 Shared-reference floor in averaged isotope stiffness ratios

A shared-reference floor in averaged isotope stiffness ratios is the calibration uncertainty that remains when many sample measurements reuse one uncertain reference. In the first-order gain model of covariance of differential spectral-calibration gains, let the error in log stiffness-ratio estimate $i$ be $2(e_i-e_r)$. Assume independent sample errors with variance $s^2$ and one independent reference error with variance $s_r^2$. Averaging $m$ estimates givesThe sample contribution decreases with replication, while the shared reference contribution is unchanged. This is the covariance propagation rule applied to the actual reuse of the reference, rather than treating every ratio as independent.^[ref]

For an original example, choose sample calibration standard uncertainty $s=0.2\%$ and reference uncertainty $s_r=0.1\%$. One ratio has approximately $0.447\%$ standard uncertainty on the log-ratio scale. Sixteen sample measurements reduce it to about $0.224\%$, but the limiting floor is $2s_r=0.200\%$. Reaching within ten percent of that floor requiresso $m\ge 20$ for these chosen numbers. Replication beyond that point has progressively less effect unless the reference uncertainty is reduced.

If a new independent reference is measured for each sample, the model changes: both error contributions can average down, giving $4(s^2+s_r^2)/m$ under the corresponding independence assumptions. Alternatively, averaging several independent reference measurements reduces their random component, while a systematic component shared by all references still remains. A reference-versus-sample allocation in repeated isotope-ratio measurements must distinguish those resources and common influences.

The floor is conditional on the measurement model. Additive spectral calibration errors, reference drift, isotope-composition uncertainty and mode-assignment error are separate contributions. A useful acquisition protocol records which sample spectra share which reference observations; numerical averaging alone cannot remove an uncertainty component reused unchanged in every estimate.