"Qiutong Cunyi" in Diplomacy: Invariance in the Topology and Geometry of Coupled Generative Systems

A diplomatic formula of the twentieth century holds that parties should seek what they have in common while preserving what they do not. Read as a claim about structure it asks what two parties can hold together without either adopting the other's terms. This paper develops that reading in the mathematics of composite systems. A pair is given a joint state on a product of their state spaces, the operations each party may perform on its own side generate a group acting on that state, and the common part is identified with the functions constant on the orbits of that action while the preserved difference is identified with the coordinates the action moves. Nothing is coarsened and no shared description is built. Five results follow. Each party can compute the complete invariant from its own description alone, so long as the pair stands in relation to nothing beyond it. The possible common parts carry an order that is partial and a distance that is total, the distance extending the order and deciding every pair it leaves open while being one extension among many that disagree about one such pair in eight. The descriptions available to a party at a given common part form a manifold of flags, whose Euler characteristic counts the distinguishable ways that party has of describing itself, and it falls to its minimum both where the common part is uniform and where it is trivial, so it is maximal strictly between the two ends. A party that enlarges its own capacity to describe leaves the common part where it stood and multiplies that characteristic. And the common part has no resting place under a generator that does not change with time, while a generator that dissipates gives it one which forgets the initial condition and moves toward the uniform end as the coupling grows against the dissipation, at the cost of the completeness of the invariants. The formalism is used for its invariant theory and no physical claim is made; the scope section states in five particulars what is used and what is withheld.