Abstract
We develop a self-contained variational calculus for the coordination of synchronised agents. An agent is modelled as an overdamped (Onsager–Machlup) dynamical system on a five-dimensional manifold M=[0,1]×[0,2π2]×[0,1]3 whose coordinates are an internal Kuramoto order parameter R, a phase variance σ2, and three S-entropy coordinates (Sk,St,Se). Crucially, and in contrast to a preliminary version of this work, every construct used here—the S-entropy coordinates, the partition potential, and the governing Lagrangian—is derived within the paper from three stated axioms (bounded phase space, categorical exclusion, finite resolution); the manuscript depends on no external or unpublished result. We prove (i) a triple-equivalence identity establishing the S-entropy coordinates from maximum-entropy counting; (ii) the explicit Euler–Lagrange equations of the agent action, given in full rather than asserted; (iii) the Kuramoto synchronisation bifurcation, with the corrected critical coupling Kc=2/[πg(0)] (for a Gaussian frequency law, Kc=2πσω(2/π)≈1.596σω), replacing an erroneous coefficient in the preliminary version; (iv) a formal, analogy-free definition of partition extinction and an observable-commutation theorem that separates functional from ontological indistinguishability; and (v) a coordination calculus for ensembles. We are explicit that the five coordination bands on R are operational classification cut-offs, not phase transitions: the only genuine transition in the model is the Kuramoto bifurcation. Rather than the self-referential numerical checks of the preliminary version, we report nine external experiments against public data (atomic shell structure, the Kuramoto onset by direct simulation, the Wiedemann–Franz constant across eleven metals, sleep-EEG order parameters from PhysioNet, enzyme kinetics, antidepressant meta-analysis, superconductor transport, and representation-disjoint classifiers on a benchmark dataset). Two of these external tests forced corrections to the theory and are reported as such. We are deliberately narrow about applicability: the S-entropy coordinates measure the shape of an agent’s state distribution, not the content or ecological salience of its information; the model is stochastic (Langevin) but single-scale and on a fixed manifold, so it does not represent the multiscale fluctuation and metastability of real biological collectives; and we exhibit a concrete interpretable Lagrangian and a real-data ensemble instance rather than claim to capture any such collective in full. A Scope and Limitations section delimits the defensible claims.