= Reference identity in algebraic uncertainty cancellation
\b[Reference identity in algebraic uncertainty cancellation] asks whether two appearances of a reference are the same measured quantity or merely readings given the same descriptive name. If a transfer is $D=B-A_1$ but a correction subtracts $A_2$, then
$$
Y=T-A_2-D=T-B+A_1-A_2.
$$
The last difference vanishes identically only when the same reference value is reused. Separate acquisitions may be correlated, independent, or affected by drift; sharing a label does not decide their <covariance>.
For independent $T$ and $B$, also independent of the reference pair, propagation gives
$$
u^2(Y)=u^2(T)+u^2(B)+u^2(A_1)+u^2(A_2)-2\operatorname{cov}(A_1,A_2).
$$
With equal reference <standard uncertainties> $s$ and correlation coefficient $\rho$, the residual reference contribution is $2s^2(1-\rho)$. It is zero for identical reused values, $2s^2$ for independent readings, and is not generally zero just because the displayed numbers agree. The GUM requires significant correlations from common inputs to be included.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.2, equation 13; §5.2.4, common instruments and standards]{ref}
An original acquisition model separates what cancels:
$$
A_1=A_0+C+\epsilon_1,\qquad A_2=A_0+C+\epsilon_2.
$$
Here $C$ is a common uncertain calibration contribution, while $\epsilon_1,\epsilon_2$ are independent repeat errors. Their difference removes both $A_0$ and $C$, but leaves $\epsilon_1-\epsilon_2$. If $u(C)=3$ mg and each repeat error has <standard deviation> 2 mg, each reading has <variance> $13\,\mathrm{mg^2}$ and their <covariance> is $9\,\mathrm{mg^2}$. The difference therefore has <variance> $13+13-18=8\,\mathrm{mg^2}$, not zero.
Adding independent $u(T)=1$ mg and $u(B)=3$ mg gives $u^2(Y)=18\,\mathrm{mg^2}$. Reusing exactly one recorded reference value in both places instead gives $u^2(Y)=10\,\mathrm{mg^2}$. These constructed cases have different data-reduction histories despite similar-looking equations.
<Reference lineage in covariance models> records acquisition identifiers, shared calibration inputs and coefficients in each derived quantity. A calibration change or time-dependent drift must appear in that model if significant. Algebraic simplification is valuable only when it preserves the identity of the underlying inputs; renaming two acquisitions with the same symbol does not make their errors cancel.
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