Canonical eventual Proposition 8.7

Fix a positive permitted-mesh constant $c_{\rm mesh}$, physical intervals $I$ with endpoints between $1$ and $U\ge1$, natural guard budgets $C_{\rm prom},C_{\rm bank}$, and a guard ledger at every integer $n$. There exist a tolerance $\tau>0$ and a prime cutoff $W_0$ with the following property. For every $W\ge W_0$, finite nonempty head-pattern family supported exactly on the primes at most $W$, nonnegative constants $C_{\rm target},C_{\rm initial},C_{\rm mass},C_{\rm fixed},C_{\rm active}$, and positive cell-mass margin $m$, there exist a positive parameter radius $a$ and $C_{\rm post}\ge0$, chosen before the mesh. For every permitted regular relative mesh of width $\delta+\eta\le\tau$ with $\delta>0$, all sufficiently large $n$ have the following property.

Every bridge built from the canonical surviving cells at $(n,W)$, with separated physical intervals, nonempty surviving cells, canonical scale-separated partition, bridge width $\delta+\eta$, and barycentric baseline of cell margin at least $m$, admits a physically centered fixed-partition fit. Here the band displacement $\Delta$ satisfies the $C_{\rm target}$ envelopes; monitored primes are primes at most $y(n)$; $N\ge0$, active mass $q\le C_{\rm mass}N$, and each initial marked error is at most $C_{\rm initial}N/(pL)$. The finite frozen layer has ambient weights in $[0,1]$, frozen and active sample weights bounded respectively by $C_{\rm fixed}/L$ and $C_{\rm active}/L$, separated head patterns, and integer quota equal to frozen mass plus $q$. In the source's notation, the conclusion is

The fit is the source's bundled path certificate, including its displacement, invariance, marked-moment and feasibility requirements. The order of choices makes the analytic constants uniform before the eventual sample size.