Coherent source and bridge families can be chosen after fixed pre-mesh data

Write $L_n=\log n$, $s_n=n/\log n$, $y_n=\lfloor n^{2/9}\rfloor$, and $\mathcal P_{n,W}=\{p\text{ prime}:W<p\le y_n\}$. Fix $c>C_0$, a paper combined-charge exponent $\delta_*$, $W>0$, protected and active coefficients $\beta_p\ge0$, $\beta_a>0$ with $\beta_p+\beta_a\le c/\rho_h(W)$, and multiplicity $K_0+1\ge1/\rho_h(W)$, where $\rho_h$ is the rough-head density; assume $W\ge2d+1$. Let $F$ be a guarded tail family whose certificates satisfy the combined anchor/bank capacity divisibility, base-bank and selector-charge divisibilities into the precharged target, and both exact target-product identities. For each prime $p\le2d+1$, require the precharged target valuation to exceed the selector-charge valuation by at least $(c-C_0)s_n/[24(p-1)]$. Then one can choose a positive head exponent $E$, positive source-cell and post-head margins, positive mass and density constants, and total scalar ledger families $\log Y,\Lambda_0,m_f$, all before the final mesh. Put $\widetilde q$ equal to the guarded smooth-mass family, $\eta_f$ equal to half the canonical physical tolerance, and $m_{\rm post}=m_{\rm head}\eta_f/\operatorname{physicalSpan}(I)>0$. These families satisfy

For every later regular relative mesh of positive width there is a total fresh bridge family satisfying the eventual coherent bridge/source obligation: eventually it arises from the literal fiber of $F$, has the canonical guarded sample, carries source and primitive-gap packages with the fixed margins, and has exactly the chosen cutoff, coefficients, smooth mass, mass bound, and cell-density bound.