 Interval propagation for bounded overlap data

Interval propagation for bounded overlap data maps an allowed measurement interval through the nonlinear inversion of a symmetric Gaussian overlap log-ratio. It exposes asymmetry and finite-model limits that a central value with a symmetric standard uncertainty can hide. A deterministic input bound and a statistical confidence interval have different meanings, even if the endpoint calculation is the same.

For known $h=\log q>0$, defineA finite positive radius ratio requires $-4h<D<4h$. Within this range $\rho$ decreases monotonically with $D$. Thus an input interval $[D_L,D_U]$ lying strictly inside that range maps exactly toThe endpoint order reverses. This is direct monotone propagation, without a small-uncertainty approximation.

An original example sets $q=2$ and $D\in [1.4,1.6]$. The inferred radius ratio lies approximately between 0.4739 and 0.5306. The endpoints should be computed from the full inverse, rather than by forcing equal absolute distances around the central inferred radius. Such asymmetry becomes much stronger near either finite-model boundary.

If an interval reaches $D=4h$ from below, it permits radius ratios arbitrarily close to zero. Reaching $D=-4h$ permits arbitrarily large radius ratios. Boundary values themselves correspond only to limiting ratios, not finite positive solutions. If the interval extends beyond the model range, some allowed observations have no solution in this forward model. Intersecting the interval with the admissible range can describe its compatible portion, but it does not justify discarding the incompatible observations or retaining the original claimed coverage without further analysis.

Uncertainty in $h$ changes the boundary too. For a rectangular set with $0<D_L\le D_U<4h_L$ and $h\in [h_L,h_U]$, $\rho$ increases with $h$. Consequently the extreme compatible radius ratios areFor negative $D$, the dependence on $h$ reverses; a sign-spanning interval requires checking that dependence instead of blindly using the positive-case formula. A correlated joint uncertainty region in $(D,h)$ need not fill the rectangle, so independent interval bounds can overstate the feasible range.

Joint radius-setting and log-ratio bounds therefore belong with the experimental calibration, not only the final arithmetic. JCGM 101 separates assigning input distributions from propagation, defines output coverage intervals and states conditions for its Monte Carlo method.^[ref] A correctly propagated deterministic set is not automatically a 95% confidence interval; any probability statement requires the input model and its calibration. Optical alignment and state-gain checks remain necessary because an interval calculation cannot repair an incorrect forward model.