HVBK Mutual Friction

Two Questions About a Boundary

Two pieces of ordinary mathematics carry this book, and each one answers a single question about a boundary in a superfluid.

What can a boundary do? Only two things — and they are exactly the two terms of the mutual friction force. When a flow meets a counter-rotating layer, the layer can take energy from it (a drag along the relative velocity: the dissipative channel, B), or it can turn the flow aside without taking any (a Hall-type deflection at right angles, which does precisely zero work: the reactive channel, B'). There is no third term in the force. Everything a boundary does to what it meets is a mixture of those two, and the mixture is one dimensionless number, \alpha_{mf} = 0.3008 — roughly thirty percent taken, seventy percent turned. That number is not fitted here; it is the measured Weinberg angle, read as a ratio.

How big can a boundary be? Only certain sizes. A circulation held inside a boundary has to oscillate on the inside and die away on the outside, and the two have to meet smoothly at the rim: Bessel J_1 within, modified Bessel K_1 without, value and slope matched at the separatrix. That is the oldest question in confined-mode physics — what fits? — and here it has no solution below one lattice cell. It fixes one more pure number, K = j_{11}^2 + 1 = 15.68, the squared wavenumber of the matched dipole mode (Bridge Equation, Photon as Modon).

One number for what a boundary does, one for how small it can be. The rest of the book is those two numbers carried to different scales.

Three Modes, One Layer

A counter-rotating wrap only ever does three things with the energy it encloses. The framework’s three headline results are those three, in order of rarity — and it is worth keeping them apart, because they are easy to run together.

It pushes back. The dissipative channel makes the wrap respond to the shape of what it wraps: it creeps down its core’s own density gradient at diffusivity D = \kappa_q/(4\pi\alpha_{mf}) = \hbar/2m. So the core never pushes on a passive wall — it pushes on a wrap that is already pushing back at the curvature of its own density. That back-pressure is the quantum potential, Q = -(\hbar^2/2m)\,\nabla^2R/R, and Planck’s constant is the receipt: \hbar = 2mD (Two Fluids → Quantum Potential).
It seeps. Read the same response function out at the lattice scale and the core is clean (\omega_0\tau \approx 400): the wrap is a near-perfect mirror, and only the reactive trickle passes — about one dc1 particle in 10^{15}. That trickle is gravity. Gravity is weak because the mirror is good (Gravity, § Gravity — the Reactive Twin).
It throws a piece of itself away. Drive the flow up to the wrap’s own circulation speed, v_\text{rot,inner} = c\sqrt{2\alpha_{mf}} = 0.776\,c, and the layer can no longer stay coherent in the lab frame. It sheds a counter-rotating pair — a modon — and that is a photon; at lightning energies, a gamma ray (Photon as Modon, Lightning).

The quantum potential is what the boundary sends back. Gravity is what it fails to hold. Light is what it throws away. One layer, two coefficients, three modes — and the Bessel match is what permits a layer of that size to exist at all.

The three are not three strengths of one leak; they are three different exits, and they scale differently. Mode 1 runs continuously on the dissipative face at the inner scale. Mode 2 runs continuously on the reactive face at the outer scale, which is why it is 10^{15} times fainter rather than merely weaker. Mode 3 does not run continuously at all — it has a threshold, and below 0.776\,c its rate is zero. A thunderstorm is not a place where the leak got bigger; it is a place where a boundary was driven past mode 3’s threshold and spent the coin instead of seeping it (Follow the Energy).

To model shear zones in a superfluid, you need a model for what happens when two vortex lines come together. Moving too fast, diffusion is not an option. Either they align and form a mutual path, or the fold into nesting eddies, a shear zone boundary with a leak fraction.

The model of the coupling between the two components of a superfluid is formalized in the Hall–Vinen–Bekarevich–Khalatnikov (HVBK) equations. It consists of a superfluid component and a normal component. Here the normal component is used as the shear zone, the counter-rotating layers meeting another balancing superfluid with oppositional energy.

After that adjustment, one dimensionless number, the mutual friction parameter \alpha_{mf} = 0.3008, threads through the quantum potential and the origin of \hbar (Two Fluids → Quantum Potential), the Weinberg angle (Weinberg Angle), the weakness of gravity (Gravity, Outer Rim Onset), and the visible fraction of each particle’s energy (Mass as Leaking Rotational Kinetic Energy).

First I’ll give an overview of HVBK mutual friction in superfluid helium, then show the adjustments made for the substrate.

Two Fluids and Their Coupling

The two-fluid model. Below the \lambda-transition at 2.17 K, liquid helium-4 behaves as an interpenetrating mixture of two fluids (Tisza 1938; Landau 1941): a superfluid component (density \rho_s, velocity \mathbf{v}_s) that flows without viscosity and carries no entropy, and a normal component (density \rho_n, velocity \mathbf{v}_n) — the gas of thermal excitations (phonons and rotons) — that behaves as an ordinary viscous fluid. The total density and momentum are

\rho = \rho_s + \rho_n, \qquad \mathbf{j} = \rho_s\mathbf{v}_s + \rho_n\mathbf{v}_n,

with \rho_n \to 0 as T \to 0 and \rho_s \to 0 at the transition. Note that the two components support two velocity fields at the same point. The same reason He-II has a second sound mode (a temperature wave in which the components oscillate in counterflow).

Quantized vortices. The superfluid component is a coherent quantum state, so its circulation is quantized in units of \kappa = h/m_{\text{He}} (Onsager 1949; Feynman 1955). A rotating bucket of He-II does not rotate rigidly; it threads itself with an array of quantized vortex lines, each a thin core around which the superfluid circulates with exactly one quantum \kappa.

Mutual friction. Hall and Vinen (1956) discovered, via the attenuation of second sound in rotating helium, that the two components are not independent: the normal fluid’s excitations scatter off the vortex lines, coupling the two velocity fields. This leads to HVBK’s two-fluid Euler/Navier-Stokes pair plus a vortex line-tension force plus the mutual friction force. In Sonin’s notation (his Eq. 6.32), the friction force per unit volume on the superfluid is:

\mathbf{f}_{fr} = -\rho_s\,\alpha\;\hat{\mathbf{s}} \times \bigl[\tilde{\boldsymbol{\omega}} \times (\mathbf{v}_n - \mathbf{v}_{sl})\bigr] \;-\; \rho_s\,\alpha'\;\tilde{\boldsymbol{\omega}} \times (\mathbf{v}_n - \mathbf{v}_{sl}),

where \tilde{\boldsymbol{\omega}} = \kappa n_v\,\hat{\mathbf{s}} is the coarse-grained vorticity (n_v = areal density of vortex lines, \hat{\mathbf{s}} = unit vector along them), \mathbf{v}_{sl} is the local superfluid velocity at the lines, and \alpha, \alpha' are the two dimensionless mutual friction parameters. The equivalent statement for the motion of the vortex lines themselves (Sonin Eq. 6.33, the “Schwarz form” used throughout the vortex-dynamics literature) is

\mathbf{v}_L = \mathbf{v}_{sl} + \alpha'\,(\mathbf{v}_n - \mathbf{v}_{sl}) + \alpha\;\hat{\mathbf{s}} \times (\mathbf{v}_n - \mathbf{v}_{sl}),

closed by the Magnus force balance \mathbf{f}_{fr} = -\rho_s\,\tilde{\boldsymbol{\omega}} \times (\mathbf{v}_L - \mathbf{v}_{sl}) (Sonin Eq. 6.30). Hall and Vinen’s original coefficients B, B' are related to \alpha, \alpha' by (Sonin Eq. 6.35):

\alpha = \frac{\rho_n}{2\rho}\,B, \qquad \alpha' = \frac{\rho_n}{2\rho}\,B'.

In the B/B' notation, the mutual friction force is:

\mathbf{F}_{ns} = \frac{B\,\rho_n\,\rho_s}{2\rho}\;\hat{\mathbf{s}} \times \bigl[\hat{\mathbf{s}} \times (\mathbf{v}_n - \mathbf{v}_s - \mathbf{v}_L)\bigr] \;+\; \frac{B'\,\rho_n\,\rho_s}{2\rho}\;\hat{\mathbf{s}} \times (\mathbf{v}_n - \mathbf{v}_s - \mathbf{v}_L),

per unit length of vortex line and per unit of coarse-grained vorticity. Other chapters (Two Fluids → Quantum Potential, Weinberg Angle, London Equations from HVBK) drop \mathbf{v}_L, use the quasi-static limit in which the vortex lines co-move.

The two channels. The force has exactly two pieces, and they do physically different things:

The B (or \alpha) term is dissipative — a drag along the relative velocity that transfers energy between the components and damps counterflow. This is what attenuates second sound.
The B' (or \alpha') term is reactive — a Hall-type force perpendicular to the relative velocity. It deflects flow without doing work: \hat{\mathbf{s}} \times (\mathbf{v}_n - \mathbf{v}_s) is perpendicular to the relative velocity, so this channel transfers zero energy. It acts like a gyroscope, not like friction.

What is measured. B(T) and B'(T) are tabulated experimental quantities in He-II (the standard compilation is Barenghi, Donnelly and Vinen 1983; Donnelly 1991). The convenient single-number summary is the dimensionless dissipative parameter \alpha: it runs from near 0 at low temperature (few excitations to scatter) toward \sim 1 near the \lambda-point (maximal two-fluid coupling), passing through \alpha \approx 0.3 at T/T_\lambda \approx 0.6 — squarely inside the robust two-fluid regime. In superfluid He-3 the coefficients depend on phase, temperature, pressure, and field, with \alpha \sim 0.1–1 typical. Keep the value 0.3 in mind; it returns below.

Note

Symbol guide. Three unrelated symbols collide across the literature and this book. (1) The HVBK dissipative coefficient B has nothing to do with the crust-profile fit parameter B \approx 0.25 in the DESI dark-energy chapters (Bridge Equation, The Universe That Boils). (2) The mutual friction parameter \alpha_{mf} (this chapter) is not the fine-structure constant \alpha_{\text{em}} \approx 1/137 (Fine Structure Constant). (3) Sonin’s \alpha is the same quantity this book calls \alpha_{mf}.

Where the Coefficients Come From: Scattering off a Vortex

The HVBK equations are phenomenological — B and B' enter as coefficients. Their microscopic origin (Sonin Ch. 8 for He-II; Ch. 9 for Fermi superfluids) is quasiparticle scattering off vortex lines, and the two channels of the force correspond directly to the two things a scatterer can do to an incident flux:

Longitudinal drag (dissipative, feeds B): the transport cross-section \sigma_\parallel for momentum transfer along the incident direction.
Transverse deflection (reactive, feeds B'): the asymmetric cross-section \sigma_\perp from the vortex’s circulating velocity field, which deflects quasiparticles preferentially to one side. This is an Aharonov–Bohm effect: a phonon passing the vortex on one side accumulates a different phase than one passing on the other, because the circulation \kappa plays the role of the flux tube (Sonin 1975; Sonin §8.5).

Averaging these cross-sections over the thermal quasiparticle distribution (Sonin Eqs. 8.38–8.39) gives the single-vortex friction coefficients. Two closed-form results anchor the low-temperature limit: the transverse coefficient is exactly D' = -\kappa\rho_n (the Iordanskii force, Sonin Eq. 8.41), and neglecting the small longitudinal drag the vortex then moves with the mass-current velocity (Sonin Eq. 8.42),

\mathbf{v}_L = \frac{\rho_s}{\rho}\,\mathbf{v}_{sl} + \frac{\rho_n}{\rho}\,\mathbf{v}_{nl},

which Sonin calls Helmholtz’s theorem for two-fluid hydrodynamics — the vortex is carried by the total momentum flux of both fluids.

Phase shifts. In partial-wave language the cross-sections are set by the scattering phase shifts \delta_l of quasiparticles on the vortex (Cleary’s formulas, Sonin Eqs. 8.81–8.82):

\sigma_\perp = \frac{1}{k}\sum_l \sin(2\delta_l - 2\delta_{l+1}), \qquad \sigma_\parallel = \frac{1}{k}\sum_l \bigl[1 - \cos(2\delta_l - 2\delta_{l-1})\bigr].

For a Fermi superfluid, the scattering is dominated by the bound quasiparticle states inside the vortex core (Caroli–de Gennes–Matricon states, spaced by \omega_0) with lifetime \tau set by impurity or quasiparticle scattering. Kopnin’s kinetic theory of these core states (Kopnin and Kravtsov 1976; Kopnin 2001, Ch. 14; Sonin §9.7) collapses both friction coefficients onto the single dimensionless control parameter \omega_0\tau, in Breit–Wigner (resonance) form:

d = \frac{\omega_0\tau}{1 + (\omega_0\tau)^2} = \tfrac{1}{2}\sin 2\delta_0 \quad (\text{dissipative}), \qquad 1 - d' = \frac{1}{1 + (\omega_0\tau)^2} = \sin^2\delta_0 \quad (\text{reactive / spectral flow}),

with \delta_0 = \operatorname{arccot}(\omega_0\tau) the effective s-wave phase shift. The dissipative channel is maximal (d = \tfrac12) at the resonance \omega_0\tau = 1 and vanishes in both the dirty (\omega_0\tau \to 0) and clean (\omega_0\tau \to \infty) limits; the reactive spectral-flow channel is fully open in the dirty limit and fully blocked in the clean limit. These are the two faces of one response function — a fact the gravity section below leans on.

Important

Provenance of \alpha_{mf} = \tfrac{1}{2}\sin 2\delta_0. This compact formula, used throughout the book, is the Kopnin vortex-core resonance form from Fermi-superfluid theory (Kopnin 2001; Sonin §9.7), not a formula that appears literally in Sonin’s He-II chapter. Sonin’s boson-superfluid treatment expresses the same physics through the Cleary partial-wave cross-sections (Eqs. 8.81–8.82) plus thermal averaging (Eqs. 8.38–8.41). The book’s shorthand attribution “Iordanskii–Sonin–Stone” (References R10) names the phase-shift formalism; the specific \tfrac{1}{2}\sin 2\delta_0 closure is Kopnin’s. A superfluid physicist checking sources will find the pieces in those two places, not in one.

Everything above this line is established physics. The substrate framework begins here.

The Substrate Identification, Stated Once

The framework’s claim (Substrate Particles, Two Fluids → Quantum Potential) is that the vacuum is a two-component quantum fluid of dc1 particles, and that every stable particle is an orbital system — a co-rotating interior wrapped in a counter-rotating boundary layer. The HVBK machinery then applies with the following dictionary:

HVBK quantity	Substrate identification
Superfluid component (\mathbf{v}_s)	Co-rotating vortex flow of dc1 — the coherent “particle” interior
Normal component (\mathbf{v}_n)	Counter-rotating boundary eddies — the diffusive, excitation-carrying layer
\hat{\mathbf{s}} (vortex line direction)	Axis of each orbital system
\mathbf{v}_L (vortex line velocity)	Drift velocity of the orbital system complexes
B / dissipative channel	Energy transfer across the boundary → hypercharge coupling g'
B' / reactive channel	Deflection without energy transfer → weak isospin coupling g
\alpha (Sonin)	\alpha_{mf} — the substrate’s mutual friction parameter

The load-bearing content is not the labels but the structure: a coherent bulk flow and a boundary excitation layer, coupled by exactly two channels — one that dissipates, one that deflects — acting on their relative velocity, with the coupling strength characterized by one dimensionless parameter,

\alpha_{mf} = \frac{\text{dissipative response}}{\text{reactive response}} = 0.3008,

fixed empirically by the Weinberg angle (next section). Roughly 30% of each boundary interaction transfers energy; 70% deflects the flow. As noted above, real He-II passes through exactly this coupling regime at T/T_\lambda \approx 0.6 — the substrate value sits comfortably inside the physical range of laboratory superfluids, not at some exotic extreme.

The identification answers a question that has been asked from the other side. Volovik’s 2024–2026 de Sitter papers ([R154]) argue on thermodynamic grounds that the vacuum is a two-fluid medium — a superfluid component playing dark energy, and a “normal” component that carries all the vacuum’s entropy — and then state three open slots in print: the microscopic identity of the normal component is “an open question,” “we do not know what are the ‘atoms of the vacuum’,” and the energy-exchange dynamics between his two components “must be supported by microscopic theory,” which he has only at the phenomenological level. Those are, respectively, this chapter’s counter-rotating boundary layer, the dc1 particle, and the HVBK mutual friction force itself — with the coupling not free but measured, \alpha_{mf} = 0.3008 from the Weinberg angle. The substrate’s dictionary above was written to map laboratory helium onto the vacuum; it happens also to be, line for line, the microscopic filling of the two-fluid vacuum Volovik’s thermodynamics requires but does not supply.

Note

Labeling convention. The assignment above — coherent co-rotating bulk = superfluid-like component, counter-rotating excitation-carrying boundary = normal-like component — is the book’s convention, chosen to match the laboratory system directly: in He-II the coherent condensate is the superfluid and the excitation gas is the normal fluid (Superfluid Helium), and in the Simeonov decomposition the counter-rotating layer is the diffusive (osmotic) one, exactly as a normal component should be (Two Fluids → Quantum Potential). The mutual friction force itself depends only on the relative velocity of the two components, so no downstream result hinges on the labels — but chapters should use this assignment consistently.

Consumer 1: The Quantum Potential and the Origin of \hbar

The full derivation is in Two Fluids → Quantum Potential; the mutual friction content is this. Simeonov’s two-fluid decomposition reproduces the quantum potential Q = -(\hbar^2/2m)\,\nabla^2 R/R exactly, provided the second fluid responds to density gradients of the first with osmotic velocity \mathbf{v}_2 = -D\,\nabla(\ln\rho_1), where D = \hbar/(2m). The substrate supplies D from vortex physics: for a superfluid with circulation quantum \kappa_q = h/m_\text{eff}, the effective diffusivity of vortex-mediated transport is

D_{sf} = \frac{\kappa_q}{4\pi\,\alpha_{mf}},

and setting D_{sf} = \hbar/(2m) yields the mass relation (Constraint C2)

\boxed{m_\text{eff}\cdot\alpha_{mf} = m.}

Planck’s constant is then not fundamental but composite: \hbar = 2mD, the particle mass times twice the diffusivity of its own boundary layer. The dissipative channel is what makes the boundary layer respond to density curvature at all — with \alpha_{mf} = 0 there would be no reaction force, no quantum potential, and no wave behavior.

Consumer 2: The Weinberg Angle

The full derivation is in Weinberg Angle; the mutual friction content is this. The two HVBK channels map onto the two electroweak gauge couplings:

g^2 \propto \text{reactive response } (B'), \qquad g'^2 \propto \text{dissipative response } (B),

because the SU(2)_L weak interaction flips states without dissipating energy (a gyroscopic deflection) while the U(1)_Y hypercharge interaction exchanges energy across the fermion boundary (a drag). Since the dissipative response is the reactive response scaled by \alpha_{mf},

\tan^2\theta_W = \frac{g'^2}{g^2} = \alpha_{mf} \qquad\Longrightarrow\qquad \sin^2\theta_W = \frac{\alpha_{mf}}{1 + \alpha_{mf}}.

Run backwards with the measured \sin^2\theta_W = 0.2312, this defines the substrate’s operating point: \alpha_{mf} = 0.2312/0.7688 = 0.30078. Through the Kopnin form this pins the core parameters \delta_0 = 18.48° and \omega_0\tau = \cot\delta_0 = 2.99 — a moderately clean vortex core, three bound-state oscillations per scattering time — which then cascade into the fine-structure constant and (g-2)/2 with no new parameters (Fine Structure Constant).

Consumer 3: Gravity — the Reactive Twin

Gravity derives Newton’s constant from a boundary-transit fraction: a tiny fraction f_\text{cross} \approx 1.1\times10^{-15} of dc1 particles leaks through each counter-rotating boundary, and G = f_\text{cross}\,v_\text{rot,outer}/4\pi. On its face that chapter never mentions mutual friction — yet several chapters call gravity “the same mutual friction mechanism.” The link is the Kopnin response function above, and it is made precise in Outer Rim Onset § Route A:

\underbrace{\alpha_{mf} = \tfrac{1}{2}\sin 2\delta_0}_{\text{dissipative face}} \qquad \underbrace{f_\text{cross} \mathrel{\widehat{=}}\ \mathcal{C} = \sin^2\delta_0}_{\text{reactive / spectral-flow face}}

One response function, two coefficients, read at two scales. At the inner (Compton) scale the boundary sits near the core resonance (\omega_0\tau = 2.99), the dissipative face is large, and its value \alpha_{mf} = 0.3008 sets the electron’s mass and the Weinberg angle. At the outer (lattice) scale the substrate sits deep in the clean limit (\omega_0\tau \approx 400), where the spectral-flow channel is almost completely blocked: \mathcal{C} \approx (\omega_0\tau)^{-2} \sim 10^{-6}, projected to f_\text{cross}\sim10^{-15} in 3D. Gravity is weak because the substrate is clean — the boundary is a nearly perfect mirror, and only the trickle of momentum that flows coherently through it (the spectral-flow fraction, the reactive face) survives as the gravitational leak. The electron’s mass and Newton’s constant are, in this reading, the two coefficients of a single vortex-friction function evaluated at opposite ends of its resonance curve.

Consumer 4: The Visible Mass Ratio

The full account is in Mass as Leaking Rotational Kinetic Energy; the mutual friction content is that \alpha_{mf} appears twice in the visibility budget:

m_e c^2 = \tfrac{1}{2}\,m_\text{eff}\,v_\text{rot,inner}^2, \qquad m_\text{eff} = \frac{m_e}{\alpha_{mf}}, \qquad v_\text{rot,inner} = c\sqrt{2\alpha_{mf}}.

Once because only the fraction \alpha_{mf} of the effective quantum’s energy reads out as observable mass (m_e = \alpha_{mf}\,m_\text{eff} — the electron shows 30% of the substrate structure that constitutes it), and once because the internal orbital speed is itself set by the coupling (v^2 = 2\alpha_{mf}c^2 \Rightarrow v = 0.776\,c). Mass is a two-sided coupling: energy must leak out across the boundary, and a probe’s energy must couple in across the same boundary, and both crossings are governed by the same dissipative channel. The proton-to-electron mass ratio 1836 is then not a stronger coupling but a seam count: the proton presents N \approx 1836 boundary seams, each with the same per-seam \alpha_{mf} = 0.3008 (Proton Core).

Note

Two “visible ratios” — do not conflate. The particle-scale visibility ratio above runs on \alpha_{mf}. The cosmological dark-to-visible ratio \Omega_\text{DM}/\Omega_b = 5.37 (The Quiet Majority) is a different mechanism entirely — its inputs are the baryon asymmetry \eta_B, the mass ratio m_1/m_p, and the boil invariant n_1/n_\gamma, and \alpha_{mf} does not enter it. Both are called “visible mass” arguments in casual summaries; they share a theme (most of what exists is hidden), not a derivation.

Two Limiting Cases

The superconductor: B \to 0. In a superconductor the Cooper-pair condensate suppresses the dissipative channel entirely — the pair’s even boundary parity presents no chirality mismatch for the B channel to grab — leaving a purely reactive (B') response. The result is the two London equations: frictionless acceleration under \mathbf{E} (first) and the Meissner screening response (second). The Meissner effect is the B = 0 limiting case of the same two-channel physics whose B/B' = 0.3008 operating point gives the Weinberg angle. Full derivation: London Equations from HVBK.

Persistence in a dissipative medium: the balance has a formal template. The framework’s standing claim that photons and modons persist indefinitely even though the medium has a dissipative channel — the balanced-boil / stealth-vacuum picture — now has a rigorous container in Zloshchastiev’s open-systems generalization of the Schrödinger equation ([R152]). His norm-conserving equation i\hbar\,\partial_t\Psi = \hat H_+\Psi - i(\hat\Gamma - \langle\hat\Gamma\rangle)\Psi formalizes exactly the distinction the substrate needs — sustainable (gain–loss balanced, normalized) versus decaying (non-normalized) states in one dissipative medium — and exhibits a regime in which the non-Hermitian and Lindblad dissipation channels cancel exactly, leaving decay-free oscillation. This chapter names the two channels; his formalism names the cancellation condition. Cited as the formal template, not yet built on.

The crust coupling: 2\alpha_{mf}^2 — asserted, not yet derived. The structure-growth suppression in the DESI analysis uses an efficiency \eta_\text{crust} = 2\alpha_{mf}^2 = 0.181 (Bridge Equation, Galactic Dynamics). The \alpha_{mf}^2 is a two-step coupling — crust energy couples into the boundary through mutual friction, and the coupled energy then disrupts the boundary’s gravitational response through the same channel. The factor of 2 is asserted as “dissipative plus reactive components at the substrate’s operating point,” which would require the reactive channel to contribute equally to the disruption efficiency — plausible for a channel that redirects the same momentum flux, but nowhere shown. This is an honest open item: deriving that factor of 2 from the HVBK force above (or replacing it) is a well-posed exercise that this chapter’s machinery makes concrete.

What Is Borrowed and What Is New

Element	Status
Two-fluid model, quantized vortices, HVBK equations, mutual friction force, B/B' ↔︎ \alpha/\alpha'	Textbook (Hall–Vinen 1956; Bekarevich–Khalatnikov 1961; Sonin Chs. 3, 6, 8)
Microscopic origin: quasiparticle–vortex scattering, Iordanskii force, phase-shift formulas	Textbook (Sonin Ch. 8)
Kopnin core-state resonance d = \tfrac12\sin2\delta_0, spectral-flow fraction \sin^2\delta_0	Textbook (Kopnin 2001; Sonin Ch. 9)
Substrate dictionary (co-/counter-rotating components, \hat{\mathbf{s}}, \mathbf{v}_L)	Interpretive identification — the framework’s central move
D_{sf} = \kappa_q/4\pi\alpha_{mf}, hence m_\text{eff}\,\alpha_{mf} = m and composite \hbar	New (framework result, C2)
B' \to g, B \to g', hence \tan^2\theta_W = \alpha_{mf}	New (framework result, C8)
f_\text{cross} as the reactive/spectral-flow twin of \alpha_{mf}	New, partially open (Outer Rim Onset, WIP-15)
Visibility ratio m_e = \alpha_{mf}\,m_\text{eff}; 1836 as seam count	New (Mass as Leaking Rotational Kinetic Energy)
Factor 2 in \eta_\text{crust} = 2\alpha_{mf}^2	Asserted, underived (open item above)

Reading List

For verification of every borrowed equation, in order of usefulness:

E.B. Sonin, Dynamics of Quantised Vortices in Superfluids, CUP 2016. Ch. 6 §6.1 (two-fluid HVBK equations; mutual friction force Eq. 6.32; Schwarz form 6.33; B/B' bridge 6.35; Magnus closure 6.30); Ch. 8 (microscopic origin: cross-sections 8.38–8.41, Iordanskii force §8.4, phase shifts 8.81–8.87, temperature dependence §8.7–8.8); Ch. 9 §9.6–9.7 (vortex-core bound states, Kopnin–Kravtsov force); Ch. 3 §3.4 (the T=0 HVBK equations and the naming). Chapter scans are archived in this repository under papers/sonin/.
N.B. Kopnin, Theory of Nonequilibrium Superconductivity, Oxford 2001, Ch. 14 — the \omega_0\tau resonance form of the friction coefficients.
W.F. Vinen and H.E. Hall, Proc. R. Soc. A 238, 204 & 215 (1956) — the discovery papers.
I.L. Bekarevich and I.M. Khalatnikov, Sov. Phys. JETP 13, 643 (1961) — the coarse-grained equations.
C.F. Barenghi, R.J. Donnelly, W.F. Vinen, J. Low Temp. Phys. 52, 189 (1983) — measured B(T), B'(T); R.J. Donnelly, Quantized Vortices in Helium II, CUP 1991 — the standard monograph.

See also References entries R9, R10, R32 for how these sources map onto the framework’s constraint system.