= Equal-product ambiguity in height-mass surface estimates
\b[Equal-product ambiguity in height-mass surface estimates] is the loss of anatomical information when the <Mosteller formula> depends on height and <mass> only through their product. In the documented centimetre-and-kilogram convention,
$$
\widehat S=\sqrt{\frac{hm}{3600}}\,\mathrm{m^2}.
$$
Thus every positive pair satisfying $hm=K$ receives the same predicted <body surface area>. The VA manual states the equation and its units.<Formulas for Body Surface Area Calculator in the VA Clinical Reminders Manager's Manual>[Printed p. 36, Mosteller equation]{ref} At $K=14{,}400$, both $(h,m)=(180,80)$ and $(160,90)$ give $2\,\mathrm{m^2}$. Equality of predictions supplies no independent evidence that the actual external surfaces agree.
The ambiguity also has a local description. Along an equal-product curve, $m=K/h$, so $dm/m=-dh/h$. Meanwhile
$$
\frac{d\widehat S}{\widehat S}=\frac12\left(\frac{dh}{h}+\frac{dm}{m}\right)=0.
$$
A change in both inputs can therefore be invisible to the predictor. This is a small <inverse problem>: the predicted area does not identify height and <mass> separately, much less reconstruct body shape.
An original geometric example shows a stronger limitation without treating a toy solid as human <anatomy>. Consider a homogeneous <cuboid> of height $H$, <mass> $M$, <density> $\rho$ and horizontal sides $a,b$. Fixing height and <mass> fixes the cross-sectional area $A=ab=M/(\rho H)$, but its total <surface area> is
$$
S=2A+2H(a+b).
$$
The square choice $a=b=\sqrt A$ and elongated choice $a=2\sqrt A$, $b=\sqrt A/2$ have the same height, <mass> and <density>. Their lateral <surface areas> differ by 25%. With stipulated $H=1.8\,\mathrm m$, $M=80\,\mathrm{kg}$ and $\rho=1000\,\mathrm{kg\,m^{-3}}$, their total areas are about $1.607$ and $1.986\,\mathrm{m^2}$. Both receive the same Mosteller prediction of $2.00\,\mathrm{m^2}$.
This <non-identifiability of shape from height and mass> is a structural limitation of the inputs, not a rounding error. Additional shape measurements or independent surface evidence are needed to assess individual prediction errors. More precise evaluation of the same formula cannot supply the missing geometry.
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