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Exact diagnostics for the research laboratory

Independent audit addition, 6 September 2026. The original model essays are unchanged. These are algebraic derivations from an optically thin Stokes sum, not measurements, cosmological forecasts, or claims of priority. Their physical setting is established by Tassis & Pavlidou (2015) and the spin-moment formalism of Vacher et al. (2023).

1. Definitions and the exact rotation condition

Use one consistent polarization convention and write

P_\nu=Q_\nu+iU_\nu=\sum_i A_i s_i(\nu)e^{2i\psi_i},\qquad A_i\geq0,\quad s_i(\nu)>0.

Here the amplitudes are polarized emission at a reference frequency, including alignment, inclination and column weighting. They are not extinction fractions. Each layer has a frequency-independent angle. The observed angle is $\psi_\nu=\arg(P_\nu)/2$ , defined modulo $\pi$ , and is undefined at $P_\nu=0$ . A 90° change in the physical angle reverses a Stokes vector; it does not leave that vector unchanged.

For two frequencies define the signed cross-product

D_{ab}\equiv\operatorname{Im}(P_bP_a^*)=U_bQ_a-Q_bU_a.

Direct multiplication gives the exact multilayer identity

D_{ab}=\sum_{i<j} A_iA_j \big[s_i(a)s_j(b)-s_j(a)s_i(b)\big] \sin[2(\psi_j-\psi_i)].

For two layers this is one term. A nonzero term requires nonzero amplitudes, a changed relative SED, and noncollinear vectors in the Q/U plane. In particular, angles separated by 90° in physical space have $\sin(2\Delta\psi)=0$ . They can change polarization amplitude and undergo a 90° flip through a zero, but do not exhibit smooth rotation away from that zero. In multilayer cases pair terms can cancel even when all three ingredients occur.

For nonzero signals,

\frac{D_{ab}}{|P_a||P_b|}=\sin[2(\psi_b-\psi_a)].

Thus $D\neq0$ detects noncollinearity; $D=0$ cannot distinguish parallel vectors from antiparallel vectors. Use the real part together with the imaginary part:

\Delta\psi_{ba}=\frac12\operatorname{atan2}\!\left(D_{ab},\operatorname{Re}(P_bP_a^*)\right).

A local angle test does not measure harmonic $BB$ decorrelation. Spatially varying real amplitude scalings can change the morphology and $BB$ cross-correlation without changing any pixel's angle. Conversely, a single sightline contains no angular power spectrum.

2. Frequency derivative and a cancellation bound

Let $x=\ln\nu$ , $a_i=A_is_i(\nu)$ , and $\alpha_i=d\ln s_i/dx$ . Differentiating the phase gives

\frac{d\psi}{dx}=\frac12\operatorname{Im}\left(\frac{P'}P\right) =\frac{\operatorname{Im}(P'P^*)}{2|P|^2}.

For two layers,

\boxed{\frac{d\psi}{d\ln\nu}= \frac{a_1a_2(\alpha_2-\alpha_1)\sin(2\Delta\psi)}{2|P|^2}.}

For a monochromatic modified blackbody in specific-intensity units,

s_i(\nu)=\left(\frac\nu{\nu_0}\right)^{\beta_i} \frac{B_\nu(T_i)}{B_{\nu_0}(T_i)},\qquad \alpha_i=\beta_i+3-\frac{y_i}{1-e^{-y_i}},\quad y_i=\frac{h\nu}{kT_i}.

A common conversion of all components to another unit adds the same slope to every $\alpha_i$ , leaving slope differences and rotation invariant. Real instruments measure bandpass integrals, so monochromatic slopes are an explanatory approximation to their channel response.

There is also a useful bound for any number of layers. Define

S=\sum_i a_i,\quad w_i=a_i/S,\quad z_i=e^{2i\psi_i},\quad m=\sum_iw_iz_i,\quad C=|m|=|P|/S,

\bar\alpha=\sum_iw_i\alpha_i,\qquad \sigma_\alpha^2=\sum_iw_i(\alpha_i-\bar\alpha)^2.

$C\in[0,1]$ is polarized-layer coherence. It is 1 for parallel Stokes vectors and can approach 0 through cancellation. It is not the observed polarization fraction $|P|/I$ .

Set

v=\sum_iw_i(\alpha_i-\bar\alpha)(z_i-m).

Since $\sum_iw_i(\alpha_i-\bar\alpha)=0$ ,

P'/S=\bar\alpha m+v, \qquad 2\frac{d\psi}{dx}=\frac{\operatorname{Im}(vm^*)}{|m|^2}.

Complex Cauchy–Schwarz and $|z_i|=1$ imply

|v|\leq\sigma_\alpha \sqrt{\sum_iw_i|z_i-m|^2} =\sigma_\alpha\sqrt{1-C^2}.

Consequently, for $C>0$ ,

\boxed{\left|\frac{d\psi}{d\ln\nu}\right| \leq\frac{\sigma_\alpha}{2}\frac{\sqrt{1-C^2}}C.}

This upper bound identifies two controls: polarized SED-slope spread and vector cancellation. Neither low coherence nor slope spread provides a lower bound on rotation. At $C=0$ the angle is undefined; the divergent bound signals loss of an angle coordinate, not infinite emitted power. To bound a finite frequency interval one must integrate the right-hand side while checking that $C$ stays nonzero throughout it. Endpoint values alone are insufficient.

The realized complex covariance $v$ can be nonzero on a finite sightline even if spectral and angular variables were sampled independently from population distributions. Therefore, saying that rotation requires a population-level causal correlation between dust properties and field angles would be too strong. This distinction is consistent with the independent-variable model and finite-sightline rotations discussed in Guillet et al. (2026), §2.1–2.3.

This bound was derived for this audit. A targeted literature search did not establish its publication priority. It should be called a derived diagnostic, not a new physical law. An independent numerical check used Python's standard library, seed 20260906, and 10,000 draws with 2–8 layers: the bound held in every draw, and the analytic phase derivative agreed with a centered finite difference away from small-coherence singularities. The largest observed ratio of actual derivative to bound was 0.99999967. This checks the implementation of the algebra, not an astrophysical population.

3. Known-angle inversion: when two layers really are separable

At one frequency, with dust-only data, exactly known angles, and exactly two layers,

\begin{pmatrix}Q\\U\end{pmatrix}= M\begin{pmatrix}a_1\\a_2\end{pmatrix},\qquad M=\begin{pmatrix}\cos2\psi_1&\cos2\psi_2\\ \sin2\psi_1&\sin2\psi_2\end{pmatrix}.

The determinant is $\sin(2\Delta\psi)$ . Because the columns have unit norm,

M^TM=\begin{pmatrix}1&\cos2\Delta\psi\\\cos2\Delta\psi&1\end{pmatrix},

so its exact Euclidean condition number is

\boxed{\kappa_2(M)=\sqrt{\frac{1+|\cos2\Delta\psi|}{1-|\cos2\Delta\psi|}}.}

Physical angle separation	0°	1°	30°	45°	60°	89°	90°
$\kappa_2$	∞	57.29	1.73	1	1.73	57.29	∞

The repository's $1/\sin(2\Delta\psi)$ expression is an inverse-determinant scaling, not this exact condition number. Distinct physical angles are insufficient: 0° and 90° are different angles but form a singular matrix. With anisotropic/correlated noise, inspect the singular values of the noise-whitened design matrix instead. Angle uncertainty, more layers, and unknown CMB/synchrotron contributions require a joint inference and invalidate a two-equation/two-amplitude claim for the full observation. Positivity helps constrain amplitudes but does not restore generic uniqueness for singular columns.

4. A genuine spectral closure test, and a false one

Suppose two specified bandpass-integrated dust SED vectors $\mathbf{s}_1,\mathbf{s}_2\in\mathbb R^N$ have rank 2. Any nonzero $\mathbf c$ in their left nullspace satisfies

\mathbf c^T\mathbf s_1=\mathbf c^T\mathbf s_2=0 \quad\Rightarrow\quad R\equiv\sum_b c_bP_b=0.

Three dust-only bands suffice to supply one such closure. Angles and layer amplitudes cancel from this test. If an unknown CMB is retained in thermodynamic units, require $\mathbf c^T\mathbf1=0$ as well: two independent dust modes plus CMB generically need at least four channels for a nontrivial null direction. Every additional independent nuisance spectral mode consumes another direction. Nearly degenerate modes may amplify noise or make the test weak.

Use the full Q/U covariance of $R$ to form a two-dimensional Mahalanobis residual. A simple chi-square reference is justified only for fixed externally specified spectra and correctly Gaussian known errors; fitted spectra, calibration uncertainty and data selection require predictive simulations or held-out data. Failure falsifies the complete model-plus-instrument assumptions. It does not uniquely establish a third physical cloud.

This is linear spectral nulling, closely related to constrained moment ILC. Calling it a closure test does not make it novel.

By contrast, $\arg(P_bP_a^*)+\arg(P_cP_b^*)+\arg(P_aP_c^*)=0\pmod{2\pi}$ is an identity for any three nonzero complex numbers. It holds for arbitrary foregrounds and for noisy observed numbers. A triangle of pairwise phase differences from the same data is therefore not a test of a two-layer dust model. Likewise, the rank of a single-pixel $2\times N$ matrix of Q/U measurements is at most 2 by dimension alone; this cannot count physical clouds.

5. What a map simulation adds

For a periodic flat patch with a consistent Fourier convention,

\widetilde E(\mathbf k)=\widetilde Q\cos2\varphi_{\mathbf k}+\widetilde U\sin2\varphi_{\mathbf k},\qquad \widetilde B(\mathbf k)=-\widetilde Q\sin2\varphi_{\mathbf k}+\widetilde U\cos2\varphi_{\mathbf k}.

The zero Fourier mode has no defined wavevector orientation and should be omitted from an E/B power comparison. Periodicity removes boundary leakage in this educational construction; a masked real-sky analysis instead needs its estimator's mask, transfer and E-to-B leakage treatment.

Given two real B maps with finite nonzero template power, the unconstrained least-squares scalar is $\widehat\alpha=\langle B_aB_b\rangle/\langle B_b^2\rangle$ . Its minimized residual power fraction equals $1-\rho^2$ , with $\rho$ computed using the same inner product and modes. This identity is specific to the fitted scalar and selected B data; it is not a generic prediction for a per-pixel MBB fit or a multifrequency likelihood. A $\delta r$ requires tensor templates, covariance, beams, mask, filtering and nuisance marginalization. The laboratory therefore reports synthetic residuals and correlations, not a numerical cosmological bias.