= Sign-sensitive measurements that resolve a covariance-axis ambiguity
{title2=$\log\Lambda=2mz/s^2$}
\b[Sign-sensitive measurements that resolve a <covariance>-axis ambiguity] add an observable whose distribution changes when the fitted axis direction is reversed. In the <global orientation ambiguity in a rank-one covariance fit>, $\mathbf u$ and $-\mathbf u$ give the same <covariance>. An independently calibrated signed projection can distinguish them.
Consider two candidates with an additional observation $z=\pm m+\eta$, where $m=a\mathbf w^T\mathbf u\ne0$ is known, $a$ has known positive sign and $\eta\sim N(0,s^2)$. Assume this calibration reading is independent of the <covariance> data conditional on the candidate. The Gaussian densities give
$$
\log\frac{p(z\mid+\mathbf u)}{p(z\mid-\mathbf u)}=\frac{2mz}{s^2}.
$$
The normal density and cumulative distribution function used here are documented in the pinned numerical reference.<SciPy 1.16.2 documentation>[scipy.stats.norm, Notes and Methods: pdf, cdf]{ref} The likelihood-ratio expression follows by expanding the two squared residuals.
For an original example, let $m=0.20\,\mathrm V$, $s=0.050\,\mathrm V$ and observe $z=0.10\,\mathrm V$. The log <likelihood ratio> is 16, strongly favoring the positive candidate under this stated model. With equal prior probabilities and equal decision costs, the sign-threshold decision has repeated-measurement error probability $\Phi(-|m|/s)=\Phi(-4)\approx3.17\times10^{-5}$. This decision-error probability is different from assigning a posterior error probability to that one observed value.
A practical calibration applies a known signed perturbation and records the signed detector-response component. Reversing the applied perturbation checks readout polarity. A calibrated phase-sensitive readout can provide such a projection; measuring only noise <covariance> does not supply its sign.
The <failure of sign recovery from magnitude-only calibration data> is important. For symmetric Gaussian noise, $|z|$ has the same distribution under means $+m$ and $-m$. If $\mathbf w$ is orthogonal to the axis, then $m=0$ and even the signed reading adds no separation. If the amplitude sign itself is unknown, reversing both amplitude and axis preserves the mean. The additional measurement resolves the ambiguity only when its reference sign and response model are independently established.
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