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The 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 $0\le \lambda \le 1$, the differential in tie-line phase-fraction sensitivity yieldsto first order. If each composition perturbation is bounded by $\epsilon$, this becomes $2\epsilon /|b-a|$. For original stipulated $\epsilon =0.001$, 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 isThe 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.^[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.

Tie-line endpoint covariance records how the measurement errors of the two measured liquid-liquid tie line endpoints vary together. Using independent mole fraction coordinates, represent the two endpoint estimates by column vectors $\mathbf{a},\mathbf{b}$. Their cross-covariance matrix isAn individual off-diagonal block need not be symmetric; the full joint covariance matrix must be a symmetric positive semidefinite matrix. The uncertainties of the two endpoint vectors alone do not determine their cross-covariance matrix.

Near an exactly balanced, interior tie-line phase-fraction fit, let $\mathbf{g}$ be the row-vector sensitivity coefficient defined in tie-line phase-fraction sensitivity. If the overall composition measurement $\mathbf{y}$ is independent of both endpoints, first-order uncertainty propagation givesThe cross term has a positive coefficient because the two endpoint perturbations enter the phase fraction with the same sign. Correlated endpoints can therefore increase the result. The general rule is to propagate the full joint covariance matrix, rather than assume that correlation always increases or always reduces uncertainty.^[ref]

For a stipulated binary mixture example, use $a=0.20$, $b=0.80$, $z=0.44$, so $\lambda =0.40$. Treat $z$ as exact and give each endpoint standard uncertainty $s=0.006$ with correlation coefficient $\rho$. The lever rule yieldsThis gives $0.002$ for $\rho =-1$, about $0.00721$ for $\rho =0$, and $0.010$ for $\rho =1$. These are first-order standard uncertainties under the stipulated joint error model, not bounds on all possible errors.

A shared calibration error affecting the overall composition as well changes the calculation. Suppose the same additive error $c$ affects $a,b,z$ identically. ThenThe shared offset cancels exactly. More generally, a common affine map $x\mapsto mx+c$ with $m\ne 0$ also cancels in this binary ratio. This common affine calibration cancellation in the lever rule concerns only a truly common contribution; separate sample errors remain. If the shared offset has standard uncertainty $0.006$, incorrectly treating its three appearances as independent would assign a spurious contribution $0.01\sqrt{1+0.6^2+0.4^2}\approx 0.01233$ instead of zero.

In general, stack the inputs as $\mathbf{x}=(\mathbf{y},\mathbf{a},\mathbf{b})$ with joint covariance matrix $C$. Let $J$ be the corresponding Jacobian matrix row from tie-line phase-fraction sensitivity, including its residual correction when needed. Then $u^2(\lambda )\approx JC{J}^T$ automatically includes correlations with the overall composition. The measurement guide specifically identifies common instruments, measurement standards and reference data as sources of such correlations.^[ref]

An instructive measurement design analyzes the two equilibrium phases and the overall mixture using one instrument calibration, then repeats the entire procedure with independent instrument calibrations. The within-run contrasts and between-run changes probe different error contributions. Repeated readings under one unchanged instrument calibration cannot by themselves establish the uncertainty of its shared offset. The measurement model must identify which influences are common before the joint covariance matrix can represent them.