Scholarly record
Research summary
Paper IV studies stationary organization and long-time function in the synthetic collective kinetic–chemical–electrical transporter source of Papers I–III. The complete position–momentum–phase kinetic response gives exactly one maintained positive stationary organization with ongoing pump, passive-return and regeneration circulation, charge within the prescribed [1, 3] band, phase–position correlation, and non-Gaussian momentum structure. The full kinetic–resource linearization and nonlinear estimates establish an explicit local attracting neighborhood, while a separate compact attracting set supports strictly positive invariant-mean circulation and quantitative long-time charge-band recurrence. Native finite-population event paths eventually exit the band with probability one for each fixed population, whereas the stationary collective limit remains in-band over every fixed finite horizon; the resulting population/time survival limits do not commute. The work does not establish global point selection or identify the compact attractor with the stationary state; these remain for Paper V.
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Scientific context
This paper belongs to the Kinetic Organization and Biological Function research program.