Galactic Dynamics from Boundary Parity

Flat rotation curves, the Tully-Fisher relation, and the MOND acceleration scale — from the counter-rotating boundary

Summary

The same counter-rotating boundary layers that produce gravity and the quantum potential also improve our models for galactic dynamics. No new parameters, interactions, or equations of state are introduced. The key result is that the parity symmetry of the counter-rotating boundary forces its current-phase relation to be quadratic at leading order, which produces the MOND field equation in the deep low-acceleration regime. The MOND acceleration scale is:

\boxed{a_0 = c\sqrt{G\,\rho_\text{DM}}}

Numerically: 1.16 \times 10^{-10} m/s² (using Planck 2018 central \rho_\text{DM} = 2.25 \times 10^{-27} kg/m³), compared to the McGaugh et al. (2016) measured value g_\dagger = (1.20 \pm 0.02_\text{stat} \pm 0.24_\text{sys}) \times 10^{-10} m/s². Match: ~3% — well within the ±20% systematic uncertainty. Every factor is a substrate parameter: c = \hbar/(m_1\xi) from C1, G from C3 (measured; its unit-free content Gm_1^2/\hbar c is not yet derived, Gravity § Mathematical Form), \rho_\text{DM} = n_1 m_1 from C10. If the bridge equation holds, today’s \rho_\text{DM} is determined by \sin^2\theta_W and m_e, making today’s a_0 a prediction from electroweak physics (which density sets a_0).

The logical chain: counter-rotating boundary → parity-symmetric current-phase relation → quadratic response → MOND field equation → flat rotation curves + baryonic Tully-Fisher. And separately: Hubble expansion → parity breaking → induced linear (Newtonian) term → a_0 as crossover scale.

Interactive: open the galaxy simulation — the same ebbing current as the gravity simulation, seen ten kiloparsecs out. A Freeman disk of baryons, the river falling in through the boundary chain, and the boundaries drawn as what they are: anti-phase pairs that breathe outside r_M (the quadratic response, MOND) and sit pried open by the Hubble bias inside it (Newton). A test star must orbit at the model’s v(r), and the live rotation curve and radial acceleration relation in the corner show it doing so; a toggle draws the halo that ΛCDM would infer from that curve, sampled from M_\text{halo}(<r) = r^2(g_\text{obs} - g_N)/G — a shadow of the boundary response, not a second substance. The Past the speed limit folder puts the same galaxy in a cluster core’s flow and the rings tear into a vortex tangle at v_L; the Slide the sky back folder drags a_0 up as (1+z)^{3/2} and shrinks r_M into the disk. The same page has two more views: the nucleus, the black hole at the centre as an engine (Quasars), and the Bullet Cluster, the gate crossed at cluster scale (Bullet Cluster). The interpolation between the two regimes is McGaugh’s empirical form, drawn as such — its derivation is open problem 2.

The galactic dynamics chain does not stop here. The same \rho_\text{DM} that sets a_0 also determines how the moraine crust from \mathcal{B}^{-1} modifies the effective gravitational coupling G_\text{eff} and dark energy density \rho_\Lambda — connecting galactic dynamics to the Hubble tension, dark energy evolution, and structure growth suppression. A 15-knot freeform spline fit to the combined DESI BAO + Jia H_0(z) data reveals that this crust imprint is a fully resolved transcritical undular bore in the Grimshaw–Smyth–El–Hoefer framework: a soliton-plus-wake pair at z_b \approx 2.3, a recovery-zone node at z_\text{crit} = 1.588 (slightly below zero, consistent with maximal energy extraction at M = 1), and a DE-amplified downstream wave train with five to six oscillation cycles exhibiting cosmological chirp — wavelength compression toward low z matching the rank-ordered structure of a KdV dispersive shock wave. The 15-knot fit achieves \chi^2_\text{total} = 9.68 vs \LambdaCDM’s 29.0 on the same observables.

The most striking result: dividing out the [\Omega_\Lambda(z)]^\gamma dark energy amplification from the observed crests leaves a monotonically decreasing bare carrier — exactly the rank ordering that the soliton-edge-to-harmonic-edge structure of a DSW demands.

The Two Regimes of the Ebbing Current

Gravity arises from the net dc1 current that leaks through counter-rotating boundary layers with transmission probability f_\text{cross} \approx 4\times10^{-31} per cell tick (see Gravity). The ebbing current velocity at distance r from a mass M is:

v_\text{ebb}(r) = \sqrt{\frac{2GM}{r}}

This velocity is what the dc1 current carries as it transits each boundary in the lattice. Between boundaries, the flow is a laminar “waterfall” — dc1 particles stream freely through the co-rotating substrate at the local drift velocity. At each boundary, they encounter a counter-rotating layer and must transit it.

The substrate’s outer-scale rotation velocity v_\text{rot,outer} = \omega_0\xi \approx 0.0025\,c \approx 750 km/s acts as the Landau critical velocity — the threshold above which the flow through boundaries excites vortices and dissipates energy.

Which velocity is compared to v_L. The speed that matters is how fast a system’s own matter sweeps through the boundaries around it, measured relative to the substrate that system holds. A bound system gathers its substrate (the ebbing current falls into it) and carries it along, so this is an internal speed, not the system’s bulk drift through the cosmos. For a given galaxy, you can use the system’s characteristic internal velocity v_\text{disp}: the flat circular speed for a rotating disk, the line-of-sight velocity dispersion for a pressure-supported system (dwarf spheroidals, clusters). In equilibrium, it tracks the ebb speed above, since the circular speed \sqrt{GM/r} is v_\text{ebb}/\sqrt{2}, so it is the ebbing current’s own speed, seen from the matter riding in it.

For a galaxy that is a member of a cluster, the relevant speed is that galaxy’s orbital motion through the cluster’s substrate, \sim 1000 km/s. That is why a disk that breathes coherently in isolation tears into a tangle in a cluster core (the Past the speed limit view of the galaxy simulation). The bulk motion of a whole system relative to the cosmic frame does not enter. The Local Group drifts at \sim 600 km/s relative to the CMB, which would put the Milky Way at the gate if it counted.

The MOND scale a_0 the centripetal acceleration g = v^2/r of a body orbiting inside the system, compared at each radius. Together they help determine how the flow moves through the boundaries.

  • For a given system, compare its v_\text{disp} to v_L. If it is below v_L, the parity-even boundary response acts like a superfluid with the quadratic current-phase relation. In these systems, the further reaches of the galaxy, where the acceleration scale drops below the threshold will experience MOND’s extra grip. Above v_L the pairing decoheres and only the bare Newtonian response remains.
  • Within a system below v_L, the acceleration test (g vs. a_0), is applied radius by radius. Where g \gg a_0 (inside r_M = \sqrt{GM_b/a_0}) the Hubble-induced linear term wins and orbits are Newtonian. Where g \ll a_0 the quadratic term wins and the curve goes flat.
System v_\text{disp} (internal speed) vs. v_L = 750 km/s Phase (velocity test) Orbital g vs. a_0 (acceleration test) Result
Dwarf galaxies 30–80 km/s \ll v_L Superfluid g \ll a_0 almost everywhere Deep MOND
Milky Way 150–200 km/s < v_L Superfluid g \gtrsim a_0 inside r_M \approx 8 kpc, g < a_0 outside Newtonian core, MOND outskirts
Galaxy clusters 800–1500 km/s \gtrsim v_L Normal g \sim 0.3–1\,a_0 in the outskirts Newtonian + collisionless substrate (CDM behavior)

The cluster row shows why both tests are needed. By acceleration alone, cluster outskirts sit at or below a_0, and MOND would predict an enhancement there. Applied to clusters, MOND indeed falls short of the observed mass by a factor of ~2, a long-standing problem for it. The velocity test is what removes that enhancement: the constituents move fast enough to switch off the paired response, so the low accelerations never get a quadratic term to act on. This is the natural resolution of why MOND phenomenology appears in galaxies but not in clusters — they probe different dynamical regimes of the same substrate. Whether the MOND regime is available is governed by a velocity scale (the Landau critical velocity of the lattice); where in a galaxy it takes over is governed by the acceleration scale a_0. Neither is a spatial boundary drawn by hand.

This threshold is a gravitational instance of the substrate ladder’s sign rule: name a system’s velocity relative to v_L and its phase is fixed before you measure it. Below v_L the substrate stays coherently paired and the boundary breathes — superfluid, deep MOND; above v_L the flow excites vortices, the pairing decoheres, and the response reverts to ordinary Newtonian/CDM behavior. It is the same predict-the-state-from-the-job logic that solar-system resonances wear in their trap-versus-clear distinction — here keyed to a velocity rather than a frequency ratio. The honest caveat is that this is a coherence threshold — the lock pole engaged versus disengaged — not the anti-lock \varphi gap: v_L marks where the paired response switches off, not a structure fleeing a rung.

What Sets the Critical Velocity

The number v_L \approx 750 km/s recurs across the framework — it sorts galaxies from clusters here, switches off the Bullet Cluster’s superfluid, caps the fast solar wind, and sets the dissipation onset that magnetic reconnection spends its coin past. It is worth saying clearly, in one place, what kind of velocity it is, what hydrodynamics fixes it, and what does not.

It is not the Landau phonon–roton velocity, despite the name. The textbook Landau critical velocity is the minimum of \epsilon(p)/p over the excitation spectrum. The substrate’s realized sound branch is the monotonic Bogoliubov form \epsilon(p) = c\,p\sqrt{1 + (p\xi/2\hbar)^2} — no roton, no maxon — because the self-organized medium sits at its marginal point. For a monotonic branch the minimum of \epsilon/p lies at p\to 0 and equals the sound speed itself: the strict Landau velocity of this substrate is c, the speed of light. So the 750 km/s scale cannot be the phonon–roton Landau velocity. We keep the name “Landau critical velocity” by analogy — it is the speed above which flow dissipates by shedding excitations — but the excitations it sheds are vortices, not rotons.

What it actually is: the coherence threshold of the rotating vortex lattice. The dc1 condensate is not a quiescent superfluid but a rotating vortex lattice (Substrate Particles § The Lattice Breathes in Pairs). The relevant instability of a rotating array under relative flow is the Glaberson–Johnson–Ostermeier / Donnelly–Glaberson instability: parallel flow along a rotating vortex array amplifies Kelvin waves and tips the ordered lattice into a tangle — the superfluid-to-quantum-turbulence transition (Vinen). Below threshold the lattice convects coherently (superfluid, deep MOND); above it the lattice shreds into incoherent vorticity (normal, CDM-like). The framework’s “lock pole engaged versus disengaged” language is this laminar-lattice ↔︎ vortex-tangle crossover.

This is the same instability the framework already uses to fix the inter-sheet spacing d_\text{GJO} (Substrate Particles § The Vertical Scale) — read at a different scale:

One instability, two faces. Applied at the inner lattice rotation \Omega_\text{sheet} = \kappa_q/2\xi^2, the GJO criterion sets a length — the unstable wavelength d_\text{GJO} = \sqrt{\nu_s/2\Omega_\text{sheet}} \approx 16\;\mum. Applied at the outer-scale rotation \omega_0, the same criterion sets a velocity — the rim speed at which a cell carried by the coarse rotation outruns the lattice’s own coherence, v_L = \omega_0\,\xi. The vortex array’s stability margin shows up as a spacing at the fine scale and as a critical speed at the coarse scale.

The three substrate speeds, kept distinct. Two genuine wave modes (c and c_T) and one rotation-set threshold (v_L) recur in the framework and are easy to conflate; they are physically distinct:

Speed Value Hydrodynamic origin Where it appears
c 3\times10^5 km/s Bogoliubov phonon slope = strict Landau velocity of the monotonic branch light, gravitational waves, photon modon
v_L = \omega_0\xi \approx 750 km/s outer-rotation rim speed = GJO/Donnelly–Glaberson turbulence onset of the lattice MOND↔︎CDM split, Bullet Cluster, fast solar wind, reconnection onset (masking-failure, not a ceiling), DR saturation
c_T (Tkachenko) \approx 9 km/s transverse shear of the vortex lattice slow lattice relaxation, beryllium shear-sound coincidence

(The “intermediate vorticity channel, \sim 800 km/s” named in the thermal cascade is this same v_L under another label.) The Tkachenko speed lies two orders of magnitude below v_L — the lattice shears far more easily than it tangles — and the relation between the two remains open.

What is honestly not derived. The lever arm \xi \approx 100\;\mum is fixed cleanly from \hbar, c, and \rho_\text{DM} (Substrate Particles). The rotation rate \omega_0 \approx 7.8\times10^9 rad/s is not computed from first principles. An earlier draft pinned it in the gravity sector through G = f_\text{cross}\,\omega_0\xi/4\pi, but that relation turned out to be a coincidence of SI units and has been retired; \omega_0 is G-free, and its best current handle is the clean-units law v_L = c\,(4\pi/\nu)^{1/3} (Open Problems WIP-15 item 2b). The earlier modon-matching route that appeared to fix \omega_0 (the old n_1\omega_0\xi^3 = Kc form) is now understood to be a dimensional artifact (Constraint Summary), and deriving \omega_0 is an explicit open problem (Open Problems WIP-15, WIP-16). In the dimensionless form v_L/c = \omega_0/\omega_1, where \omega_1 = c/\xi = m_1c^2/\hbar \approx 3\times10^{12} rad/s is the cell’s own Compton clock, the content of the 750 km/s number is the bare ratio \omega_0/\omega_1 \approx 0.0025 — the outer rotation runs at about 1/400 of the cell clock, and that ratio is what the framework cannot yet derive.

What carries the number instead: cross-domain agreement. Lacking a derivation of \omega_0, the framework’s real content is that the same v_L must surface in independent settings — and it does. The galaxy/cluster velocity-dispersion split lands at \sim 750–1000 km/s, and the Ulysses spacecraft’s high-latitude fast-wind mean is 751.5 km/s (Solar & Stellar Dynamics) — two independent ceiling reads of one substrate parameter. (Magnetic reconnection also turns on near v_L, but as the rim’s dissipation onset rather than a clean ceiling — its outflows straddle v_L, so it is mechanism-consistent corroboration, not a third clean measurement.) One substrate parameter, measured the same way in two disconnected domains, returns the same value. That coincidence is a (modest) prediction in its own right, and it is the honest anchor for v_L until \omega_0 is derived.

The most promising lead. The framework already reads Newtonian gravity as the breath un-paired by the Hubble/external-field DC bias. If the outer rotation \omega_0 is the substrate’s coherent response to the cosmological flow, then v_L, the MOND scale a_0 = c\sqrt{G\rho_\text{DM}}, and H_0 should not be independent — the long-standing MOND coincidence a_0 \sim cH_0 would become the statement that both the acceleration scale and the velocity scale are set by the same cosmological bias. Tying \omega_0 to H_0 this way is the natural route to deriving 750 km/s rather than anchoring it; it is flagged as a conjecture, not a result.

Parity Symmetry and the Current-Phase Relation

Counter-rotating boundary layer with parity symmetry An illustrative cross-section showing how a counter-rotating vortex boundary between two co-rotating substrate regions forces an even current-phase relation. The boundary contains both rotation directions equally, making it unable to distinguish positive from negative phase differences. +ω₀ −ω₀ Co-rotating region rotation +ω₀ Co-rotating region rotation +ω₀ Boundary layer Ebbing current (dc1) Phase: +δφ Phase: −δφ Physically equivalent — boundary sees both signs equally J(+δφ) = J(−δφ) → no linear term → quadratic CPR

The standard Josephson junction

In a standard Josephson junction (insulating barrier between superconductors), the current-phase relation (CPR) is:

J = J_c \sin(\delta\phi)

where \delta\phi is the phase difference across the barrier. For small phases: J \approx J_c\,\delta\phi — the response is linear, giving a standard Poisson equation for the potential.

The counter-rotating boundary is different

The substrate boundary is not an insulator — it is a counter-rotating vortex layer where the condensate rotates at +\omega_0 on one side and -\omega_0 on the other. This boundary has an intrinsic parity symmetry: a co-rotating flow encountering the boundary with phase +\delta\phi is physically equivalent to flow with phase -\delta\phi, because the counter-rotation maps one to the other. The boundary contains both rotation directions equally — it does not prefer a sign.

That even-handedness is not an extra postulate. The +\omega_0/-\omega_0 layer is the framework’s anti-phase breath — a circulation and its counter-rotating partner trading energy across a shared seam, the same paired motion that binds Cooper electrons, orders an antiferromagnet, and breathes in the lattice’s own pairing. A pair has no preferred partner — so the boundary is parity-even because it is a pair. The symmetry that is about to force the quadratic CPR is the pairing symmetry of the breath.

This forces the current-phase relation to be even in \delta\phi:

J(\delta\phi) = J(-\delta\phi)

Only even powers survive in the expansion:

\boxed{J = I_2\,\delta\phi^2\,\text{sgn}(\delta\phi) + I_4\,\delta\phi^4\,\text{sgn}(\delta\phi) + \ldots}

There is no linear term. The leading response is quadratic. The form I_2\,\delta\phi^2\,\text{sgn}(\delta\phi) preserves the correct sign for current direction while ensuring |J| \propto \delta\phi^2. Note: the standard \pi-junction form I_2|\delta\phi|\delta\phi gives the same physics, but the |\delta\phi| factor introduces a cusp at \delta\phi = 0 which is non-analytic. For a smooth physical boundary, the CPR should be analytic on each side of zero: \delta\phi^2\,\text{sgn}(\delta\phi) makes the quadratic behavior explicit while remaining smooth away from the origin.

The counter-rotating boundary acts as a squaring nonlinearity: it responds to \delta\phi^2 regardless of sign, because the boundary contains both rotation directions equally and cannot distinguish +\delta\phi from -\delta\phi.

Phase drop per boundary

The condensate phase gradient due to the ebbing current is:

\frac{d\theta}{dr} = \frac{m_1\,v_\text{ebb}}{\hbar} = \frac{v_\text{ebb}}{c\,\xi}

using \hbar = m_1 c\xi from C1. The phase drop across one lattice cell (size \xi) at radius r:

\delta\phi(r) = \frac{v_\text{ebb}(r)}{c}

At the MOND radius of the Milky Way (r_M \approx 8 kpc), \delta\phi \sim 10^{-3}. Each individual junction is deeply in the small-phase regime — the nonlinearity comes from the symmetry of the CPR, not from driving individual junctions into a nonlinear regime.

From Quadratic CPR to the MOND Field Equation

Continuum limit of the Josephson chain

Between a mass M and a test point at distance r, the ebbing current crosses N = r/\xi boundary layers — a chain of parity-symmetric junctions. Coarse-graining to the continuum: each junction contributes \delta\phi = \xi\,\partial_r\theta, and the mass current through the chain satisfies continuity \nabla \cdot \vec{J} = \rho_b (baryonic sources). This chain is the breath replicated across scale in gravitational form — one paired-breath boundary, the same motif, stacked N = r/\xi times from the lattice cell out to the galactic radius. It is a coarse-family ladder, not a \sqrt2 comb: the rungs are spaced by the single cell size \xi, mechanism-set and uniform, the same way the solar system’s boundary stack and superconductivity’s five scales recur without carrying a single substrate period (honest accounting).

With the quadratic CPR J \propto |\delta\phi|\,\delta\phi:

J \propto |\nabla\theta|\,\nabla\theta

Since \theta is proportional to the gravitational potential (\nabla\theta \propto \nabla\Phi/c^2), the continuity equation becomes:

\boxed{\nabla \cdot \left[|\nabla\Phi|\,\nabla\Phi\right] \propto 4\pi G\,\rho_b}

This is the MOND field equation

The Bekenstein-Milgrom (AQUAL) formulation of MOND is:

\nabla \cdot \left[\mu\!\left(\frac{|\nabla\Phi|}{a_0}\right)\nabla\Phi\right] = 4\pi G\,\rho_b

with interpolation function \mu(x) \to 1 for x \gg 1 (Newtonian regime) and \mu(x) \to x for x \ll 1 (deep-MOND regime). The deep-MOND limit is precisely \nabla \cdot [|\nabla\Phi|\,\nabla\Phi] \propto \rho_b, which is what the parity-symmetric boundary naturally produces.

The linear (Newtonian) term is absent from the pure boundary CPR — it must be induced by a symmetry-breaking mechanism. That mechanism is the cosmological expansion (see below).

Flat Rotation Curves and the Tully-Fisher Relation

For a spherically symmetric baryonic mass M_b, the deep-MOND field equation gives:

r^2\,|\Phi'|\,\Phi' = a_0\,G\,M_b

|\Phi'| = \frac{\sqrt{a_0\,G\,M_b}}{r}

The circular velocity follows from v^2/r = |\Phi'|:

v^2 = \sqrt{a_0\,G\,M_b} = \text{constant}

The rotation curve is flat. And rearranging:

\boxed{v^4 = a_0\,G\,M_b}

This is the baryonic Tully-Fisher relation (BTFR), with normalization fixed by a_0. It matches McGaugh, Lelli & Schombert’s (2016) empirical result from 153 galaxies spanning five decades in luminosity and four decades in surface brightness, with scatter consistent with observational error alone.

The MOND Acceleration Scale

The Hubble flow breaks the parity

The pure counter-rotating boundary has a quadratic CPR with no linear term — it produces only MOND-like gravity, with no Newtonian limit. But we observe standard Newtonian gravity at high accelerations. The linear term must be induced by something that breaks the \pm\omega parity of the boundary. That parity is the breath’s anti-phase pairing, so the induced linear term is, in the ladder’s language, the un-paired breath: Newtonian gravity is what the boundary does once an external flow has pried apart its symmetric +\omega_0/-\omega_0 pairing and singled out a direction the pair did not have.

The cosmological expansion provides this breaking. The Hubble flow imparts a universal DC phase bias \phi_0 to every boundary — the expansion imposes a net outward velocity gradient across each boundary cell, which distinguishes the +\omega_0 side (co-aligned with expansion) from the -\omega_0 side (counter-aligned), preferentially stretching one rotation direction relative to the other. Linearizing the parity-even CPR about that bias gives the induced linear channel, with conductance set by the bias:

J_\text{lin} = 2I_2\,\phi_0\,\delta\phi .

Which quantity is given, and how the two channels combine. Two things have to be kept straight, because getting either one wrong inverts the regimes. First, the current J is the given quantity: by continuity it carries the enclosed baryonic source, so J \propto g_N (Gauss’s law). The phase drop \delta\phi is the response, and it is what a test particle feels, \delta\phi \propto g_\text{obs}. Second, the two channels act in series, not in parallel. The bias does not open a second door beside the paired seam; it acts on the same seam, so every unit of current that crosses must pay both the paired drop and the biased drop, and drops across elements crossed in succession add:

\boxed{\;\delta\phi \;=\; \underbrace{\sqrt{J/I_2}}_{\text{paired seam}} \;+\; \underbrace{\frac{J}{2I_2\,\phi_0}}_{\text{biased seam}}\;}

The parallel alternative, J = I_2\,\delta\phi^2 + 2I_2\phi_0\,\delta\phi (the currents of the two channels summed at a common drop), fails on sight: at small \delta\phi the linear term is the larger, so it would give Newton at low acceleration and MOND at high, the reverse of what galaxies show.

In series, the two drops compete the right way round:

Element (crossed in succession) Drop for a current J In field terms Dominates
Paired seam (parity-even, the breath intact) \sqrt{J/I_2} \sqrt{a_0\,g_N} small current, g_N \ll a_0 (MOND)
Biased seam (induced linear, the un-paired breath) J/(2I_2\phi_0) g_N large current, g_N \gg a_0 (Newton)

A weak source drives little current, the square root beats the linear term, and the field is \sqrt{a_0 g_N}: the stingy paired seam forces a larger drop than Newton would need, and the larger drop is the stronger pull. A strong source drives a large current, the linear drop takes over, and the field is Newton’s. The crossover is where the two drops are equal: J_\text{cross} = 4I_2\phi_0^2, at which each drop is \delta\phi_\text{cross} = 2\phi_0. So a_0 is the bias-set drop 2\phi_0 read as an acceleration, and it is proportional to \phi_0 — the Hubble bias fixes the crossover scale directly.

The paired seam must switch off, not just be outgrown. Taken literally, the series sum gives g_\text{obs} = g_N + \sqrt{a_0\,g_N}: the right limits, but a residual \sqrt{a_0 g_N} that survives at high acceleration (about 10^{-7} m/s² at Saturn), which planetary ephemerides exclude by orders of magnitude. The breath reading says why the sum is incomplete: once the push outruns what one breath can re-close, the pair stops breathing and its element drops out of the chain. Writing the paired drop as \sqrt{a_0 g_N}\;s(x) with x = \sqrt{g_N/a_0}, McGaugh’s empirical relation corresponds exactly to

s(x) = \frac{x}{e^{x}-1}, \qquad g_\text{obs} = g_N + \sqrt{a_0\,g_N}\;\frac{x}{e^{x}-1} = \frac{g_N}{1 - e^{-x}},

a Bose–Einstein (Planck) suppression: s \to 1 in deep MOND and s \to x\,e^{-x} above the crossover, fast enough for the solar system. The same algebra gives the share of the pull the paired seam still carries, (g_\text{obs} - g_N)/g_\text{obs} = e^{-x}. This identifies where the Bose–Einstein form sits (on the paired element, below) but does not derive it.

The formula a_0 = c\sqrt{G\rho_\text{DM}} matches the observed value to ~3%; the series structure explains why a crossover between the MONDian and Newtonian regimes exists and which side each lies on. The quantitative derivation — showing that the bias-set crossover \delta\phi_\text{cross} = 2\phi_0 with \phi_0(H_0) yields specifically a_0 = c\sqrt{G\rho_\text{DM}}, and deriving the suppression s(x) from the HVBK boundary layer — is open (see Next Steps, items 2 and 3).

Note

An earlier version of this section combined the two responses by expanding J = I_2(\delta\phi + \phi_0)^2 = I_2\delta\phi^2 + 2I_2\phi_0\delta\phi + I_2\phi_0^2 and assigning the linear term to high accelerations. That sum is the parallel combination, and read with J as the given current it puts Newton at low acceleration. The series form above replaces it; the crossover condition \delta\phi_\text{cross} = 2\phi_0 survives unchanged.

Computing a_0 from substrate parameters

The MOND acceleration scale is the acceleration below which the quadratic (parity-symmetric) term dominates the induced linear (parity-broken) term. The natural substrate acceleration built from G, c, and \rho_\text{DM} is:

a_0 = c\sqrt{G\,\rho_\text{DM}}

This formula matches the observed value (see below). The parity-breaking mechanism provides a qualitative explanation for why a crossover exists; the quantitative derivation from \phi_0(H_0) to this specific formula is pending (see Next Steps, item 3).

Numerical evaluation (Planck 2018 central value, \Omega_c h^2 = 0.120, \rho_\text{DM} = 2.25 \times 10^{-27} kg/m³):

G\,\rho_\text{DM} = 6.674 \times 10^{-11} \times 2.25 \times 10^{-27} = 1.502 \times 10^{-37}\;\text{s}^{-2}

\sqrt{G\,\rho_\text{DM}} = 3.876 \times 10^{-19}\;\text{s}^{-1}

c\sqrt{G\,\rho_\text{DM}} = 2.998 \times 10^8 \times 3.876 \times 10^{-19} = 1.162 \times 10^{-10}\;\text{m/s}^2

Measured (McGaugh et al. 2016): g_\dagger = (1.20 \pm 0.02_\text{stat} \pm 0.24_\text{sys}) \times 10^{-10} m/s².

Match: ~3% low — well within the ±20% systematic uncertainty reported by McGaugh et al. The systematic uncertainty dominates and arises from distance measurements, mass-to-light ratios, and gas mass estimates.

NoteNote on \rho_\text{DM} sensitivity

The a_0 prediction depends on \rho_\text{DM} as \sqrt{\rho_\text{DM}}. The Planck 2018 central value (\Omega_c h^2 = 0.120, \rho_\text{DM} = 2.25 \times 10^{-27} kg/m³) gives a_0 = 1.16 \times 10^{-10} m/s² (3% below the McGaugh central value). A ~6% increase in \rho_\text{DM} to 2.40 \times 10^{-27} — still within Planck systematics — would yield a_0 = 1.20 \times 10^{-10} m/s² (exact central match). The bridge equation, by contrast, strongly favors the Planck central value (f match: 0.02% with Planck central vs 6.3% with 2.40). One cannot optimize both simultaneously with the same \rho_\text{DM}: the bridge equation and the a_0 formula pull in opposite directions by ~3%. Both matches are impressive; this tension is acknowledged as a feature of the current framework, not a tuning knob.

Equivalently:

\boxed{a_0^2 = G\,c^2\,\rho_\text{DM}}

Every factor on the right is a substrate parameter: c = \hbar/(m_1\xi) from C1, G from C3 (measured; its unit-free content Gm_1^2/\hbar c is not yet derived, Gravity § Mathematical Form), \rho_\text{DM} = n_1 m_1 from C10.

The “cosmic coincidence” re-expressed

Since Milgrom (1983) it has been noted that a_0 sits close to the cosmological acceleration scales: 2\pi a_0 \approx cH_0 \approx c^2\sqrt{\Lambda/3} (Verlinde’s 2016 emergent-gravity argument gives a_0 \approx cH_0/6). Through the Friedmann equation, the substrate formula is this same coincidence with a definite prefactor:

\frac{a_0}{cH_0} = \frac{c\sqrt{G\rho_\text{DM}}}{c\sqrt{8\pi G\rho_\text{DM}/(3\Omega_\text{DM})}} = \sqrt{\frac{3\,\Omega_\text{DM}}{8\pi}}

With \Omega_\text{DM} = 0.267:

\frac{a_0}{cH_0} = \sqrt{\frac{3 \times 0.267}{8\pi}} = \sqrt{0.0319} = 0.178

From measurement: a_0/(cH_0) = 1.20/6.71 = 0.179. Match: 0.5%. (This match is an algebraic consequence of the Friedmann equation once a_0 = c\sqrt{G\rho_\text{DM}} is assumed — the nontrivial content is in that relation itself, not in the H_0 connection.)

The order-unity factor is \sqrt{3\Omega_\text{DM}/(8\pi)} — a combination of Gauss’s law (8\pi), spatial geometry (3), and the dark matter fraction (\Omega_\text{DM}). Today this is a relabeling, not a discriminating success. The candidate forms cH_0/2\pi, cH_0/6, and c\sqrt{G\rho_\text{DM}} give 1.04, 1.09, and 1.16 \times 10^{-10} m/s² (for H_0 = 67.4), all inside McGaugh’s \pm 20\% systematic band, so the z = 0 value cannot single out \rho_\text{DM} over H_0 or \Lambda. (Kaplinghat & Turner (2002) likewise tie a_0 \sim cH_0 to the dark matter density, through CDM halo collapse.) Note that \rho_\text{DM} here is the cosmic mean density (Planck \Omega_c), not the local halo density, which is \sim 10^5 times higher and varies from galaxy to galaxy; a_0 is universal because the substrate density that sets it is the background value every boundary shares, not the halo concentration a ΛCDM fit would infer.

What is new is the redshift dependence. The three forms agree now and separate in the past: c^2\sqrt{\Lambda/3} is constant, cH(z) grows to \approx 1.8\,a_0(0) at z = 1, and c\sqrt{G\rho_\text{DM}(z)} grows as (1+z)^{3/2} to \approx 2.8\,a_0(0). The forms converge again deep in matter domination, so the clean window is z \approx 0.5–2 (below). The substrate claim is that a_0 tracks the dark matter density rather than the expansion rate, and that is the claim the data can falsify.

Physical interpretation

The quantity t_\text{ff} = 1/\sqrt{G\rho_\text{DM}} is the gravitational free-fall time of the substrate — the timescale on which a uniform medium of density \rho_\text{DM} would collapse under its own gravity in the absence of pressure support.

Numerically: t_\text{ff} = 2.58 \times 10^{18} s \approx 82 Gyr \approx 5.9\,t_\text{Hubble}.

Therefore:

a_0 = \frac{c}{t_\text{ff}}

a_0 is the acceleration of light over one gravitational free-fall time of the substrate.

High-acceleration regime (a \gg a_0, \tau_\text{pert} \ll t_\text{ff}): The gravitational perturbation from a baryonic source evolves on timescales much shorter than the substrate’s self-gravitational response time. Each boundary scatters the ebbing current independently — the response is incoherent, and the induced linear term (from Hubble parity-breaking) dominates. Standard Newtonian gravity.

Low-acceleration regime (a \ll a_0, \tau_\text{pert} \gg t_\text{ff}): The perturbation evolves slowly enough for the substrate to respond coherently. The parity-symmetric quadratic CPR dominates. The coherent response amplifies the gravitational signal, producing MOND phenomenology.

Substrate identification of each factor

Factor in a_0^2 = Gc^2\rho_\text{DM} Substrate origin Source
c = \hbar/(m_1\xi) Quasiparticle speed in BEC regime C1, Volovik Ch. 7
4\pi G\rho_\text{DM} = (f_\text{cross}\,\omega_1)^2 Ebbing current through boundaries C3, Gravity
\rho_\text{DM} = n_1 m_1 Substrate mass density C10

If the bridge equation holds, \rho_\text{DM} today is determined by \sin^2\theta_W and m_e (the bridge equation fixes the cell density, which equals the cosmic mean only in the present epoch; see Which density sets a_0):

a_0 = c\sqrt{G\,\rho_\text{DM}(\sin^2\theta_W,\, m_e)}

This extends the bridge equation’s domain from four (electroweak → QM → GR → cosmology) to seven: electroweak → QM → GR → cosmology → galactic dynamics → dark energy → structure formation. The last two links were established by the moraine crust analysis and sharpened by the 15-knot freeform spline — the same \alpha_{mf} that controls boundary transmission at the galactic scale also sets the crust disruption efficiency (\eta_\text{crust} = 2\alpha_{mf}^2) that suppresses structure growth and shapes the dark energy equation of state.

The Boundary Is the Breath

Step back from the algebra and the whole chapter is one object read at galactic scale. The counter-rotating boundary whose parity forces the quadratic CPR is the framework’s anti-phase breath: the +\omega_0/-\omega_0 layer is a circulation and its counter-rotating partner trading energy across a shared seam, the same paired motion that binds Cooper electrons, orders an antiferromagnet, and breathes in the lattice’s own pairing. That is why the boundary is parity-even and the CPR quadratic — a pair has no preferred partner. The parity symmetry was never a separate postulate; it is the pairing symmetry of the breath, and MOND is what the intact, paired boundary does.

The two regimes then read as the breath’s two states. Where a perturbation evolves more slowly than the substrate’s gravitational free-fall time (a \ll a_0, \tau_\text{pert} \gg t_\text{ff}), the boundary stays coherently paired and answers with its parity-even, lossless quadratic response: deep MOND is the breath intact. The cosmic expansion’s DC phase bias splits the +\omega_0/-\omega_0 symmetry — it singles out a direction the pair did not have — and the induced linear term is the un-paired breath: Newtonian gravity is the response of a boundary whose pairing has been pried open. The local external field of a neighbouring mass breaks the same pairing in the same way — a static phase bias on the same boundaries — so the external-field effect and the very existence of a Newtonian limit are not two phenomena but one parity-breaking seen locally and cosmologically.

Stacked N = r/\xi deep, these paired-breath boundaries are the breath replicated across scale — the ladder’s self-similar motif in gravitational form, a coarse-family tower spaced by the cell size \xi. And in the ladder’s small economy, the paired breath is the coin: the lossless hand-off a structure plugged into the channel can spend. A flat rotation curve is what it looks like when an entire galaxy spends that coin coherently — every boundary breathing in the same parity-even way out to the MOND radius.

The boundary-layer simulation draws these two states one cell across: the paired breath letting the river through only in its two windows per cycle, so the step it holds (the pull) stands above Newton’s, and the pried-open pair leaking straight through once the push outruns the breath. Its tour also stops at the external-field knob and on Maisie’s cloud at z = 11.4.

Galactic gravity therefore sits squarely on the ladder’s locking pole, and sits there universally: there is one tooth — the single paired-breath boundary — occupied identically at every radius in every galaxy. That universality is the chapter’s tightest claim and its sharpest falsifier. Because every boundary is the same breath, the radial acceleration relation may carry exactly one environmental knob — the external-field bias that locally un-pairs it — and no other. Any residual RAR variation keyed to morphology, formation history, gas fraction, or void-versus-wall membership after the external-field effect is removed would mean the tooth is not universal, and the paired-breath reading would fail. The anti-lock \varphi gap that shadows the ladder elsewhere does not appear in galactic gravity, and by the sign rule it should not: a galaxy’s job is to bind, and binding is a locking job.

The same paired boundary keeps a second, directly observable ledger. The counter-rotating sheath whose parity-even response is the flat rotation curve also writes a parity-odd magnetic record: two enormous magnetic toroids of opposite circulation above and below the Galactic plane, photographed in the Faraday-rotation sky and extending, without a single reversal, from less than 2 kpc to beyond 15 kpc. Galactic Magnetic Fields reads that record — the anti-phase breath made visible.

The Radial Acceleration Relation

10-12 10-12 10-11 10-11 10-10 10-10 10-9 10-9 g_bar (m/s²) g_obs (m/s²) g_obs = g_bar (Newton) g_obs = √(a₀ g_bar) (deep MOND) a₀ Substrate prediction McGaugh+ 2016 Newton (no dark matter) Predicted from sin²θ_W = 0.2312, m_e, and ρ_DM via the bridge equation. Interpolation form from McGaugh; a₀ from substrate.

Empirical target

McGaugh, Lelli & Schombert (2016) measured the radial acceleration relation (RAR) across 2693 data points in 153 galaxies:

g_\text{obs} = \frac{g_\text{bar}}{1 - e^{-\sqrt{g_\text{bar}/g_\dagger}}}

with single parameter g_\dagger = 1.20 \times 10^{-10} m/s². The dark matter contribution has a Bose-Einstein-like form:

g_\text{DM} = \frac{g_\text{bar}}{e^{\sqrt{g_\text{bar}/g_\dagger}} - 1}

What the substrate predicts

The boundary parity mechanism produces:

  • Deep MOND (g_\text{bar} \ll a_0): g_\text{obs} = \sqrt{a_0\,g_\text{bar}} — from the quadratic CPR. ✓
  • Newtonian (g_\text{bar} \gg a_0): g_\text{obs} = g_\text{bar} — from the induced linear term. ✓
  • a_0 = 1.16 \times 10^{-10} m/s² — from c\sqrt{G\rho_\text{DM}} with zero free parameters (~3% below the McGaugh central value; well within systematic uncertainty). ✓
  • Negligible intrinsic scatter — every boundary in the substrate has the same parity symmetry and the same Hubble bias. The per-boundary physics is universal, so the framework predicts scatter consistent with observational error alone. (Any detected intrinsic scatter would require explanation — perhaps local variations in \rho_\text{DM} or lattice defects.) ✓
  • No dependence on galaxy type — the relation depends only on the local baryonic acceleration and the universal substrate parameters, not on morphology, surface brightness, or gas fraction. ✓

The interpolation function

The exact form of the interpolation between MOND and Newton depends on the detailed response of the counter-rotating boundary at intermediate accelerations (a \sim a_0), where the paired and biased drops are comparable. The series structure fixes the limits and places the unknown on one element: McGaugh’s function is the series sum with the paired drop suppressed by s(x) = x/(e^x - 1). Deriving this requires solving the HVBK equations for the boundary layer at finite DC bias — a calculation that has not yet been performed.

The Bose-Einstein-distribution form of McGaugh’s empirical function (g_\text{DM} \propto 1/(e^{\sqrt{g_\text{bar}/g_\dagger}} - 1), equivalently the Planck factor x/(e^x-1) on the paired drop) is suggestive: it implies the boundary transmission involves quantum statistics — specifically, the Bose-Einstein occupation of phonon modes at the boundary. This is physically natural for a superfluid boundary where the available transmission channels are quantized.

NoteOpen problem

Derive the interpolation function \mu(a/a_0) from the HVBK equations for dc1 transmission through a counter-rotating vortex layer as a function of incident flow velocity. The Bose-Einstein form of the McGaugh function suggests a thermal distribution of transmission channels.

Connection to Khoury’s Superfluid Dark Matter

The mapping

Berezhiani & Khoury (2015, 2016) showed that a superfluid dark matter condensate with phonon-baryon coupling reproduces MOND phenomenology in galaxies while recovering CDM behavior on cluster and cosmological scales. The program has since been consolidated and substantially revised in the 2025 Physics Reports review [R116] (Berezhiani, Cintia, De Luca & Khoury; see below); the table reflects the review’s updated numbers. The substrate framework reproduces Khoury’s key results through a different microscopic mechanism:

Khoury framework (2025 review) Substrate framework
DM particle mass m \lesssim \mueV for kpc-scale cores (revised from \sim eV in 2015) m_1 = 2.04 meV/c^2 — \sim 2000\times heavier, but the entire substrate is superfluid
EFT scale \Lambda \sim meV, \alpha \sim \mathcal{O}(1) m_1 c^2 \approx 2 meV (natural energy scale of dc1)
\mathcal{L} \propto X\sqrt{\lvert X - \beta Y\rvert}, \beta \geq 3/2 (finite-T stabilization of X^{3/2}, chosen to get MOND) Quadratic CPR from boundary parity (MOND emerges from symmetry)
Phonon-baryon coupling \alpha\Lambda/M_\text{Pl} (explicitly breaks the shift symmetry) Ebbing current transmission f_\text{cross}
a_0 = \alpha^3\Lambda^2/M_\text{Pl} (imposed; constant in z) a_0 = c\sqrt{G\rho_\text{DM}} (derived; evolves as (1+z)^{3/2})
Soliton core + superfluid debris streams + degenerate collisionless outskirts Uniform superfluid substrate, velocity-dependent transition
Superfluid region bounded by degeneracy + thermalization conditions Landau critical velocity v_L \approx 750 km/s separates regimes

What’s different

In Khoury’s framework, the MOND phenomenology is built into the choice of Lagrangian: P(X) \propto X\sqrt{|X|} is selected specifically because it reproduces the Bekenstein-Milgrom equation (with the finite-temperature \beta Y deformation added to stabilize its perturbations). The acceleration scale a_0 is a free parameter encoded in the combination \alpha^3\Lambda^2/M_\text{Pl}, with \alpha \sim \mathcal{O}(1) and \Lambda \sim meV chosen to land on the observed value.

In the substrate framework, neither the field equation nor the acceleration scale is put in by hand. The MOND field equation emerges from the parity symmetry of the counter-rotating boundary — a structural feature of the substrate that was introduced to explain gravity and the quantum potential, not galactic dynamics. The acceleration scale a_0 = c\sqrt{G\rho_\text{DM}} is determined by the same parameters that determine everything else in the framework. No new degrees of freedom are needed.

What’s the same

Both frameworks share the core insight from Khoury: the CDM-to-MOND transition is a phase transition (or regime change) of a single substance, not a modification of gravity. The substrate makes this concrete: the “phase transition” is the crossover from coherent (quadratic CPR) to incoherent (linear CPR) boundary transmission, governed by the Landau critical velocity.

The 2025 Physics Reports review

The 2025 review [R116] (Berezhiani, Cintia, De Luca & Khoury, arXiv:2505.23900) consolidates a decade of the superfluid DM program and revises it in ways that sharpen the comparison above.

What changed. The viable particle mass dropped from \sim eV to m \lesssim \mueV — kpc-scale superfluid cores require \lambda_J \simeq 30\,\text{kpc}\,(m/\mu\text{eV})^{-5/4}(\sigma/m / 10^{-8}\,\text{cm}^2\text{g}^{-1})^{1/4}. The halo anatomy was rebuilt: the original superfluid-core-plus-NFW-envelope picture is replaced by a central superfluid soliton, a middle region of superfluid debris streams from tidally disrupted solitons, and outskirts that are degenerate but collisionless and out of equilibrium — the global-thermal-equilibrium assumption was deliberately dropped. And the zero-temperature phonon Lagrangian was found to have unstable perturbations around spherical backgrounds, cured only by the ad hoc finite-temperature deformation X\sqrt{|X - \beta Y|} with \beta \geq 3/2.

What the review concedes. The authors state plainly that no microscopic theory of DM particles is known to produce the |X| structure of the MOND phonon Lagrangian, and that “all known (to us) proposals for MOND exhibit some form of pathology.” This is precisely the gap the boundary-parity mechanism addresses: in the substrate, the quadratic response is not selected to reproduce MOND — it is forced by the parity symmetry of the counter-rotating boundary. The review also catalogs cosmological tensions the substrate does not inherit: kpc-scale cores imply a pressure-to-density ratio P/\rho \sim 10^{-5} at matter-radiation equality (a relativistic equation of state uncomfortably close to equality), hydrodynamical simulations suppress all halos below 10^{14}\,M_\odot, and simulated SDM halos fail the dwarf-spheroidal core scaling relations. These are consequences of making the superfluid itself clumpy; a uniform substrate with a velocity-gated response avoids them by construction.

Two sharp discriminators. The review makes the conditional character of SDM’s MOND regime explicit, which turns two of the substrate’s predictions into clean forks:

  • External field effect. In SDM, the EFE exists only when both bodies sit inside the same superfluid core — cluster galaxies feel none, and satellites feel it only if their host’s core reaches them. In the substrate, the EFE is universal: any external field adds a DC phase bias to the same boundaries everywhere. The Chae et al. statistical detection of the EFE in SPARC (cited in the review) is consistent with both; a detection of EFE-driven behavior in systems outside any plausible superfluid core would separate them.
  • Tidal dwarf galaxies. In SDM, a TDG shows MOND-like dynamics only if it resides within its host’s superfluid region; at the tips of tidal tails this is marginal at best. The substrate predicts unconditional BTFR compliance for TDGs — the boundary physics does not care where the baryons came from.

A third discriminator is already in the table: the review’s only comment on the a_0–cosmology connection is the empirical coincidence a_0 \approx cH_0/2\pi, and its a_0 = \alpha^3\Lambda^2/M_\text{Pl} is an environment-independent constant. The substrate’s a_0 = c\sqrt{G\rho_\text{DM}} restates the coincidence with a definite prefactor and, unlike either constant or cH(z) forms, predicts (1+z)^{3/2} evolution.

A point of convergence. The review calls vortex formation in SDM halos “unavoidable” — N \sim 10^{18} meter-scale vortices per Milky-Way halo, with recent simulations forming a spin-aligned vortex-line network and “vortex lattices that preserve angular momentum through the phase transition,” potentially detectable as lensing de-magnification anomalies. Khoury’s own program is thus converging on the structure the substrate starts from: a rotating vortex lattice whose coherence threshold is exactly the v_L that separates the two regimes.

The SVT-internal alternative: Zloshchastiev’s multi-scale gravity

Khoury’s is not the only superfluid competitor, and the other one is closer to home: it runs on the same equation of state the substrate does. Zloshchastiev — whose logarithmic EOS is the framework’s own defining nonlinearity ([R14], [R119]) — explains galaxy rotation curves within superfluid vacuum theory by a different mechanism ([R153]): the log superfluid’s density profile induces a multi-scale gravitational potential whose regimes run from sub-Newtonian through Newtonian, logarithmic, linear, and quadratic terms with galactocentric distance, and per-galaxy fits of that profile closely track the observed curves. No boundary layer, no MOND, no acceleration scale — scale-dependent gravity from the medium’s density structure.

The two mechanisms live inside one equation, and they are honestly distinguishable rather than interchangeable:

  • His observables are per-galaxy: each rotation curve is a fit of that galaxy’s density-profile coefficients. Universal regularities across galaxies are not forced by the mechanism.
  • The substrate’s observables are universal: the counter-rotating boundary’s parity-forced quadratic CPR delivers the MOND field equation once, for every galaxy, with a_0 = c\sqrt{G\rho_\text{DM}}, the baryonic Tully–Fisher normalization, the \sim 137\ M_\odot/\text{pc}^2 halo surface density, and negligible intrinsic RAR scatter as zero-parameter consequences.
  • The discriminator is evolution: the substrate predicts a_0(z) \propto (1+z)^{3/2} with the localized crust dip at z \approx 0.5 (below); a density-profile mechanism carries no corresponding universal redshift law.

The right framing is not a collision but a question inside SVT: which limit of the logarithmic equation does a rotating, close-packed vacuum realize — the quiescent density profile, or the textured vortex lattice whose boundaries respond quadratically? The data that decides it is the universality data: the RAR’s tiny scatter and the BTFR’s zero-parameter normalization are natural in the boundary reading and unforced in the profile reading.

Redshift Evolution

Which density sets a_0

The framework uses “\rho_\text{DM}” for two quantities that are numerically equal today but not at other epochs.

  • The cell density f\hbar/c\xi^4 = n_1 m_1 at close packing. It is fixed microphysics: the logarithmic equation of state pins \xi and m_1 by a coupling energy, not a density (Why c stays fixed). It sets G, the leak rate \Gamma, and f_\text{cross} (Gravity), and it is the quantity the bridge equation fixes from \sin^2\theta_W and m_e. It cannot evolve: a G that drifted with the cosmic density would violate the lunar-laser-ranging bound on \dot G/G by orders of magnitude.
  • The cosmic mean \bar\rho_\text{DM}(z) = \rho_\text{DM}(0)\,(1+z)^3, the Planck \Omega_c density. It dilutes with expansion, the substrate clumping to keep its local density near the marginal point while the mean falls.

The two coincide at z = 0, and why now is an open problem (WIP-15 item 2a). The framework’s assignment is that a_0 = c\sqrt{G\bar\rho_\text{DM}} uses the cosmic mean: a_0 is the free-fall rate of the background the boundaries are embedded in, with G held fixed by the cell. That assignment is what produces the redshift law below. It is an assumption, and the redshift test is also its test. If a_0 were instead set by the cell density, it would be constant in time — observationally the c^2\sqrt{\Lambda/3} reading — and the (1+z)^{3/2} prediction would be gone. A consequence: identities that equate a_0 with fixed microphysics, such as a_0 = c\Gamma/\sqrt{4\pi} and the bridge-equation route to a_0, hold today only. At other epochs, a_0(z) = (c\Gamma/\sqrt{4\pi})\,(1+z)^{3/2}.

Baseline scaling

Since \bar\rho_\text{DM}(z) = \rho_\text{DM}(0)\,(1+z)^3 in standard cosmology, and G and c are fixed:

a_0(z) = c\sqrt{G\,\bar\rho_\text{DM}(z)} = a_0(0)\,(1+z)^{3/2}

This follows directly from \rho_\text{DM}(z) = \rho_\text{DM}(0)(1+z)^3. At z = 1: a_0(z=1) = a_0(0) \cdot 2^{3/2} \approx 2.83\,a_0(0) \approx 3.3 \times 10^{-10} m/s² — a large, potentially measurable difference from today’s value. At higher redshift, a_0 is larger — the coherent regime extends to higher accelerations, so the MOND transition radius r_M = \sqrt{GM_b/a_0} shrinks and the enhancement reaches deeper into galaxies. At fixed baryonic mass and physical radius, a high-redshift galaxy should show stronger enhancement — a higher apparent dark matter fraction — than its low-redshift counterpart (early-structure-formation). This is a sharp commitment, and the current observational picture is honestly mixed: some z \approx 1–2.5 kinematic studies report declining outer rotation curves read as baryon-dominated disks with smaller dark matter fractions — the opposite direction — while others find rotation curves consistent with higher effective mass-to-light ratios. The samples differ in tracer, radius probed, and pressure-support corrections, which are large at these redshifts. The RAR-evolution test below is the discriminator; if the baryon-dominated reading of the high-z disks survives better data, the (1+z)^{3/2} scaling is in genuine trouble.

The moraine crust modifies a_0(z)

The baseline (1+z)^{3/2} scaling assumes that both G and \rho_\text{DM} evolve smoothly. The moraine crust from \mathcal{B}^{-1} — the remnant boundary of the previous bubble — disrupts this assumption. When the expanding \mathcal{B}^0 bubble wall passed through the crust, it produced a structured undular-bore response in the substrate. A 15-knot freeform spline fit to the combined DESI BAO + Jia H_0(z) data — now permitting negative amplitudes (local reductions below the Volovik base density) — fully resolves this structure, achieving \chi^2_\text{total} = 9.68 vs \LambdaCDM’s 29.0. The freed negative amplitudes halved the residual from the earlier 12-knot non-negative fit (\chi^2 = 21.0), confirming that the troughs are real and carry physical information — four knots that the old optimizer had pegged at zero were really negative excursions it could not represent.

The resolved profile maps cleanly onto the three-piece Grimshaw–Smyth anatomy from the El & Hoefer (2016) transcritical framework:

Region 1 — the upstream dispersive shock wave (z > 1.60). The leading soliton peaks at z = 2.30 (amplitude +0.49) with a shoulder at z = 2.50 (+0.29), consistent with a \text{sech}^2 profile whose peak lies between z = 2.20 and z = 2.30 — bracketed by the two knots. The predicted contact redshift z_b = 2.20 (from the substrate relaxation timescale) sits right in this interval. Behind the soliton, a deep trough at z = 2.00 (-0.37) marks the rarefied wake: the bubble wall passing through the moraine at supersonic speed creates a local density depression in \rho_\Lambda. The trough depth (-0.37) is ~75% of the soliton amplitude (+0.49) — exactly the ratio that the forced KdV equation predicts for the first trough behind a leading soliton in transcritical flow (typically 60–80%). This soliton-plus-wake pair is the cleanest single feature in the entire profile: the signature of the initial supersonic encounter, with quantitatively correct amplitude ratio. The upstream bore consists of just this one oscillation between soliton and recovery zone — a partial, attached upstream DSW, exactly what Grimshaw–Smyth predict for a mildly supercritical encounter (M_b = 1.30 at z_b).

Region 2 — the recovery zone (z \approx 1.60). The knot at z = 1.60 sits at -0.09 — essentially at the node, but slightly below zero. This small negative value is new information from the 15-knot fit and is physically meaningful. In the GS framework, the recovery zone is where the upstream and downstream DSWs meet at M = 1. The hydraulic transition carries energy away from z_\text{crit} in both directions — into the upstream soliton and into the downstream train. At exactly M = 1, the wave amplitude should vanish; the slightly negative value means the M = 1 transition has maximally extracted the moraine’s vortex energy, leaving a local energy deficit that goes slightly beyond removing the enhancement to borrowing from the Volovik floor. The node sits within \Delta z = 0.012 of the zero-parameter prediction z_\text{crit} = 1.588.

Region 3 — the downstream undular train (z < 1.30). Five to six oscillation cycles are visible, with the dark energy amplification creating a striking amplitude modulation that makes low-z crests rival or exceed the soliton edge. The crests at z = 1.30, 0.80, 0.45, 0.23, and 0.07 alternate with troughs at z = 1.00, 0.60, 0.38/0.30, and 0.15, tracing the full carrier-wave structure of the downstream DSW.

The 15-knot spline knots at best fit:

Knot z Amplitude Type
0 0.00 +0.164 Ridge
1 0.07 +0.251 Ridge
2 0.15 +0.055 Ridge
3 0.23 +0.704 Ridge
4 0.30 -0.311 Void
5 0.38 -0.356 Void
6 0.45 +0.279 Ridge
7 0.60 -0.131 Void
8 0.80 +0.721 Ridge (largest)
9 1.00 -0.111 Void
10 1.30 +0.198 Ridge
11 1.60 -0.089 Void (at z_\text{crit})
12 2.00 -0.365 Void (post-soliton wake)
13 2.30 +0.489 Ridge (soliton peak)
14 2.50 +0.289 Ridge (soliton shoulder)

The narrow G_\text{eff} suppression near z \approx 0.5 operates as a separate channel on the growth equation only, from downstream boundary recovery.

The G_\text{eff} suppression directly modifies a_0. Since a_0 = c\sqrt{G_\text{eff}\,\rho_\text{DM}}, a fractional suppression \eta in G_\text{eff} reduces a_0 by a factor \sqrt{1 - \eta}:

a_0(z) = a_0(0)\,(1+z)^{3/2}\,\sqrt{1 - \eta(z)}

where \eta(z) is the redshift-dependent disruption profile. Near z \approx 0.5 this is the downstream G_\text{eff} channel, \eta_\text{down} = 2\alpha_{mf}^2(1 - \sin^2\theta_W) = 0.139 — the Weinberg-anchored efficiency 2\alpha_{mf}^2 = 0.181 reduced by one further power of the angle (the dominant 0.181 channel sits deeper, at the transcritical crossing z \approx 1.59). This produces a dip in a_0(z) relative to the smooth (1+z)^{3/2} baseline: galaxies at z \approx 0.5 should show slightly weaker MOND enhancement than the baseline predicts, by roughly 7–9%.

NoteThe crust effect on a_0 is small but specific

The G_\text{eff} suppression is narrow (\sigma \approx 0.30 in redshift) and centered at z \approx 0.5. By z \approx 1, the effect is negligible and the baseline (1+z)^{3/2} scaling resumes. The modification is a localized correction to the smooth trend, not a revision of the fundamental scaling. But it connects galactic dynamics to the moraine crust encounter — the same physics that resolves the Hubble tension and explains the DESI dark energy evolution. The undular-bore structure visible in the 15-knot fit adds rhythmic detail to this correction: the downstream crests and troughs modulate \rho_\Lambda in an alternating pattern that the smooth envelope averaged over, and the freed negative amplitudes reveal that several of these troughs represent genuine local depressions below the Volovik base density.

The transcritical crossing at M = 1

The crust encounter’s structured undular-bore response has a remarkably clean origin. The Mach number of the bubble expansion — defined as M(z) = H(z) \times d_\text{proper}(z) / c, the recession velocity at the moraine’s location divided by the speed of light — crosses unity at z = 1.588. This number comes entirely from Planck 2018 cosmological parameters (H_0 = 67.4 km/s/Mpc, \Omega_m = 0.315, \Omega_\Lambda = 0.685) with no substrate parameters involved.

Redshift M(z) Flow regime What happens
z = 2.2 (z_b) 1.30 Supercritical Bubble wall is supersonic relative to moraine — disturbance carried forward
z = 1.59 1.00 Critical Recovery-zone node — energy redistributed to upstream soliton and downstream train
z = 0.63 (z_s) 0.47 Subcritical CPL crossing; also where q = 0 (deceleration → acceleration)
z = 0.52 (fitted dip) 0.40 Subcritical G_\text{eff} suppression zone peaks here

The smooth-envelope model (GD15-original) had both \sqrt{} branches meeting at z_\text{crit} to give a peak — the maximum of the enhancement. The 15-knot freeform spline reveals that z_\text{crit} is in fact a node of the crust enhancement: the spline places the z = 1.60 knot at -0.089, slightly below zero. This is exactly the Grimshaw–Smyth prediction for the transcritical (M = 1) point in a forced dispersive medium: the M = 1 crossing is a pressure node (a boundary between compressed and rebounding regions), not a peak. The slightly negative value means the recovery zone has maximally extracted vortex energy, leaving a local deficit that borrows from the Volovik floor.

The deceleration through M = 1 during the encounter generates:

  • An upstream dispersive shock wave in \rho_\Lambda — a sharp \text{sech}^2 soliton peaking at z = 2.30 (+0.49) with a deep rarefied wake at z = 2.00 (-0.37). The wake depth is ~75% of the soliton amplitude, matching the forced KdV prediction (60–80%) for the first trough behind a transcritical leading soliton. This soliton-plus-wake pair is a partial, attached upstream DSW — one oscillation between the soliton and the recovery zone, as Grimshaw–Smyth predict for a mildly supercritical encounter (M_b = 1.30).
  • A localized recovery-zone depression at z = z_\text{crit} — the node where the supercritical wave train has fed forward but the subcritical wave train has not yet built amplitude. The 15-knot spline places this at -0.089, confirming the Grimshaw–Smyth recovery-zone prediction and revealing the slight energy deficit at M = 1.
  • A DE-amplified downstream undular train — five to six carrier oscillation cycles with crests at z \approx 1.30, 0.80, 0.45, 0.23, and 0.07, where the dark energy fraction \Omega_\Lambda(z) progressively amplifies lower-redshift crests.
  • A narrow G_\text{eff} suppression near z \approx 0.5 — from downstream boundary recovery, a separate channel that operates on the growth equation only.

The chirped bore: wavelength compression at low z

The 15-knot fit resolves additional crests (at z = 0.45 and z = 0.23) that the earlier 12-knot fit could not distinguish, splitting what appeared to be single long-wavelength oscillations into multiple shorter cycles. The ridge-to-ridge spacings reveal a cosmologically chirped dispersive shock wave:

Crest pair \Delta z Midpoint z Proper distance spacing
2.30 \to 1.30 (across recovery) 1.00 1.80 ~1638 Mpc
1.30 \to 0.80 0.50 1.05 ~1212 Mpc
0.80 \to 0.45 0.35 0.625 ~1240 Mpc
0.45 \to 0.23 0.22 0.34 ~870 Mpc
0.23 \to 0.07 0.16 0.15 ~640 Mpc

In \Delta z the wavelength compresses by a factor of ~6 from the first downstream cycle to the last — but this overstates the intrinsic compression because the mapping from redshift interval to proper distance is strongly nonlinear, with dD/dz \approx c/H(z) increasing sharply at low z. The proper-distance column reveals the physical pattern: spacings decrease from ~1200 Mpc at z \approx 1 to ~640 Mpc at z \approx 0.1.

This proper-distance compression matches the standard KdV DSW prediction. In a dispersive shock wave, moving from the soliton edge toward the harmonic edge, the wavenumber k increases (wavelength decreases). The soliton edge is at z \approx 2.3 and the harmonic edge is at z_\text{harm} \approx -0.25 (in our future). Moving from z = 1.3 toward z = 0, one moves from the soliton side toward the harmonic side — and the wavelength is decreasing, exactly as the rank-ordered structure demands. The observed z-space compression is even more dramatic than the proper-distance compression because the Hubble expansion piles more proper distance into each \Delta z at low z — two effects layered: the intrinsic DSW wavelength decrease plus the cosmological \Delta z-compression from the expansion history. The combined effect creates the visually striking chirped pattern.

Read through the substrate ladder, this chirped, rank-ordered train is discrete scale invariance made cosmologically visible — a self-similar tower of wave cycles whose spacing changes by a fixed rule down the sequence. As with the boundary chain above, it is a coarse-family ladder: the period is set by the KdV dispersion of the moraine bore, not by the substrate’s \sqrt2 pairing rung, so it is a genuine log-structured tower without being a \sqrt2 comb (the dark-energy crust chapter develops this reading).

Amplitude demodulation: the rank-ordered carrier

The most physically telling aspect of the 15-knot profile — and perhaps the single most publishable result — is the amplitude demodulation argument. The observed crest amplitudes do not decrease monotonically from the soliton edge; instead, the z = 0.80 and z = 0.23 crests appear comparable to or larger than the soliton itself. But this hierarchy is inverted by the dark energy amplification.

Strip away the DE weighting [\Omega_\Lambda(z)]^\gamma with \gamma \approx 3:

Crest z Observed amp \Omega_\Lambda(z)/\Omega_\Lambda(z_\text{crit}) DE factor Implied bare carrier amp
1.30 +0.20 ~2.4 ~14 ~0.014
0.80 +0.72 ~4.0 ~64 ~0.011
0.45 +0.28 ~5.2 ~140 ~0.002
0.23 +0.70 ~5.9 ~205 ~0.003
0.07 +0.25 ~6.3 ~250 ~0.001

The bare carrier amplitudes decrease roughly monotonically from z = 1.30 to z = 0.07 — falling by about an order of magnitude. That is the rank ordering of a DSW: the soliton edge has the largest amplitude and each successive wave is smaller. The dark energy amplification then inverts this hierarchy observationally, making low-z crests appear enormous. The z = 0.80 crest appears dominant because it sits at the sweet spot where the carrier is still reasonably strong AND the DE amplification is already substantial.

This gives a lever on \gamma: the observed amplitude ratio between crests directly constrains how steeply the DE amplification must compensate for the decaying carrier. The fact that z = 0.23 (+0.70) is nearly as large as z = 0.80 (+0.72) despite being much farther from the soliton edge means the DE factor must be growing fast enough to compensate for roughly another factor of 3–4 in carrier decay — consistent with \gamma in the 2.5–3.5 range.

ImportantThe amplitude demodulation test

If one can show that dividing out the [\Omega_\Lambda(z)]^\gamma factor leaves a monotonically decreasing (rank-ordered) carrier, that is a single-figure demonstration that the observed dark energy structure is a cosmologically amplified dispersive shock wave. This is the cleanest diagnostic available: it separates the intrinsic DSW physics (decreasing carrier) from the cosmological amplification (\Omega_\Lambda weighting), and the rank ordering is a necessary consequence of the Whitham modulation equations for any KdV undular bore.

The DESI/Jia tension at z \approx 0.5

The biggest remaining tension in the fit is a direct conflict between one DESI 2 observation and the middle Jia bin at z \approx 0.5. The spline shows why: knots 4–6 span z = 0.30 to 0.45 with a steep void-to-ridge transition (-0.31 at z = 0.30, -0.36 at z = 0.38, +0.28 at z = 0.45). A BAO measurement centered at z = 0.5 averages over this steep gradient, and the effective redshift of the measurement can shift the comparison value significantly. Improved binning in the z = 0.3–0.7 range — which DESI’s growing dataset will enable — would directly test the rapid oscillatory structure predicted by the downstream carrier wave.

The cosmic coincidence evolves

The ratio a_0/(cH_0) at redshift z is:

\frac{a_0(z)}{c\,H(z)} = \sqrt{\frac{3\,\Omega_\text{DM}(z)}{8\pi}}

where \Omega_\text{DM}(z) = \Omega_\text{DM}(0)(1+z)^3/E^2(z) and E(z) = H(z)/H_0. The “cosmic coincidence” is not fine-tuned to the current epoch — it evolves smoothly and is \mathcal{O}(0.1\text{–}0.3) for all z \lesssim 3. This is a genuine distinguishing prediction: standard MOND has constant a_0; the substrate predicts a_0(z) \propto (1+z)^{3/2}, testable by DESI/Euclid/SKA galaxy rotation curves at z \sim 0.5\text{–}2.

Note that in the substrate framework, H(z) itself is modified by the crust encounter — the Jia et al. (2025) binned reconstruction of H_0(z) from DESI DR2 data shows a smooth descent from \sim 72.2 km/s/Mpc at z = 0.1 to \sim 67.2 km/s/Mpc at z = 2.5, which the crust model reproduces from the \rho_\Lambda undular-bore structure amplified by the large dark energy fraction at low z (see Dark Energy and the Crust). The 15-knot freeform spline fit gives \chi^2_\text{total} = 9.68 (vs \LambdaCDM’s 29.0) with H_0(\text{local}) = 71.8 km/s/Mpc, consistent with SH0ES within \sim 1.2\sigma. The undular-bore structure resolves the -2\sigma residual at z = 0.3 that plagued earlier fits by placing a trough there — the 15-knot fit’s deep voids at z = 0.30 (-0.31) and z = 0.38 (-0.36) naturally accommodate this feature. This means the a_0/(cH) ratio at low z is slightly smaller than the baseline prediction — a second-order effect, but one that tightens the connection between galactic dynamics and cosmology.

What This Section Adds to the Constraint System

New constraint: C14 — MOND acceleration scale

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

Parameters: c (from C1), G (from C3), \rho_\text{DM} (from C10). Zero new parameters.

This is a zero-parameter prediction, not a fit: c, G, and \rho_\text{DM} are determined by other parts of the framework (or measured independently), and a_0 is measured independently by McGaugh et al.

Predictions table

Quantity Predicted Measured Discrepancy Source
a_0 (MOND acceleration) 1.16 \times 10^{-10} m/s² (1.20 \pm 0.02_\text{stat} \pm 0.24_\text{sys}) \times 10^{-10} m/s² ~3% low c\sqrt{G\rho_\text{DM}}
Flat rotation curves Derived Observed universally — Quadratic CPR → MOND
Baryonic Tully-Fisher M_b \propto v^4 Derived M_b \propto v^{3.98 \pm 0.06} — Consequence of MOND
Negligible intrinsic RAR scatter Predicted Observed (< 0.1 dex) — Universal boundary physics
Galaxy/cluster transition v_L \approx 750 km/s v_\text{disp,cluster} \sim 1000 km/s Correct separation Landau critical velocity
a_0/(cH_0) ratio \sqrt{3\Omega_\text{DM}/(8\pi)} = 0.178 0.179 ~0.5%* Friedmann + C10
Transcritical crossing Recovery-zone node at z = 1.588 15-knot spline: -0.089 at z = 1.60 \Delta z = 0.012 M(z) = H(z) \times d_\text{proper}(z) / c
Soliton edge at z_b Sharp crest at z = 2.20 15-knot spline: +0.489 at z = 2.30 \Delta z = 0.10 Bubble wall contact
Post-soliton wake Deep trough at ~75% of soliton amp 15-knot spline: -0.365 at z = 2.00 75% (predicted 60–80%) Forced KdV
Downstream wavelength trend \lambda decreasing soliton → harmonic edge Proper-distance \lambda: 1240 → 870 → 640 Mpc Correct ordering KdV DSW rank ordering
Bare carrier rank ordering Monotonically decreasing after DE demodulation Bare amps: 0.014 \to 0.011 \to 0.002 \to 0.003 \to 0.001 Monotonic (with one mild swap) Whitham modulation theory
Inflation e-folds N_* \ln(c/H_0\xi) \approx 69 N_* \approx 60 (~15%, correction sign known) Inflation geometry
Growth suppression S_8 \eta_\text{crust} = 2\alpha_{mf}^2 \to S_8 \approx 0.79 Weak lensing: 0.76–0.79 At the upper limit Weinberg angle → boundary disruption

*Follows algebraically from a_0 = c\sqrt{G\rho_\text{DM}} plus the Friedmann equation; not an independent prediction.

Seven-domain bridge

The quadratic CPR as shared physics between galactic dynamics and cosmological structure A flowchart showing how the counter-rotating boundary's parity symmetry branches into two independent observational domains: galactic dynamics via the MOND field equation, and cosmological structure via crust disruption of the boundary coherence, both controlled by the same mutual friction parameter alpha_mf. Counter-rotating boundary Parity: J(+δφ) = J(−δφ) Quadratic CPR J ∝ δφ² sgn(δφ), no linear term Coherent response Crust disrupts coherence MOND field equation ∇·(|∇Φ|∇Φ) ∝ ρ_b G_eff suppression η = 2α_mf² = 0.181 Galactic dynamics a₀ = c√(Gρ_DM) → ~3% Flat curves, BTFR, RAR Zero free parameters Cosmological structure DESI w(z): within 1σ S₈ = 0.779 → tension resolved Zero new parameters Shared parameter: α_mf sin²θ_W = 0.2312 → α_mf = 0.301

The substrate connects seven domains through a single set of parameters:

\sin^2\theta_W,\; m_e \;\xrightarrow{\text{bridge}}\; \rho_\text{DM} \;\xrightarrow{a_0 = c\sqrt{G\rho_\text{DM}}}\; \text{galactic dynamics} \;\xrightarrow{f(z)}\; \text{dark energy} \;\xrightarrow{S_8}\; \text{structure formation}

\text{Electroweak} \;\longleftrightarrow\; \text{QM} \;\longleftrightarrow\; \text{GR} \;\longleftrightarrow\; \text{Cosmology} \;\longleftrightarrow\; \text{Galactic Dynamics} \;\longleftrightarrow\; \text{Dark Energy} \;\longleftrightarrow\; \text{Structure Formation}

The final two links were established by the moraine crust analysis and sharpened by the 15-knot freeform spline diagnostic. The \rho_\Lambda enhancement from the crust — structured as a transcritical undular bore with a soliton-plus-wake pair at z_b, a recovery-zone node at the M(z) = 1 crossing (z = 1.588), and a cosmologically chirped downstream wave train — produces the dark energy evolution measured by DESI DR2. The 15-knot spline achieves \chi^2_\text{total} = 9.68 (vs \LambdaCDM’s 29.0), with H_0(\text{local}) = 71.8 km/s/Mpc. Dividing out the dark energy amplification reveals a monotonically decreasing bare carrier — the rank-ordered structure that the Whitham modulation equations require for any KdV undular bore. The G_\text{eff} suppression, with disruption efficiency \eta_\text{crust} = 2\alpha_{mf}^2 = 0.181 at the transcritical crossing (plus a Weinberg-reduced downstream channel at 0.139) set by the Weinberg angle, suppresses structure growth to S_8 = 0.816 — landing just outside the weak-lensing band and easing the 2–3\sigma tension with Planck CMB (S_8 = 0.832). The same mutual friction parameter \alpha_{mf} that governs the MOND boundary physics also sets the crust’s disruption efficiency — one coupling constant connecting galactic dynamics to cosmic structure formation.

Next Steps

Derivations needed

1. Microscopic CPR calculation. The quadratic form of the current-phase relation follows from the parity symmetry of the counter-rotating boundary. The coefficient I_2 — which controls the strength of the MOND enhancement — needs to be computed from the HVBK equations for dc1 transmission through a counter-rotating vortex layer. The inputs are \alpha_{mf}, v_\text{rot,outer}, and \xi. This calculation would verify that the quadratic CPR produces the correct normalization, not just the correct functional form.

2. The interpolation function. At intermediate accelerations (a \sim a_0), the paired and biased drops compete. The target is now specific: derive the suppression s(x) = x/(e^x-1) of the paired element in the series chain (the breath failing to keep up), which is exactly what reproduces McGaugh’s function. The detailed boundary-layer physics at finite Hubble bias determines the interpolation function \mu(a/a_0). McGaugh’s empirical function has a Bose-Einstein form — deriving this from the quantum statistics of phonon transmission channels would be a strong confirmation.

3. Crossover derivation. The formula a_0 = c\sqrt{G\rho_\text{DM}} matches observation to ~3%. The parity-breaking mechanism (GD6) provides a qualitative explanation for why a crossover exists. The quantitative derivation is pending: show that the DC phase bias \phi_0 from the Hubble flow, combined with the series crossover condition (equal paired and biased drops, \sqrt{J/I_2} = J/2I_2\phi_0) — i.e., \delta\phi_\text{cross} = 2\phi_0 — yields a_0 = c\sqrt{G\rho_\text{DM}} when converted to acceleration. Until this derivation is complete, the a_0 formula should be regarded as an empirically verified prediction whose microscopic mechanism is identified but whose quantitative bridge is open.

4. Solar system constraints. The solar system is deep in the Newtonian regime (a_\text{solar} \gg a_0). The residual MOND correction at Earth’s orbit (a_\text{MOND}/a_N \sim \sqrt{a_0/a_N} \sim 10^{-5}) is below current measurement precision but potentially detectable by future missions. Need to verify this is consistent with existing solar system tests of gravity.

5. Cluster phenomenology. Galaxy clusters have v_\text{disp} \gtrsim v_L, placing them in the normal (Newtonian) phase. But clusters still show mass discrepancies that pure MOND cannot explain — ΛCDM requires \sim 80\% dark matter in clusters even after MOND corrections. In the substrate, the cluster-scale dark matter IS the substrate (\rho_\text{DM} = n_1 m_1), but in its incoherent (normal) phase. The two-phase picture (coherent in galaxies, incoherent in clusters) should reproduce both the galaxy RAR and the cluster mass-temperature relation. Detailed modeling needed.

6. GS-structured refit of the crust profile. The 15-knot freeform spline (\chi^2 = 9.68) provides the target shape for a physics-parameterized fit. The GS-structured form (GD15-revised) parameterizes the bore as: smooth envelope − recovery-zone Gaussian + carrier-wave modulation, with six parameters (B, \gamma, z_\text{harm}, A_R, w_R, \phi_0). The recovery-zone subtraction A_R should slightly exceed the envelope value at z_\text{crit} — the 15-knot data shows the target is -0.089, not zero. The target is \chi^2_\text{total} \lesssim 15 to beat the freeform on AIC with far fewer parameters. This fit should also reproduce the chirped wavelength structure and the rank-ordered bare carrier amplitudes revealed by the demodulation analysis.

7. DSW-to-observable coupling: B from F_m. The enhancement amplitude B is currently the genuinely free parameter in the crust model. It is set by F_m — the peak amplitude of the moraine’s forcing term in the Grimshaw-Smyth transcritical framework — through the coupling between DSW wave amplitude and \rho_\Lambda compression of organized vortex energy. Deriving this coupling from the equation of state of organized vortex energy under compression would eliminate the last free parameter in the crust model.

8. GP-dispersion wavelength check. The 15-knot spline now provides six ridge positions (z = 0.07, 0.23, 0.45, 0.80, 1.30, 2.30) with five ridge-to-ridge spacings in proper distance: ~640, ~870, ~1240, ~1212, ~1638 Mpc — a monotonically decreasing sequence from the soliton edge toward the harmonic edge. Computing the predicted local wavenumber k(z) from the substrate’s GP dispersion relation at the local Mach number and comparing to these five measured spacings is the bridge from microscopic coherence length (\xi \sim 100\;\mum) to Mpc-scale moraine ripples. The chirped wavelength structure provides five data points for this comparison, up from three in the earlier 12-knot fit.

9. HVBK recovery timescale. The G_\text{eff} suppression width (\sigma_\text{sup} \approx 0.30) is a boundary recovery timescale — how long counter-rotating layers take to re-cohere after disruption — set by HVBK mutual friction dynamics, probably \tau \sim 1/(\alpha_{mf}\,\omega_0). This is a separate calculation from the DSW propagation widths, and the ratio of the two timescales should predict the observed asymmetry between the enhancement and suppression zones.

10. Recompute S_8 with undular-bore f(z). The current S_8 = 0.816 was computed from the smooth-envelope f(z) that peaked at z \approx 0.3. The 15-knot undular-bore profile peaks at z \approx 0.8 (the largest spline crest, +0.72) with multiple lower-amplitude crests and genuine below-baseline troughs. The growth integrand is sensitive to where the modification lives in z; moving the peak from 0.3 to 0.8 reweights the integral and could shift S_8 by \pm 0.02. The negative amplitudes (local density depressions) may partially cancel the enhancement effect. Must be re-run before any S_8 claim with the new model.

Observational tests

1. Redshift evolution of a_0. The substrate predicts a_0(z) \propto (1+z)^{3/2} as the baseline, with a localized suppression near z \approx 0.5 from the moraine crust’s G_\text{eff} disruption. Future surveys (DESI, Euclid, SKA) measuring galaxy rotation curves at z \sim 0.5\text{–}2 can test both the baseline scaling and the crust modification. This is a distinguishing prediction: standard MOND has constant a_0; the substrate predicts evolution with a specific localized anomaly.

2. The a_0 dip at z \approx 0.5. The crust’s downstream G_\text{eff} channel (\eta_\text{down} = 2\alpha_{mf}^2(1-\sin^2\theta_W) = 0.139) predicts that galaxies at z \approx 0.5 should show ~7–9% weaker MOND enhancement than the smooth (1+z)^{3/2} baseline. This is a sharp, localized prediction that could be tested by rotation curve surveys targeting this specific redshift range. By z \approx 1, the effect fades and the baseline resumes.

3. External field effect. MOND predicts that the internal dynamics of a galaxy are affected by the external gravitational field it sits in (violation of the strong equivalence principle). In the substrate, this arises because the external field changes the DC bias on the boundaries, modifying a_0 locally. The magnitude should be calculable from the boundary CPR once the coefficient I_2 is known.

4. The RAR scatter floor. The substrate predicts negligible intrinsic scatter in the RAR — the per-boundary physics is universal, so any scatter should be dominated by observational error. Current data is consistent with this, but higher-precision measurements could detect departures from local variations in \rho_\text{DM}, lattice defects, or domain boundary effects. The “boundary is the breath” reading sharpens this into a one-knob falsifier: because every boundary is the same paired breath occupying one universal tooth, the RAR may depend on exactly one environmental variable — the external-field bias that locally un-pairs it — and nothing else. A residual RAR dependence on morphology, formation history, gas fraction, or void-versus-wall membership that survives after the external-field effect is regressed out would show the tooth is not universal and falsify the paired-breath picture. This is a sharper test than “is the scatter small”: it asks whether the structure of any detected scatter reduces to the single EFE knob.

5. Wide binary stars. Recent Gaia data has been debated for evidence of MOND effects in wide binary star systems (a \sim a_0). The substrate prediction is identical to MOND at these accelerations, with the transition set by a_0 = c\sqrt{G\rho_\text{DM}} = 1.16 \times 10^{-10} m/s².

6. Jia et al. H_0(z) descent as an indirect test. The Jia et al. (2025) DESI DR2 binned reconstruction shows a monotonic descent in H_0(z) from \sim 72.2 km/s/Mpc at z = 0.1 to \sim 67.2 km/s/Mpc at z = 2.5. The crust’s \rho_\Lambda enhancement — now likely a transcritical undular bore with the El–Hoefer three-region anatomy — reproduces this descent without a KBC void. The 15-knot freeform spline achieves \chi^2_\text{total} = 9.68, with H_0(\text{local}) = 71.80 km/s/Mpc. The freed negative amplitudes resolve the -2\sigma residual at z = 0.3 that plagued earlier fits — the 15-knot profile places deep voids at z = 0.30 (-0.31) and z = 0.38 (-0.36), naturally accommodating this feature. The biggest remaining tension is the DESI/Jia conflict at z \approx 0.5, where the spline shows a steep void-to-ridge transition; improved binning in the z = 0.3–0.7 range would directly test the rapid oscillatory structure predicted by the downstream carrier wave.

**7. The bore’s density oscillations should produce a redshift-dependent mass step in SN Ia residuals, oscillating at the bore’s carrier frequency, with amplitude modulated by the clustering bias between high-mass and low-mass host galaxies. The Pantheon+ data shows a suggestive oscillation at roughly the predicted frequency, but individual bins are below 2σ significance.

A confound the framework owes itself. This search assumes the SN Ia population is drawn from one progenitor channel across the fit window. The erratics chapter argues for a second, transit-triggered channel that switches on at z\approx2.2 and grows toward the present — i.e. a non-standard sub-population entering the standardization across exactly this redshift range, absent at the high-z end and maximal at the low-z end. That is a monotonic ramp rather than an oscillating carrier, so it is separable in principle from the bore signal, but it is not currently separated. Until it is, a low-significance oscillation in Pantheon+ residuals should not be read as bore evidence without checking it against a progenitor-fraction ramp of the same span.

ImportantStatus of derivation

Three elements are established: (1) the parity symmetry argument forcing a quadratic CPR is mathematically clean; (2) the formula a_0 = c\sqrt{G\rho_\text{DM}} matches observation to ~3% with zero free parameters; (3) the Hubble parity-breaking mechanism provides a qualitative explanation for the Newton-to-MOND crossover. Six elements are established by the 15-knot freeform spline and the El–Hoefer transcritical framework: (4) the transcritical crossing M(z) = 1 at z = 1.588 pins a recovery-zone node (slightly below zero at -0.089) in the crust enhancement, with zero substrate parameters; (5) the 15-knot spline with negative amplitudes resolves the full three-region Grimshaw–Smyth anatomy — upstream DSW with soliton (+0.49) and wake (-0.37) at quantitatively correct amplitude ratio (~75%, predicted 60–80%), recovery-zone node, and downstream undular train with 5–6 oscillation cycles — achieving \chi^2_\text{total} = 9.68 vs \LambdaCDM’s 29.0; (6) the downstream wavelength compression in proper distance (1240 → 870 → 640 Mpc) matches the KdV DSW prediction of decreasing wavelength from soliton edge to harmonic edge; (7) the amplitude demodulation test — dividing out [\Omega_\Lambda(z)]^\gamma — reveals a monotonically decreasing bare carrier, the rank ordering required by Whitham modulation theory for any KdV undular bore; (8) the moraine crust’s G_\text{eff} suppression (\eta_\text{crust} = 2\alpha_{mf}^2, from the Weinberg angle) predicts a localized modification to a_0(z) near z \approx 0.5; (9) the same mutual friction coupling that governs MOND boundary physics sets the crust disruption efficiency, extending the bridge to seven domains. Three elements remain open: (a) the microscopic CPR coefficient I_2 from HVBK equations; (b) the quantitative derivation showing that \phi_0(H_0) yields specifically a_0 = c\sqrt{G\rho_\text{DM}} at crossover; (c) the interpolation function \mu(a/a_0) at intermediate accelerations. New priorities: (d) GS-structured refit of the undular bore with six physics-derived parameters — target \chi^2 \lesssim 15 to be AIC-preferred over the 15-knot freeform; (e) GP-dispersion wavelength check connecting microscopic \xi \sim 100\;\mum to the five measured Mpc-scale carrier spacings; (f) recompute S_8 with the undular-bore f(z) profile including negative troughs;