FPM Foundations: Finite Local Substrates, Directed Route Ledgers, and Exact Closure

Finite Possibility Mechanics (FPM) is a mechanics of directed alternatives under finite memory, finite energy, and finite update time. Its primitive object is a local directed route ledger carried by the sites of a finite periodic cubic lattice. A positive per-tick route cost is transported through a nearest-neighbour Markov kernel and recorded in a closed signed energy ledger. This paper gives the model a precise finite-graph foundation and proves the results on which the later carrier, exact-ledger, physical-bridge, and empirical papers depend. The local transport matrix is non-negative, row-stochastic, reversible with respect to the activity measure, and supported only on graph edges and self-loops. Replenishment conserves total interior energy, admits an antisymmetric edge-flux representation, satisfies nodewise and regional continuity, and propagates support by at most one graph edge per tick. Under frozen positive weights on a connected lattice, repeated local transport converges to the activity-weighted distribution.