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.

翻译：本文引入了一种称为调整杯积神经层的结构。该神经层是一种通过高阶规范理论中的调整项硬编码杯积运算的神经基元，从而构建出具有天然规范不变性的读出机制。其主要理论结果表明，在闭合链上，输出完全取决于调整系数。将该系数设为零会完全消除输出（与其他参数无关），因此调整是规范不变信号产生的唯一来源。他们证明该可观测量是一个非零二次型，且在单次与两次规范变换下具有严格不变性。