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Boundary-sensitive Gaussian overlap inversion describes the loss of stable radius information when a symmetric log-overlap ratio approaches its finite-model limit. A central reading can remain formally invertible while small measurement uncertainty reaches values for which no finite radius ratio exists.

For known $h=\log q>0$, define $D=D_h$, $u=\log \rho$ andThe inversion of a symmetric Gaussian overlap log-ratio requires $|D|<4h$. At the upper limit $D\to 4h$, the radius ratio tends to zero; at the lower limit it tends to infinity.

An original expansion near the upper limit puts $ϵ=4h-D>0$. Expanding the hyperbolic tangent givesThus the approximate relative radius uncertainty is $u(\rho )/\rho \simeq u(D)/(2ϵ)$ when linearization is still justified. The distance to the boundary, rather than the magnitude of $D$ alone, controls this sensitivity.

For stipulated $q=2$ and a central log-ratio only 0.01 below $4\log 2$, the asymptotic radius ratio is about 0.0365. A standard uncertainty 0.02 in $D$ is twice the boundary distance, so a small symmetric error bar on the inferred radius is not credible. In a hypothetical Gaussian readout model centered there, roughly 31% of readings fall beyond the finite-inversion limit. This is a statistical illustration, not an observed receiver distribution.

Interval propagation for bounded overlap data shows the asymmetry directly. A stated interval in $D$ that reaches the upper bound maps to radius ratios arbitrarily close to zero; it does not yield a symmetric finite relative interval. An interval wholly inside the domain maps monotonically, with its endpoint order reversed.

Discarding out-of-range Monte Carlo draws would condition on a selected subset and change the distribution being propagated. Instead, examine the forward measurement model, admissible parameter range and calibration uncertainty, or construct an explicitly stated inference using the measured data. JCGM 101 separates input-distribution assignment from numerical propagation and states conditions for valid Monte Carlo use.^[ref]

Finally, proximity to the mathematical boundary is not the only possible problem. A change in shared normalization, polarization or wavefront curvature can violate the one-radius model. NIST's receiver apparatus illustrates the need for optical matching and separate electrical corrections.^[ref] Numerical clipping cannot replace those physical checks.

Joint radius-setting and log-ratio bounds retain the allowed relationship between the radius-setting parameter $h$ and measured log-ratio $D$ when inferring an optical radius ratio. Separate marginal intervals form a rectangle that may contain combinations excluded by shared calibration information. Propagating that rectangle can therefore exaggerate the feasible radius range.

For $h>0$ and $|D|<4h$, defineFor a stated joint set $\mathcal{F}$, the compatible ratios are $\{\rho (D,h):(D,h)\in \mathcal{F},|D|<4h\}$. This set definition precedes any approximation. A deterministic set is not automatically a statistical confidence region; a probability interpretation needs the corresponding input model.

An original exactly linked calibration example has $0.6\le h\le 0.8$ and $D=2h$. The marginal intervals are $D\in [1.2,1.6]$ and $h\in [0.6,0.8]$, but their extreme values cannot be chosen independently. Writing $t=\tanh (h/2)$ and using $\tanh h=2t/(1+t^2)$ givesThus the joint model gives approximately $0.5216\le \rho \le 0.5447$. Treating the two marginal intervals as an independent rectangle instead allows approximately $0.4139\le \rho \le 0.6246$. The wider result contains parameter combinations that violate $D=2h$. All values describe a stipulated calibration relation, not observed optical data.

For a positive-$D$ rectangle wholly inside the inversion range, monotonicity places the minimum ratio at $(D_U,h_L)$ and maximum at $(D_L,h_U)$. That shortcut is valid for the rectangle, but those corners may be absent from the actual joint set. With nonlinear dependencies, optimize over the stated joint set or parameterize it by the genuinely independent calibration inputs.

The feasible-set boundary in overlap inversion also matters. If part of a joint set crosses $|D|=4h$, some allowed observations have no finite positive solution in the assumed model. Report that incompatibility and examine the measurement model; silently discarding those points changes the set or probability distribution. JCGM 101 distinguishes input assignment, propagation and output summaries, and specifies conditions for its Monte Carlo approach.^[ref]

Experimentally, retain common calibration records when deriving both $h$ and $D$. A shared scale or fitted parameter should be varied consistently through both calculations. Independent draws from their marginals erase that information. Conversely, imposing a convenient correlation without physical calibration evidence can create an unjustifiably narrow range. Interval propagation for bounded overlap data is a numerical step conditional on this joint experimental model.