The Euler Ensemble as a Turbulent Attractor: Parity Sectors, Zero Modes, and a Zeta Edge

Technical Summary: The Euler Ensemble as a Turbulent Attractor

Problem Statement
This paper addresses the Lyapunov stability of the finite Euler ensembles, which are proposed as compact arithmetic fixed points of the rescaled momentum-loop equation for freely decaying incompressible Navier-Stokes turbulence. The central question is whether these ensembles, derived from the exact statistical evolution of the Navier-Stokes equations, constitute stable turbulent attractors in the continuum limit. Specifically, the work investigates the spectrum of Lyapunov exponents for linearized perturbations around these fixed points to determine if the turbulent state is stable, unstable, or marginally stable.

Methodology
The analysis proceeds through a rigorous reduction of the fluid dynamics problem to an arithmetic spectral problem:

1. Momentum-Loop Representation: The study utilizes the loop-space formulation of Navier-Stokes turbulence, where the statistical evolution is encoded in a circulation loop functional $\Psi(C, t)$. This is mapped to a momentum-loop representation involving a one-dimensional momentum loop $P_\alpha(\theta, t)$.
2. Invariant Submanifold: The analysis restricts the dynamics to a compact spherical invariant submanifold defined by the condition $P^2_{k+1} = P^2_k$. On this submanifold, the Eulerian advection term cancels identically, and the momentum-loop equation reduces algebraically to a solvable form.
3. Rescaling and Linearization: The decaying Euler solution is rescaled using $\tau = \log(\nu(t+t_0))$ to transform the time-dependent solution into a fixed point of an autonomous dynamics. Linear perturbations $F_k = f_k + \xi_k e^{\lambda \tau}$ are introduced to define the Lyapunov exponents $\lambda$.
4. Arithmetic Reduction: The finite-$N$ tangential spectral problem is shown to be exactly solvable. The full Ising history of the system ($\sigma_k = \pm 1$) enters the stability condition solely through the closure winding number $qr = \sum \sigma_k$. This reduces the stability analysis to an arithmetic problem over reduced rational angles $p/q$ and winding sectors $r$.
5. Parity Sectors and Partition Functions: The parity constraint $N - qr \equiv 0 \pmod 2$ is used to classify the continuum limit into three distinct local sectors. The partition functions $Z_o(N)$, $Z_{e,0}(N)$, and $Z_{e,\ast}(N)$ are analyzed to determine the relative weight of these sectors.
6. Transverse Perturbations: For dimensions $d > 2$, transverse perturbations are analyzed at linear and quadratic orders to determine if they generate spectral shifts or act as zero modes.

Key Contributions and Results

• Three Local Sectors: The continuum limit is not described by a single Euler ensemble but splits into three sectors based on parity and winding:

  1. $E_o$ (Odd Ensemble): $N$ is odd, implying $q$ and $r$ are odd. The zero-winding sector $r=0$ is excluded by parity.
  2. $E_{e,0}$ (Even Zero-Winding Sector): $N$ is even and $r=0$. This sector is a singular discrete zero mode. Its partition function is enhanced relative to the punctured even sector by a factor of order $N$ ($Z_{e,0}/Z_{e,\ast} \sim 3N/\pi^2$).
  3. $E_{e,\ast}$ (Punctured Even Sector): $N$ is even, but the $r=0$ sector is removed by conditioning.

• Stability of the Even Zero-Winding Sector ($E_{e,0}$):
  In this sector, the spectral angle is not quantized. The tangential spectrum forms a continuous curve $\lambda(\vartheta; p, q)$ which contains positive Lyapunov exponents. Consequently, the even zero-winding ensemble is Lyapunov unstable.

• Marginal Stability of $E_o$ and $E_{e,\ast}$:
  In both the odd ensemble and the punctured even ensemble, the spectral angle remains quantized ($\vartheta_n = \pi n / rq$).

  • For any fixed non-zero spectral label $n$, the normalized eigenvalue law collapses weakly to a delta function at zero: $\rho^{(N)}_n \Rightarrow \delta_0$.
  • These sectors are marginally stable (the spectrum concentrates at $\lambda=0$, not strictly negative).
  • Finite positive eigenvalues do not vanish pointwise but survive only as a "vanishing-weight arithmetic edge." In the odd sector, the positive moments of this edge scale as $M_{s,+} \sim \sqrt{\frac{\pi}{2N}} |n|^s \zeta(s)$ for $s > 1$, linking the stability edge directly to the Riemann zeta function.

• Transverse Modes:
  In dimensions $d > 2$, transverse perturbations are zero modes at linear order ($\lambda_\perp = 0$).

  • In the unstable $E_{e,0}$ sector, the quadratic obstruction is a genuine constraint governed by an indefinite quadratic form.
  • In the marginal sectors ($E_o$ and $E_{e,\ast}$), the quadratic obstruction is absorbed by a radial correction $c^*[\rho]$, leaving no quadratic spectral shift. Thus, the transverse sector acts as a neutral center manifold to this order.

Significance and Claims
The paper claims to provide an exact, deterministic number-theoretic solution for the turbulent attractor, contrasting with the traditional view of turbulence as purely chaotic noise amplification. The primary significance lies in the rigorous classification of the Euler continuum limit:

1. Non-Uniqueness: The turbulent attractor is not unique; it exists in two stable variations (the odd ensemble and the punctured even ensemble) and one unstable variation (the even zero-winding sector).
2. Arithmetic Control: The stability of the attractor is controlled not by a smooth continuum of modes but by the arithmetic structure of rational winding sectors. The "edge" of the spectrum is governed by the Riemann zeta function and Jordan totients.
3. Marginality: The stable attractors are "marginal," meaning their Lyapunov spectrum collapses to zero rather than being strictly negative. This suggests a specific type of stability where finite positive eigenvalues exist only as a vanishing arithmetic edge.

The authors emphasize that the physical realization of either marginal sector depends on dynamical selection and initial statistical preparation, not merely on the fixed-mode Lyapunov calculation. The work reframes the turbulence problem as an arithmetic spectral collapse, offering a mathematically exact framework for the late-time statistical state of decaying turbulence.