Substrate Particles and Properties

The Substrate Particle: dc1

The framework posits a single dark matter particle, dc1 (“dark carbon-1”), whose self-interaction is logarithmic — a Zloshchastiev superfluid-vacuum equation of state (developed below in § The Logarithmic Equation of State). dc1 forms the bulk of the substrate, the entirety of the particle physics sector, and its own scaffold:

• Mass: m_1 \approx 2.04 meV/c^2 = 3.63 \times 10^{-39} kg
• Number density: n_1 \approx 6.6 \times 10^{11} m^{-3}
• These values are not free — they follow from three relations: c = \hbar/(m_1 \xi) (the Volovik quasiparticle speed, derived in Emergent Speed of Light), n_1 m_1 = \rho_{DM} (the substrate is dark matter), and close-packing of the perturbation envelope (n_1 \xi^3 \approx 1).
• At CMB temperature, the thermal de Broglie wavelength \lambda_{dB} \sim 1.3 mm greatly exceeds the interparticle spacing n_1^{-1/3} \sim 115\;\mum: dc1 forms a quantum degenerate condensate (BEC). Individual dc1 particles are delocalized across many lattice cells.

The single particle does the work that earlier drafts split between two. Because the logarithmic self-coupling is a fixed energy (|b| = m_1 c^2), not a density, the cell scale \xi is a coupling constant of the medium rather than a function of how the condensate is loaded — so it cannot drift as the universe expands. The condensate self-binds at the healing length and self-organizes to close-packing (n_1\xi^3 \approx 1) with no external anchor. There is no need for a second particle to hold the lattice in place.

NoteHistorical note: the retirement of “dag”

Earlier drafts of this framework carried a heavy, sparse second species — “dag” (“dark silver”) — invented to pin the vortex lattice against cosmological drift. The logarithmic equation of state does that job intrinsically: its length scale is a coupling constant, not a density, so the lattice cannot drift and needs no scaffold (mechanism in § The Logarithmic Equation of State; numerical case in § Why the Scaffold Needs No Second Species). dag carried less than 3\% of \rho_{DM}, entered no particle physics constraint equation (C1–C9), and — as the bridge equation numbers show — any nonzero amount only worsened the match. It was a placeholder for the log, and it is retired. The substrate is one species, dc1. (Full record: sessions/svt-*; status in Open Problems § WIP-11.)

The Lattice Breathes in Pairs

dc1 is described above as a Bose–Einstein condensate, and at the surface that is what it behaves like — quantum-degenerate, delocalized, one wavefunction per cell. But the substrate has to do something a plain condensate cannot, and that requirement fixes its deeper structure.

The electron is a vortex of half-integer winding — its 720° rotation property (you must turn it twice to bring it home) is that half-winding, and it is why the electron is topologically indestructible (Topology as Stability). A single-component condensate cannot host such an object. Single-valuedness of \psi = |\psi|\,e^{i\theta} forces the phase to wind by whole multiples of 2\pi around any core — integer vortices only. A half-turn of phase can close on itself only if a second component flips sign to compensate. Half-integer winding therefore demands a multi-component, paired order parameter — the structure superfluid ³He-A already carries, where these half-windings are its well-studied half-quantum vortices. The framework’s own particles force the substrate into that class: dc1 is not a lone scalar field but a paired condensate, and every fermion vortex comes with a partner.

That partner is something the paper has already met. In Conductors, two electrons bind into a Cooper pair by breathing in anti-phase — one contracted to its inner scale while the other is expanded out to the envelope \xi — and the give-and-take between them organizes a shared counter-rotating vortex that locks the pair together. The same anti-phase pairing is the organizing principle of the vacuum lattice itself. Each fermion vortex is shadowed by a counter-rotating partner breathing against it, and the counter-rotating intermediate layers that sit between every pair of like-handed chirality sheets (The Vertical Scale) are those partners. The Cooper pair is not a quirk of metals; it is a glimpse, at laboratory scale, of how the substrate is built everywhere.

This pairing is not decoration — it sets the lattice’s geometry. The vortex-core size obeys the Gross–Pitaevskii balance \xi_\text{GP} = \xi/\sqrt{2}, that is \xi^2 = 2\,\xi_\text{GP}^2: the close-packed lattice of cores is exactly twice as dense as the lattice of fermions. In a scalar condensate that factor of two is just the “2” in the kinetic energy \hbar^2/2m. In the paired lattice it has a physical name — there are two cores per fermion, the vortex and its anti-phase partner. The two readings return the same number, and that agreement is the point: a factor that looked like a bare quantum-mechanical convention turns out to be the pairing count.

The same “two” then reappears wherever the lattice’s structure is probed:

• It is the 1/\sqrt{2} of the bridge equation — now derived twice, once from kinetic-energy balance and once from the pairing, meeting at one value.
• It is the \ln 2 in the chirality packing factor \varepsilon_\text{chirality} = \sqrt{2\pi\,S_M}/K, where S_M = \tfrac{1}{2}\ln 2 is the entropy of the single Majorana (fillable-or-empty) state each paired vortex carries — tying the in-plane packing fraction to the fermion sector (developed in Higgs Field).
• It is the m_1 = 2m_f that places the lattice at the marginal point where the condensate’s excitations turn massless and isotropic — the point at which the substrate emits light at a single speed c in every direction, which is the exact emergent Lorentz invariance behind bridge Step A.

One pairing, one factor of two, threaded through the cell spacing, the fermion content, and the speed of light.

The pairing runs in time as well as structure. The same anti-phase relationship is what makes a particle’s Compton breath a lossless, complete exchange — the contracting vortex hands its energy to its counter-rotating partner rather than to nothing — and it is why the radial breath runs at twice the orbital frequency (the temporal face of the pairing-two). The lattice cell’s own breath, between core (\xi_\text{GP}) and cell (\xi) at the dc1 clock \omega_1 = m_1 c^2/\hbar \approx 3\times10^{12} rad/s, is the quantum that sets the minimum photon energy E_\text{min} = 2\pi m_1 c^2: a modon cannot carry less than one full breath of one cell (Photon as Modon § Minimum Modon Energy).

Because the breath is paired and anti-phase, it is silent above the single cell. Every sheet inhales as its counter-rotating median exhales, so at any scale larger than one lattice cell the breathing sums to zero: there is no macroscopic drumbeat even though the energy is present in every cell of the vacuum, oscillating at \omega_1 \approx 3\times10^{12} rad/s. What survives the cancellation is only the per-cell residue — and that residue is exactly the infrared floor E_\text{min} = 2\pi m_1 c^2, the quantum below which no modon can exist. The substrate hums everywhere at \omega_1, and the same pairing-two that builds the lattice is what keeps the hum from ever being heard at any scale we occupy, leaving only its zero-point as the lowest note the vacuum can carry.

What survives above the cell is not the beat but its ratio. The same pairing-two that separates the core from the cell — the factor \sqrt{2} — reappears as the spacing between one preferred scale and the next, all the way up: grid-cell modules a factor of \sim 1.4 apart, cochlear octaves, vesicle coats, organelle sizes. The breath’s timing cancels above a single cell; its geometry does not, and replicated across scale it becomes a ladder of preferred sizes — the subject of The Substrate Ladder, where the same \sqrt{2} that builds the cell is shown to rule the sizes of structures that share no chemistry and no history.

Note

This paired (“³He-A class”) reading of the lattice is forced by half-integer winding: a single-component condensate cannot carry the electron’s topology. What remains open is the fully from-scratch calculation. The factor of two is currently sourced three independent ways — kinetic balance, pairing count, and Majorana state count — which agree but have not yet been derived from one Bogoliubov–de Gennes treatment; and the identification of close-packing with the marginal point (\mu \to 0) is a concrete equation-of-state calculation still owed. The full status, with sources, is in Open Problems § WIP-15.

Scaffold and Excitation

The substrate plays two roles at once — infrastructure and excitation, the stage versus the performers — and both are jobs of the same dc1 condensate. The slow, self-bound vortex lattice is the stage; the fast topological defects that propagate through it are the performers, each called here a dc1 vortex.

This separation has a precise condensed matter analog. In a crystal, phonons are excitations of the lattice — they propagate through the crystal without carrying atoms at their center. An electron in a semiconductor is a quasiparticle dressed by lattice interactions, with its effective mass and behavior set by the band structure of the crystal, not by having a silicon atom embedded in it. The lattice provides the stage; excitations are the performers.

In the dc1 substrate:

• The logarithmic EOS provides the large-scale order, organizing the dc1 BEC into a vortex lattice with perturbation envelope length \xi \approx 100\;\mum. The lattice is self-bound: its cell scale is set by the coupling |b| = m_1 c^2 — a fixed energy, not a density — so it holds its spacing without external pinning and cannot drift on cosmological timescales.

• Particles are topological defects and excitations of the same dc1 condensate. The electron is a single-quantum vortex defect — its center is a phase singularity of the BEC wavefunction, like the eye of a hurricane. It is a structural feature of the flow, not a physical object sitting at the center. The proton is a three-fold junction defect. The photon is a modon soliton that propagates through the lattice cells.

This reframe explains several features of the constraint system that were previously puzzling:

Why one medium does both jobs. The particle physics sector — constraints C1 through C9, covering the speed of light, Planck’s constant, the gravitational constant, the electron mass, the fine structure constant, the Weinberg angle, and the anomalous magnetic moment — is built entirely from dc1 properties (m_1, n_1) and the mutual friction parameter \alpha_{mf}. The lattice that hosts the particles and the excitations that are the particles are the same superfluid, so no separate infrastructure parameters ever enter the particle sector.

Why the effective quantum is universal. The mass m_\text{eff} = m_e/\alpha_{mf} = m_p/\alpha_{mf}^{(N)} \approx 1.70 MeV/c^2 is the same for electrons and nucleons. This is now literal: the effective quantum is the dc1 BEC’s fundamental vortex excitation unit, set by the condensate’s properties (m_1, \nu, \alpha_{mf}). It does not depend on which particle it’s in because it is a property of the medium, not the defect.

Why pair creation works. When a photon creates an electron-positron pair, the modon topologically splits into two opposite-chirality vortex defects — a local dc1 reconfiguration. Nothing else needs to be found, split, or captured. The pair creation is clean and local, exactly as it should be for a process that occurs in high-energy collisions everywhere in the universe.

Why particles are stable. Quantized circulation in a BEC is topologically protected — a vortex with half-integer winding cannot decay without a partner of opposite winding. The electron’s spin-½ topology (720° rotation property) is the stability mechanism. This is the superfluid helium lesson: vortex lines in He-II persist indefinitely because circulation is quantized, not because something heavy anchors their core.

The Three-Tier Hierarchy

The substrate organizes into three well-separated scales, connected by the condensation number \nu = m_\text{eff}/m_1 \approx 9.6 \times 10^8:

Tier	Mass	Spatial Scale	Physics
dc1 sea	m_1 \approx 2 meV/c^2	delocalized (\lambda_{dB} \sim 1.3 mm)	Substrate medium, dark matter
Effective quantum	m_\text{eff} \approx 1.70 MeV/c^2	r_\text{eff} \approx 150 fm	Particle physics, spin, angular momentum
Perturbation envelope	—	\xi \approx 100\;\mum	Modon/photon structure, lattice cell

The dc1 sea is the medium — delocalized, quantum-degenerate, filling all of space. The effective quantum is the vortex excitation: roughly 10^9 dc1 particles collectively carrying \hbar of angular momentum, orbiting at 0.776\,c in a 150 fm vortex core. The perturbation envelope is the propagating structure — a much larger containment scale through which modons (photons) form and the vortex lattice organizes. These are related like water molecules → ocean eddies → tsunami waves: three tiers of a single fluid, \times 10^5 from the dc1 sea to the effective quantum, and \times 10^9 from the effective quantum to it’s envelope.

Outer Scale: The Perturbation Envelope

The perturbation envelope length \xi — the characteristic size of each lattice cell is determined by three measured constants (\hbar, c, \rho_{DM}) plus the close-packing condition:

\xi = \left(\frac{\hbar}{\rho_{DM} \cdot c}\right)^{1/4} \approx 110\;\mu\text{m}

This is the close-packing route, n_1 \xi^3 \approx 1; it fixes the length top-down from cosmology. The particle-physics sector alone does not build that length — a dimensional no-go forbids it — but it does supply a scaffold: the SC2 lattice-metric condition and modon matching fix everything about the cell except one pure number, the condensation number \nu = m_\text{eff}/m_1 \approx 9.6\times10^8, the nine-decade lift from the electroweak Compton scale to the cell. So the honest structure is one length (cosmology), one scaffold (electroweak + geometry), and one open number (\nu; Open Problems § WIP-30). The scaffold’s own length recipe lands at \xi_\text{SC2} = 96.9\;\mum — within the \sim13\% that separates the two readings of \nu (9.6\times10^8 from cosmology, 8.3\times10^8 from the electroweak side, whose m_1 is read from \xi_\text{SC2}) — a mnemonic, not a second measurement. The bridge equation — the packing fraction f = n_1 \xi^3 = 4\pi/(K\sqrt{2}) = 0.5666, a zero-parameter relation, derived not fitted — connects electroweak physics (\sin^2\theta_W, m_e) to cosmology (\rho_{DM}) (see Bridge Equation for the full derivation).

With \xi in hand, the dc1 mass and number density follow immediately:

m_1 = \frac{\hbar}{c \cdot \xi} \approx 2\;\text{meV}/c^2, \qquad n_1 = \frac{\rho_{DM}}{m_1} \approx 6.6 \times 10^{11}\;\text{m}^{-3}

The minimum modon energy — the lowest-energy photon the lattice can support — is E_\text{min} = hc/\xi = 2\pi m_1 c^2 \approx 13 meV, corresponding to a wavelength of \sim 100\;\mum. Below this energy, disturbances propagate as collective lattice excitations rather than as individual modons.

The Vertical Scale: Inter-Sheet Spacing

The perturbation envelope length \xi sets the lattice’s in-plane geometry — the width of one cell. But the lattice is not a single plane; it is layered. The dc1 vortices organize into chirality-coherent sheets — 2D triangular arrays of co-rotating vortices, all of one handedness — stacked vertically, with a counter-rotating intermediate layer between each pair of like-handed sheets (the same counter-rotating boundary that stores energy in every particle; see Topology as Stability). The vortex lines themselves run perpendicular to the sheets, threading the stack from one pinning site to the next. The lattice therefore carries two geometric scales: the in-plane cell width \xi, and the vertical period d at which like-handed sheets repeat.

The vertical period is fixed — with no new parameters — by the wavelength of an instability of those threading vortex lines:

\boxed{\;d_\text{GJO} = \xi\sqrt{\frac{\ln(\xi/\xi_\text{GP})}{4\pi}} \approx 0.166\,\xi \approx 16\;\mu\text{m}\;}

where \xi_\text{GP} = \xi/\sqrt{2} is the Gross-Pitaevskii vortex-core size. The particle-physics route (\xi = 96.9\;\mum) gives d \approx 16\;\mum; the cosmology route (\xi = 112\;\mum) gives d \approx 19\;\mum.

The derivation in brief. Flow running along a rotating vortex array goes unstable above a critical speed — the Glaberson-Johnson-Ostermeier instability — and the unstable mode appears at a single wavelength d = 1/k_c = \sqrt{\nu_s/(2\Omega_\text{sheet})}. Substituting Feynman’s relation for the in-plane sheet rotation at one vortex per cell, \Omega_\text{sheet} = \kappa_q/(2\xi^2), and Saffman’s vortex-line tension cut-off \nu_s = (\kappa_q/4\pi)\ln(\xi/\xi_\text{GP}), the quantum of circulation \kappa_q enters both factors and cancels:

d^2 = \frac{\nu_s}{2\Omega_\text{sheet}} = \frac{\xi^2\ln(\xi/\xi_\text{GP})}{4\pi}

With \xi/\xi_\text{GP} = \sqrt{2}, the logarithm is \tfrac{1}{2}\ln 2 = 0.347, and the spacing is purely geometric — three independent classical-fluid results (the GJO instability wavelength, Feynman’s vortex density, the Saffman cut-off) combined with no fitted coefficient.

Why an instability sets the spacing — two stories, one medium. Two classical pictures could in principle fix d, and they come from two different laboratory superfluids. The choice between them is not cosmetic: it decides whether the substrate’s layering is imposed or self-organized.

The first is the layered-superconductor picture (Lawrence-Doniach). In a cuprate, the conducting planes are fixed by the crystal, and the theory computes the energetic response of the coupling between them. Carried over to the substrate, this gives a two-term energy e(u) = \alpha_{mf}\,u\ln(1/u) - u/2 in the anisotropy u = d/\xi, whose extremum sits at u^* \approx 0.070 — i.e. d \approx 7\;\mum. But that extremum is a maximum, an energy hilltop rather than a stable well: a structure placed there rolls off it, and rescuing 7\;\mum as an equilibrium requires a third, repulsive term that has never been derived. Its deeper problem is that it presupposes the layers — it assumes the very structure the substrate is supposed to produce on its own.

The second is the rotating-superfluid picture (Glaberson-Johnson-Ostermeier). In superfluid helium — a continuous medium with no imposed layers — flow along a rotating vortex array goes unstable and the lines ring with Kelvin (bending) waves at one selected wavelength. The substrate’s vertical mass-motion is that axial flow; the threading vortex lines ring like plucked strings, and the sheets settle at the wavelength of the ringing. The spacing is not minimized into place — it is carved by the instability, with the stack self-tuning to sit right at the marginal point of stability.

The substrate is a continuous, self-organizing, maximally stiff superfluid — Volovik’s strong-coupling limit, with vortex cores nearly touching (\xi_\text{GP} = \xi/\sqrt{2}). That is the rotating-helium regime, not the layered-crystal one. So the GJO wavelength, not the Lawrence-Doniach saddle, is what sets d. The Lawrence-Doniach machinery is not discarded: it still supplies the dimensional crossover (the Blatter screening length \Lambda = \xi) that explains why the in-plane mathematics is two-dimensional in the first place. What changes is only which mechanism fixes the spacing. The earlier 7\;\mum value is reread as an upper bound on what the two-term energy alone could support — not the equilibrium.

The half-period and its structural echo. Because the stack alternates handedness, like-handed sheets repeat at d_\text{GJO} \approx 16\;\mum, but a boundary — a counter-rotating layer — falls every half-period, at d_\text{GJO}/2 \approx 8\;\mum. Any structure that locks onto the substrate’s boundary layers (rather than onto like-handed sheets specifically) therefore feels an \approx 8\;\mum rhythm, not a 16\;\mum one. This is the scale that recurs in the cell: red blood cells at 6–8\;\mum, capillary diameters at 5–10\;\mum, mitochondrial widths at 1–10\;\mum. Whether the half-period is a genuine pinning scale for cell-spanning structures is taken up in DNA and the Living Lattice; the point here is structural — the boundary spacing is d_\text{GJO}/2, while the full sheet period is d_\text{GJO}.

Seen in the laboratory. None of this geometry is exotic — it is the ordinary behaviour of rotating superfluids, and each piece has been imaged. The in-plane triangular array is the directly photographed Onsager–Feynman vortex lattice: first resolved in rotating He-II by Yarmchuk, Gordon and Packard,¹ reproduced in rotating Bose–Einstein condensates by Abo-Shaeer et al. ([R68]), and recently visualized at high resolution in He-II by Peretti et al., who verify Feynman’s rule n=2\Omega/\kappa directly.² The instability that sets the vertical period is just as concrete: Peretti et al. drive an axial heat flux along the array and watch the lines cross the Donnelly–Glaberson (GJO) threshold into ringing Kelvin waves — below the threshold the lattice is quiet, above it a collective wave mode appears (their Fig. 4) — exactly the marginal-stability picture d_\text{GJO} rests on. The scales even line up to the eye: at 5 rpm the imaged spacing is \delta=\sqrt{\kappa/2\Omega}\approx0.3 mm, the same order as \xi — but the laboratory \delta is tunable through the rotation rate, while the substrate’s \xi is fixed by \rho_{DM},\hbar,c, so the correspondence is one of mechanism and scale, not a numerical coincidence. The closest laboratory object to the layered, alternating-handedness stack itself is the vortex sheet of superfluid ^3He-A, which in a rotating container folds into equidistant parallel layers built from an alternating chain of opposite-circulation units.³ That analogy is structural, not literal — the ^3He-A sheet carries continuous (non-singular) vorticity bound to a topological domain wall rather than a stack of discrete singular triangular lattices — but it is the nearest real medium that packs vorticity into alternating, equidistant sheets the way the substrate’s chirality stack does.

Note

The inter-sheet spacing is one result within a larger program (WIP-15) that uses the same chirality-sheet stacking to repair the dimensional bookkeeping of the bridge equation and, ultimately, to derive the Higgs vacuum expectation value. The spacing d_\text{GJO} and the in-plane packing fraction are now closed in zero-parameter form, and the Higgs VEV has gone with them: v = \sqrt{8\pi\,m_\text{eff}^2 c^4\,\nu} = 246.1 GeV against the measured 246.22 GeV (-0.06\%), with the geometric prefactor 8\pi = 2\times 4\pi_\text{SC2} now accounted for (Higgs Field). What remains is a single inter-sheet-coupling identity — the exact emergent Lorentz invariance of bridge Step A, on which both the 4\pi and the VEV’s 8\pi depend. See Open Problems § WIP-15 for the full status.

The Vertical Cone: Light at c Across the Grain

The layering raises the sharpest question the framework’s foundation has to answer. A photon travelling across the sheets — along the stacking axis — must move at exactly the same c as one travelling within a sheet, or the substrate has a preferred direction and the emergent Lorentz invariance behind bridge Step A is only approximate. The naive estimate is alarming: a disturbance bridging one inter-sheet gap would set a vertical speed \hbar/(m_1 d_\text{GJO}) = c\,(\xi/d_\text{GJO}) \approx 6\,c. Taken at face value the vertical light cone is six times too wide.

It is not, and the first reason is a number the spacing’s closed form hands us for free. Comparing the vertical period to the vortex-core size,

\frac{d_\text{GJO}}{\xi_\text{GP}} = \sqrt{\frac{\ln 2}{4\pi}} \approx 0.235 < 1,

the sheets repeat more finely than the cores are wide — vertically the cores overlap by a factor of about four. (Both lengths trace to the same \ln(\xi/\xi_\text{GP}) = \tfrac12\ln 2 and the same 4\pi; the instability that carves d_\text{GJO} automatically places it inside a core.) The substrate is therefore not a stack of weakly coupled layers in the vertical direction — it is one continuous condensate with a sub-core density ripple. At long wavelength a single condensate has one phase stiffness; the ripple only opens Bragg gaps at the zone boundary k_z \sim \pi/d_\text{GJO}, at energy \hbar c\,\pi/d_\text{GJO} \approx 19\,m_1 c^2 — far above the substrate’s own Planck energy E_\text{Pl} = m_1 c^2. The frightening 6\,c is the dispersion’s slope at the zone boundary, not at k_z\to 0: the long-wavelength vertical speed is the homogenized bulk value, isotropic up to a correction set by the (small, sub-core) density contrast and pushed above the cutoff. Whatever residual vertical Lorentz violation survives lives above E_\text{Pl} — exactly where the effective theory stops, and exactly where GW170817’s c_\text{GW}=c to 10^{-15} already tells us nothing hides below.

That settles the dc1 phonon. The deeper object is the cone the metric is built from. The substrate’s metric is fermion-induced (the paired ³He-A reading above), so the cone that matters is the Bogoliubov–de Gennes quasiparticle’s. Stacking the 2D chiral-p-wave sheets with inter-sheet coupling t_\perp and solving the BdG problem (worked through in WIP-15), the gapless relativistic fermions do not sit at the zone centre — they sit at a Bogoliubov–Weyl node on the stacking axis at k_z^\ast = \pi/2d_\text{GJO}, the point the marginal condition \mu\to 0 selects. Near it the spectrum is a clean cone,

E(\mathbf q) = \sqrt{\,c_\perp^2\,\hbar^2 q_\perp^2 \;+\; v_z^2\,\hbar^2 q_z^2\,},\qquad c_\perp = \frac{\Delta_0}{\hbar k_F},\quad v_z = \frac{2\,t_\perp\, d_\text{GJO}}{\hbar},

with the in-plane slope fixed by the gap (the framework’s c=\Delta_0/\hbar k_F) and the vertical slope by the inter-sheet coupling. By Volovik’s Fermi-point topology this cone is exactly Lorentzian however the two slopes compare — an anisotropic cone is still a metric — so the fermion sector by itself never breaks Lorentz invariance; it only chooses an effective metric g^{\mu\nu}=\mathrm{diag}(-1,\,c_\perp^2,\,c_\perp^2,\,v_z^2).

Lorentz invariance is exact when the two cones coincide — the isotropic dc1 phonon at c and the fermion’s g^{\mu\nu}. (Two distinct cones would be birefringence, the failure mode Barceló–Liberati–Visser flag as the central obstacle to exact emergent gravity the moment a medium has more than one component; their decoupling condition is precisely its absence.) Coincidence requires v_z = c_\perp = c, that is

\boxed{\;t_\perp = \frac{\hbar c}{2\,d_\text{GJO}}\;}

— exactly the inter-sheet-coupling identity bridge Step A had been reduced to (E_J = (\xi/d_\text{GJO})\,m_1 c^2, the two forms differing only by the band-versus-Josephson convention). The BdG calculation collapses two of the framework’s open items into one: “in-plane c = vertical c” and “the substrate’s emergent Lorentz invariance is exact” are the same condition, and the marginal point is what makes it reachable — it puts the node where the vertical slope is maximal and the in-plane mode is simultaneously massless.

What forces t_\perp down to that value rather than to the naive strong-overlap one — about twelve times larger (the ratio is d_\text{GJO}/2\xi \approx 0.083)? Here the gravitational waterfall supplies the template: the substrate’s habit is to store energy in its boundaries and let the interior run free, and the vertical stack is built that way. Its handedness alternates every half period — a + sheet at 0, a counter-rotating - layer at d_\text{GJO}/2, a + sheet at d_\text{GJO} — so the nearest vertical neighbours (spacing d_\text{GJO}/2) are of opposite chirality, while same-handed sheets are next-nearest (spacing d_\text{GJO}). The naive 6c assumed a quasiparticle could hop at the finest spacing d_\text{GJO}/2. It cannot: that nearest-neighbour coupling joins the two layers’ chiral order across a mismatch of \Delta\ell = 2 (from e^{+i\phi} to e^{-i\phi}), and the in-plane-rotation-invariant coupling conserves angular momentum, so the matrix element is exactly zero — \int_0^{2\pi} e^{-2i\phi}\,d\phi = 0 (verified, scripts/chirality_hop.py). The fast channel is forbidden. This is the counter-rotating intermediate layer acting as the “energetic boundary” the chirality-carrying quasiparticle cannot cross directly — the anti-phase Cooper partner of the lattice breath. (The structureless density hop is not forbidden, but that is the dc1-phonon channel the sub-core homogenization above already pins to c; the chiral, metric-inducing channel is the one the selection rule governs.)

What survives is the same-chirality hop over the full period d_\text{GJO} — the “free interior” channel — and because that step is twice as long and gated by the in-plane core scale rather than the vertical spacing, it lands the vertical cone at order c rather than 6c (the surviving slope passes through c for the natural vertical confinement w\sim d_\text{GJO}/2; scripts/chirality_hop.py). So the selection rule does the heavy lifting — it deletes the catastrophic factor of six — in exactly the way the waterfall dissolves an apparent preferred frame by reassigning the flow to the metric: a finely layered, anisotropic-looking stack is forced toward isotropy because the channel that would have broken it is the one chirality forbids. What it does not yet do is pin the exact coefficient — whether the surviving hop is precisely \hbar c/2d_\text{GJO} (the d_\text{GJO}/2\xi target) is model-dependent in this crude treatment and needs the real chiral-p-wave Bogoliubov–de Gennes profiles, the from-scratch BdG⟷GP calculation, now reduced from “explain a factor of twelve” to “fix an O(1) coefficient.” Equivalently: boson and fermion share one dc1 condensate whose phase stiffness is pinned isotropic by the homogenization above, so self-consistency should drag the fermion cone to match. The last layer, recovering Einstein dynamics rather than merely a Lorentzian metric, is the standing one-loop-dominance question (the non-covariant induced terms must stay subdominant to \int\sqrt{-g}\,R, which close-packing’s single Planck scale is the candidate to enforce); the full status is in WIP-15 §The vertical cone.

Inner Scale: The Effective Quantum

The inner scale is fully determined by Subsystem A (the electroweak sector) with zero new free parameters. The mutual friction parameter \alpha_{mf} = 0.3008 is fixed by the measured Weinberg angle (\sin^2\theta_W = 0.2312; see Weinberg Angle). From this single input:

m_\text{eff} = \frac{m_e}{\alpha_{mf}} = 1.70\;\text{MeV}/c^2

v_\text{rot,inner} = c\sqrt{2\alpha_{mf}} = 0.776\,c

r_\text{eff} = \frac{\hbar}{m_\text{eff} \cdot v_\text{rot,inner}} = \bar{\lambda}_C^{(e)} \cdot \sqrt{\frac{\alpha_{mf}}{2}} = 150\;\text{fm}

The effective quantum carries exactly \hbar of angular momentum. It is a substrate property, not a particle property: the mass hierarchy \alpha_{mf}^{(N)}/\alpha_{mf}^{(e)} = m_p/m_e guarantees that the nuclear sector produces the same m_\text{eff} \approx 1.70 MeV/c^2.

Physically, 150 fm sits between the nuclear scale (\sim 1 fm) and the atomic scale (\sim 50{,}000 fm) — about 100 times the proton charge radius. This is the scale of the electron’s internal vortex structure, where \sim 10^9 condensed dc1 particles orbit collectively around a phase singularity.

Harmonic Structure

The three tiers are not arbitrary — they are locked together by exact harmonic relations:

• \xi / \lambda_C(m_\text{eff}) = \nu/(2\pi) — the perturbation envelope length contains exactly \nu/(2\pi) effective Compton wavelengths
• \xi / r_\text{eff} = \nu \cdot \sqrt{2\alpha_{mf}} \approx 7.4 \times 10^8 — inner radii per perturbation envelope

The condensation number \nu is not a coincidence — it is the ratio of the substrate’s collective mass scale to its constituent mass, connecting the BEC ground state to the vortex excitations it supports.

Why the Scaffold Needs No Second Species

With dc1’s self-interaction logarithmic, every job the “dag” scaffold was invented to do — set the cell, set the spacing, hold the medium together, stop it drifting on cosmological timescales — the substrate does on its own. The decisive check is the bridge equation, and it does not merely tolerate the absence of a second species: it prefers it.

The bridge equation singles out zero dag. The packing fraction f = n_1 \xi_{SC2}^3 = 4\pi/(K\sqrt{2}) = 0.5666 uses the dc1 number density. Write a hypothetical dag mass fraction f_d = n_d M_d/\rho_{DM}, so that n_1 = (1-f_d)\,\rho_{DM}/m_1; the other two ingredients of the match — \xi_\text{SC2} from the particle physics route and m_1 from the Volovik relation — carry no \rho_{DM} and are therefore dag-independent. The measured packing fraction is then strictly proportional to (1-f_d):

f_\text{pack}(f_d) = (1-f_d)\,\frac{\rho_{DM}\,c\,\xi_\text{SC2}^4}{\hbar} = (1-f_d)\times 0.5657

The published value 0.5657 is the f_d = 0 value — it uses the full \rho_{DM} as dc1. Because the measurement already sits \sim 0.16\% below the geometric prediction 0.5666, every gram of dag mass pulls it further away, monotonically: there is no f_d > 0 that improves the match, and f_d < 0 is unphysical. The closest the framework can get to 4\pi/(K\sqrt{2}) is at zero dag (machine-verified, scripts/svt_dag_removal.py). The old assumption n_d M_d \ll n_1 m_1 was never merely convenient — the bridge equation was always demanding n_d M_d \to 0.

What the old two-species ledger cost — now zero. When dag was carried as a real species, its mass fraction shifted the outer-scale quantities by powers of (1-f_d): \xi \propto (1-f_d)^{-1/4}, m_1 \propto (1-f_d)^{+1/4}, n_1 \propto (1-f_d)^{+3/4}. At the former 3\% ceiling this moved \xi by +0.76\%, m_1 by -0.76\%, and n_1 by -2.3\% — all inside the \sim 1\% Planck uncertainty on \rho_{DM}, but all in the wrong direction. With dag retired, f_d = 0 holds structurally and that entire correction vanishes. The inner scale (m_\text{eff}, r_\text{eff}, v_\text{rot}), galactic dynamics (a_0), dark energy, and S_8 were always dag-independent and are untouched. Retiring dag changes no reported number — it only deletes a downside.

The Logarithmic Equation of State

The previous section settles the bookkeeping — the bridge equation prefers a single species, and every gram of a second one only widens the gap. It does not yet say why one condensate should hold itself together where the older two-species reading needed a scaffold to do it. The answer is the shape of dc1’s self-interaction. It is not the cubic (contact) nonlinearity of a dilute laboratory gas; it is logarithmic — the equation of state that Zloshchastiev’s superfluid-vacuum theory singles out — and once that is named, every job the scaffold was invented for is done intrinsically by the medium.

Why logarithmic. Start from the one demand the framework cannot give up: a photon is the substrate’s sound mode, propagating at c_s \propto \sqrt{\rho\,F'(\rho)}, and emergent relativity requires that this speed not depend on how densely the condensate happens to be loaded. Requiring c_s to be density-independent at leading order forces \rho\,F'(\rho) = \text{const}, a differential equation whose unique solution is the logarithm. Density-independent light speed is a logarithmic self-interaction. The condensate wavefunction then obeys a logarithmic nonlinear Schrödinger equation,

i\hbar\,\partial_t\Psi = \left[-\frac{\hbar^2}{2m_1}\nabla^2 \;-\; b\,\ln\!\frac{|\Psi|^2}{\rho_c}\right]\Psi,

in place of the cubic Gross–Pitaevskii term g|\Psi|^2\Psi. This is the Bialynicki-Birula–Mycielski / Rosen equation — the one local nonlinear wave equation, besides the linear one, that keeps a composite system’s wavefunction separable into uncorrelated parts.

The coupling is an energy, not a density — and that is the whole point. Writing the coupling strength in the energy form \beta^{-1} used by Avdeenkov and Zloshchastiev, the density-independent-c condition fixes it at

\beta^{-1} = m_1 c^2,

the rest energy of a single dc1 particle. Because the coupling is a fixed energy rather than a density, the length scale it sets cannot drift as the universe dilutes. This is exactly the property a cubic substrate lacks: a Gross–Pitaevskii cell scale runs as \rho^{-1/2}, a runaway of (1+z)^{3/2} \approx 3.7\times10^4 between the CMB and today, and that drift is the one thing the heavy “dag” species was invented to pin. A logarithmic substrate has nothing to pin — its scale is the coupling, and the coupling is a constant of the medium.

The gausson: a self-binding wave packet. Released into empty space, a cubic condensate either disperses (if repulsive) or collapses (if attractive); either way it needs a trap to acquire a size. The logarithmic condensate does neither. For positive \beta its ground state is a self-bound Gaussian droplet — a gausson, a Gaussian density envelope modulated by a de Broglie plane wave (Rosen 1968),

n(r) = n_0\,e^{-(r/a_\beta)^2}, \qquad a_\beta = \hbar\sqrt{\beta/2m_1},

held together with no external potential. The mechanism is the sign-change of the logarithm: where the condensate is dilute (|\Psi|^2 < \rho_c) the log term is attractive and binds, where it is dense it turns repulsive and resists — so the medium clamps itself at a finite size from the inside (Avdeenkov & Zloshchastiev 2011). A self-binding medium needs no scaffold; the scaffold was an artifact of reading a logarithmic medium as a cubic one.

One coupling, three jobs. Substitute \beta^{-1} = m_1 c^2 into the gausson width and it collapses onto a length the framework already knows:

a_\beta = \frac{\hbar}{\sqrt{2}\,m_1 c} = \xi_\text{GP},

identically the Gross–Pitaevskii healing length. So the cell scale \xi = \sqrt{2}\,a_\beta = \hbar/(m_1 c) is the Volovik/Compton length, the bridge equation’s 1/\sqrt{2} survives with both of its readings (kinetic balance and the pairing count), and the infrared floor E_\text{min} = hc/\xi = 2\pi m_1 c^2 \approx 13 meV is reproduced exactly. The single number \beta^{-1} = m_1 c^2 does three jobs at once: it fixes the density-independent speed (exact Lorentz invariance), the self-binding cell scale, and the minimum modon energy.

Why the bridge equation does not notice the change. Swapping the cubic for the logarithm costs the bridge equation nothing — and that is structural, not luck. Expanding the logarithmic potential V_\beta = -\beta^{-1} n[\ln(n a^3) - 1] to second order about close-packing, n a^3 = 1, returns

V_\beta(n) = -\beta^{-1}\!\left[\tfrac{1}{2}a^3 n^2 - n - \tfrac{1}{2a^3}\right] + \mathcal{O}\!\big((n a^3 - 1)^3\big),

whose leading term (since n = |\Psi|^2) is precisely the quartic Ginzburg–Landau interaction the Gross–Pitaevskii equation rests on. Cubic GP is the dilute-gas leading shadow of the logarithm, and the substrate sits at the close-packing point (n_1\xi^3 \approx 1) where the two coincide to first order. Every quantity the bridge equation evaluates is computed at that point, so it is invariant under the cubic→log upgrade. The logarithm changes the mechanism under the numbers, not the numbers.

What it changes is the status of Lorentz invariance. For the cubic the sound speed carries a residual density-dependence, dc^2/d\rho = g/m_1 \neq 0, marginal at one density only; for the logarithm dc^2/d\rho \equiv 0 everywhere. The log thus upgrades exact Lorentz invariance from enforced to structural — under the old reading a scaffold had to hold \rho fixed so that c could not drift; under the logarithm c cannot drift even where \rho does.

The Marginal Point: Light at the Edge of Stability

The logarithm gives one thing the cubic cannot, and it is the deepest of the three jobs above: it tells the substrate where to sit so that light is exactly massless. The Bogoliubov spectrum of the logarithmic condensate factorizes cleanly,

\epsilon(p) = \sqrt{\left(\frac{p^2}{2m_1} - \epsilon_1\right)\left(\frac{p^2}{2m_1} - \epsilon_2\right)}, \qquad \epsilon_2 = \epsilon_1 + 2\beta^{-1},

and the low-momentum (photon-like) mode is exactly massless only at one background density n_\text{tr}, the value where \epsilon_2 = 0. The single ratio n_0/n_\text{tr} rules both the photon’s mass and the vacuum’s stability at once:

Background density	Photon mode	Vacuum
n_0 < n_\text{tr}	massive (gap \Delta = \sqrt{\epsilon_1\epsilon_2})	stable
n_0 = n_\text{tr}	massless, c_0 = 1/\sqrt{m_1\beta} = c	marginal
n_0 > n_\text{tr}	tachyonic	unstable

So n_\text{tr} is the densest stable uniform vacuum, and massless light is the same condition as marginal stability — the photon travels at a single isotropic c exactly at the edge of stability. This is the explicit, dispersion-level content of the framework’s marginal (\mu\to0) point (WIP-15 § The marginal point), now carrying the word stability.

Exact Lorentz invariance is an attractor, not a setting. A hot, dense early universe sits well above n_\text{tr}, deep in the tachyonic regime — the substrate “boils” (A Universe That Boils). The instability sheds its overdensity, and that shedding is structure formation, while the smooth remainder relaxes down onto n_\text{tr}. Exact Lorentz invariance is therefore the late-time attractor of a diluting logarithmic universe rather than a tuned input: the medium falls toward the massless-photon point on its own and parks at the edge of stability.

The fall is never quite complete. Finite cosmic expansion freezes a tiny residual super-criticality in place, and read as a photon mass that residual is the Hubble energy itself, m_\gamma c^2 \sim \hbar H_0 \approx 1.4\times10^{-33} eV — a Compton wavelength of order the horizon, some 10^{15} below the laboratory bound. The Friedmann equation converts that frozen rate into the framework’s Planck-referenced weak-gravity / small-\Lambda number (m_1/M_\text{Pl})^2; the dark-energy value stays an inherited \mathcal{O}(1) initial condition. This mechanism is developed in Gravity § The residual (the photon-mass paragraph) and across sessions/svt-*.

Roton, Maxon, and the Edge of Spacetime

Below the cell scale the substrate’s sound mode is relativistic to extraordinary precision. Above it, the induced spacetime comes apart, in the sequence Zloshchastiev’s dispersion paper works out for a logarithmic vacuum. As momentum grows the sound energy climbs to a local maximum — the maxon peak, where the group velocity squared c_p^2 = dE_p/dp turns negative and there is no classical propagation at all — then dives to the roton minimum, then rises once more until the maximum attainable speed climbs back above c_0: the luminal boom, the vacuum analogue of a sonic boom. Beyond it the quantum liquid breaks down, and a fast enough observer no longer experiences the condensate or the spacetime metric it induces. In the small-momentum Taylor expansion this same structure first shows up gently, as a momentum-dependent effective photon mass \mu(p) that slows the sound mode as p grows — a post-relativistic mass generation, distinct from a superconductor’s gap, that switches on long before any of the extrema.

Two facts keep all of this from ever being seen. First, at the marginal density the branch is soft. With \epsilon_2 = 0 the factorized spectrum collapses to

\epsilon(p) = c\,p\,\sqrt{1 + \left(\frac{p\xi}{2\hbar}\right)^2},

whose slope is exactly c at low p, whose leading deviation is quadratic, and which is monotonic — no roton, no maxon. The Landau roton–maxon form is an off-critical feature; the self-organized substrate sits at criticality, so the realized sound branch is the gentle one. Second, every Lorentz-violating feature lives at the cell scale. The maxon sits at p_a = 2\sqrt{2}\,\hbar/\xi, energy E_a \approx 5.7 meV — about 3 THz, wavelength \sim\xi \sim 100\;\mum. The only scale at which the substrate’s dispersion departs from exact Lorentz invariance at all is \hbar c/\xi = m_1 c^2 \approx 2 meV.

That single number is what settles the framework’s last decision gate. Gravitational waves ride this collective sound branch and sit \sim10^{10} below the cell scale — linear to \sim10^{-20}, comfortably under GW170817’s |c_\text{GW}/c-1| < 10^{-15}. The photon, crucially, is not this sound mode: it is a modon, a topologically protected counter-rotating dipole-vortex soliton that propagates at c with no momentum dependence. Ordinary light sits 10^3 to 10^{11} times past the cell scale, and shows none of the roton structure it would carry if it were a phonon — which is precisely why optical, X-ray, and GeV photons arrive undispersed across cosmological baselines. The same cell scale that would make a phonon-photon catastrophically Lorentz-violating leaves the modon-photon untouched. The genuine, falsifiable prediction is the far-infrared / THz crossover at \lambda \sim \xi, where a solitonic modon gives way to a collective mode (the modon→phonon crossover already flagged in Photon as Modon; full confrontation against the Lorentz-invariance data in sessions/svt-11-gate2.md). The below-cell-scale side of that crossover is now pinned by data: a +\nu^2-advance refit of 896 CHIME fast radio bursts bounds any soft-branch participation of sub-floor light at \varepsilon < 6.5\times10^{-16} (sessions/frb-nu2-advance-1.md), so the crossover changes what light is — soliton above, delocalized collective mode below — without changing how fast it travels. The observable window that remains is the crossover band itself, $$0.3–10 THz, inside the terahertz gap.

Two Readings of One Medium

The two-species “dc1–dag” picture earlier drafts carried was never wrong so much as it was the cubic reading of a logarithmic medium. Reading dc1’s self-interaction as a contact nonlinearity makes the cell scale drift, which forces a heavy second species to pin it; naming the interaction logarithmic retires the scaffold and changes no reported number. The two readings line up cleanly:

	dc1 + dag (earlier reading)	dc1 + log EOS (now)
Dark species	two — BEC + scaffold	one — dc1 only
Self-interaction	cubic (Gross–Pitaevskii)	logarithmic (Zloshchastiev)
Sets the cell \xi	GP healing length; dag pins it	gausson width = \xi_\text{GP}, self-bound
Stops cosmological drift	the dag scaffold	the coupling is an energy, not a density
Marginality / exact LI	enforced (scaffold holds \rho)	structural (dc^2/d\rho \equiv 0) + self-organized attractor
Bridge equation (0.16%)	the signature result	unchanged (GP = log at close-packing)
1/\sqrt{2}, \xi, E_\text{min}	as published	identical (gausson = healing length)
Photon mass	not addressed	floor m_\gamma \sim \hbar H_0/c^2
Free parameters	one measured (\sin^2\theta_W) + dag	one measured; dag removed

What matches between the columns — the bridge equation, the 1/\sqrt{2}, the cell scale, the infrared floor, the leading Dirac cone — matches because the cubic is the logarithm’s close-packing expansion. What changes is the column on the right’s mechanism: one species instead of two, marginality made structural and then made an attractor, and a new falsifiable handle in the photon-mass floor. The full investigation that established this — three decision gates, all concordant — lives in sessions/svt-*; its status in the live paper is tracked at Open Problems § WIP-11.

Topology as Stability

Stability comes from topology — the mathematical property that certain vortex configurations cannot unwind without cutting.

Quantized circulation in superfluid helium is stable for exactly this reason: a vortex line in He-II persists indefinitely because the circulation integral \oint \mathbf{v} \cdot d\mathbf{l} = n\kappa is quantized by the single-valuedness of the BEC wavefunction. No impurity is needed at the core. The vortex is a topological defect of the condensate — it exists because the phase of \psi winds by 2\pi n around the core, and that winding number is an integer that cannot change continuously.

The same mechanism operates in the dc1 substrate:

• The electron carries a half-integer winding (spin-½). Its vortex core is a phase singularity where the BEC wavefunction’s phase is undefined — exactly as in a superfluid vortex line. The 720° rotation property (two full rotations to restore the state) follows from the half-integer winding number, not from any classical mechanical property. This topological protection makes the electron indestructible except by annihilation with a positron — an anti-vortex with opposite winding that unwinds the singularity.

• The proton is a Borromean three-fold junction — three interlocking vortex channels that cannot be unlinked without cutting. This is the topological origin of quark confinement: pulling two channels apart stretches the counter-rotating boundary between them until enough energy accumulates to create a new channel pair (vortex reconnection), producing a meson rather than a free quark. The junction topology is the confinement mechanism.

• Photons (modons) are topologically simpler: counter-rotating dipole pairs with zero net winding. They can be created and absorbed freely because zero-winding configurations can form and dissolve without topological obstruction.

The pattern is consistent: stable particles have nontrivial topology (nonzero winding or linking number); unstable particles or radiation have trivial topology. Stability is not mechanical — it is topological.

Material Properties

The dc1 substrate has three defining material characteristics:

Near-perfect elasticity. Collisions between dc1 particles are elastic to extraordinary precision: \varepsilon = 1 - \delta, where \delta \sim 10^{-40} per collision is the “universe decay constant” (a free parameter). This tiny inelasticity is what makes the universe not quite eternal — energy leaks away at a rate far too slow to detect, but sufficient to guarantee that the substrate is not in exact equilibrium. The residual disequilibrium \delta T/T_c \sim 10^{-61.5} is what produces the observed cosmological constant (see Gravity).

Primordial angular momentum. All dc1 carries angular momentum from the creation event. At the outer scale, this manifests as a lattice rotation rate \omega_0 \approx 7.8 \times 10^9 rad/s, giving a rotation velocity v_\text{rot,outer} = \omega_0 \xi \approx 0.0025\,c. This value is not free — it is fixed in the gravity sector, via the boundary-transit fraction f_\text{cross} in G = f_\text{cross}\,v_\text{rot,outer}/(4\pi) (see Gravity; the modon dispersion does not fix it — see Emergent Speed of Light). The outer rotation velocity matches the Landau critical velocity for the CDM-to-MOND transition.

Quantum degeneracy. With the thermal de Broglie wavelength about ten times the interparticle spacing, the dc1 sea is deep in the BEC regime. The gap energy \Delta_0 greatly exceeds the Fermi energy E_F, placing the substrate in Volovik’s strong-coupling limit. This is the physical origin of Lorentz invariance: in the BEC regime, quasiparticle excitations propagate with a single isotropic speed c, regardless of direction — no anisotropy to tune away.

Energy Budget of a Topological Excitation

The energy of any particle — electron, proton, or otherwise — is entirely rotational:

m_\text{particle} \cdot c^2 = \tfrac{1}{2}\, m_\text{eff}\, v_\text{rot,inner}^2

For the electron: \tfrac{1}{2}(m_e/\alpha_{mf})(2\alpha_{mf}\,c^2) = m_e c^2. The kinetic energy of the effective quantum at peak contraction equals the full rest energy. This is one extremum of the Compton oscillation — energy shuttles between the contracted phase (maximum internal rotation at r_\text{eff}) and the expanded phase (maximum ripple in the substrate at \xi) at the Compton frequency \omega_c = m_e c^2/\hbar. Mass is not a static property; it is the time-averaged energy of a breathing vortex (see Mass as Leaking Rotational Kinetic Energy).

The boundary energy stored in counter-rotating layers dominates in heavier particles: the proton’s 938 MeV is 99% boundary energy from counter-rotating vortex sheets, with only \sim 9 MeV from bare quark orbital energies (see Proton Core).

The pattern at every tier is “dc1 flow organized into topological structures, with mass arising from the energy of circulation and the boundaries between flow regions.” The building block — the effective quantum at m_\text{eff} \approx 1.70 MeV/c^2 — is a property of the dc1 BEC, universal across all particles. The difference between an electron and a proton is not what the building blocks are, but how many are organized and how tightly the counter-rotating boundaries confine them.

Footnotes

1. Yarmchuk, E.J., Gordon, M.J.V. & Packard, R.E., “Observation of Stationary Vortex Arrays in Rotating Superfluid Helium,” Physical Review Letters 43, 214, 1979 — the first direct images of the quantized-vortex array, via electron bubbles trapped on the cores. [R123]

2. Peretti, C., Vessaire, J., Durozoy, É. & Gibert, M., “Direct visualization of the quantum vortex lattice structure, oscillations, and destabilization in rotating ^4He,” Science Advances 9(30), eadh2899, 2023. [R124]

3. Volovik, G.E., “Superfluids in rotation: Landau–Lifshitz vortex sheets vs Onsager–Feynman vortices,” arXiv:1504.00336, 2015 — fifteen figures contrasting the layered-sheet and triangular-line ways a rotating superfluid stores vorticity; folding and meron-chain structure also described by the Aalto Low-Temperature Laboratory page, Vortex sheet in superfluid ^3He-A. [R125]