The Failure Calculus of Coherent Measurement Design

The experimenter’s recurring question is not “which named modality is best” but “what operator should I use to estimate this physical quantity in this sample?” Every named modality is a particular setting of that operator, and the operator itself can be any point in a continuous space. The question therefore becomes an optimization over that space: solve for the operator that minimizes the estimation error for the sample and parameter, then implement it with real optical components. Because the pre‑detector field is a sufficient statistic, only three mechanisms can compromise such a measurement design: ill‑conditioning of the information form, misspecification bias, and global likelihood ambiguity. The deployed estimator’s leading‑order risk at photon dose N decomposes as b² + V/N + r_alias(N), corresponding to the three mechanisms, and each mechanism has a computable discriminant. The conditioning discriminants in particular are closed forms in optical angles. Let Δψ be the angle between two parameters’ optical signal phases. The information surviving a jointly estimated nuisance is then sin²Δψ. Reading both field quadratures simultaneously incurs a penalty 2/(1 + sinΔψ·H) versus the optimal homodyne sequence, where H = 2ab/(a²+b²) and a,b are the sensitivity magnitudes. That optimal homodyne sequence attains the Holevo bound of the pure‑state displacement model, so it is quantum‑optimal at every angle and magnitude pair, with heterodyne exactly 3 dB above the quantum Fisher trace bound. Beyond the two‑parameter laws, a single spatial mode supports at most two jointly identifiable field‑property parameters. For orientation estimation, the unobservable rotations of a rigid particle are the symmetries of the modality’s contrast functional, not of its geometry, so fusing modalities recovers exactly the rotations whose symmetry is not shared. On the bias axis, a deployed model is structurally blind to its own displacement (the residual is orthogonal to the model’s score), so bias can only be seen as a derivative along a model‑fidelity axis: two forward models of different fidelity on one shared scene. The dose‑independent bias b defines a trust dose N* = V/b² past which error is bias‑dominated, so the optimal operator migrates with dose (e.g., analyzer → polarimeter). Likelihood aliases set an estimator‑free per‑operator floor N_amb, infinite for an exact alias. The feasible window [N_amb, N*] is computable pre‑instrument. When a sample symmetry leaves every candidate’s detected distribution invariant, the window is empty and the parameter is certified unmeasurable at any dose. This is a symbolic check on mean intensities that also names the escape: an operator whose contrast breaks the symmetry. The premise that the model’s bias lower‑bounds the true bias reduces, via Cauchy–Riemann, pair‑order cancellation, and endpoint analysis, to a sign criterion on the static structure factor, G[g] = Σ_2k − w²Σ_k > 0. Here Σ_2k = ∫g(r)W(r)sin(2k n_med r + 2δ_2k)dr and Σ_k = ∫g(r)W(r)sin(k n_med r + 2δ_k)dr are the sine transforms of the pair‑distance distribution (Born limit; §7 carries the general amplitude phase), and w = |f(π/2)/f(π)| is the transverse‑to‑backscatter scattering‑amplitude ratio. The criterion is measurable by small‑angle scattering before any instrument exists and holds in the majority sense with a computable exception probability.