Incidence budgets and completion thresholds in finite progression systems

How do simple counting budgets constrain completion in finite progression systems? This companion presents exact centre-accounting contracts, an interval bound and supporting polynomial identities, with statement-level mathematical evidence. Saved native certificate admissions and explicit external proof acceptance remain separately labeled; unresolved inputs are retained rather than counted as proved results. These supporting contracts do not establish the open incidence bridge, a density theorem, or upper or asymptotic bounds for progression-free sets. No fresh proof replay or private checker distribution is claimed.