= Formazan recovery identifiability
\b[<Formazan recovery identifiability>] asks whether measurements distinguish the amount of <formazan> formed from the fraction recovered in the optical reading. Better repeatability does not necessarily separate those quantities.
Let $n$ be total product formed, $r\in[0,1]$ the <dissolved formazan recovery fraction>, and $K=\epsilon\ell/V$ a known optical conversion factor in a suitable fixed-path solution model. Then net <absorbance> is
$$
B=Krn.
$$
The manufacturer's procedure separates formation of crystals, dissolution and measurement, motivating these distinct model quantities.<Cell Proliferation Kit I MTT instructions by Roche>[Section3.1 and Figure4, printed p.9]{ref} Knowing $B$ and $K$ determines the dissolved amount $d=B/K=rn$, but not the two parameters separately.
For any admissible scale $a>0$, replacing $(r,n)$ by $(ar,n/a)$ leaves $d$ unchanged, provided $ar\leq1$. For observed $d=6\,\mathrm{\mu mol}$, both $(r,n)=(1,6\,\mathrm{\mu mol})$ and $(1/2,12\,\mathrm{\mu mol})$ fit. In this conserved-product model the bound $r\leq1$ implies $n\geq d$, with no finite upper bound supplied by that measurement alone. This is an <identifiability> limitation, rather than merely a large <measurement uncertainty>.
Changing volumes or optical paths can provide several known factors $K_j$, but measurements $B_j=K_jrn$ still determine only the same product if $r$ and $n$ stay fixed. Such changes can test the optical law without resolving the underlying ambiguity.
An original matched-addition model illustrates what extra information could help. Suppose a known amount $\Delta n$ is added before the recovery step and experiences exactly the same unknown fraction $r$ as the native product. Converting both readings to dissolved amounts gives
$$
d_0=rn,\qquad d_1=r(n+\Delta n),\qquad
r=\frac{d_1-d_0}{\Delta n},\qquad
n=\frac{\Delta n\,d_0}{d_1-d_0}.
$$
If $d_0=6\,\mathrm{\mu mol}$ and adding $2\,\mathrm{\mu mol}$ increases the dissolved amount by $1\,\mathrm{\mu mol}$, the model gives $r=0.5$ and $n=12\,\mathrm{\mu mol}$. Adding already dissolved product after the recovery step would instead test later optical response and would not identify earlier losses.
The shared-recovery assumption must be justified: a dissolved spike can behave differently from product crystals formed inside a preparation. A second measurement helps only if its relation to the unknown quantities is independently established. Repeated measurement of the same product $rn$ cannot supply that missing relation.
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