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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.

An ellipsoidal admissibility test for overlap data checks whether every point of a specified joint calibration ellipsoid permits finite Gaussian radius inversion. For the inversion of a symmetric Gaussian overlap log-ratio, write $x=(D,h{)}^{\mathsf{T}}$ with $D$ the measured logarithmic ratio and $h$ the positive radius-setting parameter. The admissible set is $D-4h<0$ and $-D-4h<0$. Together these strict inequalities already imply $h>0$.

Suppose the joint set isFor nonsingular $\Sigma$ this is equivalently $(x-\mu {)}^{\mathsf{T}}{\Sigma }^{-1}(x-\mu )\le k^2$. The parameterized form also accommodates a singular covariance matrix or a lower-dimensional deterministic set. Its support function in direction $l$ isIndeed, $l^{\mathsf{T}}Ay=(A^{\mathsf{T}}l{)}^{\mathsf{T}}y$, so the Cauchy-Schwarz inequality gives the bound, attained by a parallel $y$ when $A^{\mathsf{T}}l\ne 0$.

Let $\mu =(D_0,h_0{)}^{\mathsf{T}}$, marginal scales $s_D,s_h$, and off-diagonal entry $c_{Dh}$. Substituting $l_{+}=(1,-4{)}^{\mathsf{T}}$ and $l_{-}=(-1,-4{)}^{\mathsf{T}}$ gives two exact tests:Both must hold. Equality means that the closed joint set touches a boundary where no finite positive radius ratio exists; testing only the center would miss that point.

For an original example, take $D_0=2.4$, $h_0=0.7$, $s_D=0.1$, $s_h=0.03$ and $k=2$. Zero covariance gives upper-boundary excursion $2\sqrt{0.0244}\approx 0.3124$, smaller than the central margin $0.4$. The lower-boundary test also passes. With correlation coefficient $-0.8$, however, $c_{Dh}=-0.0024$ and the upper excursion becomes $2\sqrt{0.0436}\approx 0.4176$: the joint set crosses the forbidden region despite identical marginal scales.

The offending point need not be found by grid search. When $l^{\mathsf{T}}\Sigma l>0$, the maximizing point isFor the correlated example and $l=l_{+}$, this gives approximately $(D,h)=(2.5877,0.6425)$, with $D-4h\approx 0.0176>0$. It is a concrete incompatible calibration pair.

The ellipsoid here is a stated joint set. Interpreting it as a confidence region requires a separate sampling or uncertainty model; choosing $k=2$ alone supplies no universal coverage probability. JCGM 101 distinguishes assignment of joint input distributions from propagation and summaries.^[ref] If the set fails, report the incompatible region and reconsider the input model or measurement. Clipping it before inversion changes the set being propagated.

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.

A symmetric beam-radius perturbation for overlap branch selection compares two controlled settings of one Gaussian beam radius: $q{w}_2$ and $w_2/q$, with known $q>1$. Under the centred, matched-wavefront curvature model of reciprocal size-ratio ambiguity in centred Gaussian overlap, writeThe hyperbolic secant form makes the symmetry explicit: the two settings evaluate the same even function at $u+h$ and $u-h$. Their ordering identifies the sign of $u$, as shown in branch selection by a controlled beam-radius change.

An original extension removes an unknown common scale and recovers the radius ratio itself. Suppose the two measured positive efficiencies satisfyThe factor $A$ is a shared multiplicative nuisance parameter, such as an unchanged response normalization. The symmetric log-overlap ratio eliminates it exactly:For the last equality, use the addition formula for hyperbolic cosine to write the fraction as $(1+t)/(1-t)$, where $t=\tanh u\tanh h$. The defining logarithmic identity for the inverse hyperbolic tangent completes the derivation. Because $h>0$, $D_h$ is positive for the initially narrower beam, negative for the wider beam, and zero at matching.

The inversion of a symmetric Gaussian overlap log-ratio now givesA finite radius ratio exists exactly when $|D_h|<4h$. This condition follows from $|\tanh u|<1$; equality is approached only as one radius ratio tends to zero or infinity. A measured log-ratio outside that interval has no finite solution in this ideal two-setting model.

For an original numerical example, take $q=1.5$, true $\rho =0.8$ and unknown scale $A=0.75$. The ideal readings are $V_{+}=0.7256114$ and $V_{-}=0.5172352$. They give $D_h=0.3385169$, recovering $\rho =0.8$ without supplying $A$ to the inversion. The physical bound is $4h=1.6218604$.

Finite measurement noise still limits branch selection. For small relative errors, the delta method givesA perfectly shared multiplicative fluctuation cancels sample by sample, paralleling common-gain cancellation in heterodyne overlap uncertainty. Independent relative standard uncertainties of $1\%$ in the two readings instead give $u(D_h)\simeq 0.014142$. At the worked point, the derivative of the inverse gives $du/d{D}_h=-0.6780385$, hence about $0.959\%$ relative standard uncertainty in $\rho$. At exact matching the two branches coincide; a noisy sign cannot identify a distinct physical branch there.

These conclusions require $A$ and the other optical mode matching conditions to remain shared across the settings. Changing a lens can alter wavefront curvature and polarization as well as radius. NIST's physical heterodyne apparatus explicitly controlled polarization and spatial matching.^[ref] Larger radius changes may strengthen the log-ratio response while lowering either measured efficiency and changing its noise. Step choice therefore depends on the apparatus and noise model.