TREP:Topological–Renormalization and Endpoint Persistence. A Conditional Ledger Calculus for Critical-Channel Exclusion

We formulate TREP (Topological-Renormalization and Endpoint Persistence) as a ledger calculus for filtered systems in which local reductions, endpoint limits, boundary routes, and defect rows must be accounted for within a common primitive budget. The basic object is a critical channel: a family of local states or defects whose synchronized persistence through a stopped tree would produce a terminal obstruction. The main quantitative result is a persistence bound for ledgered passive stopped Hilbert systems. If synchronized witnesses below a fixed parent carry a uniform positive lower charge, if the witness family has same-parent overlap at most M, and if all defect and port rows are dominated by primitive ledger rows, then the length of the synchronized tube is bounded by the ratio of the total primitive parent ledger to the uniform lower witness charge, scaled by the overlap factor M. Thus, infinite synchronized persistence is excluded whenever the parent ledger is finite. The proof is a finite-budget argument combining same-parent Carleson packing with passive or defect-positive visibility. The paper also records the associated passive terminal-exclusion theorem, the synchronized no-tube theorem, an instantiation criterion for concrete models, and stability under quotients and finite ledger extensions. The framework is conditional in the precise sense that, in each application, the same-parent witnesses, uniform lower charge, primitive domination, and no-late-endpoint rows must be proved independently. Model cases include Carleson ledgers on stopped trees, scalar passive colligations, and defect-visible stopped channels.