Spatial resource bounds for classical-communication simulation of controlled momentum-kick statistics

We study classical-communication simulation of the joint source and probe statistics of a qubit-controlled momentum kick. Competitors retain quantum local matter, arbitrary causal classical memory and two-way communication, but no shared entanglement or quantum communication. For fixed Gaussian probe preparation and specified source preparations and readouts, an explicit local instrument approximates the complete continuous momentum statistics arbitrarily well while preserving branch-control distributions exactly. An independently bounded full output position second moment gives a strictly positive total-variation lower bound against the entire competitor class; the explicit simulator attains the same asymptotic order. At fixed nonzero kick separation, the minimum RMS spatial resource for error d scales as Theta(d^(-1/2)); at fixed relative size greater than one, the minimum error scales as Theta(eta^2) in the weak-kick parameter eta. Fixed finite bins admit exact simulation at sufficiently large finite resource. These are statements about specified readout statistics, not full-channel approximation, a Newtonian realization of the simulator, or demonstrated experimental feasibility. The manuscript discusses conditional routes to independent resource certification. This record contains the preprint, editable LaTeX source and an offline reproducibility archive. The archive reproduces Table 1 and runs 22 computational tests; three project-history tests requiring undistributed materials are explicitly excluded. The preprint has not undergone external peer review. Licensing: the manuscript, documentation and numerical records are licensed under CC BY 4.0; the Python software and tests are licensed under MIT. The archive specifies these separate scopes.