Cutoff-aware analytic completion at every selected capacity depth

For every $c>C_0$ and natural depth $d\ge201$, there exists a cutoff $W_0$ such that every $W\ge W_0$ satisfying $W\ge2$, $W\ge2d+1$, and the public reciprocal-sum and actual-moment cutoffs has the following property. For any $1<r_0<3/2$ and paper combined-tangent exponent $\delta_*$ for $(c,W,r_0)$, a combined-charge terminal at depth $d$ yields the synchronized top-frozen Section 9 analytic completion at the same $(c,d,W,r_0,\delta_*)$:

The completion consists of a regular mesh, a family of bridges eventually at index $n$ and cutoff $W$, a multiplicity, positive tangent and reserve constants, and a synchronized post-height fitting input. Its active masses satisfy $q_n\le C_qs_n$, its post-fit bound satisfies $(2/9)C_{\rm post}C_q\le C_{\rm tan}$, and its mesh obeys the ratio-cell width bound for ratio $21/20$. The mesh tolerance is chosen before $W$, and the numerical and analytic constants are chosen before the final mesh.