The Canonical Society Theorem

Under specified source laws and complete responsibility contracts, we construct a family of inhabited canonical societies and prove that actor generation, root-law completeness, real exit, and all-depth regeneration can hold together. Actors arise from the sources and lineages of activity. Every complete installed institutional execution receives presentation, refusal retaining the bearer, action, or revision grounded in exact failure; actual writing occurs precisely in the last two cases. General exhaustiveness and inhabitation at one constitutional point are proved separately. At every natural-number depth, the same actual payment jointly generates six regeneration guarantees, and its actual successor is the entry to the next depth. More strongly, at every position in every cycle of the original type, each lawful payment carries the complete canonical contract into its actual target and generates there a new service or exit occurrence outside the entire old authority image. Even when instantaneous exit capacity remains one throughout, the frontier of fulfilled occurrences grows strictly with depth. Beyond every finite stage, actual revision and a paid successor birth also occur together. "Canonical" has its mathematical meaning: a standard construction generated from specified sources and rules, with its parts connected by compatibility relations. These positive results and a family of impossibility results jointly characterize what institutions can accomplish. A worker's inability to exit may arise from missing payment incidence, an interrupted route, or captured continuation; all three yield zero capacity. Two actual sources with capacity one can likewise have, respectively, an inhabited and an empty complete capability portfolio. Accurate capability counts are therefore insufficient to identify failure sources, determine domain completeness, or select exact repair. In the constructed source families, this obstruction persists even when all domain-reachability readings are retained; two actual branches even share a root event, whole-ledger write-back, and next state while requiring different repairs. Conversely, the failure kind retained in the source suffices to recover canonical repair and its validity. The economic runs further separate legal recognition, settlement of old debt, and continuing payment. Original responsibility passes through revision to actual performance; for an exhausted source ledger, every same-source sequential target that adds no initial rows and increases no row's budget has zero progress budget at its first whole-ledger successor. Institutions must thus account for the bearer and consequences of every exercise of power, and supply actual sources for further continuation.