Canonical actual covariance row with separate sharp remainders

Fix a finite nonempty head family $P$, physical intervals $I$ with lower endpoints at least one and upper endpoints at most $C_{\max}$, and a guard ledger at every $n$ with fixed census constants $C_{\rm prom},C_{\rm bank}$. For a valuation law $\lambda$, put $\Delta_\lambda(p,q)=\operatorname{Cov}_\lambda(v_p,v_q)-\operatorname{Cov}_\lambda(\mathbf1_{p\mid m},\mathbf1_{q\mid m})$. Let $C_*$ be the fixed box-independent prime-power constant. After choosing $W>1$ above all head primes, then $A_c,A_p\ge0$, there exist a nonnegative $\epsilon_n$ and threshold $N_0$ such that $\epsilon_n\to0$ and $\epsilon_n\log\log n\to0$. Write $G=\mathrm{canonicalPhysicalPowerCorrectionConstant}(P,I,C_{\max},W,A_c)$, $e_n=A_p\log C_{\max}/\log n$, and $R_n=\mathrm{physicalPowerCorrectionRowError}(e_n,G,n,W)$. Put $C_e=\mathrm{valuationLogCoefficient}(C_{\max},W)$, $D=\mathrm{canonicalGuardPerturbationConstant}(P,I,A_cC_e)$, and $T_n=R_n+\mathrm{guardPowerCorrectionWeightedMajorant}(C_{\rm prom},C_{\rm bank},C_e,D,n)$. Then $T_n\log\log n\to0$. The estimates hold uniformly for every later finite band partition, bridge $B$, and tilt parameter $\xi$ at ambient $n\ge N_0$, provided its cutoff is $W$, its sample data equal the canonical constructor with separated intervals and nonempty surviving cells, every effective prime coefficient is at most $A_c$ in absolute value, and the physical coordinate has absolute value at most $A_p$.

The raw-law remainder and the actual/reference correction remain separate, with their precise rates exposed for the moving-low argument.