Cross-Method Consistency and Systematics

Cross-method agreement is informative only after identifying what the methods share. A dynamical mass may set the length and luminosity scales used by continuum fitting; distance and inclination can enter both continuum and reflection analyses; the same response calibration can shape spectrum and lag; and several spin methods identify a fitted disk edge with the ISCO. Multiplying published posteriors as though they were independent counts these assumptions repeatedly. The correct object is a joint generative model with one copy of each shared quantity, method-specific nuisance parameters, and predictions in every method’s native data space.

Required background. Probability and Inference for Gravity supplies hierarchical likelihoods, covariance, posterior prediction, selection effects, and model comparison.

Helpful background. Black-Hole Geometry and Mechanics supplies the common Kerr spin and ISCO mapping; Relativistic Matter and Compact Objects supplies disk and atmosphere systematics; and Black-Hole Astrophysics supplies source states and observing context.

Build the dependency model before combining results

Let the shared physical vector be

$$z=(M,D,i_{\rm orb},i_{\rm disk},a_*,r_{\rm in},\eta_{\rm cal,shared}),$$

and let $\eta_k$ contain the nuisance parameters unique to method $k$. If the raw data blocks $d_k$ and method-specific nuisance priors are conditionally independent once all shared variables $z$ and declared selections $S_k$ are specified, the joint posterior is

$$p(z,\{\eta_k\}\mid\{d_k,S_k\}) \propto p(z)\prod_k \mathcal L_k(d_k\mid z,\eta_k,S_k)\, p(\eta_k\mid z).$$

Conditional independence is an assumption to check, not a consequence of naming different methods. Cross-calibration, simultaneous background, or a common atmosphere table can introduce dependence between data blocks. Include their shared uncertain parameters in $z$; if residual dependence remains, replace the likelihood product by the appropriate joint likelihood. Likewise, correlated nuisance priors need a joint prior. Orbital inclination and inner-disk inclination should remain distinct until alignment is tested. The inner radius and spin should also remain distinct until each method has supported $r_{\rm in}=r_{\rm ISCO}(a_*)$ in its source state.

A published posterior $q_k(z)\propto \mathcal L_k(z)\pi_k(z)$ is not a likelihood. Dividing samples by a stated prior can recover a likelihood only when the full prior and marginalized nuisance structure are available. A contour plot, symmetric error bar, or posterior generated with an undocumented external mass likelihood cannot be safely “combined.” The truthful alternatives are to reanalyze the native data, obtain a likelihood object from the authors, or report a qualitative consistency check.

Selection also enters twice. Within a source, disk-dominated spectra, low-absorption intervals, QPO detections, or high-polarization bins are selected for method validity. Across a source population, brightness, inclination, state duty cycle, absorption, and the availability of dynamical parameters determine which objects receive a spin estimate. A cross-method sample is therefore not a random sample of black holes.

Shared covariance changes both tension and precision

Suppose two methods estimate the same parameter vector $x$ with covariances $C_1$, $C_2$, and cross-covariance $C_{12}$. The difference has covariance

$$C_\Delta=C_1+C_2-C_{12}-C_{12}^{\mathsf T},$$

so a Gaussian consistency statistic is

$$T=\Delta^{\mathsf T}C_\Delta^{-1}\Delta, \qquad \Delta=\hat x_1-\hat x_2.$$

Positive shared errors can cancel in a difference; ignoring them may hide tension. The opposite error occurs when combining estimates: treating positive correlation as independence gives too much weight and an interval that is too narrow. A calibration or distance parameter should preferably be represented explicitly rather than compressed into $C_{12}$, because its effect is often nonlinear and asymmetric near the extremal-spin boundary.

The strongest check is predictive, not a tension number. Leave out method $k$ and compute

$$p(d_k\mid d_{-k})= \int p(d_k\mid z,\eta_k)\, p(\eta_k\mid z)\, p(z\mid d_{-k})\,d\eta_k\,dz.$$

Then compare predicted counts, cross-spectra, Stokes parameters, velocities, or timing peaks with the held-out data. Agreement in a derived spin can coexist with failure in observable space if two models compensate differently.

A concrete cross-method bundle

XTE J1550-564 provides an instructive historical example because continuum fitting used several dozen RXTE spectra while reflection used ASCA spectra, with a common dynamical mass, distance, and inclination. The data selection, continuum models, reflection analysis, and combined result can be traced through Steiner et al. 2011, §§ 2–4 and 6–7.1. Calling the two results independent is only approximate: they use different X-ray data, but share binary parameters, disk-alignment assumptions, a Kerr transfer framework, and the ISCO identification.

Cygnus X-1 shows why a result must remain connected to its external likelihood. VLBA astrometry and optical modeling revised its distance and mass; the measurements and propagated binary solution are given in Miller-Jones et al. 2021, pp. 1046–1049 and Supplementary Materials. Any continuum result that used the earlier mass–distance–inclination distribution should be rerun, not adjusted by replacing one number in a table. Reflection, timing, and polarization analyses from other epochs add disk and corona information, but source-state variability prevents them from being treated as one static observation.

A reproducible bundle therefore freezes one source, method, epoch, event identifier, response and calibration release, energy or frequency selection, software and model version, likelihood, prior, and posterior representation per row. It stores the shared dynamical likelihood once. For XTE J1550-564, reproduce the published RXTE and ASCA selections before trying a modern response comparison. For Cygnus X-1, retain the Miller-Jones astrometric and binary samples and the public IXPE/NICER/NuSTAR identifiers described in Krawczynski et al. 2022, pp. 650–654 and Supplementary Materials. A new joint inference should use overlapping states or explicitly model temporal change.

Results and limitations through 2026-08-18

The evidence does not establish that electromagnetic spin methods are universally interchangeable. The unresolved question is whether apparent agreement in each source survives shared-prior removal, method-family changes, calibration covariance, state variability, and selection.

Evidence or result	Shared and method-specific axes	Predictive check and scope or limitation
Dynamical stellar or binary data constrain $M$, $D$, and orbital inclination	Shared scale and geometry; tracer anisotropy, irradiation, veiling, astrometry	Predict velocities and light curves; orbital inclination need not equal inner-disk inclination
Continuum fitting constrains a thermal emitting area	Shares $M$, $D$, inclination, ISCO mapping; unique atmosphere, torque, thickness, state cuts	Predict radius constancy and held-out count spectra; fails outside the selected thin-disk regime
Reflection constrains relativistic broadening and an inner edge	Shares inclination, ISCO mapping, calibration; unique density, ionization, absorption, emissivity	Predict the Compton hump, lines, and other epochs; alternative physical families can shift the edge
XTE J1550-564 continuum and reflection results overlap	Shares dynamical priors and disk geometry but uses distinct RXTE and ASCA epochs	Refit with one dependency model; overlap alone does not measure the cross-covariance
Timing, reverberation, and polarization constrain frequencies and geometry	Shares mass, inclination, inner edge, response or state; unique mode and transfer models	Predict phase, cross-spectra, and Stokes observables; no unique mode or corona is established
Discordant or null cross-method results expose model failure	Compare source, epoch, signal-to-noise, calibration domain, and inference prior	Publication and detectability selection make the observed agreement rate a biased population statistic

Credible competing positions are that all methods measure one common $(M,a_*)$ under compatible geometry; that agreement is partly induced by shared priors and the common ISCO assumption; that discordance reveals calibration or source-model failure; or that state variability makes a static combination ill posed. Reynolds compares the physical dependencies and current samples in Reynolds 2021, §§ 3.2–3.4 and 5.2. No universal verdict follows from the number of overlapping error bars.

The interpretation could change after a revised mass, distance, inclination, alignment, or shared calibration likelihood; after an independent method with genuinely different systematics becomes available; or after a leave-one-method-out posterior prediction fails at a predeclared significance in the method’s native data space.

Adversarial checks and reproduction boundary

Remove shared priors. Reconstruct each likelihood under a broad common prior and compare it with the published posterior. If the apparent agreement disappears, the result was prior consistency rather than independent data consistency.

Separate state from spin. Permit corona, ionization, truncation, thickness, and alignment to change between epochs while keeping $M$ and $a_*$ fixed. A static joint model that fails this test should not average the epochs.

Publish the dependency map. List which data constrain every shared node and which methods consume it. This prevents a parallax posterior, for example, from entering once through a dynamical mass chain and again as an apparently independent continuum prior.

Exact reproduction may be impossible when event selections, calibration covariance, atmosphere or reflection tables, dynamical likelihood samples, or posterior weights were not released. In that case report the missing object and perform sensitivity analyses; do not digitize contours and claim a joint posterior. Statistical intervals remain conditional on a method family, while the spread across atmosphere, disk, corona, mode, tracer, and calibration families is systematic model dependence.

Exercises

Two analyses give $a_1=0.40\pm0.20$ and $a_2=0.55\pm0.15$, with Gaussian errors and correlation $\rho=0.60$ from shared distance and inclination. Compute the tension of their difference and the best linear combined estimate. Compare with the result obtained by falsely assuming independence.

Solution

The shared covariance is $\rho\sigma_1\sigma_2=0.018$. Thus

$$\sigma_\Delta^2=0.20^2+0.15^2-2(0.018)=0.0265, \qquad \frac{\lvert a_1-a_2\rvert}{\sigma_\Delta} =\frac{0.15}{0.1628}\simeq0.92.$$

For covariance matrix

$$C= \begin{pmatrix} 0.0400 & 0.0180\\ 0.0180 & 0.0225 \end{pmatrix},$$

the generalized least-squares weights are proportional to $C^{-1}(1,1)^{\mathsf T}$, giving weights about $(0.170,0.830)$. Therefore $a_{\rm comb}\simeq0.525$ with $\sigma_{\rm comb}=[(1,1)C^{-1}(1,1)^{\mathsf T}]^{-1/2}\simeq0.147$. Assuming independence instead gives $a_{\rm comb}\simeq0.496$ and $\sigma_{\rm comb}=0.120$, an unjustified gain in precision. The correlated model also reveals more difference tension, $0.92$ rather than $0.60$, because the shared error cancels in the comparison.

References

• Krawczynski, Henric, et al. 2022. “Polarized X-Rays Constrain the Disk–Jet Geometry in the Black Hole X-Ray Binary Cygnus X-1.” Science 378:650–654. DOI.
• Miller-Jones, James C. A., et al. 2021. “Cygnus X-1 Contains a 21-Solar Mass Black Hole—Implications for Massive Star Winds.” Science 371:1046–1049. DOI.
• Reynolds, Christopher S. 2021. “Observational Constraints on Black Hole Spin.” Annual Review of Astronomy and Astrophysics 59:117–154. DOI.
• Steiner, James F., et al. 2011. “The Spin of the Black Hole Microquasar XTE J1550-564 via the Continuum-Fitting and Fe-Line Methods.” Monthly Notices of the Royal Astronomical Society 416:941–958. DOI.