Dynamics from Closure and Stabilization in Finite-Resolution Physics: Why Hamiltonians Are Conditional

We examine the structural role traditionally played by dynamical laws and ask whetherthat role can be reconstructed without taking an equation of motion, a Hamiltonian, or anaction principle as primitive. The argument begins from the historical separation betweenkinematics and dynamics and reframes the problem as one of coherent composition. At afixed finite resolution, this leads naturally to an internal closure of admissible operationalcomposition. Closure alone, however, is not yet law-like: the closed structure may changewhen the operational description is refined. A second requirement is therefore needed.When closure relations persist across admissible refinements, a stabilized plateau is formed.Dynamical laws can then be understood as representations of continuation within such aplateau. Hamiltonian evolution is a further conditional specialization, available only whenthe stabilized continuation admits an appropriate reversible continuous representation. Thepresentation is deliberately didactical and proceeds from familiar dynamical language towardclosure and stabilization rather than assuming the framework in advance.