Street Spin Foam: A Preliminary Formulation

Movement through a city is usually modelled by fixing a geometry, placing agents inside it, and treating the worth of a location as a number attached to that location. This paper develops a representation in which relations carry the state instead. A street district is written as a compact lattice gauge theory of valuation on a two-complex, in the spin-foam representation of covariant loop quantum gravity. Persons, places and stores become one-cells, that is, worldlines, and the two-cells become the histories of the relations among them, so that being somewhere is a relation with a history rather than a coordinate a person carries. An entity is then a constraint on how the relations meeting at it may combine, rather than a bearer of state. Two consequences follow. Supplying a group and an action makes the spins, the intertwiners and the vertex amplitude follow from a character expansion rather than from stipulation, so that a large collection of local weights reduces to a small number of global choices; the expansion is verified against an independent numerical evaluation. Pedestrians then enter as charged matter, minimally coupled, and Kolmogorov's criterion yields a prediction sharper than the statement that valuation affects movement: circulation is sourced by the topological part of the valuation field alone, so that a region of large local disagreement and zero winding carries zero net circulation. A model whose forces are gradients of potentials leaves that pattern inaccessible, which makes the difference testable rather than interpretive. A case study on the street network of Ginza, Tokyo, with constructed commercial content, illustrates what the formalism does. The gauge structure changes simulated behaviour against a flat-connection control; circulation tracks the winding number while leaving a matched non-topological predictor inert; an effective distance read off correlation decay departs from street distance; encounter order is inert under an abelian connection and becomes material as soon as the valuation axes separate; and a walk on the complex sees an effective two-dimensional geometry over about a decade and a half of scale. One prediction fails, and its diagnosis is the most transferable result: it was imported from a confining gauge theory that the model leaves uninstantiated. The cell assignment admits two orientations, with entities on edges or objects on faces, and a counting argument bears on which of them leaves a configuration space to sample. Under the second, persons enter the spin foam directly, on condition that the one-cells are occasions gathering several parties rather than pairwise relations. The contribution is a representation in which relations rather than sites carry the state, a route by which local weights follow from a group and an action, a falsifiable point of difference from gradient-based models, and a criterion for choosing the cell assignment. Estimating the couplings from observation is identified as the next stage.