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Angular resolution set by transverse spectral noise quantifies how sharply an observed spectral vector constrains the orientation of an anisotropic two-coordinate noise model. Use the covariance from transverse-noise limit of a rotating rank-one covariance model, with known longitudinal standard deviation $\sigma$ and transverse standard deviation $0<ϵ<\sigma$. For an observation of length $R>0$ aligned with orientation ${\theta }_0$, the Gaussian density givesThe determinant is independent of orientation, so the angular dependence comes entirely from the covariance-weighted residual.^[ref]

Define a local half-width by a chosen log-density drop $c>0$. For $2c\le K$, the lobe around ${\theta }_0$ has half-widthWhen this angle is small and $ϵ\ll \sigma$, it is approximately $\sqrt{2c}ϵ/R$. If $2c>K$, even an orthogonal orientation fails to reach that drop, so this local-width criterion does not exclude any orientation. At $R=0$ or equal noise scales, the observation has no angular preference in this model.

For an original example, choose $R=2$, $\sigma =0.8$ and $ϵ=0.04$ in consistent signal units. Then $K=2493.75$. A drop $c=1/2$ gives half-width about $0.0200$ radians, or $1.15^{\circ }$. Reducing only the transverse standard deviation to $0.02$ approximately halves the width. Increasing a numerical regularization parameter is not an experimental improvement in this noise scale.

The global orientation ambiguity in a rank-one covariance fit remains: orientations separated by $\pi$ give the same covariance. A narrow local lobe does not distinguish those two orientations. Nor is a fixed log-density drop automatically a confidence interval with a stated coverage probability. Multiple observations, unknown noise scales and experimental nuisance parameters require their corresponding inference model.

Repeated reference spectra under stable conditions can estimate the longitudinal and transverse noise components. Their covariance must include shared calibration effects, as required by the measurement-propagation rule.^[ref] The numerical example is a model calculation in spectral coordinates, rather than a specification of a particular optical instrument.

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 $m=a{\mathbf{w}}^T\mathbf{u}\ne 0$ 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 giveThe normal density and cumulative distribution function used here are documented in the pinned numerical reference.^[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 $\Phi (-|m|/s)=\Phi (-4)\approx 3.17\times 10^{-5}$. 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.