Agentic workflows are commonly evaluated by whether they reach the correct outcome. That is insufficient in institutional settings, where a correct action may rely on the wrong authority, an unsupported completion claim, or work made stale by a later change. We define governed execution as work whose decisions, completion, and response to change are supported by inspectable provenance. We present Matrix, a deterministic causal-state layer that records authority and fact dependencies, verifies completion evidence, and selectively invalidates affected work. Across controlled comparisons, governed and direct workflows often reached the same outcomes, but only the governed path consistently preserved governing evidence, refused unsupported closure, and limited recovery to dependent tasks. A role-separated transfer challenge then failed: a deterministically enforced completeness contract severely over-blocked synthetic packets produced outside its authoring context. These results do not establish Matrix as a general accuracy enhancer; they support its primary role as an institutional integrity layer for making agentic work auditable and independently verifiable. Blogger's Review: This paper proposes a new method for managing agentic workflows by recording and verifying the provenance of the workflow to ensure its integrity and reliability, which is of great significance for workflow management in institutional environments.