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An integer allocation near a continuous isotope-reference optimum converts a real-valued budget solution into feasible counts of sample and reference observations. For the model in reference-versus-sample allocation in repeated isotope-ratio measurements, minimizewith positive integer counts. The shared component $s_c^2$ does not affect the choice when it is genuinely independent of the allocation. Counting new reference readings does not make a shared calibration error independent.^[ref]

For equal unit costs and a budget of $N$ readings, put $n_r=N-n_s$. With $s,r>0$, the count-dependent objective $f(n)=s^2/n+r^2/(N-n)$ hasIts continuous minimum is $n_{\ast }=Ns/(s+r)$. Therefore an integer minimum lies among the feasible floor and ceiling of $n_{\ast }$, clipped to $1,\dots ,N-1$. Comparing those candidates is sufficient because the function decreases before its minimum and increases after it. Rounding both continuous counts independently can violate the budget.

For an original example, let $N=17$, $s=0.003$ and $r=0.001$. The continuous sample count is12.75. Counts $(n_s,n_r)=(12,5)$ give variance $3.8000\times 10^{-6}$ before a common component, while $(13,4)$ give approximately $3.7692\times 10^{-6}$. The latter is preferable, with log-ratio standard uncertainty about0.1941%. The nearby $(14,3)$ gives approximately $3.9048\times 10^{-6}$.

Unequal costs introduce a stepped feasible boundary. One exact finite search enumerates positive $n_r$ and, for each, useswhen that value is positive. For fixed $n_r$, using the largest feasible $n_s$ improves this stated objective. For an original $T=20$, $c_s=3$, $c_r=2$, $s=3$ and $r=1$ in consistent planning units, the best pair is $(5,2)$, consuming19 units. Its count-dependent expression $s^2/n_s+r^2/n_r$ is2.3; $(6,1)$ and $(4,4)$ both give2.5. An unused budget unit can be a legitimate integer remainder.

Acquisition ordering, paired exposures, minimum replicate counts or drift constraints can restrict the feasible set further. The continuous optimum remains a useful guide, but only comparisons among genuinely feasible integer designs establish the chosen discrete optimum.

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.