提出可自动保持规范不变性的神经层，专用于物理与几何中的上积运算。
Adjusted Cup-Product Neural Layer
- 通过高阶规范理论的修正项硬编码上积运算
- 在闭合循环上输出仅依赖修正系数，零值时信号完全消失
- 对一阶和二阶规范变换严格不变，适合物理模拟建模
物理学与几何学中许多重要可观测量是链上积的体现。本文提出调整上积神经层，该神经原语将上积运算与来自高阶规范理论的修正项相结合，天然具备规范不变性。理论证明：在闭合圈上，输出完全依赖于修正系数；当该系数为零时，无论其他参数如何，输出均归零。因此，修正项是唯一产生规范不变信号的来源。进一步证明该可观测量为非零二次型，且对一阶和二阶规范变换保持精确不变。
原文摘要 · Abstract (English)
Many important observables in physics and geometry are cup products of cochains. The adjusted cup product neural layer has been introduced in this paper. It is a neural primitive that hard wires the cup product with an adjustment term from higher gauge theory. This creates a readout that is gauge invariant by design. Their main theoretical result shows that on a closed cycle the output relies entirely on the adjustment coefficient. Setting this coefficient to zero removes the output completely regardless of other parameters. Thus the adjustment is the only source of gauge invariant signal. They prove this observable is a nonzero quadratic form and is exactly invariant under one and two gauge transformations.
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