作为湍流吸引子的欧拉系综：宇称扇区、零模和一个ζ边缘
Number Theory of Decaying Turbulence 1: Operator Representation and Universality
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中文总结 AI 辅助
研究有限欧拉系综的李雅普诺夫谱，通过求解切向线性化问题简化为算术谱问题。发现偶数 \(N\) 时零缠绕扇区有奇异离散零模，其配分函数有特定关系。奇数和穿孔偶数系综是边缘固定模李雅普诺夫极限，有限正特征值由特定函数控制，还分析了不同维度下扰动情况。
中文摘要 AI 辅助
我们计算了有限欧拉系综的李雅普诺夫谱，它是自由衰减不可压缩纳维 - 斯托克斯湍流的重标动量 - 环方程的紧致算术不动点。在有限截止 \(N\) 时，切向线性化问题可精确求解：完整的伊辛历史 \(\sigma_k = \pm1\) 仅通过闭合缠绕 \(qr=\sum_{k=1}^N\sigma_k\) 进入。稳定性问题因此简化为关于约化有理角 \(p/q\) 和缠绕扇区 \(r\) 的算术谱问题。连续极限分为三个局部扇区。对于奇数 \(N\)，\(q\) 和 \(r\) 均为奇数，所以 \(r = 0\) 被宇称排除。对于偶数 \(N\)，零缠绕扇区 \(r = 0\) 是允许的，且必须与穿孔扇区 \(r\neq0\) 分开。它们的配分函数满足 \(Z_{e,0}(N)/Z_{e,*}(N)\sim 6N/\pi^2\)，所以零缠绕扇区是奇异离散零模，不属于高斯 \(r\) 连续统。偶数零缠绕系综具有连续切向谱且李雅普诺夫指数为正，是不稳定的。在奇数和穿孔偶数系综中，谱角保持量子化，对于每个固定谱标签 \(n\)，归一化特征值定律弱收敛到 \(\delta_0\)。因此这两个扇区是边缘固定模李雅普诺夫极限。它们有限的正特征值仅作为由互质余切和、约当函数、狄利克雷卷积和 \(\zeta(s)\) 控制的消失算术边缘存在。对于 \(d>2\)，横向扰动在一阶是零模；在两个边缘扇区，它们二次阻碍被径向修正吸收，没有二次谱移。
英文摘要
We derive a formal statistical solution of freely decaying incompressible turbulence in arbitrary dimension \(d>1\) using Navier--Stokes loop equations. The loop Fourier transform maps smooth deterministic Cauchy data in infinite space to an oscillatory amplitude of a one-dimensional momentum-loop quantum field theory. In bounded-variation calculus the nonlinear advection term becomes a closed-loop total derivative and cancels on the compact spherical target, leaving a diffusive momentum-loop evolution. Its exact decaying solution is the planar Euler ensemble of rational star-polygon walks, whose continuum limit splits into parity classes, \(η=N\bmod 2\). In logarithmic time the Euler ensemble is a fixed point of the compact momentum-loop dynamics. The even ensemble carries an alternating unstable mode with Lyapunov exponent \(λ=\cot^2(πp/q)>0\), while the odd representatives have no local shape instabilities. For \(d>2\) the planar ensemble is a slice of a degenerate manifold of equal-step spherical polygons. The transverse deformations along this manifold are exact zero modes; integrating over them gives a singular Wilson-loop functional, so they are projected out of the admissible ensemble. The normal edge-length defects are strictly stable, with the universal angular Laplacian as the leading continuum operator. The odd Euler ensemble is therefore the locally stable turbulent attractor in every dimension. From the velocity correlation of the ensemble we prove that its energy spectrum does not depend on the dimension \(d>1\): two- and three-dimensional decaying turbulence share one scaling function. The spectrum and its Riemann-zeta structure are derived in the second paper of this series, and the comparison with simulations and experiments is given in the third.
