This is the fourth paper in the CayleyPy project, which applies AI methods to the exploration of large graphs. In this work, we suggest the existence of a new discrete version of holographic string dualities for this setup, and discuss their relevance to AI systems and mathematics. Many modern AI tasks -- such as those addressed by GPT-style language models or RL systems -- can be viewed as direct analogues of predicting particle trajectories on graphs. We investigate this problem for a large family of Cayley graphs, for which we show that surprisingly it admits a dual description in terms of discrete strings. We hypothesize that such dualities may extend to a range of AI systems where they can lead to more efficient computational approaches. In particular, string holographic images of states are proposed as natural candidates for data embeddings, motivated by the "complexity = volume" principle in AdS/CFT. For Cayley graphs of the symmetric group S_n, our results indicate that the corresponding dual objects are flat, planar polygons. The diameter of the graph is equal to the number of integer points inside the polygon scaled by n. Vertices of the graph can be mapped holographically to paths inside the polygon, and the usual graph distances correspond to the area under the paths, thus directly realising the "complexity = volume" paradigm. We also find evidence for continuous CFTs and dual strings in the large n limit. We confirm this picture and other aspects of the duality in a large initial set of examples. We also present new datasets (obtained by a combination of ML and conventional tools) which should be instrumental in establishing the duality for more general cases.

翻译：本论文为CayleyPy项目的第四篇，该项目致力于运用人工智能方法探索大规模图结构。本研究提出在该框架下存在一种新型离散弦全息对偶，并探讨其与人工智能系统及数学领域的关联。现代人工智能中的诸多任务（如GPT类语言模型或强化学习系统所处理的问题）可被视为图结构上粒子轨迹预测问题的直接类比。我们针对一大类Cayley图系统研究了该问题，并发现其意外地允许一种离散弦对偶描述。我们推测，此类对偶性可能适用于更广泛的人工智能系统，从而催生更高效的计算方法。具体而言，受AdS/CFT中"复杂度=体积"原理启发，我们提出将状态的弦全息像作为数据嵌入的自然候选方案。对于对称群S_n的Cayley图，研究结果表明对应的对偶对象为平面多边形（即平坦多边形）。图的直径等于多边形内部经n缩放后的整点数量。图的顶点可全息映射为多边形内的路径，而通常的图距离对应路径下方的面积，从而直接实现了"复杂度=体积"范式。我们还发现了大n极限下连续共形场论与对偶弦存在的证据。我们通过大量初始实例验证了该图景及对偶性的其他方面。此外，我们发布了由机器学习与传统工具联合生成的新数据集，这些数据集应有助于在更一般情形下确立该对偶性。