Eventual rounded source state from finite residual inputs

Fix a finite set $P$ of primes containing every prime at most $W$, positive exponent $E$, physical intervals with lower endpoints at least one and upper endpoints strictly below two, and a guard-ledger family. Fix depth, cutoff, rough level $K$, $c>0$, active coefficient $\beta_a>0$, positive target margin floor, and protected coefficient $\beta_p\ge0$. Suppose the Section 8 analytic ledger holds for the raw smooth base mass and scalar families $\widetilde q,A_0$. Then eventually, uniformly over canonical bridges with these sample data and compatible guarded bank certificates, every target with cell-mass margin above the floor yields a source state for its top-frozen rounded selector, provided the bridge guard agreement, exact equality of $m_{\rm frozen}$ with its concrete frozen-top mass, and the finite top-frozen residual-input predicate hold. The latter predicate supplies feasibility, charged nonsmooth rows, and prime-band support for the unrounded source. The conclusion is

Only the stated source-specific identities and residual inputs remain at the synchronized finite index.