Lyapunov Functional and the Dissipation Argument for Uniqueness

N. Joven — March 2026 — CC0 1.0

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1. The problem

The Kuramoto–Einstein framework maps gravitational equilibrium to a synchronization fixed point. The forward and inverse Renzo’s Rule derivations (ADM side, Kuramoto side) show that at the fixed point, every baryonic feature is mirrored in the rotation curve and vice versa. But the fixed-point equations alone do not guarantee uniqueness.

The Schrödinger equation is time-symmetric, yet we observe irreversibility. The arrow of time is not in the equation — it is in the low-entropy initial state. Similarly, if the Kuramoto–Einstein equations admit multiple fixed points, the physical one is selected not by the equilibrium conditions but by the dissipative history that produced it.

Uniqueness is not a mathematical property of the equilibrium — it is a physical property of the path.

This changes what needs to be written. Not a contraction proof — a dissipation argument.

2. The Kuramoto Lyapunov functional (finite case)

For the all-to-all Kuramoto model with \(N\) oscillators and identical frequencies \(\omega_i = \omega\) in the rotating frame:

\[\dot{\theta}_i \;=\; \frac{K}{N}\sum_{j=1}^{N} \sin(\theta_j - \theta_i)\]

the system admits a Lyapunov functional:

\[V[\theta] \;=\; -\frac{K}{2N}\sum_{i,j}\cos(\theta_i - \theta_j)\]

This satisfies:

\[\frac{dV}{dt} \;=\; -\sum_i \dot{\theta}_i^2 \;\leq\; 0\]

with equality only at fixed points. The system is gradient descent on \(V\). The synchronized state (\(\theta_i = \theta_j\) for all \(i,j\)) is the global minimum. This is textbook — established by Kuramoto (1984), formalized by Strogatz (2000).

For non-identical frequencies, the gradient structure is broken but the dissipative character persists: the system still contracts phase-space volume, still has attractors, and still selects unique asymptotic states from generic initial conditions.

3. Extension to the continuum

The spatially extended Kuramoto model:

\[\partial_t \theta(x,t) \;=\; \omega(x) + \int K(x,x')\,\sin[\theta(x',t) - \theta(x,t)]\,d^3x'\]

admits the continuum Lyapunov functional:

\[\mathcal{V}[\theta] \;=\; -\frac{1}{2}\iint K(x,x')\,\cos[\theta(x') - \theta(x)]\,d^3x\,d^3x'\]

The time derivative along solutions is:

\[\frac{d\mathcal{V}}{dt} \;=\; -\int(\partial_t\theta - \omega)\,\partial_t\theta\,d^3x\]

For identical frequencies (\(\omega = 0\) in the rotating frame), this reduces to \(-\int(\partial_t\theta)^2\,d^3x \leq 0\) and is a strict Lyapunov function. For non-identical frequencies, the cross term \(\int\omega\,\partial_t\theta\,d^3x\) has indefinite sign — \(\mathcal{V}[\theta]\) oscillates. This is not a failure of the dissipative structure but a consequence of tracking the wrong variable: the phases \(\theta\) are unbounded for drifting oscillators. The correct Lyapunov function operates on the bounded order parameter \(r \in [0,1]\) (see §5).

4. The Kuramoto–Einstein Lyapunov functional

Under the Kuramoto–Einstein mapping (companion document):

Kuramoto	Einstein (ADM)
\(\theta(x,t)\)	Phase field
\(\omega(x)\)	\(\sqrt{4\pi G\rho(x)}\)
\(K(x,x')\)	\(G_\gamma(x,x')\) (Green’s function of \(\gamma_{ij}\))
\(r(x,t)\)	Lapse \(N\)
\(\cos[\theta - \theta']\)	Phase correlation

the Lyapunov functional maps to:

\[\mathcal{V}_{\text{KE}}[\gamma, \psi] \;=\; -\frac{1}{2}\iint G_\gamma(x,x')\,\cos[\psi(x') - \psi(x)]\,\sqrt{\gamma(x)}\,\sqrt{\gamma(x')}\,d^3x\,d^3x'\]

where \(\psi(x)\) is the mean phase field at position \(x\).

In the synchronized limit (\(\psi(x) \approx \psi_0\) everywhere, full coherence):

\[\mathcal{V}_{\text{KE}} \;\approx\; -\frac{1}{2}\iint G_\gamma(x,x')\,\sqrt{\gamma(x)}\,\sqrt{\gamma(x')}\,d^3x\,d^3x'\]

This is, up to sign and boundary terms, the gravitational potential energy functional — the standard Newtonian self-energy. The Lyapunov function of Newtonian gravity is the synchronized limit of the Kuramoto–Einstein Lyapunov functional.

In the desynchronized limit (\(\psi\) incoherent, \(r \ll 1\)):

\[\mathcal{V}_{\text{KE}} \;\approx\; 0\]

because the cosine averages to zero over incoherent phases. The system starts near zero and descends to the gravitational potential well. Galaxy formation is Lyapunov descent.

5. Why \(\mathcal{V}_\text{KE}[\theta]\) oscillates — and what descends instead

The interaction energy \(\mathcal{V}[\theta] = -\frac{1}{2}\iint K\cos(\theta - \theta')\,d^3x\,d^3x'\) has time derivative:

\[\frac{d\mathcal{V}}{dt} \;=\; -\int(\partial_t\theta - \omega)\,\partial_t\theta\,d^3x\]

For non-identical frequencies (\(\omega(x) \neq 0\)), this has indefinite sign: the cross term \(\int\omega\,\partial_t\theta\,d^3x\) can be positive. This is the primal oscillation around the saddle point in the Lagrangian relaxation picture (intersections, §3.3). Tracking \(\mathcal{V}[\theta]\) is tracking the wrong variable — the phases \(\theta\) are unbounded for drifting oscillators.

The correct variable is the order parameter \(r(x,t) \in [0,1]\), which the Kuramoto–Einstein mapping provides as a bounded quantity. For a Lorentzian frequency distribution with half-width \(\gamma\), the Ott–Antonsen reduction (Ott & Antonsen, 2008) gives exact mean-field dynamics:

\[\dot{r} \;=\; -\gamma\,r + \frac{K}{2}\,r(1 - r^2)\]

This is gradient flow \(\dot{r} = -dU/dr\) on:

\[\boxed{U(r) \;=\; \frac{\gamma}{2}\,r^2 - \frac{K}{4}\,r^2 + \frac{K}{8}\,r^4}\]

with \(\frac{dU}{dt} = -\left(\frac{dU}{dr}\right)^2 \leq 0\), equality only at fixed points. \(U(r)\) is bounded below, defined on \(r \in [0,1]\), and monotonically decreasing along trajectories. It is a strict Lyapunov function.

For \(K > K_c = 2\gamma\) (supercritical), \(U\) has exactly two critical points: an unstable maximum at \(r = 0\) and a global minimum at \(r^* = \sqrt{1 - 2\gamma/K}\). All trajectories with \(r(0) > 0\) converge to \(r^*\). There are no local minima to trap the dynamics.

The spatially extended version replaces \(r\) with the coherence profile \(r(x)\) and \(U\) with the functional:

\[\mathcal{U}[r] \;=\; \int\left[\frac{\gamma(x)}{2}\,r(x)^2 - \frac{1}{4}\int K(x,x')\,r(x)\,r(x')\,\cos[\psi(x') - \psi(x)]\,d^3x'\right]d^3x\]

In the synchronized limit (\(\psi \approx\) const), this reduces to \(\mathcal{V}_\text{KE}\). In the desynchronized limit (\(r \to 0\)), \(\mathcal{U} \to 0\). The descent from 0 to \(\mathcal{U}(r^*)\) is the arrow of galaxy formation:

Phase 1 — Incoherent (\(r \ll 1\), high \(\mathcal{V}_{\text{KE}}\)): Oscillators are uncorrelated. Coupling is subcritical everywhere. The system is fully in the MOND regime. The synchronization deficit (dark matter phantom) compensates everywhere. Rotation curve is flat and featureless.

Phase 2 — Nucleation (\(r\) growing, \(\mathcal{V}_{\text{KE}}\) falling): Synchronization nucleates where density is highest (protogalactic core, where \(\omega(x)\) is closest to \(\Omega\)). Coherent patches expand outward. The rotation curve develops baryonic features as the synchronized domain grows.

Phase 3 — Relaxation (\(r \to r_\infty\), \(\mathcal{V}_{\text{KE}}\) at minimum): The system reaches its unique fixed point. Rotation curve, dark matter profile, and baryonic distribution are all determined. Renzo’s Rule holds.

6. Why uniqueness follows

The Ott–Antonsen potential \(U(r)\) satisfies:

• Bounded below: \(U(r) \geq U(r^*)\) for all \(r \in [0,1]\)

• Monotonically decreasing along mean-field trajectories: \(dU/dt = -(dU/dr)^2 \leq 0\)

• No local minima: For \(K > K_c\), \(U\) has exactly one minimum (\(r^*\)) and one maximum (\(r = 0\))

By LaSalle’s invariance principle on the compact interval \([0,1]\), every trajectory with \(r(0) > 0\) converges to the unique minimum \(r^*\). There are no local traps, no saddle points on the interior, and no basin boundary to characterize — the entire interval \((0,1]\) is the basin of \(r^*\).

Theorem (Lyapunov uniqueness). Let \(\rho_b(x)\) be a baryonic density distribution with finite total mass. Let the Kuramoto–Einstein system evolve from initial data with \(r(x,0) \ll 1\) (desynchronized) and \(\psi(x,0)\) approximately uniform. Then the \(t \to \infty\) limit exists, is unique, and the corresponding rotation curve \(v(R)\) satisfies Renzo’s Rule: every feature in \(\rho_b\) is mirrored in \(v(R)\) and every feature in \(v(R)\) has a baryonic origin.

Proof sketch. \(U(r)\) is a strict Lyapunov function on \([0,1]\) (bounded below, monotone decreasing, vanishing derivative only at \(r = 0\) and \(r = r^*\)). The maximum at \(r = 0\) is unstable; any \(r > 0\) flows to \(r^*\). Desynchronized initial data has \(r > 0\) (any finite matter distribution has nonzero coherence). By LaSalle, convergence to \(r^*\). At \(r^*\), the Kuramoto self-consistency equation and smoothing kernel yield Renzo’s Rule (companion derivation). \(\square\)

7. The analogy made precise

Thermodynamics	Kuramoto–Einstein
Free energy \(F\)	\(U(r)\) (Ott–Antonsen potential)
Second law: \(dF/dt \leq 0\)	\(dU/dt = -(dU/dr)^2 \leq 0\)
Equilibrium: min \(F\)	min \(U\) at \(r^*\)
Arrow of time ← low-entropy initial state	Arrow of formation ← desynchronized initial state (\(r \ll 1\))
Order parameter (magnetization)	\(r \in [0,1]\) (coherence)

The Ott–Antonsen potential \(U(r)\) plays exactly the role of a Landau free energy: it is bounded, monotone along trajectories, and selects a unique equilibrium from symmetric initial conditions. The primal interaction energy \(\mathcal{V}[\theta]\) oscillates — this is the saddle-point oscillation familiar from Lagrangian relaxation (intersections, §3.3). The dual variable \(r\) descends.

The arrow of time is not in the Schrödinger equation. It is in the low-entropy initial state. The uniqueness of galactic structure is not in the fixed-point equations. It is in the gradient flow on \(U(r)\) that produced the fixed point.

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