A Universe of Vortices
Eight domains of physics from one superfluid - with zero adjustable parameters
Status: This site is a work in progress. The paper is the best place to start for the math.
The Elephant in the Room
Science does not come easy, and there’s an unbelievable amount of content here for a new unproven theory, a side project. Here’s how this happened. It started with an understanding about the energy a superfluid exhibits in shear zones - counter-spinning layers. After that, one additional conjecture: the photon is a modon in a superfluid of vortices, spinning at close to c that fills the universe. From there, the model snowballed with the speed Claude allows when you have a new fundamental idea that can be applied to existing research.
I’m a curious old programmer who has spent decades reading to understand more, and stumbled on this by looking for evidence that the universe is not weird. If you are a scientist, you know that Quantum Field Theory and General Relativity are not intuitive or connected, the Hubble and S_8 tensions, the Cosmological Constant Problem. These results suggest there’s an intuitive understand that you might find interesting and predicts the expected constants. The paper has what I believe are the solid parts. This site paints a more expansive, still fuzzy picture.
For a long time, I was sure the lattice size and energy of the vortices were way too large. My original model was “dark carbon buckeyballs” - hence the name “dc1” for the superfluid - but instead of wavering, the model crystallized and keeps making more and more sense.
Still, this is a side project and I have a lot more research to validate and prose to clean up.
Exploring these topics helped build the richer picture. The substrate fits in perfectly, and ties up so many loose ends in science. In hindsight, it’s not surprising that the discovery of a highly energetic substrate below it all explains so much. While this is still a broad brush, my goal is to sharpen it. If you are skeptical, wait till it’s refined, but if you are curious I could use your help and it feels to me like it’s a science gold mine right now.
About This Project
This is a compilation of the work of many great scientists. I’m a 61-year-old Engineering Fellow at Posit PBC — a code-first data science company that supports open source. I’m pretty good at software, but a novice when it comes to physics, and this is an open source side project.
In late June 2025, after seeing one of the first Vera Rubin photos on a hi-res monitor, I had a realization that dark matter had to be dark material vortices and orbital systems spinning at close to the speed of light. They had to be in the atom and in space. In my mind’s eye, I saw why it has to be this way. The universe is not weird, and there’s an explanation for cellular dynamics, the Gulf Stream, magnetic fields.
I eventually found these five amazing theories that paved the last mile to the math that realized that vision:
- Pilot wave hydrodynamics by Bush, Oza, et al.1
- Volovik’s theories from “The Universe in a Helium Droplet”2
- Simeonov’s mapping of the Madelung equations as a fluid form of the Schrödinger equation3
- Khoury’s papers on Dark Matter Superfluidity4
- Larichev & Reznik, Saffman, Aftalion, Sonin, Blatter, Fetter, El & Hoefer for the math behind modons, vortices, and shock waves5
From this foundation I built the substrate framework, based on the idea that there is an active substrate some particle small enough it would form superfluid vortices holding enormous energy, forming counter-spinning layers at shear boundaries. That model explains the dynamics of the atom, an electron racetrack carved in a superfluid, and how gravity flows in space — leaking through boundaries, then a waterfall. The photon is a modon travelling in this substrate, a dipole vortex of opposite spinning energy that self-propels with a least energy path in the substrate, with no effective mass because the opposite spinning momentum of the substrate balances.
I refined this model till the equations first bridged particle physics with general relativity, then predicted JWST’s “impossibly early” galaxies, then DESI 2 (dark energy density), then the S_8 tension (clumpiness), then the Hubble tension (how energy expands) matched by the shape of the dispersive shock wave formed when Our Big Bubble hit the moraine crust from previous big bubbles at supersonic speed.
Beyond the fact that the substrate model makes more sense than Quantum Field Theory and General Relativity, it also makes the right accurate predictions adding evidence that we live in a fluid universe, with fluid atoms. The hidden counter-spinning layer, a rapids of vortices spinning at \approx 0.776c, removes the uncertainty behind the mysterious Q and explains open mysteries of atoms, space, geology, and biology.
Why you should keep reading
Are you curious how the substrate framework explains how the universe began, how Gaia evolved, how the Moon’s tidal lock helped it along, how the Solar System connects us to the outside, or how it shapes the Sun?
How about understandable explanations for mass, electrons, photons, quantum potential, super conductivity, crystals, higgs field, spin statistics, quarks, gluons all based on fluid dynamics?
See how the substrate influences geology and the elements.
When applied across the board, it lands with explanations. This energetic superfluid has evaded our detection, but the math is clear with predictions like the Higgs VEV, the fine structure constant, MOND acceleration, and cosmic coincidence.
The story that emerges is compelling too: the recurrent feedback topology, topological modons, and the ladder form a grammar, and the story fills in the gaps for cells, perception, brain, and how the substrate allows chemistry to hold information, and how that helps organize the mind.
These results connect through a single chain. The equation computes the lattice cell size in the substrate in two ways. From the top down it uses the number of particles per unit volume and the model for gravity. And from the bottom up it finds the same value using the mass of the electron with the angle that measures how electromagnetism interacts with the weak nuclear force (the Weinberg angle).
Those same parameters also predict the mysterious pitch angle of DNA and why microtubules have 13, not 14 protofilaments.
And to add three more domains, using that same angle, the substrate model predicts the location and shape of the moraine crust from previous Big Bubbles (\mathcal{B}^{-n}), and adds a solid prediction for the cosmological constant.
One chain of equations:
\begin{aligned} \sin^2\theta_W &\;\xrightarrow{\text{scattering}}\; \alpha_{mf} \;\xrightarrow{\text{bridge}}\; \rho_\text{DM} \;\xrightarrow{a_0}\; \text{galactic dynamics} \;\xrightarrow{f(z)}\; \text{dark energy} \\ &\;\xrightarrow{S_8}\; \text{structure formation} \;\xrightarrow{DNA}\; \text{molecular biology} \end{aligned}
All connected through one superfluid, with one geometric backbone (f = 4\pi/(K\sqrt{2})) and one dynamical thread (\alpha_{mf}). Zero adjustable parameters, off by 1-3%. A mathematical model that explains the Hubble tension, recent data from DESI 2, the S_8 tension, tidal dwarf galaxies, the bullet cluster, Maisie’s galaxy’s unusual redshift, the helical twist angle in DNA, and the wall geometry of the 13-protofilament microtubule.
One mind-bending result is the size of the lattice cell — \xi \approx 100\;\mum, about the width of a human hair. It is the wake distance of a photon/modon in the dark matter particle vortex sea. The massive scale difference between the moving object and its wake creates a distorted, confusing view of the lattice from the outside. The lattice’s vortex cores spin at roughly three-quarters the speed of light, which is what makes the substrate stiff enough to propagate waves cleanly across those hundred-micron cells. The cells breathe in pairs, joined by an opposite-spinning core, organized in layers ~16 μm apart, with a counter-spinning layer in the middle at ~8 μm, forming a bi-directional transport channel, a median strip in the substrate highway. The opposite phase breathing gives the lattice a scale invariant ladder for an in-sync latch at the half-octave, or an out-of-sync anti-latch at the golden-ratio. The lattice is an organizational scaffolding that shows up throughout.
As I dig deeper into the substrate framework, it continues to find ways to use existing science to expand our understanding.
What it predicts
See all of the predictions across particle physics, cosmology, chemistry, geology, and biology, sorted into three tiers.
Free parameters: The Standard Model + ΛCDM requires ~25 free parameters. The substrate framework has 3 truly unconstrained parameters (\delta, B, z_s), and \delta appears in no current prediction and the other two (B, z_s) describe the previous cycle’s crust — inherently cycle-dependent. All quantitative results above use only measured constants as inputs.
See the paper for an introduction to the math that ties it all together across eight domains.
Framework Thesis
If there were a tiny particle, that collides elastically, after the big bang, it would create vortices in a superfluid that spin frictionlessly with enormous energy and nothing to slow it down. This would create a superfluid, vortices spinning as topologically protected, persistent storms in the substrate. All particles are created from persistent vortices in this substrate, and there’s a background balancing energy between boundary layers. The framework nicknames that particle, dc1 (“dark carbon”) that makes up this substrate. The background substrate organizes into sheets, with a localized planar order, a lattice size of ~100 μm. This is seen as the perturbation envelope for vortices in the substrate. The sheets themselves are stacked ~16 μm apart with a counter-spinning layer at every half-step (~8 μm). Electrons and other particles are persistent dc1 vortex storms — topologically protected circulation patterns, not objects with a center. The breath like a storm with a contracted and expanded phases. Atomic nuclei are vortex cores, or possibly orbital system complexes, stabilized by counter-spinning dc1 vortex layers at every turbulent boundary.
Like electrons, the dc1 vortex cores also breath, and organize in opposite phase pairs. This gives the lattice a structure that chemistry can lock onto or push away, or some combination. This allows chemistry to use substrate energy to create structure, and hold patterns that persists with substrate energy.
There is no mechanism that could have removed this material from the atom, or from space, and the existence of a superfluid substrate of this nature would create these exact conditions based on the math. And the math predicts that the substrate would create protons, electrons, baryonic matter all formed as vortices, wrapped in flywheels, tied in unbreakable three-way knots.
Volovik, Bush, Simeonov and others provided the foundation and vision. The roll of the opposite spinning layers, and photon as a modon provided the missing link.
How light travels through the substrate, and crossing boundaries without losing energy:
Energy transfer follows the boundary equations for modons — dipole vortex streams that travel long distances against the flow in ocean currents. These photon/modons are vortex pairs absorbed and emitted between systems, conveying packets of energy with large wakes of interference due to the speed, stiffness, and energetic nature of the superfluid. They are transmitted freely in (and propelled by) the energy of the substrate, crossing boundary layers frictionlessly by flip-flopping spin direction, producing the lowest-energy transmission: Lorentz Invariance.
The substrate framework asks: how do co-rotating systems in low-dissipation environments create and maintain boundary layers, and what is the energy budget of those boundaries? In all analogous macro-scale systems — binary star collisions, vortex streets, Jupiter’s atmosphere, modons, pilot wave pairs — counter-rotating structures form spontaneously at boundaries between co-rotating regions. They self-organize into the lowest-energy configuration consistent with the boundary conditions, and they transport excess energy away as propagating vortex pairs (modons). The math is the same across scales: Euler equations with a vorticity source term at boundaries.
This same mechanism operating at the Planck scale in a dc1 superfluid, organized by the lattice reproduces quantum mechanics à la Madelung/Bohm6, where the quantum potential is the reaction force of the counter-rotating boundary layer. Quantized states emerge from wave interference in the dc1 medium, not from imposed quantum rules. Quantization is a geometric consequence of oscillatory solutions enclosed by decaying solutions, joined at a boundary.
Spacetime is the acoustic geometry of the dc1 substrate. Curvature is the gradient of the current that ebbs through the boundary, then falls in the laminar stream in the substrate by the energetic system in between. And Einstein’s equations are understood at a deeper level. The substrate reframes General Relativity as the low-energy effective theory of quasiparticle propagation.7 Einstein’s equations are the self-consistency condition for the substrate’s response to organized energy. And the features of our universe that ΛCDM takes as given — \Lambda, dark matter, flatness, inflation — emerge from the material properties of the substrate.
The equations show how effective mass disappears, and how a dipole vortex spinning at 0.776c will flow through a superfluid substrate at a constant c.
The dc1 substrate includes what we consider dark matter. It’s the missing mass in galaxy rotation curves, cluster dynamics, and CMB anisotropies.
- Collisionless: counter-rotating boundary layers do not interact with electromagnetic modons
- Pressureless on galactic scales: bulk flow is coherent, P_\text{eff} \approx 0 for structure formation
- Self-gravitating: vortices and orbital system rotational energy generates ebbing currents
The key identifications:
| Standard Physics | Substrate Framework |
|---|---|
| Quantum vacuum | dc1 vortices organized by the lattice |
| Photon | Modon (counter-rotating vortex dipole) |
| Electron | One effective quantum (\sim 10^9 dc1, mass m_\text{eff} = 1.7 MeV/c^2) — a large dc1 vortex circulating at r_\text{eff} = 150 fm with \hbar angular momentum, enclosed in a perturbation envelope with radius \xi \approx 100\;\mum |
| Quantum potential Q | Reaction force from counter-rotating layer |
| Planck’s constant \hbar | 2m \cdot D (diffusion constant of counter-rotating layer) |
| Speed of light c | \hbar/(m_1 \cdot \xi) — ratio of Planck’s constant to dc1 mass times coherence length (Volovik quasiparticle speed in BEC regime) |
| Gravitational constant G | Parametrizes dc1 leak current through boundaries |
| Wave function \psi | \sqrt{\rho} \cdot \exp(iS/\hbar) — amplitude + phase of co-rotating layer |
| Spin | Angular momentum of effective quantum about its axis |
| Chirality | Direction of spin relative to direction of motion |
| Higgs field | Local chirality state of the dc1 substrate |
| Measurement | Interaction that couples particle vortex/orbital system to detector field |
| Wave function collapse | Orbital system reorganization upon boundary interaction |
Fermions, including the electron, proton, and quark, are polarized orbital systems or dc1 vortices — with an unbalanced topological charge and an odd number of counter-rotating boundary layers separating their internal co-rotating flow from the external substrate. They repel each other according to the Pauli exclusion principle because two same-state fermions would create an irreconcilable boundary conflict.
Bosons, including the photon, W, Z and Higgs, are balanced opposite-spinning pairs or larger systems that have formed an orbital system complex. They move through the fluid with zero forward momentum — massless — due to the counter-balancing energetic effects and even boundary parity.
Mass is rotational energy - the orbital kinetic energy of substrate particles spinning in organized systems. An electron’s 0.511 MeV is entirely accounted for by one effective quantum (~9.6 × 10⁸ condensed dc1 particles) orbiting at 0.776c.
The speed of light is a substrate property: c = \hbar/(m_1 \cdot \xi), the ratio of Planck’s constant to the dc1 mass times the coherence length. It is the maximum speed at which organized disturbances propagate through the superfluid — set by the medium.
Planck’s constant is the minimal action of a counter-rotating boundary layer: \hbar = 2m \cdot D, where D is the diffusion constant of the counter-spinning eddies. Quantization is not imposed — it emerges because only discrete standing-wave patterns survive the boundary matching between co-rotating interior and decaying exterior.
Photons are modons — counter-rotating vortex dipoles ejected when an orbital system reorganizes across boundary layers. They form from the electron’s coherence dress, ejected as a compact vortex dipole that propagates through the ξ-scale perturbation envelope, which provides the boundary-matching domain for quantization and speed c.
In standard QM, the wavefunction \psi is a probability amplitude with no agreed-upon physical meaning. In the substrate framework, \psi = \sqrt{\rho}\,\exp(iS/\hbar) decomposes into two measurable quantities — \rho is the co-rotating substrate density, and S is the phase of the pilot wave flow. The quantum potential Q emerges from the interaction between the co-rotating flow and the counter-rotating eddies at its boundaries. Nothing is mysterious. Nothing requires interpretation. It’s fluid dynamics.
And this results in these phenomena:
- Free-flowing, Lorentz Invariant energy transport; gravitational lensing based on pressure/energy changes in the substrate; a way to model subtle observed CMB effects without warping space-time or invoking nonlocality — instead creating hidden pathways of wave energy in an active dynamic substrate that reacts like pilot-wave hydrodynamics.
- Gravity acts like an “ebbing” or tidal force applied to boundaries, not individual particles. It applies to the contained mass of the orbital system enclosed by that boundary. The theory predicts gravity’s weakness as a consequence of boundary layer efficiency. And once the boundary layer has been penetrated, it accelerates through the boundary to the next layer.
- QFT is incredibly accurate but combines two dynamical layers into one effective description, producing terms (Q, \mathbf{A}) whose physical origin is obscured because the counter-rotating layer’s degrees of freedom have been integrated out. This framework proposes the microscopic content that would make QFT a complete theory — analogous to how kinetic theory provides the microscopic content behind thermodynamics.
- Tunneling occurs because the counter-rotating layer at the turning point isn’t a perfect wall. It’s a dynamic, fluctuating boundary whose eddies occasionally create momentary gaps.
- Entanglement occurs because a singlet’s topology guarantees signal fidelity through a topologically protected vortex channel — the half-integer winding protects information in transit the same way it protects a qubit in a topological quantum computer. The substrate predicts a drop-off in Bell correlations beyond long distances.
- The spin-statistics connection — fermion/boson distinction, Pauli exclusion, 720° rotation — emerges as a topological consequence of how many counter-rotating boundary layers separate a particle’s internal flow from the external substrate.
- The Higgs mechanism is the local chirality ordering of the dc1 substrate, with spontaneous symmetry breaking arising from same-chirality clustering.
- The Aharonov-Bohm effect shows that the substrate field polarity is an almost immeasurable difference in a field that makes a big difference in electron spin.
- The Big Bang as classically conceived is a singular origin with no place to be — spacetime expanding everywhere all at once. The substrate framework’s bang is a local bubble popping in a universe that boils. It pops wherever the pressure has built up enough to nucleate a pocket of the other phase. There is no contradiction between this and what we observe. There is, instead, a different relationship between the observer and the cosmos: we are not surveying the aftermath of a unique creation. We are inside one of the substrate’s normal relaxations, looking outward at the wall that made us, and asking — reasonably, but mistakenly — where it came from. It came from the substrate boiling. That is the whole story. The rest is hydrodynamics.
- Dark energy density changes represent Our Big Bubble’s interaction with previous ones.
- The substrate organizes with the same topology with nested layers as increased organization, and life.
- The substrate lattice breathes with a pattern that gives chemistry either a lock or an anti-lock at any scale giving the universe a hidden layer of organizational scaffolding.
The framework has a consistent story from the Planck scale (substrate particles) through nuclear physics (quark confinement), atomic physics (hydrogen orbitals), condensed matter (conductivity, superconductivity), particle physics (electroweak symmetry breaking, Higgs mechanism, weak force chirality), galactic dynamics (flat rotation curves, the MOND acceleration scale, Tully-Fisher, tidal dwarf galaxies — all from boundary parity with zero new parameters), dark energy evolution, the growth of cosmic structure, and molecular biology. Eight domains. One superfluid. Zero adjustable parameters. Lots of predictions.
What changes
At the scale of the universe:
| Standard physics | Substrate framework | |
|---|---|---|
| The Big Bang | Singular origin, no location, expanding into itself | Our Big Bubble \mathcal{B}^{0} — a nucleation event in a substrate that boils |
| Spacetime | Warped geometry (no medium) | Acoustic geometry of a superfluid |
| Time dilation | Fundamental, unexplained | Pressure-dependent clock rate in the medium |
| Gravity | Curvature of spacetime | Ebbing current through counter-spinning vortex boundaries; its weakness is boundary-layer efficiency |
| Dark matter | Unknown particle (not yet detected) | The substrate itself (n_1 m_1 = \rho_\text{DM}) |
| Galaxy rotation curves | Dark-matter halos (not detected) or modified gravity (MOND, ad hoc) | Boundary parity → MOND field equation; a_0 = c\sqrt{G\rho_\text{DM}}; Tully–Fisher M_b\propto v^4 |
| Tidal dwarf galaxies | Should be halo-free → purely baryonic dynamics | The medium, not a halo — immersed like everything else → same MOND, same a_0, zero new parameters |
| The bullet cluster | Cited as proof dark matter is a collisionless particle | The substrate flows through; only the baryonic gas shocks — lensing tracks the medium, not the gas |
| “Impossibly early” galaxies (JWST) | Too massive, too soon for ΛCDM | a_0(z)\propto(1+z)^{3/2} — structure assembles faster in the denser young substrate |
| Dark energy | Fine-tuned to 10^{-122} | Zero at equilibrium; the observed \Lambda is residual disequilibrium; DESI-confirmed C = 1 |
| The cosmological constant | QFT vacuum off by 10^{120} — “the worst prediction in physics” | An \mathcal{O}(1) disequilibrium times the weak-gravity hierarchy (m_1/M_\text{Pl})^2 — \Lambda and Newton’s G are the same problem; dark-energy scale \rho_\Lambda^{1/4}\approx m_1 c^2 |
| DESI’s “phantom” dark energy | w(z) wiggles beyond ΛCDM, unexplained | The wake of Our Bubble hitting the previous cycle’s crust at Mach 1.3 — a chirping undular bore, the universe’s fingerprint |
| S_8 tension | Unexplained 2–3\sigma gap between Planck and weak lensing | Crust disruption from \mathcal{B}^{-1}: \eta_\text{crust} = 2\alpha_{mf}^2 \Rightarrow S_8 = 0.816 (zero new parameters) |
| Hubble tension | 5\sigma early-vs-late H_0 disagreement | A real gradient: c\propto\rho^{1/3} → a hemispheric H_0 dipole ({\sim}9\% already seen in X-ray clusters) |
| Inflation | A postulated inflaton with a tuned potential | A sound-speed phase transition in the substrate; n_s\approx0.968, r\approx0.01–0.02 |
At the microscopic scale:
| Standard physics | Substrate framework | |
|---|---|---|
| Mass | Yukawa couplings to the Higgs (free parameters) | Rotational energy of organized substrate; the electron’s 0.511 MeV = one effective quantum at 0.776c |
| The electron | A structureless point | A persistent dc1 vortex storm — \sim10^9 particles, a 150 fm core inside a 100\;\mum envelope |
| The photon | An excitation of the EM field | A modon — a counter-rotating vortex dipole that flips spin to cross boundaries frictionlessly |
| Speed of light c | A postulated constant of geometry | A substrate property, c = \hbar/(m_1\xi) — the medium’s signal speed |
| Planck’s constant \hbar | A fundamental quantum of action | 2m\cdot D — the action of the counter-rotating boundary layer |
| Electroweak / Higgs scale | v = 246 GeV, an input | v = \sqrt{8\pi\,m_\text{eff}^2 c^4\nu} = 246.1 GeV from the same Weinberg angle (-0.06\%) |
| Fine-structure constant | \alpha\approx1/137, a measured input no one can derive | \sin^2\!\delta_0\,\sin^2\theta_W/\pi = 1/135 (+1.5\%) |
| Quarks & confinement | Charges +\tfrac23,-\tfrac13 postulated; confinement put in by hand | Charges from the vortex-junction solid angle; confinement = boundary seams that cannot be cut |
| Proton mass | {\sim}99\% QCD binding energy (lattice) | Counter-rotating boundary layers — the same number, now a picture |
| Nuclear binding & the iron peak | Liquid-drop model, fitted coefficients | Boundary-seam saturation vs. \alpha-set Coulomb → peak in the iron–nickel region; pairing = anti-phase breathing (Cooper/BCS) |
Phenomena:
| Standard physics | Substrate framework | |
|---|---|---|
| Quantum vacuum | Abstract field with 10^{122}\times too much energy | Superfluid at the measured \rho_\text{DM} |
| Wave–particle duality | Complementarity principle | A vibrating particle in a responsive medium |
| The wavefunction \psi | A probability amplitude with no agreed meaning | \sqrt{\rho}\,e^{iS/\hbar} — measurable density and pilot-wave phase |
| Wavefunction collapse | Measurement problem (unresolved) | Orbital-system reorganization at a boundary |
| Quantization | Postulated (Born rule) | Standing-wave boundary matching (geometric) |
| Tunneling | Probabilistic barrier penetration | A fluctuating boundary layer whose eddies open momentary gaps |
| Spin | “Intrinsic” (no classical analog) | Counter-rotating boundary-layer angular momentum |
| Pauli exclusion | A rule (antisymmetric states) | Two same-state fermions = an irreconcilable boundary conflict |
| Entanglement | Nonlocal correlations | A topologically protected vortex channel — predicts a drop-off at extreme distance |
Chemistry and biology:
| Standard physics | Substrate framework | |
|---|---|---|
| Why structures cluster at set sizes | Contingent — no unifying reason across cells, organelles, vesicles | The substrate ladder: discrete, log-spaced rungs set by the \sqrt2 pairing factor (discrete scale invariance) |
| DNA’s geometry | 10.5 bp/turn — measured, no first-principles reason | 2\pi r f/h = 10.47 from the lattice packing fraction (0.3\%) |
| Microtubules’ 13 protofilaments | An evolutionary accident | The unique paraxial integer; wall ratio 3/(2f) = 2.648 |
| The golden angle in plants | A Fibonacci curiosity / packing heuristic | The ladder’s anti-lock gap (\varphi, the most-irrational angle) — a flux-lattice ground state, realized even magnet-free |
| Brain rhythms | Descriptive band names | Octave-nesting rungs on the comb when the cortex binds; the \varphi gap when it rests |
Footnotes
Bush, J.W.M. & Oza, A.U., “Hydrodynamic Quantum Analogs,” Annual Review of Fluid Mechanics 52, 2020. A vibrating particle in a responsive medium reproduces quantized orbits, tunneling, and interference — the template for the electron. See also Dagan & Bush, “Hydrodynamic quantum field theory: the free particle,” Comptes Rendus Mécanique, 2020. [R1, R1b]↩︎
Volovik, G.E., The Universe in a Helium Droplet, Oxford University Press, 2003. The single most important source: emergent speed of light (Ch. 7), two-fluid model (Ch. 4–5), gauge fields from vortex cores (Ch. 22–25), and cosmological constant self-tuning (Ch. 29–30). [R2]↩︎
Simeonov, L., “Quantum mechanics as a two-fluid stochastic theory,” arXiv:2509.02868, 2025. Shows that the osmotic velocity of a counter-rotating fluid derives from the HVBK mutual friction force, formally bridging superfluid hydrodynamics to quantum mechanics. [R3]↩︎
Khoury, J., “Dark Matter Superfluidity,” arXiv:1507.01860, 2015; Berezhiani, L. & Khoury, J., “Theory of Dark Matter Superfluidity,” Phys. Rev. D 92, 103510, 2015. Dark matter as a superfluid on galactic scales, with a MOND-like phonon-mediated force and CDM-to-MOND transition at the Landau critical velocity. [R4, R26]↩︎
Larichev, V.D. & Reznik, G.M., “Two-dimensional solitary Rossby waves,” Doklady Akademii Nauk SSSR 231, 1976 (modon boundary matching, K = j_{11}^2+1); Saffman, P.G., Vortex Dynamics, Cambridge, 1992 (dipole dynamics, lattice stability); Aftalion, A. et al., Phys. Rev. A 71, 023611, 2005 (vortex lattice energy functional); Fetter, A.L., Rev. Mod. Phys. 81, 647, 2009 (GP healing length, 1/\sqrt{2} factor); El, G.A. & Hoefer, M.A., “Dispersive shock waves and modulation theory,” Physica D 333, 2016 (dispersive shock waves). [R5, R6, R7, R8, R80]↩︎
Nelson, E., “Derivation of the Schrödinger Equation from Newtonian Mechanics,” Phys. Rev. 150, 1079, 1966; Bohm, D. & Vigier, J.-P., Phys. Rev. 96, 208, 1954. The formal bridge from superfluid hydrodynamics to QM is completed by Simeonov [R3]. [R18, R19]↩︎
Barceló, C., Liberati, S. & Visser, M., “Analogue gravity,” Living Reviews in Relativity 8, 12, 2005. Any barotropic, irrotational, inviscid fluid produces an acoustic metric formally identical to curved Lorentzian spacetime. The substrate satisfies this. [R13]↩︎