Goal: convert raw measurements (clock frequency ratios or interferometer visibilities) into a single number that either (a) estimates the flux→drift amplitude or (b) upper-bounds the lapse-noise spectrum \(S_\Phi\).
You measure a time-series of redshift \(y(t)=\ln(\nu_\infty/\nu_r)\) and estimate its derivative \(\dot y(t)\) (or fit \(y\) directly). The model from Step 2 predicts a template
\[ h(t)\;=\;-\frac{G}{c^4}\frac{L(t)}{r}\,, \quad\text{so}\quad \dot y(t)\;=\;\alpha\,h(t)\;+\;n(t), \]
with \(\alpha\approx1\) in GR and \(n\) noise (colored). The generalized least squares (GLS) / matched-filter estimate is
\[ \hat\alpha \;=\; \frac{h^\top C^{-1} \dot y}{h^\top C^{-1} h}\,, \qquad \sigma^2_{\hat\alpha} \;=\; \frac{1}{h^\top C^{-1} h}, \]
where \(C\) is the noise covariance (from your clock’s PSD after detrending). Sign-lock test: outgoing flux implies \(h(t)<0\Rightarrow \hat\alpha>0\). A negative \(\hat\alpha\) is a null-test failure (or a pipeline bug). 1/r validation: repeat at two radii \(r_1,r_2\); the best-fit amplitudes must scale as \(\hat\alpha_1/\hat\alpha_2 \approx r_2/r_1\).
From Step 5,
\[ -\ln V \;=\; \frac{\omega^2}{2}\!\int_0^\infty \frac{d\Omega}{2\pi}\,S_\Phi(\Omega)\,\big|F(\Omega)\big|^2 . \]
Two practical routes:
Flat (low-\(\Omega\)) bound: if \(|F|\) lives at \(\Omega\!\lesssim\!\Omega_c\) and \(S_\Phi\!\approx\!S_\Phi(0)\),
\[ S_\Phi(0) \;\lesssim\; \frac{2\,|\ln V|}{\omega^2\,W},\qquad W\equiv\int_0^{\infty}\!\frac{d\Omega}{2\pi}\,\big|F(\Omega)\big|^2 . \]
Combine several runs \(k\) by inverse-variance weighting of the individual bounds.
Power-law fit: assume \(S_\Phi(\Omega)=A(\Omega/\Omega_0)^{-p}\). Then
\[ -\ln V_k \;=\; \frac{\omega_k^2}{2}\,A \int_0^\infty \frac{d\Omega}{2\pi}\left(\frac{\Omega}{\Omega_0}\right)^{-p}\!\big|F_k(\Omega)\big|^2 \;\equiv\; A\,\omega_k^2\,\Xi_k(p), \]
which is linear in \(A\). Solve \(A\) by least squares over runs \(k\), scanning a few \(p\) (e.g., \(0\)–\(3\)). Quote \(A(p)\) or turn it into \(S_\Phi(\Omega)\) at a reference band.
Controls & sanity checks: (i) software injections of known \(\delta\Phi\) phases; (ii) jackknife/time-shift tests; (iii) environmental veto channels; (iv) verify the \(\omega^2\) scaling by comparing two internal lines.
| Symbol | Name | Meaning (units) | Typical value/example | Metaphor |
|---|---|---|---|---|
| \(y(t)\) | Log redshift | \(\ln(\nu_\infty/\nu_r)\) (–) | Fit slope \(\dot y\) vs template | “Pitch log” of a note |
| \(h(t)\) | Drift template | \(-\tfrac{G}{c^4}L(t)/r\) (s\(^{-1}\)) | Set by source flux & distance | “Predicted tilt of time” |
| \(C\) | Noise covariance | From clock PSD (var matrix) | Built after detrending | “Map of noise wrinkles” |
| \(V\) | Visibility | Fringe contrast \([0,1]\) | Measured per run | “Stripe sharpness” |
| \(F(\Omega)\) | Sequence filter | FT of path indicator \(f\) (s) | Rectangular ⇒ sinc² | “Sieve selecting tones” |
| \(S_\Phi(\Omega)\) | Lapse PSD | One-sided spectrum (s) | Bounded via \(-\ln V\) | “Noise color of time” |
Ready for Step 7 — Clock-native null tests & 1/r sign-locked templates (how the paper validates pipelines with injections and geometry-driven scaling).