CayleyPy-4: AI-Holography. Towards analogs of holographic string dualities for AI tasks
This paper proposes a bold interdisciplinary bridge between holographic string dualities and artificial intelligence, hypothesizing that AI tasks such as language modeling can be viewed as particle trajectory prediction on graphs admitting a holographically dual "string" description. Drawing on the AdS/CFT correspondence, the authors conjecture that word metrics on $S_n$ Cayley graphs correspond to areas under lattice paths in dual planar polygons, verified computationally via their CayleyPy library.
The paper presents an ambitious but highly speculative research program connecting AdS/CFT holography to AI embeddings, with rigorous mathematical content in combinatorial examples (particularly $S_n$ Cayley graphs) but significant unverified leaps in applying these insights to general AI systems. While the quasi-polynomiality conjectures for graph diameters are grounded in computational evidence and Ehrhart theory, the central claim that discrete strings provide efficient embeddings for language models or reinforcement learning remains a hypothesis supported primarily by formal analogy rather than empirical validation on benchmarks.
The mathematical analysis of specific Cayley and Schreier graphs demonstrates concrete instances of the "complexity = area" principle, where word metrics are computed via lattice path areas and Ehrhart quasi-polynomials. The connection between ROC curves (Dyck paths) and Grassmannian graphs is rigorous, showing that distance equals area, and the Stanley-Edelman-Greene correspondence is elegantly reinterpreted as a bijection between particle extremals on graphs and string worldsheets represented by Young tableaux with tropical (ReLU) actions.
The leap from mathematical combinatorics to AI applications relies on analogical reasoning rather than demonstrated utility, with claims that holographic duals provide "good embeddings" lacking experimental validation on actual AI tasks. The paper admits its preliminary status, and many conjectures (e.g., the $(k-1)$-shrinkage principle) rely on pattern-matching from small-$n$ computations without proof. The extension from the $S_n$ case to "English and other natural languages" or "protein sequences" constitutes an enormous extrapolation unsupported by technical derivation, risking circular reasoning where the ability to learn a language is taken as evidence for the duality's existence.
The evidence supporting the quasi-polynomiality of diameters for $S_n$ Cayley graphs is computational (BFS and AI-assisted pathfinding for $n \leq 30$), fitting Ehrhart polynomials to observed data. However, the comparison to AdS/CFT remains metaphorical; while the authors correctly note that "complexity = volume/action" maps to word metrics equaling path areas, they do not derive this duality from first principles or demonstrate it for non-group-theoretic AI tasks. The related work covers relevant physics and mathematics but omits critical discussion of why discrete tropical string actions should outperform existing embedding methods in NLP or RL.
The paper references the CayleyPy GitHub repository and provides explicit quasi-polynomial formulas for various graph families (e.g., consecutive $k$-cycles), enabling reproduction of the combinatorial results. However, as a preliminary version, it lacks detailed hyperparameters for the AI-based pathfinding components, and the methodology for "guessing the longest elements" depends on heuristic pattern recognition without algorithmic specification. The computational results for larger $n$ (approaching $n=40$) are described as work in progress, limiting full reproducibility of the quasi-polynomial conjectures.
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.
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