= Shared-reference cancellation in transferred blank corrections
\b[Shared-reference cancellation in transferred blank corrections] occurs when the same numerical reference enters a correction with opposite coefficients. It is an algebraic property of the measurement equation. Its uncertainty counterpart is a <covariance> contribution, not permission to discard arbitrary background uncertainties.
Let a specimen reading be $T$, a source blank $A$ and a target blank $B$. If the estimated transfer is $D=B-A$, the corrected result is
$$
Y=T-A-D=T-B.
$$
The source blank cancels exactly. Even when $T,A,B$ are independent, $A$ and $D$ are not: $\operatorname{cov}(A,D)=-\operatorname{Var}(A)$. Applying the correlated-input rule gives
$$
\operatorname{Var}(Y)=\operatorname{Var}(T)+\operatorname{Var}(A)+\operatorname{Var}(D)+2\operatorname{cov}(A,D)=\operatorname{Var}(T)+\operatorname{Var}(B).
$$
The GUM requires correlations from shared inputs to be retained and describes equivalent formulations in which the common input is introduced explicitly.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.2.2 and 5.2.4]{ref}
For stipulated independent <standard deviations> $u(T)=1$ mg, $u(A)=2$ mg and $u(B)=3$ mg, $u^2(D)=13\,\mathrm{mg^2}$ and $\operatorname{cov}(A,D)=-4\,\mathrm{mg^2}$. Thus $u^2(Y)=1+4+13-8=10\,\mathrm{mg^2}$. Incorrectly treating $T,A,D$ as independent gives 18 instead. The same result follows from the <covariance matrix>
$$
\Sigma=\begin{pmatrix}1&0&0\\0&4&-4\\0&-4&13\end{pmatrix}\mathrm{mg^2}
$$
and coefficient vector $(1,-1,-1)^{\mathsf T}$. These numbers illustrate error accounting, not a measured carbon-recovery test.
<Reference identity in algebraic uncertainty cancellation> is essential. If a transfer used an earlier reading $A_1$ but the correction subtracts a later reading $A_2$, then $Y=T-B+A_1-A_2$. Their difference does not vanish merely because both were called the source blank. Correlation may reduce or increase its uncertainty depending on how the readings were obtained.
Preserve reference identifiers, timestamps and coefficients in the data reduction. Simplifying the measurement equation before propagating uncertainty can expose exact cancellations; a transfer estimated independently or containing additional drift needs its own remaining uncertainty terms.
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