Common affine calibration cancellation in the lever rule

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 $m\ne 0$,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 $x\mapsto 1.08x+0.03$ 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 $0.002/0.648\approx 0.00309$. 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 isIt 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 $JC{J}^T=0$. The measurement guide explicitly identifies common instruments and reference data as sources of correlation that must be retained.^[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.