Finite Operational Structure Theory: A Mathematical Theory of Finite Operational Cuts

Finite Operational Structure Theory (FOST) develops a mathematical framework for finite operational reasoning. It does not assume a hidden true state, a complete view of reality, a global probability model, or an objective standard as primitive. Instead, it studies what a finite process can operationally support when it evaluates a finite expression under a declared mode of use. The core object of the theory is an evaluated ledger: a finite record of claims, support, certificates, omissions, unresolved issues, obligations, environment conditions, and transition witnesses. FOST treats a ledger not as a proof of truth, but as an accountability structure that shows what was checked, what was assumed, what remains unavailable, what is blocked, and what must be rechecked later. The theory introduces finite operational cuts, staged ledger construction, validation kernels, object kernels, certificate targets, support graphs, typed frontier closure, status records, admissibility records, transition validity, and obligation tracking. It also includes safeguards against vacuous reasoning, unsupported scope narrowing, missing external criticism, weak impact classification, environment omission, root-debt hiding, and misuse of certificates beyond their valid target. FOST is intended for settings where intelligent systems, AI agents, research workflows, safety cases, governance processes, or automated decision systems must act without access to final truth or complete information. Its purpose is to preserve finite operational support and make omissions, uncertainty, obligations, and changes visible rather than erased. The appendices connect the formal system to broader philosophical and methodological concerns, including appearance and reality, non-self, horizon and residue, fallibilism, constructivism, approximation, bounded rationality, argumentation, safety assurance, and runtime monitoring. These sources are used as motivation and comparison, while the formal theory remains based on finite records and operational checks.