What this section does (plain English): Use a known, modulated power source \(L(t)\) at distance \(r\) and two co-located optical clocks to look for the sign-locked \(1/r\) drift
\[ \dot y(t)\equiv \frac{d}{dt}\ln\!\frac{\nu_\infty}{\nu_r} =\partial_t\Phi(t,r)\;\approx\;-\frac{G}{c^4}\,\frac{L(t)}{r}. \]
This gives a phase-locked template to cross-correlate against the clock data; the prediction has a fixed sign (outgoing flux → negative drift) and exact \(1/r\) scaling.
Estimator you actually run: Model the (whitened) clock series as \(y(t)=-\lambda s(t)+n_y(t)\) with \(s(t)=\int^t L(t')dt'\) and \(\lambda=G/(c^4 r)\). In frequency space \(s(f)=L(f)/(i2\pi f)\). The GLS / matched-filter variance is
\[ \mathrm{Var}(\hat\lambda)=\Big[\,2\!\int_0^\infty\!\frac{|s(f)|^2}{S_y(f)}\,df\Big]^{-1}. \]
Because \(s(f)\propto 1/f\), most SNR comes from sub-Hz where state-of-the-art optical clocks are quiet; stacking \(M\) identical flares gives \(\mathrm{SNR}\propto\sqrt{M}\), and two-clock cross-correlation gains a further \(\sqrt{2}\) on uncorrelated noise.
Uncertainty propagation (what really limits you): Treat range and calibration as fractional nuisances \(r\!\to\!r(1+\epsilon_r)\), \(L\!\to\!L(1+\epsilon_L)\); to first order
\[ \frac{\delta\hat\lambda}{\hat\lambda}\simeq -\epsilon_r-\epsilon_L+O(\epsilon_S), \]
so the total variance is
\[ \mathrm{Var}_{\rm tot}(\hat\lambda)\approx \mathrm{Var}_{\rm stat}(\hat\lambda) + \hat\lambda^2\!\left(\sigma_r^2+\sigma_L^2\right)+\Delta_S, \]
with \(\Delta_S\) from PSD mis-model tests. In lab practice, calibration on \(r\) and \(L\) dominates over the statistical term.
Registered controls you must pass: Pre-register band, vetoes, stacking rule, and success criteria. Demonstrate sign-lock, zero-lag template correlation, and strict \(1/r\) by repeating at \(r=\{5,10,20\}\,\mathrm{m}\). Use shuttered-source and cold-dump nulls, time-slides, and off-band checks.
Why this is a validation channel (not a detection one): In ground labs, near-field EM/thermal couplings dwarf the tiny gravitational signal, so controlled-flux runs are for pipeline validation with software injections (including forced \(1/r\) scaling). The actual flux-drift from realistic lab sources is 12–22 orders below today’s clock stability.
| Symbol | Name | Meaning (units) | Typical value/example | Metaphor |
|---|---|---|---|---|
| \(L(t)\) | Luminosity | Power crossing sphere at \(r\) (W) | Modulated lamp/beam-dump | “Brightness knob” |
| \(r\) | Areal distance | Sets the \(1/r\) falloff (m) | Repeat at 5,10,20 m to test | “Distance lever” |
| \(y(t)\) | Log-redshift | \(\ln(\nu_\infty/\nu_r)\) (–) | Differentiate to get \(\dot y\) | “Pitch log” |
| \(s(t)\) | Integrated flux | \(\int^t L(t')dt'\) (J) | \(s(f)=L(f)/(i2\pi f)\) | “Cumulative push” |
| \(\lambda\) | Gain to drift | \(G/(c^4 r)\) (s² kg\(^{-1}\) m\(^{-1}\)) | Predicted slope in fit | “Gear ratio” |
| \(S_y(f)\) | Clock PSD | Frac-freq noise (Hz\(^{-1}\)) | Used to whiten/weight | “Noise map” |
Next: Step 9 — Lever B (quantum): visibility loss and how the paper turns \(V(T)\) into bounds on \(S_\Phi(\Omega)\).