= Common affine calibration cancellation in the lever rule
\b[Common affine calibration cancellation in the lever rule] is the exact invariance of the <binary mixture> <phase fraction> under the same <affine map> applied to the overall composition and both endpoints. For $m\ne0$,
$$
\frac{(mz+c)-(ma+c)}{(mb+c)-(ma+c)}=\frac{z-a}{b-a}.
$$
Both a shared offset and a shared scale cancel. The map must be common to all three measurements; separately fitted calibration curves or sample-dependent effects do not satisfy that assumption.
An original example has $a=0.10$, $b=0.70$, $z=0.25$ and $\lambda=0.25$. Applying $x\mapsto1.08x+0.03$ gives measured values $0.138$, $0.786$ and $0.300$, still yielding $0.162/0.648=0.25$. If only the overall-composition reading then receives an additional $0.002$ error, its inferred fraction changes by $0.002/0.648\approx0.00309$. Common calibration cancellation leaves independent sample errors intact.
The <covariance matrix> formulation explains why treating the shared errors as independent gives a false uncertainty. In input order $(z,a,b)$, the <Jacobian matrix> row is
$$
J=\frac{(1,-(1-\lambda),-\lambda)}{b-a}.
$$
It annihilates both the common-offset direction $(1,1,1)^T$ and the common-scale direction $(z,a,b)^T$, since $z=(1-\lambda)a+\lambda b$. A joint <covariance matrix> representing only these shared affine influences therefore gives $JCJ^T=0$. The measurement guide explicitly identifies common instruments and reference data as sources of correlation that must be retained.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.2, equation13; §5.2.4, common influences]{ref}
Analyzing the overall sample and separated phases under one unchanged <instrument calibration> can realize this common contribution; repeating only one sample under another calibration breaks it. The cancellation is a property of this binary ratio. It does not remove nonlinear response, composition closure errors, endpoint misidentification or uncertainty in whether the samples establish <liquid-liquid equilibrium>.
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