Structural Limits of AI Accountability: Institutional Rank and the Geometry of Governance

As artificial intelligence systems are deployed into complex human institutions, the standard reflex of the technology industry and its regulators has been to treat evaluation as an epistemic problem—a lack of knowledge that can be solved with more data or rigorous A/B testing. This manuscript argues that the problem is instead ontological. When human users and latent institutional incentives endogenously adapt to the presence of an algorithm, the system absorbs the intervention. Using the framework of Structural Causal Models (SCMs) and the do-calculus, this work proves a Structural Limit Theorem: under the constrained intervention sets typical of real-world deployment, the causal attribution of AI use is fundamentally structurally indeterminate. The true causal effect cannot be identified because the institution's adaptation absorbs the intervention, resulting in a Jacobian rank deficiency in the intervention mapping. To solve this, the manuscript introduces the concept of Institutional Rank—the effective dimensionality of causal experimentation permitted by an institution's governance structures. It demonstrates that transparency and auditing are mathematically insufficient to resolve attribution without expanding the admissible intervention set. Finally, the manuscript introduces the Cognitive-Structural Field (CSF-2 deprecated) newly named to Authority Bounded Containment Field (ABCF-v1) and the Structural Probe Domain (SPD), a rigorous institutional architecture designed to safely execute the geometric expansion of interventions (probes) required to restore identifiability, without allowing algorithmic metrics to silently usurp human authority.