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

We compute the Lyapunov spectrum of the finite Euler ensembles, compact arithmetic fixed points of the rescaled momentum-loop equation for freely decaying incompressible Navier--Stokes turbulence. At finite cutoff \(N\), the tangential linearized problem is exactly solvable: the full Ising history \(\sigma_k=\pm1\) enters only through the closure winding \(qr=\sum_{k=1}^N\sigma_k\). The stability problem therefore reduces to an arithmetic spectral problem over reduced rational angles \(p/q\) and winding sectors \(r\).
The continuum limit splits into three local sectors. For odd \(N\), both \(q\) and \(r\) are odd, so \(r=0\) is excluded by parity. For even \(N\), the zero-winding sector \(r=0\) is allowed and must be separated from the punctured sector \(r\ne0\). Their partition functions satisfy \(Z_{e,0}(N)/Z_{e,*}(N)\sim 6N/\pi^2\), so the zero-winding sector is a singular discrete zero mode, not part of the Gaussian \(r\)-continuum.
The even zero-winding ensemble has a continuous tangential spectrum with positive Lyapunov exponents and is unstable. In the odd and punctured even ensembles, the spectral angle remains quantized, and for every fixed spectral label \(n\) the normalized eigenvalue law converges weakly to \(\delta_0\). Thus these two sectors are marginal fixed-mode Lyapunov limits. Their finite positive eigenvalues survive only as a vanishing arithmetic edge governed by coprime cotangent sums, Jordan totients, Dirichlet convolution, and \(\zeta(s)\). For \(d>2\), transverse perturbations are zero modes at linear order; in the two marginal sectors their quadratic obstruction is absorbed by a radial correction, leaving no quadratic spectral shift.