Programming with Geometry: Synchronous Spatial Execution and Host-Order Erasure in Operator-Witness Fabrics

Canonical Geometric Programs defined a static program object in which textual and visual editing surfaces converge on one normalized FabricSpec, semantic placement is coordinate-free, and operator-witness typing determines interface topology. This companion paper defines how that object computes. A run is modeled as a sequence of synchronous global states in which the canonical program is fixed and dynamic state records payload custody, pending input frames, admission candidates, obligations, and receipts. Each governance tick performs snapshot, cell evaluation, route propagation, runtime governance, and simultaneous commit. The execution model introduces a four-state payload machine, canonical role-keyed input frames, partial-input custody, and durable egress dispositions. It distinguishes successful termination from nonterminal fixed points through HALTED and BLOCKED. Four quiescence results and the central HOST_ORDER_ERASURE theorem establish that valid host schedules produce the same canonical successor state. Consequently, run status, blocked reasons, and canonical tick evidence are independent of host iteration order. A reference implementation passes a frozen 70-vector corpus. Repeated runs reproduce all 71 evidence artifacts byte-for-byte. The result is a software-FPGA execution semantics in which computation remains spatial and synchronous when realized on a conventional sequential or multithreaded host.