= Conditioning of tie-line phase-fraction inference
{title2=$1/|b-a|$}
The \b[conditioning of tie-line phase-fraction inference] describes how strongly composition perturbations change an inferred <phase fraction>. For a <binary mixture> with distinct endpoints, the <lever rule> gives $\lambda=(z-a)/(b-a)$. Its absolute sensitivity to $z$, with endpoints fixed, is $1/|b-a|$. This is an absolute amplification coefficient; a relative <condition number> additionally depends on the chosen input and output scales.
For an interior $0\leq\lambda\leq1$, the differential in <tie-line phase-fraction sensitivity> yields
$$
|\delta\lambda|\leq
\frac{|\delta z|+(1-\lambda)|\delta a|+\lambda|\delta b|}{|b-a|}
$$
to first order. If each composition perturbation is bounded by $\varepsilon$, this becomes $2\varepsilon/|b-a|$. For original stipulated $\varepsilon=0.001$, endpoint separations $0.50$ and $0.05$ give local worst-case bounds $0.004$ and $0.040$. Tenfold closer endpoints make the same composition precision tenfold less informative about phase amount. Correlations can change the attainable combinations of errors, as <common affine calibration cancellation in the lever rule> illustrates.
For fixed multicomponent endpoints and overall-composition <covariance matrix> $\Sigma$, the <tie-line phase-fraction uncertainty> is
$$
u(\lambda_0)=\frac1{\sqrt{\mathbf d^T\Sigma^{-1}\mathbf d}}
=\frac1{|\Sigma^{-1/2}\mathbf d|}.
$$
The denominator is endpoint separation after a <whitening transformation>: separation must be compared with measurement precision, not only plotted distance. This follows from covariance-based <uncertainty propagation> and the linear fit.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.1–5.2, sensitivity coefficients and covariance propagation]{ref}
Near coincident endpoints the inverse problem becomes poorly informative, and at coincidence it is undefined. Near the physical bounds, a large propagated uncertainty also warns that the interior linear approximation may fail. More numerical digits in the fitted answer do not resolve this measurement limitation.
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