Why Pressure-Test First
The heavy-molecule corner is the surviving remnant of a no-go theorem; remnants of no-gos die more often than they mature. Rewriting the manuscripts’ experimental sections around a model that fails its own consistency checks would compound the original error of building on the untested \(\alpha_\text{eff}\) assumption. This note therefore stress-tests the swept-medium model on four independent fronts before any manuscript work proceeds. Throughout, the coupling is held at its atom-interferometry ceiling \(X \equiv \varphi_\text{rms}^2\,\xi = 2.8\times 10^{-54}\) m (the maximum the corner allows), \(\xi = 100\) nm, and the reference species is \(10^4\) amu at beam velocity \(150\) m/s with \(\tau = 10\) ms transit.
Test 1: Does the Channel Destroy Its Own Signal?
A channel that dephases also kicks (information–disturbance, Lemma 1 of the attack note). If the momentum diffusion accompanying \(\Gamma \approx 3.8~\text{s}^{-1}\) blurred the molecule’s transverse momentum by more than the grating momentum \(\hbar k = 2\pi\hbar/d\), the interference pattern would be destroyed independently of dephasing and the “detection” would be meaningless. At maximum coupling, \[\sigma_p = \sqrt{D_p \tau} = \frac{\hbar}{\xi}\sqrt{\Gamma\tau} \approx 2.1\times 10^{-28}~\text{kg\,m/s} \qquad\text{vs.}\qquad \hbar k \approx 2.5\times 10^{-27}~\text{kg\,m/s},\tag{1}\] a ratio of \(0.08\). Pass. The pattern survives, and the \(8\%\) momentum blur is itself a secondary observable: at couplings near the ceiling, fringe visibility loss should be accompanied by slight envelope broadening with the same \(M^2/v\) scaling—a built-in cross-check no background mimics.
Test 2: Optical Clocks
The medium couples to \(T^{00}\), so it is a fluctuating redshift field: an atom’s clock states, differing in energy by \(\hbar\omega_\text{opt} \sim 1\) eV, accumulate relative phase \((\hbar\omega_\text{opt}/\hbar)\!\int\!\varphi\,dt\) from the uniform component of \(\varphi\)—no spatial separation needed. This evades the EP-gauge lemma (the baseline is in energy, not space) and could have killed the corner from a direction the attack note never considered. It does not: the required field amplitude is \(\varphi_\text{rms} = \sqrt{X/\xi} \approx 5.3\times 10^{-24}\), five orders of magnitude below the \(10^{-18}\) fractional instability of the best optical clocks. Pass, and for a structural reason worth recording: molecular interferometry beats clocks at probing this channel by the hierarchy \((Mc^2/\hbar\omega_\text{opt})^2 \approx 10^{26}\)—a \(10^4\) amu molecule is a “clock” whose tick energy is its entire rest mass. This is the sharpest statement yet of why heavy-molecule platforms are the unique probe.
Test 3: The Falling-Medium Dichotomy
“Comoving with the laboratory” was left vague in the attack note. It cannot remain so: a medium comoving with local free-fall frames accelerates downward at \(g\) relative to any surface laboratory and sweeps past it at up to escape speed. At \(v_\text{sweep} \approx 11\) km/s, the rate dilutes to \(\Gamma_{10^4} \approx 0.05~\text{s}^{-1}\)—below threshold. The corner is therefore forced to a pressure-supported, Earth-bound, static halo (atmosphere-like, not raining). One residual choice remains: whether the halo co-rotates with the surface.
| Halo state | effective sweep (equator) | \(\Gamma_{10^4\,\text{amu}}\) |
|---|---|---|
| Co-rotating | beam only, \(150\) m/s | \(3.8~\text{s}^{-1}\) |
| Non-rotating (inertial) | \(\sqrt{150^2 + 465^2}\) m/s | \(1.2~\text{s}^{-1}\) |
Both survive above the \(0.6~\text{s}^{-1}\) threshold, and the split is itself a signature: a non-rotating halo predicts latitude dependence and a diurnal modulation as the beam orientation rotates relative to the sweep—measurable with the same apparatus run at different orientations. Pass with a forced specification, recorded in the cost ledger.
Test 4: GRACE-FO and the Scale-Height Question
The attack note implied the halo must be tuned to evade orbital accelerometry. Computing rather than assuming: if the halo fills low Earth orbit, GRACE-FO’s \(600\) kg satellites sweep it at \(7.7\) km/s, but self-averaging over \((L/\xi)^3 \sim 3\times 10^{19}\) correlation cells suppresses the resulting acceleration noise to \(\sim 5\times 10^{-15}\) m s\(^{-2}/\sqrt{\text{Hz}}\)—five orders below the laser-ranging interferometer’s \(\sim 10^{-10}\) sensitivity. Pass, and a tuning dies: the halo needs no engineered scale height; it may extend through LEO freely. (LISA Pathfinder at L1 remains the binding spaceborne instrument, and any Earth-bound halo plausibly thins long before \(1.5\times 10^6\) km; this is the one residual geometric assumption, now mild rather than tuned.)
Updated Cost Ledger and Signature Set
Costs (complete list).
A preferred frame (Lorentz violation); a pressure-supported Earth-bound halo with no proposed microphysics; correlation length \(\xi \lesssim\) path separation; intrinsic dynamics slow enough that fountain-regime bounds dominate (\(\tau_c\) freedom); halo thins before L1. Five items: one fewer than feared (scale height removed), one forced (static halo).
Signatures (complete list, all with no conventional mimic).
\(\Gamma \propto M^2\) at fixed velocity—the original anchoring signature, now at \(10^3\)–\(10^4\) amu.
\(\Gamma \propto 1/v_\text{beam}\)—slower beams decohere faster per unit time; testable by velocity selection in one apparatus.
Orientation/latitude/diurnal anisotropy if the halo is non-co-rotating; null anisotropy plus full rate if co-rotating. Either outcome is informative.
Envelope broadening accompanying visibility loss at \(8\%\) of grating momentum near maximum coupling, with the same \(M^2/v\) scaling.
Verdict
The swept-medium corner survives four independent kill-tests, loses a fine-tuning, gains a forced specification, and emerges with four quantified signatures spanning a predicted rate window \(\Gamma_{10^4\,\text{amu}} \in [1.2,\, 3.8]~\text{s}^{-1}\) at the coupling ceiling—all within reach of a LUMI-class platform running \(10^4\) amu species at \(50\)–\(150\) m/s. The model is now specified sharply enough that a single campaign kills it or makes a discovery; it can no longer retreat. On that basis—and only on that basis—it is now safe to rewrite the manuscripts’ experimental sections around the heavy-molecule program, with this note and the attack note’s cost ledger incorporated verbatim. The honest sequence of the past five notes (derive, evade, close, attack, pressure-test) is itself the strongest evidence of the program’s integrity and should be presented as such.