Emergent Speed of Light

The Claim

The speed of light is not a fundamental constant — it is a property of the substrate. It emerges as the maximum propagation speed of quasiparticle excitations in the dc1 condensate, determined by the dc1 mass and the coherence length:

\boxed{c = \frac{\hbar}{m_1 \cdot \xi}}

The Volovik Route

This result comes from Volovik’s analysis of quasiparticle spectra in BCS-BEC superfluids (Ch. 7, eq. 7.51). In the strong-coupling (BEC) limit — where the gap energy \Delta_0 greatly exceeds the Fermi energy E_F — the quasiparticle spectrum is automatically Dirac-like:

E^2 = \mu^2 + c^2 p^2

with a single isotropic speed c = \hbar/(m_1 \xi). The speed is set by the interaction energy of the condensate, not by any constituent particle velocity. The inner-scale orbital velocity v_\text{rot,inner} = 0.776\,c and the outer-scale lattice rotation v_\text{rot,outer} = 0.0025\,c are both well below c — just as the speed of sound in a superfluid can far exceed the velocity of individual atoms.

Combined with the dark matter density relation n_1 m_1 = \rho_{DM} and close-packing (n_1 \xi^3 \approx 1), the Volovik formula determines both the coherence length and the dc1 mass from measured constants:

\xi = \left(\frac{\hbar}{\rho_{DM} \cdot c}\right)^{1/4} \approx 110\;\mu\text{m}, \qquad m_1 = \left(\frac{\hbar^3 \rho_{DM}^3}{c^3}\right)^{1/4} \approx 2\;\text{meV}/c^2

These are the values quoted in Substrate Particles — they follow from this route.

Why the BEC regime is guaranteed. The dc1 thermal de Broglie wavelength (\sim 1.3 mm at CMB temperature) exceeds the interparticle spacing (\sim 115\;\mum) by an order of magnitude, placing the substrate deep in the quantum-degenerate regime. The BEC condition \Delta_0 \gg E_F is automatically satisfied. No fine-tuning is required.

Lorentz invariance from the BEC regime. In the strong-coupling limit, the quasiparticle spectrum is automatically isotropic: c_\parallel = c_\perp = c. There is no anisotropy to tune away. This is the substrate’s physical origin of Lorentz invariance — the BEC regime produces a single light cone with no preferred direction. Compare He-3-A (weak coupling), where c_\perp/c_\parallel \sim 10^{-5}. The substrate must be in the opposite regime, and it is.

All low-energy excitations share this speed. Scalar modes (phonons), vector modes (modons/photons), and tensor modes (gravitational wave metric perturbations) all inherit the same isotropic c from the BEC medium. GW170817 confirmed |c_\text{GW}/c - 1| < 6 \times 10^{-15}; the substrate predicts exact equality.

The Modon Existence Condition

The Volovik route determines c. A second, independent condition — the Larichev-Reznik modon dispersion relation — then determines the outer-scale lattice rotation \omega_0.

The physical picture: the dc1/dag orbital systems create a background vorticity field throughout the substrate, analogous to a planetary atmosphere where the Coriolis effect gives rise to Rossby waves. Modons (counter-rotating vortex dipoles) propagate against this background vorticity gradient, just as oceanic modons propagate against the planetary vorticity gradient on a beta-plane.

The Larichev-Reznik modon exists in a medium with a background vorticity gradient \beta. Its propagation speed is:

U_{LR} = \frac{-\beta}{p^2 + \kappa_\text{ext}^2}

where \beta is the vorticity gradient (the rate at which the substrate’s orbital angular momentum density changes with position), p is the interior wavenumber satisfying J_1(p \cdot a) = 0 at the modon boundary, and \kappa_\text{ext} is the exterior decay rate set by K_1 matching conditions.

The matching condition at the modon boundary (r = a) couples interior and exterior:

p \cdot \frac{J_0(p \cdot a)}{J_1(p \cdot a)} = -\kappa_\text{ext} \cdot \frac{K_0(\kappa_\text{ext} \cdot a)}{K_1(\kappa_\text{ext} \cdot a)}

This transcendental equation has discrete solutions — the modon speed is quantized by boundary matching. The lowest-energy solution gives the propagation speed.

From Dispersion to \omega_0

Identifying the substrate parameters: the vorticity gradient is \beta \approx n_1 \cdot \omega_0 \cdot \xi, the exterior decay rate is \kappa_\text{ext} \approx 1/\xi (the modon’s influence decays over one coherence length), and the ground-state interior wavenumber is p \approx j_{11}/a where j_{11} \approx 3.83 is the first zero of J_1 and a \approx \xi is the modon radius. Substituting:

c = U_{LR} = \frac{n_1 \cdot \omega_0 \cdot \xi}{(j_{11}/\xi)^2 + (1/\xi)^2} = \frac{n_1 \cdot \omega_0 \cdot \xi^3}{j_{11}^2 + 1} \qquad

Still only a numerical recipe

The formula above has LHS [m/s] and RHS [s⁻¹] — off by a factor of [m]. Here n_1 is a 3D number density [m⁻³], but the L-R modon matching operates on a 2D vorticity gradient \beta [m⁻¹s⁻¹]. The substrate’s vortex lattice is organized into chirality-coherent 2D sheets, and the correct 3D→2D projection involves solving the math behind WIP-15 - essentially, how lattices layers offset, polar jets from one dc1/dag system in layer one exchange energy with a vortex joining multiple dc1/dag systems in layer layer n-1. It makes sense that it’s the Lorentz invariant - the least energy path due to the nature of the system but the math behind it is probably new.

Define K = j_{11}^2 + 1 = 15.67. Since c is already determined by the Volovik route, this equation is solved for the outer-scale rotation:

\omega_0 = \frac{K \cdot c}{f \cdot \xi} \approx 7.8 \times 10^9\;\text{rad/s} \qquad\text{(from dimensionless form, using $f = 0.5666$)}

The outer-scale rotation velocity follows:

v_\text{rot,outer} = \omega_0 \cdot \xi \approx 0.0025\,c \approx 7.6 \times 10^5\;\text{m/s}

This is the velocity that appears in the gravitational constant (G = f_\text{cross} \cdot v_\text{rot,outer} / (4\pi); see Gravity) and sets the Landau critical velocity for the CDM-to-MOND transition. Below v_\text{rot,outer}, the substrate responds as a superfluid; above it, as collisionless dark matter — matching Khoury’s dark matter superfluidity prediction (v_L \sim 10^{-3}c). The modon existence condition does not merely reproduce this transition velocity — it determines it.

Two Speeds, One Substrate

The substrate now has two well-separated rotational velocities, both derived rather than assumed:

Scale	Velocity	Origin	Role
Outer (\xi)	v_\text{rot,outer} = \omega_0 \xi \approx 0.0025\,c	Modon existence condition	Gravity, CDM-MOND transition
Inner (r_\text{eff})	v_\text{rot,inner} = c\sqrt{2\alpha_{mf}} \approx 0.776\,c	Electroweak sector (Subsystem A)	Particle mass, spin

The inner velocity determines how much energy is stored in each orbital system — and therefore the mass of every particle. That connection is the subject of the next chapter.