Unifying Equations
HVBK mutual friction, a ~2 meV logarithmic BEC with quantized vortices at the speed of light
1Mass is a leak · E = mc2 from flywheel physicsWeinberg angle → mutual friction

\underbrace{\tfrac{1}{2}\,m_\text{eff}\,v_\text{rot,inner}^2}_{\substack{\text{the vortex's full spin energy}\\[1pt] \text{at the inner speed}}} \;=\; E \;=\; \underbrace{m_e\,c^2}_{\substack{\text{Einstein's rest energy}\\[1pt] \text{at the rim speed}}}

\alpha_{mf} = \tan^2\theta_W = 0.3008

the visible mass fraction

m_e = \alpha_{mf}\,m_\text{eff} \qquad\qquad {v_\text{rot,inner} = \sqrt{2\alpha_{mf}}\;c = 0.776\,c}

\tfrac{1}{2}\,m_\text{eff}\big(2\alpha_{mf}\,c^2\big) \;=\; \big(\alpha_{mf}\,m_\text{eff}\big)\,c^2 \;=\; \boxed{m_e\,c^2}

a counter-rotating seam hides 70% of the vortex energy; the visible 30%, at the rim speed c, is the rest energy — mass is what leaks through
plain kinetic energy of a vortex at 0.776 c, no relativity assumed — and E = mc2 falls out
2The quantum potential is the reaction force

Q = -\frac{\hbar^2}{2m}\,\frac{\nabla^2 R}{R}\,, \qquad R = \sqrt{\rho}

\nabla \cdot \mathbf{F}_{ns} \;\propto\; \nabla^2\!\left(\frac{\nabla^2 R}{R}\right)

HVBK mutual friction in steady state — integrating back gives Q as the reaction force

\text{\footnotesize vortex diffusivity}\quad D = \frac{\kappa_q}{4\pi\alpha_{mf}} \quad \text{\footnotesize set equal to} \quad \frac{\hbar}{2m}\,:

\hbar = 2mD \;\;\Longrightarrow\;\; \mathbf{m_\text{eff}\,\alpha_{mf} = m}

Simeonov’s two fluids are HVBK’s co- and counter-rotating components; Planck’s constant is one seam’s diffusivity, the wrap for single vortex particles like the electron
3Gravity is the ebbing leak that falls

v_\text{ebb} = \sqrt{\frac{2GM}{r}} \;\;\Rightarrow\;\; ds^2 = -c^2 dt^2 + \big(dr - v_\text{ebb}\,dt\big)^2 + r^2 d\Omega^2

Schwarzschild, exactly — space as a river - the rare leak through the stack of boundary layers

f_\text{cross} = \frac{4\pi G}{v_L} \approx 10^{-15}

weight is the ebbing leak, the weak nature of gravity, one boundary crossing per quadrillion

a_0 = c\sqrt{G\rho_\text{DM}} = 1.16\times10^{-10}\ \text{m/s}^2

the MOND acceleration scale from the substrate density

\sqrt{\rho_\Lambda/\rho_\text{Pl}} \;\approx\; \big(\ell_\text{Pl}/\xi\big)^2 \;\approx\; 10^{-61.5}

the cosmological-constant number and weak gravity are the same geometric ratio

the metric, MOND, and Λ from one bookkeeping: substrate falling through counter-rotating boundaries
4The vacuum’s superfluid lattice

{\color{#184d85} c} = \frac{\hbar}{m_1\,\xi}, \qquad \xi = \frac{\hbar}{m_1\,{c}}

Volovik’s quasiparticle speed, equivalently \xi is the vacuum particle’s Compton wavelength

\xi = \left(\frac{\hbar}{\rho_\text{DM}\,c}\right)^{\!1/4} f^{\,1/4} \approx 97\ \mu\text{m}, \qquad f = \frac{4\pi}{K\sqrt{2}} = 0.5666

Planck’s dark-matter density using the vortex to lattice cell occupancy fraction finds the lattice cell size

m_1 c^2 = \hbar c/\xi \approx 2.0\ \text{meV}

the cell size as the vacuum particle’s Compton wavelength shows its visible mass

\nu = m_\text{eff}/m_1 = 8.35\times10^8

the very large effective mass ratio finds the measured Higgs VEV to ~0.04% — the vacuum’s energy expressed
it hides with seams of counter-rotating layers, anti-phase oscillating pairs, nesting eddies of vortex lines
From HVBK mutual friction, constants and standard observations. mass’s leak \alpha_{mf} (dissipative) and gravity’s leak f_\text{cross} (reactive) are two coefficients of one vortex response — both from \hbar = 2mD. One particle (dc1, ~2 meV) and its condensation number (\nu \approx 8.35\times10^8), the ratio of hidden to visible mass energy and the number of dc1 particles in one effective quantum. Along with the expected predictions, these results find an energetic topology that influences boundaries at all scales.