(app:d)=
# Reading Routes and Validity Boundaries for Core Relations
Several relations appear repeatedly in the main text. This appendix places them side by side, with their physical assumptions, first use, and common failure modes. The complete formula index is in Appendix {ref}`app:a`.

(appsec:d-siegert)=
## The Siegert Relation
The physical picture behind the Siegert relation is introduced in Chapter {ref}`chap:01`, developed in quantum-optics language in Chapter {ref}`chap:04`, and used for spatial intensity interferometry in Chapter {ref}`chap:05`. When reading those chapters, check the following sequence.

1.  First decide whether the optical field can be approximated as thermal light, chaotic light, or a Gaussian random field. This condition allows the fourth-order moment to factor into combinations of second-order moments.

2.  Normalize the first-order coherence function to obtain $g^{(1)}$ or the spatial coherence degree $\gamma_{12}$.

3.  For ideal single-mode thermal light, the zero-delay second-order correlation excess is set by $|g^{(1)}|^2$.

4.  For a real instrument, write finite bandwidth, finite timing response, polarization averaging, spatial modes, and background as explicit contrast factors.

| Check | How it can be used when satisfied | What goes wrong when it fails |
|:-----------------|:------------------------|:----------------------|
| Thermal-light or Gaussian-field approximation | Second-order correlation can be connected to the squared modulus of first-order coherence. | Stimulated emission, a small number of emitters, or strong non-stationarity can change $g^{(2)}$. |
| Single mode, or known mode number | Bunching-peak contrast can be interpreted as physical coherence information. | Multimode averaging suppresses the peak and can be mistaken for source incoherence. |
| Known time response | The theoretical peak can be connected to the observed peak. | ns-scale electronics can dilute an fs-scale optical coherence peak to $10^{-6}$--$10^{-5}$. |
| Separated background | The correlation excess can be converted into the target $|V|^2$. | Night sky, companion stars, or continuum light dilute the signal by the squared flux fraction. |

Whenever a later chapter writes $g^{(2)}-1$, it should also state the coherence time, correlation bin, effective mode number, background flux fraction, and null test. A zero-delay peak height alone does not establish astrophysical bunching.

(appsec:d-vcz)=
## The van Cittert--Zernike Theorem
The VCZ theorem connects the sky brightness of an incoherent far-field source to first-order spatial coherence. Chapter {ref}`chap:01` defines complex visibility, Chapter {ref}`chap:05` uses it for intensity interferometry, and Chapter {ref}`chap:18` turns it into observing design. Use the relation through the following route.

1.  Write the astrophysical surface brightness as a function $I(\boldsymbol\theta)$ of angular coordinates.

2.  Divide the projected baseline by wavelength to obtain the spatial frequency $\boldsymbol u=\boldsymbol B_\perp/\lambda$.

3.  Take the normalized Fourier transform of the brightness distribution to obtain $V(\boldsymbol u)$.

4.  Intensity interferometry directly measures only $|V|^2$, so phase, mirror degeneracy, and asymmetric structure require extra information.

| Condition | Use in the main text | Typical ways it fails |
|:-----------------|:------------------------|:----------------------|
| Far-field approximation | Angular structure of stars, binaries, disks, and transients can be described in the $u,v$ plane. | Near-field experiments, extreme space-baseline geometry, or unmodeled wavefront curvature. |
| Narrow band, or corrected bandwidth | Each channel can be approximated with a single $\lambda$. | Broad-band averaging washes out high-spatial-frequency visibility structure. |
| Source stable during integration | Each epoch has one visibility model. | Flares, rotation, pulsation, and rapid structural evolution mix different models. |
| Nonnegative brightness with a reasonable model | Phase retrieval can use prior constraints. | $|V|^2$ alone has difficulty separating mirror images and complex asymmetry. |

(appsec:d-atmosphere)=
## Atmospheric-Phase Immunity in Intensity Interferometry
Intensity interferometry is insensitive to atmospheric piston phase because it correlates intensity fluctuations rather than coherently combining two electric fields. That advantage has clear boundaries.

- Transparency variations remain. They change photon rates, accidental coincidences, and background normalization.

- Scintillation remains. It introduces intensity correlations on slower time scales and must be checked with time shifts and blockwise analysis.

- Electronic crosstalk remains. It can produce a zero-delay feature narrower and higher than the astrophysical peak.

- Background dilution remains. Even uncorrelated background lowers the target signal through the flux fraction.

When using atmospheric-phase immunity as a selling point, also state how non-phase errors are monitored. Typical checks include calibrator stars, off-source data, off-band data, crossed polarizations, dark fields, time shifts, and injection recovery.

(appsec:d-rayleigh)=
## Rayleigh Limits and Fisher Information
The Rayleigh criterion is an imaging scale, not the information limit for every parameter-estimation problem. Chapter {ref}`chap:08` contrasts direct imaging with mode measurements, and Chapter {ref}`chap:20` turns that contrast into a computational experiment. Start by specifying the model elements.

1.  State the parameter: two-source separation, centroid, flux ratio, angular diameter, or a more complex image.

2.  State the data: pixel intensities, mode counts, event times, or $|V|^2$.

3.  Write the likelihood or Fisher information; do not rely on the word "superresolution" alone.

4.  Include background, finite photon number, mode mismatch, aberrations, and centroid errors.

SPADE is cleanest for weak, equal-brightness, mutually incoherent point-source pairs with a known Gaussian PSF. For galaxies, accretion disks, strong-lensing arcs, or sources containing coherent components, the source model and measurement basis must be written again.

(appsec:d-error-budget)=
## Error Budgets and Roadmap Logic
Observing error budgets appear in Chapter {ref}`chap:18`; readiness and milestones appear in Chapter {ref}`chap:22`. The error budget answers whether the target quantity can be measured. The roadmap answers when the project is mature enough to justify the next level of investment.

| Error term | Example | Mitigation to write down |
|:-----------------|:------------------------|:----------------------|
| Statistical error | Shot noise that depends on photon number, integration time, and bandwidth. | More collecting area, longer integration, channel coaddition, or brighter targets. |
| Calibration error | Zero-baseline contrast, timing response, polarization efficiency, and filter shape. | Calibrator stars, laboratory response measurements, and nightly calibration. |
| Background error | Night sky, companion stars, continuum light, and dark counts. | Off-source data, line/continuum separation, background windows, and flux-fraction estimates. |
| Model error | Limb darkening, asymmetry, velocity model, and radiative transfer. | Model comparison, posterior predictive checks, and external observational constraints. |
| Selection error | Triggers, weather, target samples, and post-selected windows. | Pre-registered target criteria, global significance, and failure criteria. |