The Scattering Ring
The same lattice scale ξ that fixes the 3 THz frequency floor also fixes a single diffuse scattering ring at q ≈ 2π/ξ. The floor and the ring are one point on the photon’s dispersion relation, seen in two conjugate domains — two velocity limits and one lattice scale from this model.
Interactive: open the scattering-ring simulation — reciprocal space, drawn. New to structure factors? Start with the guided tour (G): twelve narrated stops, about three minutes, from “why is a dense lattice invisible?” to the one experiment that would test it. A slice of the lattice and its structure factor computed live: a crystal’s comb, a gas’s speckle, the domain glass’s one ring; then the probe, whose Ewald circle sits inside the dark hole until \lambda=2\xi and reaches the ring at 60° by \lambda=\xi; and the light cone with the lattice’s one point projected onto both axes.
The Claim
This chapter shows the scattering ring as a spatial twin of the modon floor:
\boxed{\ \nu_\text{floor}=\frac{c}{\xi}\approx3\;\text{THz}\quad\text{and}\quad |\mathbf q_\text{ring}|\approx\frac{2\pi}{\xi}\ \ \text{are the same point on}\ \ \omega(k)=ck.\ }
The stealth-vacuum chapter computed the ring: the substrate’s texture — a domain glass of triangular vortex crystallites — scatters light into a single diffuse powder ring at |\mathbf q|\simeq2\pi/\xi around a dark S(\mathbf q)\to0 hole, and nothing below it (Stealth Vacuum § Computed). The ring’s edge and the modon floor must coincide, because \xi is the only length the lattice owns. The substrate’s one lattice scale can be read at d\approx0 in two conjugate domains at once.
The Floor and the Ring Are One Point on the Light Cone
The substrate’s photon obeys the simplest dispersion relation there is — the emergent light cone \omega=ck, the Volovik speed shared by every low-energy excitation. A dispersion relation is a curve in the (k,\omega) plane, and the lattice’s single length \xi marks exactly one point on it: the place where the wavelength equals the cell size. At that point the wavevector and the frequency are both fixed,
k_\xi=\frac{2\pi}{\xi}\approx 6.3\times10^{4}\;\text{m}^{-1},\qquad \omega_\xi=c\,k_\xi=\frac{2\pi c}{\xi}\approx 1.9\times10^{13}\;\text{rad/s}\ \ (\nu_\xi\approx3\;\text{THz}).
The difference between the floor and the ring is only which axis you project onto:
- Project onto the frequency axis and you get the modon floor: \nu_\text{floor}=\omega_\xi/2\pi=c/\xi\approx3 THz, the frequency at which light changes quantization character (a compact countable soliton above, a stretched delocalized winding below). This is what a spectrometer reads — a temporal instrument, watching the field oscillate in time. The modon-floor chapter is its full treatment.
- Project onto the wavevector axis and you get the scattering ring: |\mathbf q_\text{ring}|\simeq k_\xi=2\pi/\xi, the single momentum transfer at which the hyperuniform vacuum finally scatters. This is what a scattering or diffraction experiment measures using a spatial instrument that watches the field’s structure in space. The stealth-vacuum chapter is its full treatment.
The lattice has one length — observed through two instruments that are conjugate to each other, time versus space.
The Backscatter Factor of Two
The floor sits at \lambda=\xi exactly; the ring’s far-infrared scattering onset sits at \lambda\simeq2\xi (Stealth Vacuum § far-IR opacity). The factor of two comes from the kinematics of scattering. A probe of wavevector k can transfer at most |\mathbf q|=2k (backscatter, \theta=180°, since |\mathbf q|=2k\sin(\theta/2)). To reach the ring at |\mathbf q|=2\pi/\xi in a single scattering event therefore needs 2k\ge2\pi/\xi, i.e. \lambda\le2\xi. So the vacuum is rigorously transparent for \lambda>2\xi\approx200\,\mum (\nu<1.5 THz — the stealth window), scattering turns on at \lambda=2\xi, and the ring reaches full strength at the cell scale \lambda=\xi, which is the floor. Floor and ring coincide where they should — at \xi — with the scattering channel opening one octave below in wavelength because a back-scatter doubles the momentum transfer.
This becomes a falsifiable prediction - a scattering edge at 200\,\mum, a full ring at 100\,\mum, transparency below 1.5 THz.
The Ring’s Status
The ring has the same caveats as the reading of the modon floor:
| Modon floor (frequency domain) | Scattering ring (wavevector domain) | |
|---|---|---|
| Reads | \xi as \nu_\text{floor}=c/\xi\approx3 THz | \xi as |\mathbf q|\simeq2\pi/\xi |
| Clean (d\approx0) probe | laboratory THz-TDS / photonic-crystal band edge | laboratory far-IR scattering off the cell-scale texture |
| Clean detection | unrun | unrun |
| Cosmological probe | FRB (from below), FIRAS (at the crossing) | far-IR/submm sky transparency (Herschel, ALMA) |
| Cosmological result | null bounds — transparency confirmed | transparency confirmed to z\sim4–6 |
| Cosmological detection of \xi | unable to confirm | unable to confirm |
Both are seen in the laboratory, both have a cosmological cousin that is propagation-degraded over gigaparsecs, and in both cases the cosmological data already in hand confirm transparency — not detection. The far-IR sky’s transparency bounds the per-cell cross-section below \sim10^{-31}\xi^2 (Stealth Vacuum § falsify), exactly as FIRAS bounds the floor’s dissipation to \sim10^{-28} and CHIME/FRB bounds sub-floor dispersion to \varepsilon<6.5\times10^{-16} (Modon Floor § data). The two channels have run the same experiment — is the vacuum transparent in the band where it should be stealthy? — once in frequency and once in space, and both answered yes. Neither has yet run the in-band detection: a sharp, transparent, period-independent feature at the cell scale, read in the one window — the terahertz gap — humanity built instruments for last.
The discriminators carry over with one addition. The floor’s three discriminators — fixed at the substrate scale not the engineered one, exponential not power-law dispersion, transparent not absorptive — are frequency-domain statements. The ring adds a spatial discriminator the floor cannot make: the texture must be a single diffuse ring, not a Bragg comb and not flat speckle (Stealth Vacuum trilemma). A comb would mean a periodic aether (and a preferred frame, breaking Michelson–Morley); flat speckle would mean a random gas (and a foggy sky). Only the single ring at 2\pi/\xi around a dark hole is the substrate. Reading the lattice scale in both domains therefore tests more than reading it in one — the ring’s “what shape is the scattering” is independent of the floor’s “what shape is the dispersion.”
What the Catalogue Actually Reads
With the ring counted, the zero-depth catalogue’s structure becomes cleaner than “two velocities and a frequency,” not messier. The substrate has, at bottom, one scale — the lattice cell, equivalently the dc1 mass, since \xi=\hbar/m_1c ties them with no freedom — plus two rotations. The clean reads sort onto exactly these:
| Substrate quantity | Read as | Domain | Anchor | Status |
|---|---|---|---|---|
| outer rotation \omega_0\xi | velocity limit v_L | velocity | fast solar wind | measured, 0.3\% |
| inner circulation c\sqrt{2\alpha_{mf}} | velocity limit 0.776\,c | velocity | γ shoulder at 300 keV | predicted, unrun |
| lattice scale \xi (\Leftrightarrow m_1) | frequency c/\xi | frequency | modon floor, 3 THz | predicted, unrun |
| lattice scale \xi (\Leftrightarrow m_1) | wavevector 2\pi/\xi | wavevector | scattering ring | predicted, unrun |
The two rims are the velocity limits of the flow — how fast organized motion can ride the lossless channel before it saturates or sheds. The floor and ring come from the lattice scale of the dispersion — the one length where light’s character and the vacuum’s transparency both turn over. The first pair lives on the flow’s velocity ceilings; the second pair lives on the photon’s dispersion curve. That is the real organizing principle the catalogue was reaching for: the substrate hands over its scale wherever an instrument can be laid against the medium with nothing in between — and the cleanest reads come in conjugate pairs, because a single physical scale always projects onto more than one measurement axis. The bridge equation’s cell occupancy is the dimensionless echo of the same fact — the lattice scale read yet again, this time as a pure number that needs no instrument at all.