AI 中文总结
本文提出相角框架，逐维度分类Navier-Stokes方程n元周期解，证明n=3,4存在无外力解，n=5-8受阻但存在闭式强迫解，并解决三维完备性问题。
AI 中文摘要
我们引入了一个相角框架，用于构造和分类不可压缩Navier-Stokes方程的n元周期解，并利用该框架逐维度确定此类解何时存在以及为何失败。速度场是两个n重循环三角乘积之差，通过一个伸缩恒等式，对于每个相位和每个n≥3，该速度场均满足无散度条件。问题随后简化为关于n个相角的两个显式条件：一个约化条件（必要且代数）和一个压力可积性条件（充分且微分）。将两者分离，可以定位失败的构造，而不仅仅是记录失败。与早期三周期解所依赖的流函数路径不同，该框架不限于三维，也不预设Beltrami性质。对于3≤n≤8，我们证明无外力解在n=3和n=4时存在，而在其他维度不存在。在三维中，可容许相位恰好构成两个族，它们是互为镜像的，以螺旋度的符号区分；这些族是已知的，我们证明该构造不承认其他族，从而解决了这些族被发现时遗留的完备性问题。在四维中，可积性将一条单参数曲线约化为一个单族，因此四分之一周期偏移是强制的，而非假设的。障碍按奇偶性划分：在n=5,7时为算术障碍，此时每个可容许相位向量都会湮灭场；在n=6,8时为微分障碍，此时场存在但其对流项不是梯度。尽管如此，每个受阻维度都承载一个闭式形式的精确强迫解。这些不仅仅是自平衡：这些相位是应变和旋转在逐点精确平衡的相位，因此根本不产生压力。阻碍无外力问题的因素，在强迫问题中变成了可精确预设的输入。
英文摘要
We introduce a phase-angle framework for constructing and classifying n-tuple-periodic solutions of the incompressible Navier-Stokes equations, and use it to determine, dimension by dimension, when such solutions exist and why they fail. The velocity field is a difference of two n-fold cyclic trigonometric products, made divergence-free for every phase and every n >= 3 by a telescoping identity. The problem then reduces to two explicit conditions on the n phase angles: a reduction condition, necessary and algebraic, and a pressure-integrability condition, sufficient and differential. Separating them allows a failed construction to be located rather than merely recorded. Unlike the stream-function route on which earlier tri-periodic solutions rest, the framework is not confined to three dimensions, and does not presuppose the Beltrami property. For 3 <= n <= 8 we prove that unforced solutions exist at n=3 and n=4 and nowhere else. In three dimensions the admissible phases form exactly two families, mirror images distinguished by the sign of the helicity; these families are known, and we establish that the construction admits no others, settling a completeness question left open when they were found. In four dimensions integrability reduces a one-parameter curve to a single family, so the quarter-period offset is forced, not assumed. The obstructions divide by parity: arithmetic at n=5,7, where every admissible phase vector annihilates the field; differential at n=6,8, where the field survives but its convective term is not a gradient. Every obstructed dimension nevertheless carries an exact forced solution in closed form. These are not mere self-balancing: the phases are those at which strain and rotation are in exact pointwise balance, so that no pressure is generated at all. What obstructs the unforced problem becomes, in the forced problem, an exactly prescribable input.
Comments105 pages. Verification scripts and the complete constraint systems are included as ancillary files
