The Lattice Cell Size
Every way the framework knows its one length — the ~100 μm dark-matter lattice cell — sorted by confidence
What this page is
The substrate framework the size of a single lattice cell in the dark-matter superfluid, \xi \approx 100\;\mum — about the width of a human hair. Everything else follows from \xi: the dc1 mass m_1 = \hbar/(c\xi) \approx 2 meV/c^2, the emergent speed of light c = \hbar/(m_1\xi), the infrared photon floor E_\text{min} = hc/\xi \approx 13 meV, and — through the bridge equation — the electroweak scale itself. Because so much rides on this single number, it deserves its own honest scorecard.
Here are the places the framework’s prediction match to $ tiered by confidence.
- Tier 1 — Independent determinations. Two separate bodies of data — cosmology and electroweak physics — each return a value for \xi with no tuned parameters. Their agreement is the bridge equation.
- Tier 2 — The substrate’s own dynamics. The framework’s own excitations (the smallest photon, the vacuum’s scattering pattern) land on the same \xi from the bottom up. These are internal cross-checks and live, falsifiable predictions — not new external measurements.
- Tier 3 — External anchor & the wide net. An independent-literature reason the 100\;\mum scale is natural (the dark-energy length), plus the order-of-magnitude and structural matches — cell size, the BCS gap, the neutrino mass — where the same scale reappears across domains. Suggestive of reach, not decisive.
The last section, What is not the cell size, is the honesty guardrail: the substrate carries other locked lengths (\xi_\text{GP}, d_\text{GJO}, the 8\;\mum half-period) that are close to 100\;\mum but set by different physics and must not be conflated with the cell.
Tier 1 — Two independent determinations
Two unrelated bodies of data each return a value for \xi with no adjustable parameters. That they agree — to 13\% as raw lengths, and to better than 0.2\% once written as the dimensionless packing fraction — is the framework’s central cross-check.
| Reading | Expression | Returns | Uses | Status |
|---|---|---|---|---|
| Cosmology / close-packing | \xi=\left(\hbar/\rho_\text{DM}c\right)^{1/4} | \mathbf{111.8\;\mu}m | \rho_\text{DM}, \hbar, c | The clean determination — dimensionally airtight, zero parameters |
| Higgs VEV | v=\sqrt{8\pi\,m_\text{eff}^2c^4\,\nu}, \;v\propto\sqrt{\xi} | \mathbf{{\sim}97\;\mu}m | measured v, m_e, \sin^2\theta_W | The second anchor — restored on a measured input, conditional on the VEV derivation |
| Bridge equation (their agreement) | f=\rho_\text{DM}c\,\xi^4/\hbar=4\pi/(K\sqrt2) | \mathbf{0.5666} | both sides above | Zero-parameter match to \sim0.2\% — Tier 1 in spirit |
The cosmology route is the determination. Combine the measured dark-matter density with the substrate’s close-packing condition (n_1\xi^3\approx1) and the Volovik speed relation (c=\hbar/m_1\xi), and a single length falls out: \xi_\text{CP}=(\hbar/\rho_\text{DM}c)^{1/4}=111.8\;\mum, from \rho_\text{DM}, \hbar, and c alone. Every step balances dimensionally; there is no free knob. This is the value the rest of the framework inherits. And the length is not merely fit to the density — dc1’s logarithmic equation of state makes \xi a coupling constant of the medium (the gausson self-binding width set by the fixed energy \lvert b\rvert=m_1c^2), so the cell holds its spacing without an external scaffold and cannot drift as the universe expands.
The Higgs VEV restores the second, independent anchor. The equilibrium chirality amplitude obeys v=\sqrt{8\pi\,m_\text{eff}^2c^4\,\nu} with \nu=m_\text{eff}c\,\xi/\hbar, so v\propto\sqrt\xi. Read forward it predicts v=246.1 GeV against the measured 246.22 (-0.06\%); read backward, the measured VEV plus m_\text{eff} (from \sin^2\theta_W and m_e) solves for the length — and the chain never touches \rho_\text{DM}. It lands on the electroweak-side length, {\sim}97\;\mum (\nu\approx8.3\times10^8), not the cosmology 112\;\mum. So this is not a third number but a genuinely independent re-measurement of the same electroweak leg, now driven by a hard measured input rather than the demoted dimensional mnemonic of the old Route 2. Two honest caveats keep it out of unqualified Tier 1: its 8\pi prefactor is traced (2\times4\pi_\text{SC2}) but rests on the exact-Lorentz-invariance pillar, and its quadrature law v^2\propto\nu is now read as the standard relativistic Bose-field amplitude relation but not yet derived from first principles — the same conditional status as the cosmology route leaning on close-packing.
Their agreement is the bridge. As raw lengths the two routes sit 13\% apart; as the dimensionless packing fraction f=\rho_\text{DM}c\,\xi^4/\hbar they match the purely geometric 4\pi/(K\sqrt2)=0.5666 to better than 0.2\% — inside the \sim1\% Planck uncertainty on \rho_\text{DM}. That two disjoint bodies of data (cosmological density and the electroweak Weinberg angle) meet in one number is the bridge equation, and it is the strongest single reason to take the 100\;\mum scale seriously.
Tier 2 — The substrate’s own dynamics land here
The framework’s own excitations reach the same \xi from the bottom up — no cosmological or electroweak data required. These are internal cross-checks that double as live, falsifiable predictions: they can only fall on \xi because it is the one length the lattice owns.
| Reading | Expression | Lands at | What it is | Status |
|---|---|---|---|---|
| Photon infrared floor | E_\text{min}=hc/\xi=2\pi m_1c^2 | \approx13 meV, \lambda\sim100\;\mum | smallest modon the lattice can hold | derived; FRB-tested (sub-floor light =c to 7\times10^{-16}) |
| Vacuum scattering ring | \lvert\mathbf q_\text{ring}\rvert\simeq2\pi/\xi | ring at \sim100\;\mum | the texture’s structure factor S(\mathbf q) | computed from the framework’s own texture; far-IR edge untested |
The smallest photon is one cell wide. A modon (the framework’s photon) cannot be smaller than one lattice cell: the Bessel boundary-matching that lets a dipole-vortex exist has no solution below \xi. So the lattice has a hard infrared floor, E_\text{min}=hc/\xi=2\pi\,m_1c^2\approx13 meV (\lambda\sim100\;\mum, \nu\sim3 THz) — reached entirely from the excitation side, with no density and no VEV. This floor is already tested: a +\nu^2 refit of the CHIME/FRB catalog bounds sub-floor light to travel at c to a few parts in 10^{16}, so the floor is a change in the quantization character of light, exactly as the modon picture requires (see Photon as Modon and the dedicated Modon Floor chapter; the same E_\text{min}=hc/\xi reappears as the blackbody infrared cutoff).
The vacuum should scatter at one wavelength only. The substrate’s texture — a domain glass of triangular vortex crystallites — is disordered hyperuniform, so its structure factor S(\mathbf q\to0)\to0: no Bragg comb, no fog, just a single diffuse powder ring at \lvert\mathbf q\rvert\simeq2\pi/\xi. The ring’s edge coincides with the modon floor because both read the same \xi — they are one point on the dispersion \omega=ck projected onto two conjugate axes (The Scattering Ring). The falsifiable prediction is sharp: the far-infrared sky is transparent for \lambda>2\xi\approx200\;\mum, a scattering edge switches on near 1.5 THz, and the ring reaches full strength at the cell scale \lambda=\xi\approx100\;\mum (the stealth edge). This is “seeing the lattice a third way” — the size read directly off how the vacuum transmits light.
Tier 3 — External anchor & the wide net
One reason from outside the framework that 100\;\mum is a natural scale, plus the order-of-magnitude and structural matches where the same length (or its conjugate \sim2 meV energy) reappears across domains. These show reach; they are not decisive, and the home chapters are candid about which are suggestive.
The external anchor — the dark-energy length
| Reading | Expression | Value | Why it matters |
|---|---|---|---|
| Dark-energy length | \xi_\text{DE}=(\hbar c/\rho_\Lambda)^{1/4} | \approx85–88\;\mum | the scale torsion-balance gravity tests already probe |
The cosmology cell is not our invention writ large: \xi_\text{DE}=(\hbar c/\rho_\Lambda)^{1/4}\approx85\;\mum is the canonical dark-energy length (Beane 1997; Kapner & Adelberger 2007 title their sub-millimetre-gravity paper “Tests of the Gravitational Inverse-Square Law Below the Dark-Energy Length Scale”) — and it is why Eöt-Wash torsion balances probe \sim100\;\mum at all. The lattice cell exceeds it by exactly the dark-sector ratio, \xi_\text{DM}/\xi_\text{DE}=(\Omega_\Lambda/\Omega_\text{DM})^{1/4}=1.27, so the long-noted “\rho_\Lambda^{1/4}\sim m_1c^2\sim meV” coincidence is one identity carrying a single measured cosmological number. An established experimental frontier already sits where the framework says the cell is.
The conjugate energy — one \sim2 meV scale, three of the lowest masses in physics
| Reading | Where | Value | What the match adds |
|---|---|---|---|
| dc1 particle mass | dark matter | m_1c^2=\rho_\text{DM}^{1/4}=1.77 meV | the Compton energy conjugate to \xi — same number, energy side |
| Dark-energy scale | cosmology | \rho_\Lambda^{1/4}=2.24 meV | the substrate’s single density scale, seen as vacuum energy |
| Lightest neutrino | particle physics | m_{\nu,1}\approx m_1\approx2 meV | the barest knot sits at the substrate’s own quantum |
Reading \xi as an energy, m_1c^2=\hbar c/\xi\approx2 meV, ties three of the lowest scales in physics — the dark-matter constituent, the dark-energy scale, and the lightest neutrino — to one substrate constant, the same one the cell size is (Neutrino Mass Scale).
Structural & order-of-magnitude matches
| Pattern | Substrate reading | Status |
|---|---|---|
| Cell size \lesssim\xi | most eukaryotic cells fit in one lattice bubble (10–100\;\mum); a cell larger than \xi cannot hold coherent internal coupling | scale coincidence — the framework’s most speculative extension, not a derivation |
| Structured-water width | exclusion-zone widths 133–142\;\mum near hydrophilic surfaces (Wang & Pollack 2024) read as the static \xi boundary | concrete external measurement landing at \xi; framework interpretation untested |
| BCS gap \sim2 meV | \Delta_\text{BCS}\sim m_1c^2, the dc1 rest energy as the natural pairing scale (the gap, not the \sim40 nm coherence length) | order-of-magnitude coincidence |
| Vacuum transparency | far-IR universe transparent for billions of light-years — a “fog” vacuum could not be | already-passed half of the ring prediction |
What is not the cell size
A length is understood by its boundaries. The substrate carries three other locked lengths that sit near 100\;\mum and are easy to mistake for the cell — but each is set by different physics and must be kept distinct, exactly as the ladder index keeps the scale rungs distinct from the spatial template.
| Length | Value | What it actually is | Why it is not \xi |
|---|---|---|---|
| GP healing length \xi_\text{GP} | \xi/\sqrt2\approx79\;\mum | the vortex-core / gausson self-binding width | one \sqrt2 rung below the cell — the pairing factor, not the cell |
| Inter-sheet period d_\text{GJO} | \approx16\;\mum | vertical spacing of chirality sheets (GJO instability wavelength) | a different direction — set by an axial instability, not close-packing |
| Counter-rotating boundary | d_\text{GJO}/2\approx8\;\mum | the boundary layer between like-handed sheets | the half-period any boundary-locked structure feels — still not the cell |
Keeping these apart is what keeps the cell size falsifiable: a match at 8\;\mum or 16\;\mum is a match to the vertical geometry, and only a match at \sim100\;\mum tests the cell itself.
What would settle it
The two Tier-1 determinations already agree to 0.2\% as a packing fraction; the decisive open test is the Tier-2 scattering ring — a far-infrared scattering edge near 200\;\mum (1.5 THz) with transparency below it and a diffuse ring peaking at the cell scale 100\;\mum. Torsion-balance gravity tests already probing the dark-energy length sit at the same scale from the other side. If the far-IR sky shows no scattering edge and sub-millimetre gravity shows no oscillatory deviation near 0.5–1 mm, the 100\;\mum cell is in trouble. Until then, the honest summary is the one at the top: one length, reached independently from cosmology and from the electroweak scale, cross-checked by the substrate’s own light, and anchored to a frontier experiments already probe.
The companion indexes read the lattice two other ways: the ladder index catalogs the \sqrt2 spacing between scales, and the modon index catalogs the dipole-vortex topology — where this page catalogs the one size they are all built on.