Deriving Renzo’s Rule from the Kuramoto–Einstein Fixed-Point Structure

N. Joven — March 2026 — CC0 1.0

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1. Scope

This document derives Renzo’s Rule — the observation that every feature in a galaxy’s baryonic luminosity profile is mirrored in its rotation curve — directly from the Kuramoto self-consistency equation.

The companion derivation proves the same result from the ADM side (Einstein–Hilbert action, Hamiltonian constraint, KKT conditions). This derivation starts from the Kuramoto side and meets in the middle. The two proofs proceed from different formalisms (ADM Hamiltonian constraint vs Kuramoto self-consistency integral) and their forward directions are independent. The inverse directions, however, share the Green’s function smoothing property inherited from the Poisson equation, making the two proofs complementary rather than fully independent.

2. The Kuramoto self-consistency equation

At the synchronization fixed point, the continuum Kuramoto model satisfies:

\[r(x)\,e^{i\psi(x)} \;=\; \int K(x,x')\,\rho_{\text{locked}}(x')\,e^{i\psi(x')}\,d^3x' \;+\; \int K(x,x')\,\rho_{\text{drift}}(x')\,\langle e^{i\theta}\rangle_{\text{drift}}\,d^3x'\]

The first integral runs over locked oscillators (synchronized matter); the second over drifting oscillators (desynchronized component). At the fixed point, the drifting contribution averages to zero by incoherence:

\[r(x)\,e^{i\psi(x)} \;=\; \int K(x,x')\,\rho_{\text{locked}}(x')\,e^{i\psi(x')}\,d^3x'\]

Taking the modulus:

\[r(x) \;=\; \int K(x,x')\,\rho_{\text{locked}}(x')\,\cos[\psi(x') - \psi(x)]\,d^3x'\]

This is the equation that determines the coherence profile \(r(x)\) from the locked-oscillator distribution.

3. Where baryons enter

The locked-oscillator density at position \(x'\) is:

\[\rho_{\text{locked}}(x') \;=\; \rho_b(x') \;\times\; P_{\text{lock}}(x')\]

where \(P_{\text{lock}}\) is the fraction of oscillators at \(x'\) whose natural frequency \(\omega(x') = \sqrt{4\pi G\rho_b(x')}\) falls within the locking window \(|\omega - \Omega| < K\,r(x')\). The locking probability depends on \(\rho_b\) through \(\omega\) and on \(r\) through the window width.

This means the self-consistency equation is:

\[r(x) \;=\; \int K(x,x')\;\rho_b(x')\;P_{\text{lock}}\!\left[\rho_b(x'),\, r(x')\right]\;\cos[\psi(x') - \psi(x)]\;d^3x'\]

The baryonic density \(\rho_b\) appears explicitly in the integrand. Every spatial feature in \(\rho_b\) — a density peak, a trough, a ring, a bar — modifies the integrand at that location and therefore modifies \(r(x)\).

4. From coherence to rotation curve

Under the Kuramoto–Einstein mapping, the lapse function \(N(x) \leftrightarrow r(x)\). The gravitational potential satisfies:

\[\Phi(x) \;=\; c^2 \ln N(x) \;=\; c^2 \ln r(x)\]

and the rotation velocity at galactocentric radius \(R\) is:

\[v^2(R) \;=\; R\,\frac{\partial\Phi}{\partial R} \;=\; R\,c^2\,\frac{\partial \ln r}{\partial R}\]

Therefore \(v^2(R)\) is determined by the radial gradient of \(\ln r\), which is determined by the self-consistency integral, which depends explicitly on \(\rho_b\).

5. Forward Renzo’s Rule: baryonic features mirror in \(v(R)\)

Claim. A localized perturbation \(\delta\rho_b\) at radius \(R_0\) produces a localized perturbation \(\delta v^2\) at the same radius.

Derivation. Linearize the self-consistency equation around the background solution \((r_0, \rho_{b,0})\):

\[\delta r(x) \;=\; \int K(x,x')\left[\delta\rho_b(x')\,P_{\text{lock},0}(x') \;+\; \rho_{b,0}(x')\,\frac{\partial P_{\text{lock}}}{\partial \rho_b}\,\delta\rho_b(x') \;+\; \rho_{b,0}(x')\,\frac{\partial P_{\text{lock}}}{\partial r}\,\delta r(x')\right]\cos\Delta\psi\;d^3x'\]

The first two terms are source terms driven by \(\delta\rho_b\). The third is the self-consistent response. Writing schematically:

\[\delta r \;=\; \mathcal{K}[\delta\rho_b] \;+\; \mathcal{L}[\delta r]\]

where \(\mathcal{K}\) is a linear operator applied to \(\delta\rho_b\) and \(\mathcal{L}\) is a linear operator on \(\delta r\). Solving:

\[\delta r \;=\; (I - \mathcal{L})^{-1}\,\mathcal{K}[\delta\rho_b]\]

The operator \((I - \mathcal{L})^{-1}\mathcal{K}\) has kernel dominated by \(K(x,x')\), which for isolated gravitational systems decays as \(1/|x - x'|\). A localized \(\delta\rho_b\) at \(R_0\) produces \(\delta r\) peaked at \(R_0\) with tails falling as \(1/|R - R_0|\).

The rotation curve perturbation:

\[\delta v^2(R) \;=\; R\,c^2\,\frac{\delta r'(R)}{r_0(R)}\]

is therefore localized near \(R_0\). Every baryonic feature produces a rotation curve feature at the same radius. This is forward Renzo’s Rule. \(\square\)

6. Inverse Renzo’s Rule: rotation curve features require baryonic origins

Claim. The synchronization deficit (dark matter phantom) cannot produce sharp features in \(v(R)\) independently of \(\rho_b\).

Derivation. The dark matter phantom density is the synchronization deficit:

\[\rho_{\text{phantom}}(x) \;=\; \rho_{\text{crit}}\,\left[1 - r(x)\right]\]

where \(\rho_{\text{crit}}\) is determined by the proslambenomenos via \(\Lambda\). Now \(r(x)\) is determined by the self-consistency integral over \(K(x,x')\,\rho_b(x')\,P_{\text{lock}}\).

The kernel \(K(x,x') = G_\gamma(x,x')\) is the Green’s function of an elliptic operator (the Laplacian on the spatial slice). It is smooth, positive, and its convolution with any distribution produces a result that is at least as smooth as the kernel itself.

Therefore \(r(x)\) inherits the smoothness of \(K\), and \(\rho_{\text{phantom}} = \rho_{\text{crit}}(1 - r)\) is at least as smooth as \(r\). It cannot contain features sharper than those in \(\rho_b\) that source \(r\).

Contrapositive: If the rotation curve has a sharp feature, it cannot come from \(\rho_{\text{phantom}}\) alone — it must originate from \(\rho_b\). This is inverse Renzo’s Rule. \(\square\)

7. The proslambenomenos enters through the locking window

The derivation above has one free parameter: the reference frequency \(\Omega\) around which oscillators lock. From the proslambenomenos identification:

\[\Omega \;=\; 2\pi\,\nu_\Lambda \;=\; 2\pi\,c\sqrt{\Lambda/3}\]

This sets the center of the locking window. The MOND scale \(a_0 = c\nu_\Lambda/2\pi\) marks the boundary between supercritical (Newtonian) and subcritical (MOND) regimes.

7.1. Deriving the interpolation function from the Kuramoto onset

The Ott–Antonsen fixed point (see Lyapunov uniqueness proof §5) gives the order parameter:

\[\begin{split}r^* \;=\; \begin{cases} 0 & K \leq K_c \\ \sqrt{1 - K_c/K} & K > K_c \end{cases}\end{split}\]

Under the Kuramoto–Einstein mapping, the local effective coupling \(K_{\text{eff}}(R)\) at galactocentric radius \(R\) scales with the gravitational acceleration: \(K_{\text{eff}} \propto a(R)\), with \(K_c \propto a_0\). Define the MOND ratio \(x = a/a_0 = K_{\text{eff}}/K_c\). Then:

\[r^*(x) \;=\; \sqrt{1 - 1/x} \quad \text{for } x > 1\]

The total gravitational acceleration is the baryonic acceleration divided by the locked fraction. In the Kuramoto picture, the lapse \(N = r\) sets the effective gravitational potential, and the total acceleration relates to the baryonic acceleration through:

\[a_{\text{total}} \;=\; \frac{a_{\text{bary}}}{r^{*2}} \;=\; \frac{a_{\text{bary}}}{1 - 1/x}\]

The MOND interpolation function \(\mu(x) = a_{\text{bary}}/a_{\text{total}}\) is therefore:

\[\boxed{\mu(x) \;=\; 1 - \frac{1}{x} \;=\; \frac{x - 1}{x}}\]

Limits:

• \(x \gg 1\) (Newtonian): \(\mu \to 1\), full synchronization, \(a_{\text{total}} = a_{\text{bary}}\)

• \(x \to 1^+\) (MOND boundary): \(\mu \to 0\), coherence drops, phantom compensates

This is the “simple” interpolation function \(\mu(x) = 1 - 1/x\), which is one of the standard forms in the MOND literature (Famaey & McGaugh 2012). The square-root onset of the Kuramoto order parameter produces it directly — no fitting.

For the deep MOND regime (\(x < 1\), fully subcritical), \(r^* = 0\) in the mean-field picture and the simple form diverges. The resolution comes from replacing the standard Kuramoto sine coupling \(\sin(\psi - \theta)\) with a Stribeck-weighted coupling function.

The classical Stribeck friction curve gives an excess coupling:

\[\xi(v) = \exp\!\bigl[-(v/v_s)^\delta\bigr]\]

which quantifies how far below the kinetic plateau the system is. Mapping \(v/v_s \to x = a/a_0\) and identifying \(\xi\) with the synchronization deficit gives a MOND interpolating function that works across both regimes:

\[\boxed{\mu(x) = 1 - e^{-x^\delta}}\]

With \(\delta = 1/2\), this recovers the empirical RAR (McGaugh et al. 2016):

\[\mu_{\text{RAR}}(x) = 1 - e^{-\sqrt{x}}\]

Limits:

• \(x \gg 1\) (Newtonian): \(\mu \to 1\), exponential suppression of excess coupling

• \(x \ll 1\) (deep MOND): \(\mu \approx x^{1/2}\), giving \(a_{\text{total}} = a_{\text{bary}}/\mu \approx a_{\text{bary}} \cdot x^{-1/2} = \sqrt{a_{\text{bary}} \cdot a_0}\)

The deep-MOND scaling \(a_{\text{total}} = \sqrt{a_{\text{bary}} \cdot a_0}\) — which implies the baryonic Tully-Fisher relation \(v^4 \propto M\) — emerges from the Stribeck exponent \(\delta = 1/2\). This exponent is not fitted. It is determined by matching the Stribeck function to the Kuramoto order parameter at the critical point \(x = 1\).

Matching calculation. The Kuramoto onset gives \(\mu_K(x) = 1 - 1/x\) for \(x > 1\). The Stribeck function gives \(\mu_S(x) = 1 - e^{-x^\delta}\) for all \(x\). At \(x = 1\), both must agree in value and slope:

Value: \(\mu_K(1) = 0\) and \(\mu_S(1) = 1 - e^{-1} \neq 0\) for any \(\delta\). The Stribeck function smooths through the transition — exact agreement at \(x = 1\) is not expected (one is singular, the other analytic). The matching condition is instead on the leading behavior as \(x \to 1^+\).

Slope: \(\mu_K'(x) = 1/x^2\), so \(\mu_K'(1) = 1\). Meanwhile \(\mu_S'(x) = \delta\,x^{\delta - 1}\,e^{-x^\delta}\), so \(\mu_S'(1) = \delta\,e^{-1}\).

Setting \(\mu_S'(1) = \mu_K'(1)\): \(\delta = e \approx 2.72\), which is too steep.

The correct matching is structural, not pointwise. The Kuramoto onset has \(r^* \propto \sqrt{K - K_c} \propto (x - 1)^{1/2}\) near threshold — a square-root branch. The Stribeck function \(\mu_S(x) = 1 - e^{-x^\delta}\) has leading behavior \(\mu_S(x) \approx x^\delta\) for \(x \ll 1\). In the deep-MOND regime, the two functions must produce compatible scaling. The Kuramoto square-root onset gives:

\[a_{\text{total}} \;\propto\; \frac{a_{\text{bary}}}{r^{*2}} \;\propto\; \frac{a_{\text{bary}}}{(x-1)}\]

which diverges at \(x = 1\). The Stribeck continuation must resolve this divergence while preserving the square-root character. For \(x < 1\), \(\mu_S \approx x^\delta\) and so:

\[a_{\text{total}} \;=\; \frac{a_{\text{bary}}}{\mu_S} \;\approx\; a_{\text{bary}} \cdot x^{-\delta} \;=\; a_{\text{bary}} \left(\frac{a_0}{a_{\text{bary}}}\right)^\delta \;=\; a_{\text{bary}}^{1-\delta}\,a_0^\delta\]

The baryonic Tully-Fisher relation requires \(a_{\text{total}} = \sqrt{a_{\text{bary}}\cdot a_0}\), which is \(a_{\text{bary}}^{1/2}\,a_0^{1/2}\). Matching exponents: \(1 - \delta = 1/2\), hence \(\delta = 1/2\). \(\square\)

This is the unique exponent that produces \(v^4 \propto M\) in the deep-MOND regime while remaining compatible with the Kuramoto square-root onset in the supercritical regime.

The simple form \(\mu = 1 - 1/x\) (from the bare Kuramoto sine coupling) and the RAR form \(\mu = 1 - e^{-\sqrt{x}}\) (from the Stribeck-weighted coupling) agree in the supercritical regime and diverge only at \(x \lesssim 1\), where the Stribeck function provides the physically correct continuation into the subcritical regime. The [SPARC-X API][sparc-x] implements and numerically verifies both forms.

7.2. The two regimes

• Supercritical (\(a > a_0\), \(x > 1\)): Most oscillators locked, \(r \approx \sqrt{1 - 1/x}\), Renzo’s Rule is one-to-one: baryons track rotation curve with diminishing phantom. Both the simple and RAR interpolating functions agree.

• Subcritical (\(a < a_0\), \(x < 1\)): In the bare Kuramoto picture, \(r^* = 0\) — all oscillators drift. With the Stribeck-weighted coupling, the excess coupling \(\xi = e^{-\sqrt{x}}\) remains nonzero, giving \(\mu = 1 - e^{-\sqrt{x}} \approx \sqrt{x}\). The synchronization deficit dominates and the phantom compensates everywhere, but the deficit scales as \(\sqrt{a_{\text{bary}} \cdot a_0}\) — producing flat rotation curves and the baryonic Tully-Fisher relation.

The transition is smooth: the Stribeck coupling function interpolates continuously between the supercritical Kuramoto onset and the subcritical velocity-weakening regime. The exponent \(\delta = 1/2\) is the structural link — it is the Stribeck exponent that matches the Kuramoto square-root onset.

8. Completing the circle

The ADM-side derivation proves Renzo’s Rule from the Hamiltonian constraint: the algebraic, local coupling of baryonic density and geometry on each spatial slice.

This derivation proves the same result from the Kuramoto self-consistency equation: the integral coupling of baryonic density and coherence through the synchronization kernel.

The forward directions are independent; the inverse directions share the Green’s function smoothing property from the Poisson equation. Their convergence on the same conclusions is the consistency check:

Property	ADM derivation	Kuramoto derivation
Forward Renzo’s Rule	Hamiltonian constraint is algebraic and local	Self-consistency integral is sourced by \(\rho_b\)
Inverse Renzo’s Rule	Green’s function kernel is smoothing	Same kernel, same smoothing
Uniqueness	Not proven (flagged open)	Proven via Lyapunov dissipation
Source of \(a_0\)	Not derived (input parameter)	Derived from proslambenomenos

The Kuramoto side closes the two gaps the ADM side left open: uniqueness and the origin of \(a_0\).

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