The rounded smooth source-to-guarded defect has the paper rate

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 head patterns and physical intervals, a guard ledger, $W>1$ containing all head primes, natural $K_0,d$, $c>0$, a real active coefficient $\beta_a$, and $A_z\ge0$. Consider any later bridge $B$ at cutoff $W$ with exactly the canonical guarded sample, bank realization, and guarded anchor certificate. Require the structured active values to lie in the guarded smooth row, $\delta_*\le1$, a nonempty guarded broad smooth pool, and exact synchronization of $\widetilde q$ with the guarded smooth-base mass. For each prime under consideration assume both zero-head-cell valuation means are at most $A_z/p$. For an arbitrary barycentric source target and protected coefficient, use the canonical post-fit balanced coefficient and top-frozen nearest-integer rounding. There exist $C_{\rm source}>0,N_0$, depending only on the fixed data above, such that for every such construction at $n\ge N_0$,

This is the complete defect predicate of the source, including rounding. The active coefficient is fixed before the uniform quantifiers, so the constant may depend on $|\beta_a|$.