjī) and 氣化 (qìhuà), the work develops a modern mathematical account of reciprocal conditioning between organized movement and material transformation. Its results establish a source-specific framework for collective kinetic dynamics, quantitative predictive sufficiency, and controlled functional approximation, providing a foundation for subsequent investigations of stationary and long-time collective organization.