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Flow-history ambiguity of a timed expiratory volume arises because integrating volumetric flow rate retains the total gas moved but discards its detailed time ordering. If a timed measurement is $A={\int }_0^{T}Q(t)dt$, then every admissible perturbation $h$ satisfyingproduces the same $A$ from $Q+h$. Nonnegative flow restricts the possibilities, but generally does not leave a unique waveform. This is an integral null-space ambiguity in expiratory measurements.

An original example preserves both FEV1 and FVC. Let a constant baseline flow $Q_0=1\mathrm{L}{\mathrm{s}}^{-1}$ continue for 4 s. Modify only the first second, with $T=1$ s, byand set $h=0$ afterward. The modified flow stays between 0.5 and 1.5 L/s during the first second. Its cumulative difference from the baseline isAt one second the difference returns to zero. Both traces therefore have FEV1 of 1 L, FVC of 4 L and FEV1 to FVC ratio of 0.25. Their peak expiratory flows differ: 1 versus 1.5 L/s. At half a second, the modified trace has expelled approximately 0.159 L more than the baseline, before the later flow difference cancels it.

These artificial waveforms isolate information loss; they are not physiological reference curves. A timed volume is not interchangeable with a peak value or an entire flow-volume history. The 2019 technical statement requires inspection of both volume-time and flow-volume displays because the displays expose different features of a maneuver.^[ref]

Adding more timed integrals can constrain the waveform more strongly, but finitely many such summaries still leave many possible continuous histories unless a restrictive model is imposed. Report which waveform features were actually measured rather than inferring all of them from one scalar volume.