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Separate spectral-calibration gains in isotope stiffness inference arise when the two Raman spectra retain different multiplicative calibration errors. Let the corrected wavenumbers satisfy ${\hat{\nu }}_i=(1+g_i){\nu }_i$. For $K=({\mu }_2/{\mu }_1)({\nu }_2/{\nu }_1{)}^2$, small residual gains giveThe relative calibration error in an isotope-corrected stiffness ratio article treats the accompanying offset terms. Here the important quantity is the gain difference, not either gain alone.

If the two residual gains have standard uncertainties $u_1,u_2$ and correlation $\rho$, their contribution isThis follows by applying the full covariance propagation rule.^[ref] A shared scale component cancels; independent session errors do not. The covariance of differential spectral-calibration gains therefore determines whether combining calibrations improves the ratio.

For an original example, let $u_1=u_2=0.2\%$. Independent calibrations give relative uncertainty about $0.566\%$ in $K$. A justified correlation $\rho =0.9$ reduces it to about $0.179\%$; a correlation $-0.9$ instead gives about $0.780\%$. These are different calibration models, not alternative ways of adding the same independent errors.

A practical comparison can measure the same known reference line during each sample session. If raw readings are $y_i=c_i{\nu }_i$ and reference readings $r_i=c_i{\nu }_{\mathrm{ref}}$, a purely multiplicative model givesBoth session gains cancel, but reference-reading noise, offsets and drift between reference and sample readings remain. Repeated readings do not automatically reveal an unknown common bias.

The primary graphene isotope experiment compared Raman bands over isotope concentrations and excitation energies, finding the harmonic-oscillator mass dependence.^[ref] The calibration model above concerns precision of the inferred residual stiffness ratio; it does not validate mode assignment or the effective-mass approximation.

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.