AI 中文总结
研究4维N=1超杨-米尔斯理论的圈方程，通过路径排序算子演算消除虚假奇点，构造超对称霍奇对偶曲面泛函，得到精确非微扰禁闭面积律因子及梯度流不动点。
AI 中文摘要
我们将纯 $4d, N=1$ 超杨-米尔斯（SYM）理论的圈方程表述为有限几何形式。通常的等点圈导数是奇异的，因为当围道点重合时，规范场插入的顺序会丢失。我们的路径排序算子演算（POOC）将插入项保持在不同的有序槽中，形成阶化对易子和雅可比组合，之后才取重合极限。这在不引入截断的情况下消除了虚假的运动学奇点；真正的紫外接触项仍是一个独立的物理分布。\n 我们将 POOC 应用于精确的 Itoyama-Takashino 超圈层级结构，并构造了一个洛伦兹超对称霍奇对偶（SHD）曲面泛函。其霍奇分解后的面积导数被局域手征和反手征圈算子湮灭。曲面测地线缝合下的精确可加性表明，该零模的指数可乘以完整有限 $N$ 层级结构的任意解，且不会改变方程。对于平面围道，SHD 泛函就是几何面积。特别地，对于长矩形威尔逊圈，$W[C_{T,L}]\text{\textasciitilde}\exp[-iσ_kLT]$，且 $E_k(L)=σ_kL>0$。因此，该构造在 $N=1$ SYM 中给出了一个精确的非微扰禁闭面积律因子。\n 欧几里得延拓后，同一个 SHD 因子是确定性零噪声 SYM 梯度流的精确不动点。将其解释为动力学选择的平衡态仍需要长流时间、大体积和零噪声极限下的稳定性，但这并不影响精确零模定理和有限 $N$ 修饰定理。完整的平面威尔逊圈解（包括未修饰的涨落因子和激发谱）将在本系列的后续论文中展开。
英文摘要
We formulate the loop equations of pure $4d, N=1$ super Yang--Mills (SYM) theory in a finite geometric form. The usual equal-point loop derivatives are singular because the order of gauge-field insertions is lost when contour points coincide. Our path-ordered operator calculus (POOC) keeps the insertions in distinct ordered slots, forms the graded commutators and Jacobi combinations, and only then takes the coincidence limit. This removes the spurious kinematical singularities without introducing a cutoff; the genuine ultraviolet contact remains a separate physical distribution. We apply POOC to the exact Itoyama--Takashino superloop hierarchy and construct a Lorentzian supersymmetric Hodge-dual (SHD) surface functional. Its Hodge-resolved area derivative is annihilated by the local chiral and anti-chiral loop operators. Exact additivity under surface-geodesic sewing shows that the exponential of this zero mode multiplies any solution of the complete finite-$N$ hierarchy without changing the equations. For planar contours the SHD functional is the geometric area. In particular, for a long rectangular Wilson loop, $W[C_{T,L}]\sim\exp[-iσ_kLT]$ and $E_k(L)=σ_kL>0$. Thus the construction gives an exact nonperturbative confining area-law factor in $ N=1$ SYM. After Euclidean continuation, the same SHD factor is an exact fixed point of the deterministic zero-noise SYM gradient flow. Its interpretation as a dynamically selected equilibrium still requires stability in the long-flow-time, large-volume, and zero-noise limits, but this does not affect the exact zero-mode and finite-$N$ dressing theorems. The complete planar Wilson-loop solution, including the undressed fluctuation factor and excitation spectrum, will be developed in the next papers of this series.
Comments53 pages, 1 figure, Supplementary Material pdf file plus 12 Mathematica files validating the algebra derived in Supplementary Material
