= Calibrated photometer model
Use the triangular $w$ in <equal radiant power under visual weighting> and stipulate the detector's <spectral responsivity>
$$
r(\lambda)=(0.5\,\mathrm{A\,W^{-1}})w(\lambda)
\left(1+\frac{\lambda-555\,\mathrm{nm}}{150\,\mathrm{nm}}\right).
$$
This model has nonnegative response: $w$ is zero outside $405$–$705\,\mathrm{nm}$. For a $1\,\mathrm{mW}$ ideal calibration line at $555\,\mathrm{nm}$, $i_A=0.5\,\mathrm{mA}$ and $\widetilde\Phi_{v,A}=0.683\,\mathrm{lm}$, so $C_A=1366\,\mathrm{lm\,A^{-1}}$. An equal-<radiant flux> test line at $630\,\mathrm{nm}$ gives $w=1/2$, $r=0.375\,\mathrm{A\,W^{-1}}$ and $i_T=0.375\,\mathrm{mA}$. The unchanged conversion indicates $0.51225\,\mathrm{lm}$, whereas the model's intended weighted result is $0.3415\,\mathrm{lm}$. The required <spectral mismatch correction> is $F=2/3$: the uncorrected result is 50% too large. All these numbers are declared model outputs, not instrument measurements or standard <photometry> data.
\Image[/photometry-calibrated-detector-spectral-mismatch.png]
{title=Calibration at one <wavelength> leaves a spectral mismatch at another}
{id=photometry-calibrated-detector-spectral-mismatch}
{description=Original schematic and analytic model. The apparatus shows the signal path of a filtered <photodiode> <photometer>, with shutter-closed subtraction; it is not a drawing to scale of a specific instrument. The lower panels use the declared triangular weighting and detector response, not CIE tables. The two ideal lines carry the same collected <radiant flux>; the source-dependent factor $2/3$ corrects the test indication within this model only.}
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