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CayleyPy-4: AI-Holography. Towards analogs of holographic string dualities for AI tasks

A. Chervov, F. Levkovich-Maslyuk, A. Smolensky, F. Khafizov, I. Kiselev, D. Melnikov, I. Koltsov, S. Kudashev, D. Shiltsov, M. Obozov, S. Krymskii, V. Kirova, E. V. Konstantinova, A. Soibelman, S. Galkin, L. Grunwald, A. Kotov, A. Alexandrov, S. Lytkin, D. Fedoriaka, A. Chevychelov, Z. Kogan, A. Natyrova, L. Cheldieva, O. Nikitina, S. Fironov, A. Vakhrushev, A. Lukyanenko, V. Ilin, D. Gorodkov, N. Bogachev, I. Gaiur, M. Zaitsev, F. Petrov, L. Petrov, T. Gaintseva, A. Gavrilova, M. N. Smirnov, N. Kalinin, A. Khan, K. Jung, H. Mousset, H. Isambert, O. Debeaupuis

Year: 2026Venue: arXiv preprintArea: hep-thType: PreprintEmbeddings: 293

Intelligence

Status: succeeded | Model: google/gemini-3.1-flash-lite-preview | Prompt: intel-v1 | Confidence: 93%

Last extracted: 3/26/2026, 2:48:01 AM

Summary

The paper introduces a novel framework for AI-holography, proposing that AI tasks (like language modeling and reinforcement learning) can be modeled as particle trajectories on Cayley graphs. It hypothesizes a discrete holographic string duality, where graph nodes are mapped to lattice paths within planar polygons, and graph metrics (word metrics/diameters) correspond to geometric quantities (areas/lattice point counts) in the dual space, aligning with the 'complexity = volume/action' principle from AdS/CFT.

Entities (5)

AdS/CFT correspondence · theoretical-physics-principle · 100%CayleyPy project · research-project · 100%S_n Cayley graphs · mathematical-structure · 100%Ehrhart quasi-polynomials · mathematical-concept · 95%ROC curves · data-representation · 90%

Relation Signals (3)

CayleyPy project → appliesto → Cayley graphs

confidence 95% · The CayleyPy project, which applies AI methods to the exploration of large graphs.

Word metrics → correspondsto → Area under lattice paths

confidence 90% · word metrics (“gate complexities”) are equal to the areas under the corresponding paths

S_n Cayley graphs → dualto → planar polygons

confidence 90% · For Cayley graphs of the symmetric group S_n, our results indicate that the corresponding dual objects are flat, planar polygons.

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Abstract

Abstract: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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CAYLEYPY-4: AI-HOLOGRAPHY. TOWARDS ANALOGS OF HOLOGRAPHIC STRING DUALITIES FOR AI TASKS. (PRELIMINARY VERSION) A. CHERVOV, F. LEVKOVICH-MASLYUK, A. SMOLENSKY, F. KHAFIZOV, I. KISELEV, D. MELNIKOV, I. KOLTSOV, S. KUDASHEV, D. SHILTSOV, M. OBOZOV, S. KRYMSKII, V. KIROVA, E.V. KONSTANTINOVA, A. SOIBELMAN, S. GALKIN, L. GRUNWALD, A. KOTOV, A. ALEXANDROV, S. LYTKIN, D. FEDORIAKA, A. CHEVYCHELOV, Z. KOGAN, A. NATYROVA, L. CHELDIEVA, O. NIKITINA, S. FIRONOV, A. VAKHRUSHEV, A. LUKYANENKO, V. ILIN, D. GORODKOV, N. BOGACHEV, I. GAIUR, M. ZAITSEV, F. PETROV, L. PETROV, T. GAINTSEVA, A. GAVRILOVA, M. N. SMIRNOV, N. KALININ, A. KHAN, K. JUNG, H. MOUSSET, H. ISAMBERT, AND O. DEBEAUPUIS ABSTRACT. This work is the fourth paper in the CayleyPy project, which aims to apply AI-based methods to large graphs. Here, we propose connections to a novel discretized analogue of holo- graphic string dualities originating in theoretical physics. We argue that this perspective can lead to more efficient approaches to a wide range of AI tasks. Many modern AI problems—such as those addressed by GPT-style language models or rein- forcement learning systems—can be viewed as direct analogues of predicting particle trajectories on graphs. We hypothesize that these tasks admit a holographically dual description in terms of discrete strings, and that working in this dual representation can provide a more tractable formu- lation of the original problems. In particular, strings - holographic images of states are proposed as natural candidates for embeddings, motivated by the “complexity = volume/action” principle in AdS/CFT. In a simple illustrative example, the ROC curves serve as holographically dual strings to the nodes of some graphs. From a mathematical standpoint, we expect that all properties of graphs can be expressed entirely within this duality framework, yielding nontrivial identities. Furthermore, for Cayley graphs of the symmetric group S n , we conjecture that the corresponding dual objects are planar polygons and present various examples. Graph diameters equal the number of integer lat- tice points in the n-scaled polygon (Ehrhart quasi-polynomials). Vertices of graphs can be mapped (“holography”) to lattice paths inside the polygon in such a way that word metrics (“gate com- plexities”) are equal to the areas under the corresponding paths in accordance to “complexity = volume/action” principle. This thus provides an explanation for the quasi-polynomiality conjecture regarding diameters and word metrics from our previous paper. Stanley-type formulas for count- ing shortest paths can be reinterpreted as identities relating particle extremals on Cayley graphs to string extremals on the associated polygons. In some cases, the graph Laplacian coincides with integrable spin-chain Hamiltonians and yields conformal field theories in the large-size limit. We also study the corresponding H-polynomials and their properties, including positivity, unimodality, duality, and analogues of the Riemann conjecture. Project page: https://github.com/CayleyPy/CayleyPy Key words and phrases. Machine learning, reinforcement learning, Cayley graphs. 1 arXiv:2603.22195v1 [hep-th] 23 Mar 2026 CayleyPy-4: HolographyCayleyPy collaboration CONTENTS 1. Introduction3 1.1.Main hypothesis: particle-string holographic duality for AI-tasks5 1.2.Guiding principle: “complexity = volume/action”5 1.3. S n -Cayley graphs to planar polygon duality: lattice paths as discrete strings holographically dual to graph nodes6 1.4.List of contributions8 1.5.Organization of the paper9 2. Simplest Examples. ROC curve = string dual to Gr(k,n) graph node11 3. AI tasks as predictions of particle trajectories. General AI-holography expectations14 3.1. Particle trajectories as texts or action sequences14 3.2. AI-holography18 3.3. Further remarks.19 4. Case studies19 4.1. S n Cayley graphs and polygon duality19 4.2. CayleyPy AI methodology for assisting in determining the duality23 4.3. SL(2,Z) and the Farey graph24 5. Neighbor transpositions Cayley and Schreier graphs26 5.1. Multiset0 n 0 1 n 1 ... m n m 26 5.2. Cosets with 0,1,2 components - alternative representation27 5.3. Box-Ball System evolution as a deterministic walk on a Schreier/Cayley graph and the AUC–AAC duality29 5.4. Schreier graphs for orbits of tuples31 5.5. Large size limits, towards ”macroscopic” descriptions33 5.6. Vershik’s (1996) limit shapes for rectangular Young diagrams, general q33 5.7. Vershik’s (1985) limit shapes for rectangular Young diagrams q = 134 5.8. Limit shape for ROC-curves (Dyck paths)35 5.9. Dynamics in large size limits. TASEP-Burgers correspondence. KPZ.36 5.10. Spanning trees for finite spin-up sector, 1/n expansion, Benjamini-Schramm limit to lattices, Mahler measures38 5.11. Limit to field theory string-like model. Spanning trees for many spins up and down sector45 6. Neighbor transpositions extended by (0,n− 1) (“wrapped” or “affine” case)49 6.1. Duality for the Wrapped Case49 6.2. Diameters for the Schreier coset graphs “few-coincide”: S n /S d 51 7. Bethe ansatz and graph spectrum52 7.1. Spectral gap53 7.2. Relation to Laplacian spectrum53 7.3. Schreier graph with non-wrapped 2-cycles53 7.4. Relation between Schreier and Johnson graphs54 8. Consecutive-(k)-cycles. Quasi-polynomiality, etc.55 8.1. Section outline55 8.2. (k− 1)-Shrinkage heuristics: Results and Difficulties55 8.3. Theoretical lower and upper bounds on the diameters. Cayley graphs.55 8.4. Theoretical diameters estimate. Schreier coset graphs S n / S ⌊n/2⌋ × S n−⌊n/2⌋ 56 8.5. Quasi-polynomials for diameters. Cayley graphs.57 8.6. Schreier coset graph: S n / S ⌊n/2⌋ × S n−⌊n/2⌋ (not inverse-closed)62 8.7. Schreier coset graph: S n / S ⌊n/2⌋ × S n−⌊n/2⌋ (inverse-closed)65 8.8. Schreier coset graph: S n / S l × S n−l (“k-shrunken” Grassmannian Gr(l,n,k))74 8.9. Schreier coset graph: ”few coincide”76 8.10. Schreier coset graph: ”L-Different” inverse closed82 8.11. Word-metrics for [012]-repeated full flips, 3-cycles86 9. Wrapped (”affine” or ”periodic”) (k)-consecutive cycles. Quasi-polynomiality, etc.88 9.1. Section outline88 9.2. 2(k− 1)-Shrinkage heuristics: Results and Difficulties88 9.3. Theoretical diameters estimate. Schreier coset graphs S n / S ⌊n/2⌋ × S n−⌊n/2⌋ 88 2 CayleyPy-4: HolographyCayleyPy collaboration 9.4. Schreier coset graph: S n / S l × S n−l (“k-shrunken” affine Grassmannian Gr aff (l,n,k))89 9.5. Schreier coset graph: S n / S ⌊n/2⌋ × S n−⌊n/2⌋ (not inverse-closed)91 9.6. Schreier coset graph: S n / S ⌊n/2⌋ × S n−⌊n/2⌋ (inverse-closed)98 9.7. Some eccentricities for Schreier coset graph: S n / S ⌊n/2⌋ × S n−⌊n/2⌋ 101 9.8. Word metrics to full flips for Cayley and Schreier (S n /S d ) graphs106 9.9. Coset 2-different115 9.10. Coset 3-different116 9.11. Coset 4-different121 10. Reminder. Background and related works122 10.1. Cayley and Schreier graphs, diameters, growth122 10.2. Quasi-polynomial functions124 10.3. Ehrhart polynomials124 10.4. ROC curves and AUC125 Acknowledgments128 References129 1. INTRODUCTION The deep learning revolution is one of the most exciting scientific breakthroughs of our time. The second superstring revolution [Witten1995; Schwarz1996] was one of these in the past. The present paper takes a step toward merging these two. The theme of duality is one of the central ideas in theoretical physics: the belief that for many complex systems there exists a “dual” descrip- tion in which questions that are difficult in the original (“strong-coupling”) formulation become simpler in the dual (“weak-coupling”) one (e.g. [Polyakov1987]). In particular, the AdS/CFT holographic duality [Maldacena1998] marked the culmination of the second superstring revolu- tion with the discovery of the long-sought duality between gauge theory and string theory, and has since become the most cited paper in high-energy theoretical physics. Here, we present evidence that a similar principle may apply to artificial intelligence tasks, arguably leading to more efficient approaches than those currently available. We draw a parallel between dualities in physics and the concept of embedding (“latent represen- tation”) in cognitive theory, both for natural and artificial neural networks. The role of embeddings in AI is analogous to the notion of duality in physics: they transform an input representation, in which semantic relations are difficult to analyze, into a representation where these relations be- come more tractable. A well-known example in ML is the relation ”king - man + woman = queen” (Word2vec [Mikolov2013]), where a nontrivial semantic relationship is mapped to a simple vec- tor arithmetic operation. Finding appropriate embeddings is a central problem in modern AI. We argue that the perspective of the AdS/CFT correspondence offers a new conceptual framework for approaching this problem. Brief outline of the main ideas. (1) Setup. The starting point is that AI tasks such as language or RL modeling can be viewed as particle trajectory prediction tasks on edge-labeled graphs. (2) We hypothesize that in many cases such particle systems admit a holographically dual string de- scription (in the spirit of AdS/CFT), converting difficult problems into more tractable ones. We outline how this can be used to build more effective AI systems: embeddings (”latent representa- tions”) are strings, training is based on the ”complexity = volume” principle in AdS/CFT. From the mathematical viewpoint, we expect that all properties of graphs can be seen from the dual side, as is typical in string dualities, leading to new mathematical insights. (3) For the case of S n Cayley graphs, we present significant evidence that the dual objects are planar rational polygons, and that holography maps nodes of the graphs to lattice paths (discrete strings) inside the poly- gons. We also present connections to various deep questions and conjectures in mathematics. (4) As simplest examples, ROC curves (Dyck paths) can be used to illustrate the ideas in a clear and straightforward manner. 3 CayleyPy-4: HolographyCayleyPy collaboration Surprisingly, the simplest example – which is just the Cayley (Schreier) graph generated by neighboring transpositions – is connected to several advanced areas of research: quantum inte- grable systems and the Bethe ansatz (since the graph Laplacian coincides with the Hamiltonian of the Heisenberg spin chain), Mahler measures (related to supersymmetric Landau–Ginzburg models and mirror symmetry for Fano varieties ) for counting spanning trees, Burgers and Kar- dar–Parisi–Zhang (KPZ) equations for description of geodesic flow in a dual formulation for large size, limit shapes of Young diagrams, Beilinson’s conjectures on special values of L-functions, and more. Moreover, we propose that conformal field theory arising in large size limit for such graphs is dual (in AdS/CFT manner) to a simple free scalar with radius of compactification related to normalized area of the dual polygon (in this case it plays the role of the dual ”string”) and propose conjectures on spectrum of conformal dimensions. Let us present a more detailed outline. (1) AI as particle on edge-labeled graphs. AI tasks involving languages or reinforcement learning can be naturally interpreted as problems of predicting particle dynamics on edge- labeled graphs (for instance, Cayley graphs). The data — texts or sequences of actions — correspond to particle trajectories, that is, sequences of traversed edges. Such edge (”token”) sequences naturally define the corresponding “texts.” Typical tasks then amount to determining the trajectory given initial (the ”prompt” for generative models) or bound- ary conditions (the ”masked language modeling objective” for e.g. BERT or ”win the game/reach the goal” in RL). (2) Particle-string holographic duality. We hypothesize that many such tasks admit a holo- graphically dual equivalent description in terms of discrete string theory, transforming a difficult problem on the original graph-side (”CFT-side”) into a more tractable one on the dual string-side (”AdS-side”), in the spirit of strong–weak coupling duality, thus giving a key to build more efficient AI-systems. This perspective is inspired by the landmark AdS/CFT correspondence. We further argue that strings (viewed as holographic duals of states) provide natural candidates for embeddings (latent representations), and that un- conventional (“tropical”) string actions may be required to capture phenomena intrinsic to discrete settings. We remark that “embeddings,” whether in natural or artificial neural systems, play a role analogous to “duality” in physics: they transform input data into repre- sentations that are more tractable than the original formulation. The celebrated AdS/CFT principle ”complexity = volume/action” can be used as a training objective for string em- beddings, and also plays many other key roles. From a mathematical point of view, we expect that all properties of the graph admit a complete description in terms of the dual theory, as it is typical in string dualities. This dual perspective may provide new results and structural insights, some of which are presented here. We suggest various connec- tions with integrable systems, matrix models, conformal field theory, cluster algebras, the thermodynamic Bethe ansatz, and related structures. (3) S n Cayley graphs / planar polygon duality. In the case of Cayley graphs of the per- mutation group S n , we propose that the holographically dual objects are planar rational polygons. The holographic correspondence maps graph nodes to lattice paths (discrete strings). Diameters and word metrics can then be described in terms of Ehrhart quasi- polynomials of the associated polygons or of their subregions (e.g. areas under lattice paths), thus explaining the quasi-polynomiality conjecture from our previous paper. This is consistent with the “complexity = volume/action” principle from AdS/CFT: complex- ities coincide with word metrics in Cayley graphs, while counting lattice points corre- sponds to discretized volumes. Using the CayleyPy AI-based library and methodology, we obtained various results and conjectures concerning the corresponding quasi-polynomials and related H -polynomials. That is, we compute “complexities” and demonstrate their agreement with the corresponding “volumes.” 4 CayleyPy-4: HolographyCayleyPy collaboration (4) Simplest examples. ROC-curves (Dyck like paths) as strings. As a starting point, we present simple and explicit examples in which classical ROC curves can be interpreted as discrete strings holographically dual to nodes of an appropriate graph. Several combina- torial facts then admit a natural interpretation in terms of discrete string duality. Surpris- ingly, even this simple example is connected to deep recent mathematical results, and in fact touches upon several open problems. 1.1. Main hypothesis: particle-string holographic duality for AI-tasks. The AdS/CFT holo- graphic duality [Maldacena1998] predicts that difficult questions on the CFT side (”strong cou- pling”) can be computed via more tractable geometric methods on the AdS side (where string theory at ”weak coupling” reduces to gravity). We hypothesize the existence, outline expecta- tions and elaborate several examples of a similar holographic duality in the context of graphs, languages, and reinforcement-learning “environments” — systems of interest in artificial intelli- gence, mathematics, and physics. First, we emphasize that many such settings can be viewed as particle trajectories on edge-labeled graphs (e.g. Cayley graphs), where the trajectories define a “language,” i.e., sequences of admissible tokens/moves. Second, we hypothesize that a “particle” moving on such graphs may admit a holographically dual description as a (discrete) “string” living on an appropriate dual object. If such a duality exists, the string description may be more tractable (analogous to a weak-coupling regime) and could provide a key to constructing more powerful AI models. In particular we expect that strings understood as holographic images of states may provide ”good embeddings”. Our examples suggest that one should consider unusual (“tropical”) analogues of discrete string actions that involve functions of the form max(·, 0) (also known as ReLU ). Such terms ensure that, even after fixing initial or boundary conditions, the equations of motion retain substantial local freedom in their solutions — a feature that appears necessary for describing discrete setups. From a mathematical perspective, we expect that the properties of a graph (or the associated language, etc.) may be computable from the dual object, potentially leading to non-trivial identities and new structural insights. In particular, various combinatorial results might be reinterpreted as manifestations of string dualities. If indeed true as stated, our proposal would imply that string-theoretic holographic dual de- scriptions, in the sense outlined above, may exist for a wide range of AI systems. Examples of this could potentially include even such settings as English and other natural languages; programming languages; robotic manipulator systems; mathematics viewed as a formal proof system; games such as Go, chess, etc.; the languages of life (admissible protein and DNA sequences); chemical molecules encoded in databases; and so on. One may even argue that one reason a duality or sim- plification of this kind might exist for e.g. English language is that we indeed can learn it! Both humans and machines master it, which would be impossible without ability to convert information to some latent representation where complex semantic relations can be seen via simple operations (which is the basically the duality: complex to simple). Our current proposal is more subtle: we expect that duality may be holographic in the spirit of AdS/CFT and that these efficient latent representations (”embeddings”) could be interpreted as strings – that is, as holographic images of states (sentences in this case). Downstream computations performed on these embeddings may naturally involve Riemannian metrics –reminiscent of the AdS metric – while more subtle phe- nomena could correspond to finite-size corrections. More broadly, this perspective may even lead to a deeper geometric understanding of key concepts in AI. 1.2. Guiding principle: “complexity = volume/action”. The celebrated AdS/CFT principle “complexity = volume/action” (L. Susskind et al: [Stanford2014; Brown2016]), together with subsequent works (including [Lin2019] and work by one of the present authors, D. Melnikov, [Camilo2019]) that bridge this idea with Cayley graphs, serves as one of the key insights and guiding frameworks for what follows. In the graph-theoretic setting, complexity admits a very simple interpretation. The complexity of one node with respect to another is defined as the length of the shortest path between them in the graph. Then the ”complexity=volume/action” (abbrevi- ated to ”C=V/A” below) principle becomes a particular manifestation of particle-string duality. 5 CayleyPy-4: HolographyCayleyPy collaboration Indeed, the extremal action of a particle – namely, the length of the shortest path – can be directly viewed as ”complexity”. According to the duality principle, it should be equivalent to the extremal action of the corresponding string worldsheet, which is naturally related to a volume-type quantity. Let us list a few other reasons this principle is important for us. One key point is that estab- lishing the duality itself is not expected to be a trivial task, even for relatively simple graphs. In contrast, results supporting the ”complexity = volume/action” principle appear to be more acces- sible, and in some cases even numerical simulations can provide valuable insight. In particular, the primary goal of the CayleyPy project is to develop AI-based tools to estimate “complexities” (i.e. lengths of shortest paths) using modern machine learning methods. Previous papers in the project have already achieved state-of-the-art results in this direction. Moreover, one of the key conjectures formulated in our earlier work is that the “complexities” (diameters and word metrics) for S n -Cayley graphs are quasi-polynomials in n. In the present paper, we propose an extension of this conjecture: we observe that these quasi-polynomials appear to be closely related to Ehrhart quasi-polynomials of certain planar polygons. This conjecture can be viewed as a manifestation and refinement of the “C=V/A” principle for S n -Cayley graphs, since it identifies complexities with the number of lattice points in corresponding polygons. These lattice point counts may be interpreted as discrete “volumes,” or “volumes” with finite-size corrections. Another reason is that the principle “C = V/A” can itself be viewed as a manifestation of a strong–weak coupling duality. Computing complexity for large systems (such as graphs) is typ- ically extremely difficult—often NP-hard—whereas the computation of geometric quantities like volumes is comparatively more tractable. In this way, a complicated problem is translated into a simpler one through a dual description. This perspective also underlies our proposal to in- terpret strings (i.e., holographic images of states) as good embeddings (latent representations). The primary goal of embeddings—whether in natural neural networks or artificial systems—is to transform an original representation of information into a more tractable format, so that questions which are difficult in the original representation become easier in the embedding space. From this viewpoint, the principle “C = V/A” can be seen as predicting precisely such a mechanism: by representing states through strings, one effectively moves to a dual geometric description where complexity-like quantities become accessible through simpler geometric computations. We pro- vide a more detailed discussion of this perspective in the main text. In the present work, we use the term “area” rather than “volume”, since our examples are pla- nar. We also typically use other terms instead of “complexity,” which are more standard in graph theory. Often, a distinguished reference node is chosen—such as the identity element in the case of a Cayley graph—and the complexity of all other nodes is measured relative to this reference. In group-theoretic language, this notion is known as the word metric. In coding theory and bioinfor- matics, it is related to metric codes and evolution metrics and mutations, respectively (see [Kon- stantinova2008] for more details). In computer science, it is closely related to circuit complexity, while in quantum computing it appears as quantum gate complexity, quantifying the minimal num- ber of elementary gates required to generate a given element. Complexity optimization is one of the central challenges in quantum computing [Nam2018; Ruiz2025], while the computation of word metrics is a fundamental problem in computational group theory. Previous papers from the CayleyPy project [Chervov2025c; Chervov2025a; Chervov2025b] have provided state-of-the-art solutions to this problem. 1.3. S n -Cayley graphs to planar polygon duality: lattice paths as discrete strings holo- graphically dual to graph nodes. The present paper elaborate the conjectures for S n -Cayley and Schreier graphs. We propose that the corresponding dual objects are rational planar polygons (or line segments in degenerate cases). The holographic map associates each node of the graph with a lattice path on the dual polygon. Our central conjecture is that the diameter of the graph equals the number of integer lattice points in the polygon, while more generally, the word metric of any node corresponds to the area under its associated lattice path. Consequently, both diameters and word metrics can be described as Ehrhart quasi-polynomials of the n-rescaled polygon, providing a 6 CayleyPy-4: HolographyCayleyPy collaboration natural explanation for the quasi-polynomiality conjecture in our previous work [Chervov2025b]. These conjectures can be viewed as refinements of the “complexity = area” principle, since word metrics—interpreted as computational complexities—are predicted to correspond directly to areas under the dual lattice paths. The planarity of those polygons is conditioned to the celebrated 50+ years old open problem [Rubtsov1975] (reviews [Glukhov1999], [Helfgott2013]) that diameters of S n Cayley graphs are bounded by n 2 . The conjecture can be a seen a special case of the L.Babai conjecture on diameter of any finite simple group, for which certain progress has been achieved by T.Tao et.al. Still both conjectures are widely open, even establishing a polynomial bound is not achieved. For further discussion and refinements, we refer to [Chervov2025b] and discussions below. Each quasi-polynomial has a naturally associated H -polynomial, that is simply the numerator of the generating function, which coincides with the Poincar ́ e polynomial of the corresponding toric variety under appropriate smoothness assumptions. Important questions in combinatorics are whether these H -polynomials satisfy the same properties as those arising from ”good” toric varieties, namely: positivity, duality (Poincar ́ e symmetry), unimodality, and an analogue of the “Riemann conjecture” (i.e., that all roots lie on the unit circle). In the present paper, we compute various H -polynomials associated with diameters and word metrics, and investigate their prop- erties, formulating a number of conjectures. In some cases, all of the expected properties hold; however, in other situations we observe that each of them may be violated. The cases of the Cayley graphs with neighbor transposition (Coxeter) generators of S n are the most classical ones. We present quite an detailed picture of the dualities here. The holography map can be described by e.g. Lehmer codes or by related constructions e.g. ROC-curves. ”Complex- ity = Area” reduces to known statements that areas under the ROC-curves correspond to Mann- Whitney statistics, and similar statements for the Lehmer codes. Bijective description of shortest paths on these graphs [Stanley1984; Edelman1987] can be interpreted as bijections between the extremals of a particle and the corresponding strings, in accordance with the conjectured dual- ity. In this framework, extremal string worldsheets are naturally associated with Young tableaux. String action is ”tropical” (or ”ReLU”) analogue of the conventional string action. Graph Lapla- cians can be identified with the Hamiltonians of Heisenberg X spin chains. Etc. The other cases studied in the present paper are generalizations of neighboring transpositions to k-neighbor versions, such as k-consecutive cycles of the form (i,i+1,i+2,...,i+k−1) and their variations. We present evidence that, in this setting, the results can be described by a (k− 1)-shrinkage prin- ciple: namely, the quantities (in particular dual polyygons) in the standard case should be rescaled by (k− 1) to obtain the corresponding results in the general case, at least at the level of leading terms. We present various conjectures on diameters, word metrics, as well as the corresponding quasi-polynomials and H -polynomials in these generalized settings, obtained with our CayleyPy library. In general, determining the dualities and the corresponding dual polygons is a highly nontrivial task. Already the computation of diameters and word metrics is well known to be difficult in its own right. Our approach is based on the CayleyPy library, in particular on its AI component. The heuristic strategy for identifying dual polygons is as follows. First, we attempt to determine quasi-polynomials for the diameters. Second, we try to identify polygons whose Ehrhart quasi- polynomials match the observed data. To determine diameters, we proceed in several steps. Using efficient implementations within CayleyPy, we perform brute-force computations of diameters for the first several values of n, and we identify the corresponding longest elements (states). Next, we attempt to detect patterns in these longest states and formulate conjectural descriptions valid for all n. If this step is successful, we then use the AI component of CayleyPy to compute word metrics of these candidate longest states, which allows us to reach significantly larger values of n than are accessible by brute force alone. With sufficiently extensive data at hand, we fit quasi-polynomials for the diameters and compare them with Ehrhart quasi-polynomials of suitable polygons. We demonstrate that this approach is successful in a number of cases. 7 CayleyPy-4: HolographyCayleyPy collaboration 1.4. List of contributions. Let us outline contributions of the present paper in the itemized form, which mainly follow the order of the exposition: • Simplest examples. ROC curve (Dyck like paths). Surprisingly, the basic quality met- rics in machine learning such as the ROC curve and ROC-AUC score can be used to illustrate our holographic duality for graphs in a very simple and explicit way. In this framework, ROC curves (which are quite similar to Dyck paths) are strings “holograph- ically dual” to nodes of a certain class of graphs. (S n /(S k × S n−k ) or ”Grassmanians” over ”field with one element.”) Within this setting, the analogue of the AdS/CFT relation “complexity = area” reduces to the familiar equality between the Mann–Whitney statistic and the area under the ROC curve. Bijective description of shortest paths on these graphs [Stanley1984; Edelman1987] can be interpreted as bijections between the extremals of a particle and the corresponding strings, in accordance with the conjectured duality. In this framework, extremal string worldsheets are naturally associated with Young tableaux. String action is ”tropical” (or ”ReLU”) analogue of the conventional string action. Graph Laplacians can be identified with the Hamiltonians of Heisenberg X spin chains. The number of spanning trees is related to the Mahler measure, in accordance with general expectations. We also present an analysis of the limit shapes of ROC curves and outline several open questions. • Charting general holographic duality for AI. We hypothesize that for many graphs and corresponding AI-systems (languages, and RL-environments) there are dual objects whose features include: – Holography. [Hooft1993; Susskind1995] Meaning that d-dimensional objects on graph side are mapped to (d + 1)-dimensional ones on the dual side. I.e. graph nodes to paths (”strings”) on the dual side, paths on graph to 2-dimensional objects - like Young tableaux (”string worldsheets”). We expect that the strings which are holographic images of states and ”good embeddings”. – Particle on the graph side (“CFT”-side) = Discrete String on dual side (“AdS”- side). As is typical in string dualities, we expect that a theory defined on one side is equivalent to a theory defined on the other side, and our proposal consists of duality between particle theory and string theory. – Corollary: Complexity = Area/Action. This is a consequence of the previous prin- ciple: values of action for particles on extremals are lengths of the shortest path, while for the string these are related to certain areas. – Strong coupling to weak coupling. As in conventional string theory, we expect that difficult problems on the original side are converted into more tractable problems on the dual side. In particular we expect that strings dual to nodes of the original graph provide ”good” embeddings (from AI point of view). • Case studies and mathematical conjectures. We outline several concrete examples of duality for Cayley graphs. We explain its relation to quasi-polynomiality hypothesis and the n 2 conjectural bound for diameters of S n Cayley graphs, which is a celebrated open problem in mathematics for more than 50 years. We explain how CayleyPy AI methodol- ogy assists in determining the duality. We discuss other examples related to SL(2,Z) and Farey graphs, etc. – S n -Cayley to planar polygon duality. For S n -Cayley and Schreier graphs, we con- jecture that the dual objects are rational polygons in the plane (or, in degenerate cases, line segments) such that the graph diameter equals the number of integer lattice points in the n-scaled polygon. This provides an illustration of a refined version of the “complexity = area” principle in this context. In other words, diameters are given by the Ehrhart quasi-polynomials of the polygons, explaining the conjecture from our 8 CayleyPy-4: HolographyCayleyPy collaboration previous work. Such a relation is far from trivial: in particular, it implies the cel- ebrated open problem predicting that the diameters of these graphs are bounded by n 2 , a conjecture that has resisted the efforts of leading mathematicians for decades [Rubtsov1975] (reviews [Glukhov1999], [Helfgott2013]). – Holography: nodes to lattice paths; word-metrics quasi-polynomiality. In this framework, ”holography” predicts that vertices of the graph can be mapped to lattice paths inside the polygon in such a way that word metrics (or “gate complexities”) coincide with the areas under the corresponding paths. This again mirrors the “com- plexity = area” principle familiar from AdS/CFT, and implies that word metrics are also described by Ehrhart quasi-polynomials, explaining the conjecture from [Cher- vov2025b] that word metrics are quasi-polynomials in n for S n . – Holography = Lehmer code (for Coxeter generators graphs). For Cayley graphs generated by neighboring transpositions, such a mapping is closely related to the Lehmer code. – CayleyPy AI-methodology to find diameters and wordmetrics. We present a methodology how we can find diameters using our AI-based CayleyPy library, and present multiple successful examples of computations discussed before are based on it. – Behavior under taking G/H . Empirically, we observe the following pattern: the polygon associated with the Schreier coset graph of G/H appears as a subpolygon of the polygon corresponding to Cayley graph of the full group G. – Consecutive k-cycles and variations — (k− 1)-shrinkage principle. We present several computations of diameters and word metrics for generators given by con- secutive k-cycles, explicitly obtaining the corresponding quasi-polynomials. Em- pirically, we observe that these formulas approximately correspond to shrinking the graph (k − 1) times, a behavior that is particularly reflected in the associated dual polygons. – SL(2,Z) and Farey graphs. We put into the framework of the present paper results from the previous paper by one use: (D.Melnikov et.al. [Camilo2019]). – Properties of the corresponding H -polynomials. To each quasi-polynomial, one naturally associates an H -polynomial, and we study their properties. In many cases, we observe positivity, unimodality, Poincar ́ e duality, and analogues of the Riemann conjecture. While some examples satisfy all of these properties, there are cases in which one or more of them are violated. – Limit shapes. We study the limit shape of the corresponding ”strings” in large size limit, on the example of the Coxeter group, we proposed exact formulas for limit shapes with fixed areas under the curves, which are consistent with previous results by A.M.Vershik et.al. – Evolution in large size limit, Burgers equation and KPZ. We study numerically the geodesic flow on the graph in the dual picture and conjecture its relation with Burgers and KPZ equations. – Spanning trees, Mahler measures, string-like model with polygonal worldsheets, spin chains in thermodynamic limits, spectra of conformal dimensions. We study Laplacian and spectral properties in large size limits. We propose various conjectures on numbers of spanning trees. We also propose identification of limitings theories with a free bosonic model (playing here the role of a string-like dual) which surpris- ingly may have polygonal worldsheets. The conformal dimensions of primary fields are conjectured to be eigenvalues of the Laplacian on the dual planar polygon. 1.5. Organization of the paper. The first five sections present the main examples and core ideas, while the remaining sections provide further details and technical computations. Although the overall length of the paper is substantial, we hope that the first five sections are sufficient for the 9 CayleyPy-4: HolographyCayleyPy collaboration reader to grasp the central concepts and motivation. We put reminders on background material to the last section. AI, mathematics and physics The second superstring revolution and related developments revealed highly nontrivial du- alities among string-related theories and uncovered far-reaching consequences [Candelas1991; Seiberg1994; Witten1995; Polchinski1995; Schwarz1996; Strominger1996; Vafa1996; Strominger1996; Banks1997; Connes1998; Maldacena1998; Gubser1998; Witten1998; Kontsevich2003; Catta- neo2000; Gross2000; De Boer2000; Dijkgraaf2002; Nekrasov2003; Minahan2003; Okounkov2003; Ooguri2004; Gukov2005; Dabholkar2005; Ryu2006; Kapustin2007; Pestun2012; Mironov2010; Alday2010; Verlinde2011; Gaiotto2013; Stanford2014; Grassi2016; Gaiotto2015; Brown2016; Hijano2016]. These developments have profoundly influenced the evolution of modern mathe- matics in a variety of directions. We hope that the dualities proposed here may extend the scope of string-theoretic ideas to new domains, including artificial intelligence, as well as to such areas of mathematics such as graph theory, group theory and combinatorics. One of the ”pre-AdS/CFT” ex- amples of dualities was proposed for two-dimensional Yang-Mills (its string description, ”Gross- Taylor formula”) [Witten1992; Gross1993b; Gross1993a; Cordes1995], recently it has been con- nected to Cayley graphs via ”Yang-Mill/Hurwitz correspondence” [Novak2024], moreover quasi- polynomial expressions which play key role in the present paper, also appear in Hurwitz theory [Norbury2010; Andersen2018; Kramer2020], it would be tempting to understand relations to the present paper. The present time is characterized by growing interest in and number of applications of deep learning methods to mathematics and physics: machine learning has been emerging as “a tool in theoretical science” [Douglas2022]. In recent years, this has led to several noteworthy ap- plications to mathematical and physical problems: [Lample2019; Davies2021; Bao2023; Romera- Paredes2024; Coates2023; Alfarano2025; Charton2024; Shehper2024; Swirszcz2025; Hashemi2025; He2024; Lal2024; Lal2025; Douglas2025; Georgiev2025; Berczi2026; Ju2026; Guevara2026; El- lenberg2026; Chen2026; Knuth2026; Morozov2026]. Seewoo Lee created a repository that col- lects papers in AI for mathematics, Awesome AI for Math. The present paper can be seen as an attempt at a dual-sided application of AI to mathematics and physics, and vice versa. 10 CayleyPy-4: HolographyCayleyPy collaboration 2. SIMPLEST EXAMPLES. ROC CURVE = STRING DUAL TO Gr(k,n) GRAPH NODE FIGURE 1. Left to right: the graph Gr 2,5 with marked vertices (red and green), and a polygon illustrating the corresponding paths (red and green). FIGURE 2. Left to right: the graph Gr 2,6 with marked vertices (red and green), and a polygon illustrating the corresponding paths (red and green). Interactive widget (F.Khafizov) available at link. The distance between nodes on graph is equal to area between the paths - ”complexity = area” principle. Which in that case is equality of the Mann–Whitney statistic and the area under the ROC curve. Section Outline. Here we present the basic example for the main ideas of the present paper. We consider a simple Cayley graph and describe the duality for it. ROC curves (Receiver Operating Characteristic curves) and area under the ROC curves are widely used quality metrics in ML. (See e.g. the exposition by A. G. Dyakonov, first Kaggle top-1 Grandmaster.) Surprisingly, these examples can be used to illustrate the idea of holographic string duality, as well as to highlight various non-trivial results and open questions. The “complexity = area” principle reduces here to the familiar equality between the Mann–Whitney statistic and the area under the ROC curve. Figures 1, 2 illustrate the discussion below. So we describe below two sides of the correspondence: the graph (“CFT-side”) and the rectangle polygon (“AdS-side”); the map from graph nodes to paths on rectangle (“holography map”). We demonstrate its simple, but non-trivial properties which can be summarized a “particle on a graph dual to a string on the rectangle”. The discrete string has an simple action which is a ”tropical” (”ReLU”) analogue of the conventional action. The explanations how it is applied in ML tasks as a quality metric are also provided. 11 CayleyPy-4: HolographyCayleyPy collaboration Graph. “CFT-side”. Graph nodes correspond to vectors of length n with entries k zeros and n− k ones. Two nodes are connected by an edge if there is a transposition of some neighbor elements (i,i + 1) which sends one vector to the other one. This graph is the Schreier coset graph for S n with neighbor-transposition generators; it is a quotient of the permutohedron graph by S k ×S n−k and should be thought of as the Grassmanian Gr(k,n) = GL(n)/(GL(k)×GL(n−k)) over the field with one element, by the usual analogy S n = GL n (F 1 ). Polygon = rectangle. “AdS-side”. Consider the rectangle of the size k× (n− k) on the plane with integer coordinates. Set of our paths (“strings”) are paths making steps right and up going from (0, 0) to (k,n− k). These are similar to Dyck paths, but there is no restriction for them to be under the diagonal. “Holography map” from graph nodes to paths (“strings”). Take a vector of 0’s and 1’s and associate to it a path by the rule: each “1” is a step up, each “0” is a step right. Clearly it is a bijection from graph nodes to paths described above. As we will discuss below these are precisely the ROC curves for certain machine learning models. “Complexity = area”. Mann-Whitney = area under the ROC curve. Take two nodes on the graph and consider the distance between them, i.e., length of the shortest path, or, equivalently, the minimal number of neighbor transpositions which are needed to transform one vector to the other one (“gate complexity”). One can check: Proposition 1. For any two nodes the distance between them on the graph (i.e. the complexity of one with respect to the other) is equal to the area between corresponding paths. Typically as one of the nodes we take the sorted vector 0...01...1, which corresponds to the path going along the bottom-right border. So in that case we get that the area under the curve equals the complexity of the other node. Use in ML. Notation abuse of “under” vs. “above” the curve. Consider a binary classifica- tion task with n observations, of which k have ground-truth label 0 and n− k have label 1. A machine-learning model assigns a probability score to each observation. Sorting the observations by these scores produces an ordered binary vector consisting of zeros and ones (ground truth la- bels). For a perfect model, this vector is 0 k 1 n−k (i.e. it is sorted). An imperfect model produces a non-trivial interleaving of zeros and ones. Any metric that quantifies deviation from the per- fectly sorted vector therefore can serve as a natural measure of model quality. One such metric is obtained by encoding the binary vector as a lattice path, as described above. The area under this path provides a quantitative measure of deviation from the perfectly sorted vector 0 k 1 n−k . Indeed, the lattice path corresponding to that vector is the bottom-right boundary of the corresponding rectangle and hence has zero enclosed area. Proposition 1 shows that this measure has a natural combinatorial interpretation: it equals the minimal number of neighboring transpositions required to transform the given vector into the sorted one. This is precisely the statistic introduced by Mann, Whitney, Wilcoxon, and Kendall in classical rank-based hypothesis testing, and it admits modern group-theoretic interpretations as developed in foundational work by P. Diaconis (see, e.g., [Chat- terjee2016] and references therein). We note a minor abuse of terminology: in our convention, the area under the path measures deviation from the perfectly sorted vector, whereas in the standard ROC-curve exposition, the area under the curve measures similarity to the ideal classifier. Our choice of convention aligns naturally with the guiding principle “complexity = area”. Stanley–Edelman–Greene correspondence as a bijection of extremals for a particle on a graph and strings on a polygon. [Stanley1984] computed the number of shortest paths between the two most distant vertices of the permutohedron graph; such paths are known as sorting networks. As discussed in the influential paper Random Sorting Networks [Angel2006]: “another breakthrough was achieved by Edelman and Greene” [Edelman1987], who constructed a bijection between sorting networks and staircase-shaped standard Young tableaux of size n. We now formulate an analogue of this result for our graph and provide its string theory interpretation. 12 CayleyPy-4: HolographyCayleyPy collaboration Proposition 2. Let A and B be two vertices of the graph above. Then the shortest paths between A and B are in bijection with Young-type tableaux associated with the region bounded by the corresponding holographically dual lattice paths. Discrete string action whose extremals are Young tableaux: standard + ReLU. The simplest continuum string action is R R X 2 a + X 2 b dadb. In the discrete setting, deriva- tives are replaced by finite differences, X a = X(a,b)−X(a + 1,b),X b = X(a,b)−X(a,b + 1). Consider the discrete action: P a,b ReLU(X a ) + ReLU(X b ) , where ReLU = max(·, 0) is ReLU function. It follows that the minima of this action are precisely Young tableaux, i.e. they provide solutions for equations of motion; at the same time, the action closely resembles that of a conventional string theory. Corollary. (Stringy interpretation of analogue of Stanley–Edelman–Greene correspondence). Extremals of a particle moving on a graph (i.e. shortest paths) between vertices A and B are in bi- jection with extremals of the discrete string action (i.e. Young tableaux) with boundary conditions defined by the holographic images of A and B. FIGURE 3. The Young tableau (upper left corner) provides a solution to the string equations of motion. It encodes the motion of the string as depicted on the right. In the right panel, each position of the string at times t = 1,..., 6 is shown in a different color, and the number in each box indicates the step at which the string passes through it. Each obtained string is a non-decreasing path, thus it belongs to the image of the holography map. In the holographically dual picture, the tableau encodes one of the shortest paths on the graph – indicated by wavy edges. It would be natural to consider other activation functions, and also other distances on graphs, e.g. diffusion distances (see e.g. the second paper of the project) related to supersymmetric LG models. Further results. We discuss further results on this example in the next section ”Neighbor transpo- sitions Cayley and Schreier graphs”. Open questions. We hope that the answers can be naturally formulated using duality. The spectral density of the eigenvalues (as opposed to Bethe roots) is, to the best of our knowl- edge, unknown. Similarly, the properties of the resolvent—such as the equations it satisfies—are not known. The Jacobian (also called the sandpile, critical, or Picard) group of these graphs is also un- known. While its order, given by the number of spanning trees, is known, the group structure itself has not been determined. Graph invariants like Tutte polynomial are not known. In large n limit CFT description is not well-understood, duality may might be related to a version of AdS/CFT. In particular, we may expect the spectrum of conformal dimensions can be related to the eigenvalues of the ordinary Laplacian on the plane in the rectangle, by analogy with AdS/CFT principles: Casimir of conformal algebra maps to Laplacian in the bulk. 13 CayleyPy-4: HolographyCayleyPy collaboration 3. AI TASKS AS PREDICTIONS OF PARTICLE TRAJECTORIES. GENERAL AI-HOLOGRAPHY EXPECTATIONS Section Outline. Here we first discuss an analogy in which common tasks in AI are viewed as predicting particle trajectories. Then outline the idea of “AI holography”: rather than working directly with a particle (difficult, ”strongly coupled” regime), it is advantageous to seek a dual string description, which provides a more tractable (”weakly coupled”) formulation. We also em- phasize that, in a certain sense, modern AI systems—as well as natural neural networks—already operate according to a somewhat similar principle through the use of embeddings. However, we believe that fully exploiting the power of string dualities can lead to a deeper and more systematic understanding of these mechanisms, enabling the design of more effective AI systems. But to work in discrete setting it is necessary to consider unusual ”tropical” string actions which involve e.g. ”ReLU” ( max(·, 0)) functions, and lead to some unexpected, but desirable properties of equations of motion. 3.1. Particle trajectories as texts or action sequences. The goal of this subsection is to draw the attention of the physics community to the fact that many core aspects of AI — including input data, prediction objectives, and even methodological approaches — are closely analogous to the study of particle dynamics on graphs with labeled edges. Cayley graphs serve as particularly natural and representative examples. Such questions are not uncommon in mathematical physics, and their conceptual and technical tools may prove beneficial for the development of AI. Texts / robot manipulations / game play / theorem proofs as discrete particle trajectories on edge-labeled graphs. Let us bridge the settings and tasks commonly used in AI—such as text corpora or action sequences in reinforcement learning—with more traditional problems in physics, namely particle trajectories on graphs and the study of their dynamics. We also provide a brief review of several AI concepts from this perspective, which may be of interest to physicists. Modern large language models operate on texts, while other powerful systems—such as those based on reinforcement learning—work with sequences of actions: robot manipulations, moves in games, and so on. In all these cases, the primary data consist of sequences of discrete tokens, for example ABBCADB... These tokens may represent letters of the English alphabet, commands for a robotic manipulator, moves in a game such as chess, or the names of theorems and lemmas in a mathematical proof. To connect this viewpoint with frameworks commonly used in physics, it is natural to interpret such sequences as trajectories of a particle, where each token specifies an elementary increment of the particle’s motion. Pushing the analogy further, one may regard all possible states (for instance, sentences or configurations) as nodes of a graph, with tokens labeling the edges. Appending a new token then corresponds to moving from the current node to a neighboring node along the edge labeled by that token. Thus toke sequences can be view as trajectories of particle on graph where edges are marked by the token labels. An archetypal example of this consideration is provided by Cayley graphs in group theory. There, nodes correspond to the elements of a group, and edges correspond to a chosen generating set: two nodes are connected if a = gb for some generator g. Thus, edges labeled by group generators and thus paths (particle trajectories) on the Cayley graph correspond to words formed from these generators. One may further restrict attention to those words that correspond to shortest paths in the graph, thereby defining the geodesic language of the group. Thus, a quite standard task in physics and mathematics – to study a free particle on a graph, appears to be essentially equivalent to studying the graph’s geodesic language. More generally, one can consider a variety of related languages, such as words whose associated paths deviate from geodesics by at most a prescribed amount, or that satisfy other geometric constraints. These choices lead to different classes of admissible trajectories, interpolating between strictly geodesic motion and more flexible, near-geodesic dynamics. The considerations above do not require working with groups. Similar arguments apply to arbitrary state-transition graphs whose edges are labeled by tokens, and where edge weights can be naturally incorporated as probabilities of selecting a given transition. In this 14 CayleyPy-4: HolographyCayleyPy collaboration perspective, recorded sequences of moves in games such as Go or chess play a role analogous to words in group-theoretic settings: each game record corresponds to a path in the underlying state- transition graph, while individual moves act as tokens labeling the edges. Collections of such records therefore form a “language” of admissible trajectories, shaped by the rules of the game. Let us rephrase the same idea as above with different emphasizes and provide more examples. What is a “language”? One fixes a set of tokens (i.e., an alphabet) and considers a collection of admissible sequences built from these tokens — this collection is, essentially, the language. In other words, a language is a rule (explicit or implicit) that selects, among all possible sequences, those that are allowed. For example, the set of all texts written in English defines the English language. There are many other such languages: admissable sequences of moves in games such as Go or chess define the corresponding game languages; all possible protein sequences over the 20–amino-acid alphabet define the language of proteins; all possible DNA sequences define the DNA language; all valid SMILES encodings define a language for chemical molecules, etc. As discussed above, one can interpret these examples as collections of particle trajectories on a graph whose edges are labeled by elements of the alphabet. In this sense, the current AI paradigm is closely analogous to experimental physics: one observes a collection of trajectories of a parti- cle — here represented by a language, that is, a set of admissible sequences — and attempts to uncover the hidden laws governing its dynamics. It is remarkable that essentially the same tech- niques originally developed for natural language processing, namely transformer-based models, can be transferred to other “languages,” such as protein language models (e.g. ESM2 [Lin2023]) or chemical languages based on SMILES representations, (e.g. ChemBERTa [Ahmad2022]), etc. This transfer has led to the creation of tools that have become indispensable in modern bioinfor- matics and cheminformatics, achieving top results on a wide range of benchmarks, for example in the CAFA protein properties prediction challenge [Chervov2024]. Let us reemphasize that nothing more than a set of sequences—a “language”—is required to begin AI modeling. I.e., having a collection of texts—where “texts” may be completely arbitrary sequences over any alphabet—is already sufficient to apply AI techniques. Modern AI techniques are remarkably successful across a variety of such “languages,” many of which are quite different from natural ones. To some extent, this reflects a simple principle: what the human brain can do, artificial neural networks may also learn to do. Since humans can master natural languages, programming languages, and games such as Go or chess, it is perhaps not entirely surprising that similar AI techniques are capable of mastering these domains as well. Thus, texts in natural languages or sequences of actions in reinforcement learning can be viewed as collections of particle trajectories on a graph whose edges are labeled by tokens, with admissible words corresponding to trajectories that satisfy constraints imposed by both the graph structure and the particle dynamics. Studying dynamics on graphs is a well-established problem in physics and mathematics. AI tasks as prediction of particle trajectories with prescribed initial or boundary condi- tions. Let us discuss the close analogy between typical AI tasks and computing particle trajectories with given initial or boundary conditions. The basic training paradigms of modern LLM systems can be broadly divided into two modes. The first is the GPT-style setting of generative modeling, where one is given the beginning of a text and the task is to generate—or predict—its continuation. The second is the masked mode (as in classical Word2vec and related models), where both the beginning and the end of a sequence are provided, and the goal is to infer the missing middle portion. Closely related reinforcement-learning systems—such as those used for robot manipulation, gameplay, or automated theorem proving—are typically generative as well: at each step, the ob- jective is to predict the next action in a sequence. For example, the task of finding a path on a Cayley graph—which is mathematically equivalent to decomposing a group element into a prod- uct of generators—can be viewed as finding a trajectory with prescribed boundary conditions: given element and identity of the group. In the case of the Rubik’s cube group, this corresponds to solving the cube. For games such as Go or chess, the objective is to find a path from an initial 15 CayleyPy-4: HolographyCayleyPy collaboration position to a position labeled “victory” in an environment where a second player is simultane- ously attempting to achieve the same goal. Despite this added complexity, the problem can still be framed as finding a particle trajectory with boundary conditions on a state-transition graph, with the additional complication that the state evolves in response to the opponent’s moves (a kind of randomized environment). Similarly, proving a mathematical theorem can be seen as finding a particle trajectory on the state-transition graph of all admissible proofs. Overall, these examples are representative of a broad class of problems in AI: generating a sequence of tokens such that the resulting sequence satisfies specified constraints or desired properties. From the physical perspective suggested above, these learning tasks admit a natural interpreta- tion in terms of particle dynamics. The generative setting corresponds to predicting a trajectory given fixed initial conditions, whereas the masked or infilling setting corresponds to predicting a trajectory subject to fixed boundary conditions. Counterintuitive - multiple local classical trajectories with fixed initial/boundary condi- tions. To what extent are discrete systems—such as particles moving on graphs, or more gener- ally symbolic systems like languages—similar to classical physical systems? In particular, can one meaningfully apply the standard physical language of actions and equations of motion to such settings? At first sight, there appears to be a serious obstacle to such a description. In classical mechanics, actions are smooth functionals of coordinates and their derivatives, leading to equations of motion in which fixing the initial position and momentum (and possibly higher derivatives) uniquely deter- mines the trajectory of a particle. (In a discrete setup, this is analogous to fixing several positions in the history, which should completely determine the continuation.) By contrast, in typical discrete systems like on graphs, fixing initial conditions generally allows for a large number of possible continuations. For example, a particle moving on an infinite tree (such as the Cayley graph of a free group) may move in essentially any direction except immediately backtracking (i.e. previous history almost have no effect on next moves), and each such choice produces a geodesic. In this sense, initial conditions impose only weak constraints on the subsequent motion. A similar phenomenon occurs in languages: the beginning of a sentence often does not uniquely determine its continuation. This apparent non-uniqueness may give the impression that the classi- cal framework of actions and equations of motion is inapplicable to discrete systems. Unusual ”tropical/ReLU” discrete string actions - for the rescue. Our observation is that modifying classical string actions so as to include tropically inspired expressions such as max(·, 0) (also known as the ReLU function) leads to string equations of motion that are well suited to dis- crete settings. In particular, the resulting equations admit solutions in the form of Young tableaux, which are highly non-unique even fixing initial conditions, yet still subject to nontrivial global constraints. The example has been decsribed in the previous section. In a broader sense, this behavior might be consistent with the perspective of tropical geometry. Graphs can be viewed as tropical limits of Riemann surfaces, and it is therefore natural to expect that the corresponding action functionals should also involve operations characteristic of tropical geometry, such as the max operation. Although such analogy may not be fully correct. Embeddings: LLMs and natural brains are already performing a form of “duality.” In a nutshell, the idea of duality in physics is to replace a given description of a system by a dual one in which previously difficult questions become unexpectedly tractable. This closely parallels a central paradigm of modern AI: before solving a task, one first seeks a more convenient repre- sentation of the data called —an ”embedding” or ”latent space representation”—and then operates primarily within that representation. Even more striking is the parallel with the AdS/CFT principle that “complexity = area”: in modern AI, a key feature of embeddings is that notions of similar- ity (difficult to compute in original setting) are reduced to simple geometric quantities, such as dot products or distances. While these ideas are not identical, they share the same essential fea- ture: transforming a hard-to-compute notion of complexity or similarity into an easily computable geometric measure—whether an area, a distance, or a scalar product. 16 CayleyPy-4: HolographyCayleyPy collaboration Let us reiterate the points made above, adding some details and providing perspective from both natural brains and artificial neural networks. From the perspective of natural brains, all incoming information—whether visual, auditory, or tactile—ultimately affects the neurons, leading to their activation or deactivation. The degree of activation of a neuron can be approximated by a number, so the state of the brain at any moment can be represented as a vector of real numbers indexed by neurons. In other words, the brain essentially converts all incoming information into a vector rep- resentation, or an embedding. The goal of the brain is to operate effectively on these neural states. One can reasonably expect that, through evolution, brains have been optimized to find the most suitable representations of any incoming information. Effective cognitive operations can then be thought of as relatively simple operations on these vectors. In this sense, the brain naturally per- forms a process analogous to the concept of duality in physics: transforming a complex problem into a representation where simple operations suffice to solve it. Modern artificial intelligence follows the same principle. Regardless of the task—whether lan- guage translation, summarization, sentiment analysis, prediction of protein properties or of chemi- cal molecules, or other applications — the first step is typically not the task itself, but the construc- tion of high-quality vector representations (embeddings). Once these embeddings are obtained, all subsequent operations are performed on them, without directly interacting with the original data. Consequently, designing “good” embeddings becomes a central problem in AI. Analogous to the AdS/CFT principle in physics, where “complexity = area,” embeddings in AI often trans- form computationally difficult similarity measures of raw information into simple, tractable oper- ations—such as dot products—allowing complex problems to be solved efficiently. The famous examples on Word2vec [Mikolov2013] word embeddings which are sometimes shortened (not quite accurately) to ”King - Man + Woman = Queen” (i.e., the corresponding vector embeddings approximately satisfy this equality) - represents the same phenomena: semantic and not trivial relations in the language are transformed to simple operation on good embeddings (just the addi- tion of vectors in that example). Which is intriguing, as the constraint was not explicitly imposed during training, yet it emerged naturally. Despite the success of modern AI, most embedding constructions have been achieved in an ad hoc manner. There is still no clear understanding of how to internally characterize “good” em- beddings, nor how to systematically improve them beyond trial-and-error methods. The currently dominant approach in AI is “scaling”: training ever-larger neural networks on ever-larger volumes of data. This strategy indeed works. However, it is somewhat akin to trying to approximate a highly non-trivial function using a trivial one. Increasing the number of parameters and data can improve the fit, but a better approach may be to understand the underlying nature of the function from first principles and to choose an approximation strategy more wisely. The perspective offered by string dualities may provide valuable insight into these questions, potentially guiding the design of more principled and effective representations. In our idea, good embeddings are holographic dual strings corresponding to the original particle states. To compute downstream quantities from embeddings one should take into account possibly not flat Riemannian metric (similar to AdS metric), which is expected to be a part of the dual description, more subtle effects should take into account finite-size corrections. Let us recall the difference between older Word2Vec-style embeddings and newer context- dependent, transformer-based embeddings. Word2Vec produces embeddings for words them- selves, but not for words in context. In direct analogy with our picture, words may be viewed as nodes of a graph (states), and the goal is to construct embeddings for these nodes. (Moreover, Word2Vec admits a direct generalization to graph embeddings, as demonstrated in the well-known works DeepWalk and Node2vec [Perozzi2014; Grover2016]). In contrast, modern transformer- based architectures generate embeddings sequentially, so that the representation of a word depends on the preceding text—that is, it is context-dependent. In this sense, embeddings are effectively constructed for sequences rather than isolated tokens. This perspective is fully consistent with our framework: paths on a graph can be identified with their terminal states, and therefore embed- dings of these states naturally correspond to context-dependent embeddings of words. Our setup 17 CayleyPy-4: HolographyCayleyPy collaboration encompasses both scenarios: one in which multiple paths can lead to the same state, and another in which this is not allowed (as in the case of a tree graph). 3.2. AI-holography. Here we outline our expectations for analogues of holographic string duali- ties in AI tasks. In brief, we expect that for broad classes of systems there exists a dual formulation in which states of the original system are mapped (‘holographically’) to paths (strings) in a dual space. In this picture, the particle system on one side (the ‘CFT side’) is equivalent to a string theory on the dual side (the ‘AdS side’). This dual description may offer a more tractable frame- work; in particular, these paths (strings) can serve as ‘good’ embeddings. Thus, providing new approaches for improving AI methods. As described in the previous subsection, we may view AI systems of interest as defined by a set of admissible token sequences (‘language’), which can be interpreted as paths (‘particle tra- jectories’) on an edge-labeled graph and basic AI questions can be view as predicting particle trajectories with initial or boundary conditions. The nodes of this graph represent the states of the original system. (In AdS/CFT terminology that is ‘CFT side’). We expect the existence of a dual object equipped with a holographic map that sends nodes of the original graph to paths (strings) in the dual space. Such that ‘particle theory’ on the original graph would be equivalent to ‘string the- ory’ on the dual object: any question about particles could be reformulated and computed via their holographic images, with the expectation that the dual description is more tractable. Moreover, based on examples and general considerations, we expect that unconventional, ‘tropical’ actions for discrete strings are necessary to describe AI-related systems. In suitable large-size limits, however, these discrete models may converge to more familiar geometric (gravitational) descrip- tions. Thus, questions that are especially difficult when original system is large could translate into geometric (gravitational) computations that are more tractable. This perspective parallels the original AdS/CFT proposal, where quantum observables such as Wilson loops on CFT-side admit dual gravitational descriptions on AdS-side, as well as subsequent developments, in particular the ‘complexity = volume/action’ principles. Let us summarize expectations in the itemized form: • Holography. [Hooft1993; Susskind1995] Meaning that d-dimensional objects of the orig- inal system are mapped to (d + 1)-dimensional, i.e. nodes of graph are mapped to paths (strings) on the dual object. Similarly, paths on graphs (i.e. 1-dimensional objects) are mapped to 2-dimensional surfaces (string worldsheets). We expect that the strings which are holographic images of states and ”good embeddings” for the states in original system. • Particle on the graph side (“CFT”-side) = Discrete String on dual side (“AdS”-side). As is typical in string dualities, we expect that a theory defined on one side is equiva- lent to a theory defined on the other side. On one side, the dynamics describe a particle moving on a graph, while on the other side they are captured by a discrete string theory defined on the dual object. Holography provides a map between the two descriptions, under which the quantities computed in one theory are equal to corresponding quanti- ties computed in the other one. We illustrate what we mean by discrete string theory through concrete examples, e.g. worldsheets are naturally associated with Young di- agrams, while their images in target space correspond to Young tableaux. Moreover, based on examples and general considerations, we expect that unconventional, ‘tropi- cal’ actions for discrete strings are necessary to describe AI-related systems. E.g. such as P a,b ReLU(X a ) + ReLU(X b ) , where ReLU = max(·, 0) is ReLU function and X a = X(a,b)− X(a + 1,b),X b = X(a,b)− X(a,b + 1) - common analogs of discrete derivatives. • Corollary: Complexity = Area/Action. (Similarity = Geometric Measure). Similar to the principle discovered in AdS/CFT correspondence (L. Susskind et. al. [Stanford2014; Brown2016]), we expect that lengths of paths on the graph side—serving as measures of complexity— are mapped to areas under the corresponding curves on the dual side. That is a consequence of the previous principle: values of action for particles on extremals are 18 CayleyPy-4: HolographyCayleyPy collaboration lengths of the shortest path, while for the string these are related to certain areas. However results in that direction are typically far more accessible than establishing the duality. • Strong coupling to weak coupling. As in conventional string theory, we expect that difficult problems on the original side are converted into more tractable problems on the dual side. For example, the computation of complexity is typically NP-hard in general and remains difficult even in specific cases, with the difficulty growing rapidly as the size of the system increases. The key idea of the duality is that it maps this hard computational problem to the evaluation of a geometric quantity—namely, an area—which is often much easier to compute. 3.3. Further remarks. Let us comment on further analogies with, as well as differences from, the AdS/CFT correspondence. In string theory, graphs (e.g., Feynman diagrams) can be viewed as degenerate string worldsheets; in more mathematical terms, graphs arise as tropical limits of Riemann surfaces. From this perspective, particle theories on graphs may be regarded as degen- erations of particle theories on Riemann surfaces. Which are closely related to conformal field theories (for metrics of constant curvature on Riemann surfaces). One may therefore speculate that the conventional AdS/CFT correspondence for CFT related to Riemann surfaces, in an appro- priate tropical limit, could be connected to the considerations proposed in the present paper. Secondly, in the original AdS/CFT correspondence, the CFT side lives on the boundary of the AdS space, whereas in our proposal, based on the examples considered, there is no direct relation between the graph and its dual object. In particular, there is no bulk/boundary correspondence. To our mind, this is an advantage, indicating that holographically dual theories may arise in more gen- eral setups than the conventional bulk/boundary scenario. We also expect that holographic duality is not restricted to conformal field theories, which is natural from a general duality perspective. Moreover, for the S n -Cayley graph, the dual polygon emerges from an abstract mathematical ex- istence conjecture, rather than from a geometric picture. In this sense, the approach is reminiscent of S. Wolfram’s ideas [Wolfram2002], suggesting that cellular automata might provide insight into physics on the Planck scale, and conventional theories may emerge from such microscopic descriptions. Modern AI systems are data-driven, as reflected in the well-known phrase ‘fire the linguist — the language system starts working better.’ Meaning that language models can learn to solve tasks on their own, without explicitly encoding structural linguistic knowledge. Indeed, modern LLMs learn entirely from data and have achieved remarkable success. At the same time, it is widely argued that the human brain learns far more efficiently (even —orders of magnitude more). From our perspective, these may be two sides of the same coin. Languages are, of course, constrained by grammatical rules, ignoring that may partly explain why current models require more training ef- fort than one might ideally expect. However, the question of how to incorporate these grammatical rules in the most efficient way and combine with successful AI approaches might not be trivial. From our point of view structural linguistic knowledge may provide insight into understanding the holographically dual description of languages, thereby offering a path toward building more efficient and powerful AI systems than those currently available. In this sense, the goal is not to ‘fire the linguist,’ but to bring linguists together with string theorists in pursuit of further progress. 4. CASE STUDIES 4.1. S n Cayley graphs and polygon duality. 4.1.1. General idea: graph-polygon duality and the quasi-polynomiality hypothesis. Here we ar- gue that the holographic duals of graphs associated with S n are planar polygons. This perspective provides a natural—and essentially unique—explanation for the quasi-polynomiality conjectured in our previous paper. The construction of polygons and holography maps from graphs is not ex- pected to be easy in general, since it is related to the computation of diameters and word metrics, which are known to be NP-hard. This task is difficult even in specific cases; for example, it took 19 CayleyPy-4: HolographyCayleyPy collaboration over 30 years of continuous effort to determine the diameter of the Rubik’s cube [Rokicki2014]. In subsequent sections we work out some explicit examples. Hypothesis: S n -Cayley to polygon duality. ForS n -Cayley and Schreier graphs, we conjecture that the corresponding dual objects are rational polygons in the plane (or, in degenerate cases, intervals), such that the diameter of the graph equals the number of integer lattice points in the n-scaled polygon. Moreover, we expect the existence of a holography map such that vertices of the graph are associated with lattice paths inside the polygon in a way that word metrics (or “gate complexities”) coincide with the areas under the corresponding paths. So both diameters and word-metrics are Ehrhart quasi-polynomials associated with certain rational polygons. Both expectations can be viewed as refinements and concrete realizations of the AdS/CFT principle “complexity = area”. Quasi-polynomiality hypothesis. The motivation for these conjectures is as follows. In earlier work within the CayleyPy project, we computed diameters and word metrics for a large class of Cayley and Schreier graphs. Empirically, these quantities were observed to be eventually quasi- polynomial functions of n, of degree at most two. We further hypothesized that this behavior is generic under conditions, such as when the generating sets are Presburger-definable. Ehrhart quasi-polynomials. Quasi-polynomials arise most naturally as Ehrhart quasi-polynomials associated with rational polygons, which count the number of integer lattice points inside n-scaled polygons. This observation motivates the search for polygons whose Ehrhart quasi-polynomials coincide with those obtained from the corresponding graphs. In the present paper, we consider multiple examples and observe that such a correspondence can indeed be established. Moreover, this analysis reveals a clear connection with AdS/CFT ideas and holographic string dualities, pro- viding new examples and insights. Planarity from the n 2 conjecture. The appearance of plane polygons, rather than higher- dimensional analogues, is a consequence of the fact that the observed quasi-polynomials are al- ways of degree at most two. This behavior is conditional on a celebrated open problem predicting that the diameters of these graphs are bounded by n 2 . This conjecture has resisted the efforts of leading mathematicians for several decades. This statement can be viewed as a special case of the Babai conjecture [Babai1988], for which partial progress has been achieved (notably by B. Green, T. Tao, and collaborators [Breuillard2011; Breuillard2012]). Nevertheless, the specific n 2 bound for these graph diameters has remained open for at least fifty years [Rubtsov1975] (re- views [Glukhov1999], [Helfgott2013] ). For refinements and further discussion of this conjecture, we refer to our previous work [Chervov2025b]. Behavior under taking G/H . Empirically, we observe the following pattern: the polygon associated with the Schreier coset graph of G/H appears as a subpolygon of the polygon corre- sponding to Cayley graph of the full group G. H-polynomial properties. Given any quasi-polynomial, it is natural to associate to it what is known as the H-polynomial. For Ehrhart polynomials of integral polytopes, the corresponding H-polynomial coincides with the Poincar ́ e polynomial of the associated toric variety if this toric variety is smooth and projective (equivalently, the polytop admits a unimodular triangulation, which is always the case for polygons). As a result, it enjoys several strong properties, such as non-negativity of coefficients, Poincar ́ e duality, and unimodality. A deep question in combinatorics is whether analogous properties continue to hold for more general quasi-polynomials. We observe that in many examples the H-polynomials arising from graphs exhibit these desirable properties. Nevertheless, there also exist examples in which all of them are violated. Of particular interest is a combinatorial analog of the Riemann hypothesis, which predicts that all roots of the H-polynomial have modulus equal to one. We find that this phenomenon occurs frequently in our examples, although counterexamples do exist. Bulk-boundary is unnecessary. In contrast to conventional AdS/CFT, the graph and its dual object are not related by the requirement that the boundary of the dual object coincide with the 20 CayleyPy-4: HolographyCayleyPy collaboration graph. Instead, the polygon arises from a purely mathematical existence hypothesis and has no immediate geometric relation to the original graph. 4.1.2. Permutohedron and Lehmer code. Here we discuss the well-known example of the permu- tohedron graph, argue that its holographic dual is an isosceles right triangle in the plane, identify the holography map with the Lehmer code, and demonstrate that the desired properties indeed hold. 5 serves as an illustration. Permutohedron graph. Here we discuss another example: the Cayley graph of S n with respect to the neighbor transposition generators (i,i + 1), which are the Coxeter generators of S n . The nodes of the graph are all the n! permutation vectors, and an edge exists between two nodes if they differ by a transposition of neighbors. This graph is known to be the 1-skeleton (i.e., the set of vertices and edges) of the permutohedron polytope. It is also called the bubble-sort graph, since the bubble-sort algorithm operates exactly by neighbor transpositions (i,i + 1) and, in fact, provides an optimal path-finding method for traversing the graph. Holography dual polygon is a triangle. We will argue that the dual polygon is just the isosceles right triangle with nodes (0, 0), (n, 0), (0,n). The number of possible lattice paths (without backtracking) that can move right, up, or down starting at (0, 0) with ending in (n, 0) and are restricted to the triangle is exactly n! which matches the graph size. Holography map via Lehmer code. The Lehmer code maps a permutation to a tuple of n integers such that they satisfy L(k) < n− k, for k = 0,...,n− 1. That is, the image belongs to the triangle above. The map is known to be a bijection. One can think of the image as a lattice path tracing the upper boundary of the resulting region. Such lattice paths lie inside the triangle and consist only of right, up, and down steps, as illustrated in Figure 5. FIGURE 4. Lehmer-code representation of the permutations (3, 2, 1, 0) (cyan) and (1, 0, 2, 3) (magenta). Their Lehmer codes are L(3, 2, 1, 0) = (3, 2, 1, 0) and L(1, 0, 2, 3) = (1, 0, 0, 0), respectively. On the left, the corresponding ver- tices of the permutohedron are marked in matching colors. On the right, each permutation is represented by a lattice path given by the upper boundary of its Lehmer diagram inside the staircase Young diagram. The complexities of these permutations are 1 and 6, respectively, which coincide with the areas under the corresponding paths. “Complexity = area”. The analogue of that AdS/CFT principle is known to experts in this context. For any permutation, one can compute its complexity, defined as the number of neighbor transposition generators (i,i + 1) in its shortest decomposition — in other words, the number of steps required by bubble sort to transform it to the sorted form, or equivalently, the number of inversions in the permutation. (Example: 5). It turns out that the number of boxes under the Lehmer code path, or equivalently the sum P i L(i), exactly equals this quantity. This means, that the complexity equals the area under the holographic image of the node - in accordance with general expectations. Bijection between particle extremals and string extremals on a polygon. Similar to the pre- vious section on ROC curves, the Stanley–Edelman–Greene correspondence can be interpreted as 21 CayleyPy-4: HolographyCayleyPy collaboration a bijection between shortest paths on a graph and Young tableaux in the triangle, which represent string extremals. Open questions. This case appears to be more difficult than the previous ROC curve case, and even results that are known for ROC curve scenarios remain open here. In addition to the open questions in the ROC case. For example, the Laplacian of the graph also serves as the Hamiltonian of the X-Heisenberg spin chain. However, it is not in the spin-1/2 representation, making it less tractable from the Bethe ansatz viewpoint. Moreover, the number of spanning trees is not known, even asymptotically. Lehmer code and adjacent transpositions. Fix a permutation σ ∈ S n and its Lehmer code L(σ) = (k 0 ,k 1 ,...,k i ,k i+1 ,...,k n−1 ), where k i = #j > i : σ(i) > σ(j),0≤ k i ≤ n− 1− i. Consider the action of the adjacent transposition (i,i + 1) on σ, for i = 0,...,n− 2. Each component k i is the number of inversions of the element σ(i). When two elements of the permutation are swapped, we obtain a new permutation σ ′ . The new Lehmer code L(σ ′ ) can be obtained from the old one L(σ) as follows. The inversion numbers move together with the elements. After the transposition, the inversion numbers are written again according to the positions, as in the definition of the Lehmer code. In particular, the coordinates of the Lehmer vector stay in the same places; only the data attached to the elements move. Let a = σ(i), b = σ(i + 1). The next step is to check the following cases. • If σ(i) > σ(i+1), then the pair (a,b) is an inversion. After applying (i,i+1), the element a loses exactly one inversion. Hence the Lehmer code changes as L7−→ (k 0 ,k 1 ,...,k i+1 ,k i − 1,...,k n−1 ). • If σ(i) < σ(i + 1), then the pair (a,b) is not an inversion. After applying (i,i + 1), the element a gains exactly one inversion. Hence the Lehmer code changes as L7−→ (k 0 ,k 1 ,...,k i+1 + 1,k i ,...,k n−1 ). In both cases, all other components of the Lehmer code with index̸= i,i + 1 stay unchanged. FIGURE 5. Example of the adjacent transposition (0, 1). Top: σ = (0, 3, 1, 2), σ ′ = (3, 0, 1, 2); one inversion is added. Bottom: σ = (2, 0, 1, 3), σ ′ = (0, 2, 1, 3); one inversion is removed. Only the components k 0 and k 1 of the Lehmer code change. All other components remain unchanged. 22 CayleyPy-4: HolographyCayleyPy collaboration 4.2. CayleyPy AI methodology for assisting in determining the duality. To determine the dual polygon, we currently follow the procedure below. First, we attempt to determine the diameters, then guess quasi-polynomials for them, and finally fit polygons whose Ehrhart quasi-polynomials match. Determining the diameters is the most difficult step. Our currently successful cases use the following workflow, based on the AI-assisted CayleyPy library. (1) BFS (Brute force). Compute the entire Cayley graph for small n using brute-force BFS (CayleyPy currently allows up to n≤ 15). (2) Guess the longest elements. Examine the elements in the last layer (i.e., those whose word metric equals the diameter) and identify patterns that may generalize to arbitrary n. (3) AI pathfinding. For larger n, take these candidate elements and apply AI-based algo- rithms to estimate their word metrics. The current version of CayleyPy allows exact computations (optimal paths) up to around n = 30 (graph size∼ 10 30 ). Extending this to n = 40 (graph size∼ 10 50 ) is a work in progress. (4) Fit quasi-polynomials. Fit quasi-polynomial expressions to the resulting data. In partic- ular, one needs to determine both the starting value of n from which the formula becomes valid and the period of the quasi-polynomial. The most difficult part of the pipeline is Step 2, where one must identify a pattern for the longest elements. Whether this can be done in general remains unclear, but there are examples where it is possible, such as for consecutive cycles and related generators – discussed below. 4.2.1. k-Consecutive cycles (i,i+1,i+2,...,i+k−1). Fix an integer k. For n > k, we consider elements of S n given by the cyclic permutations (i,i + 1,...,i + k− 1) for i = 0,...,n− k. For k = 2, these are the neighbor transpositions (Coxeter or bubble sort generators) considered previously. For odd k, they generate A n inside S n , for even k they give S n . There are two natural options: whether or not to include inverses in the generating set. We consider both cases. Here we present quasi-polynomial formulas for some of the diameters associated with these generators and briefly outline the shape of the dual polygon, leaving the determination of the holography map for future investigation. To the best of our knowledge, formulas for the diameters are new for k > 3. From the physical point of view, the Laplacian of such graphs corresponds to the situation when k neighboring spins interact, spin chains of that sort appear e.g. in AdS/CFT. Informally. k leads to (k − 1) shrinkage. Various effects for general k compared to basic k = 2 can be informally summarized as shrinkage (k− 1)-times. For example, the diameter of the Cayley graph for k = 2 is n(n− 1)/2, while the leading term for the diameters for general k is n(n− 1)/(2(k− 1)), the same concerns various Schreier coset graphs and not only diameters, but also word metrics. Dual polygon as a (k− 1)-shrinkage. For k = 2, the Cayley graph has as its dual polygon the triangle with vertices (0, 0), (n, 0), and (0,n). For general k, we expect the dual polygon to be close to the triangle with vertices (0, 0), (n/(k− 1), 0), and (0,n), although exact results have yet to be obtained. Similarly, for various Schreier coset graphs associated with these generators, we expect a (k− 1)-fold shrinkage comparing to k = 2 case, along the x-axis. Conjecture on diameters of the coset graph. Consider the generating set given by consecutive k-cycles (without inverses), acting on0, 1-vectors with⌊n/2⌋ zeros and n−⌊n/2⌋ ones. We conjecture that the diameters are as follows for large enough n: n≡ 0 (mod (2k− 2)) : D k (n) = n 2 + 4(k− 1) 4(k− 1) , n≡−1 (mod (2k− 2)) : D k (n) = n 2 + 4(k− 1)− 1 4(k− 1) , 23 CayleyPy-4: HolographyCayleyPy collaboration and else, denoting p = n (mod (2k− 2)), D k (n) = n(n + 2k− p− 2) 4(k− 1) ,p is even D k (n) = (n− 1)(n + 2k− p− 2) 4(k− 1) , p is odd . “Riemann conjecture” forH -polynomials. We computed quasi-polynomials andH -polynomials for the coset graphs discussed above, as well as for their modifications with inverse-closed gen- erating sets. We observe that for small k (up to k = 6) all zeros of the H -polynomials in the inverse-closed case lie on the unit circle, whereas this is not the case for the non–inverse-closed generators. In this sense, the “Riemann conjecture” [Rodriguez-Villegas2002],[Bump2000] holds in the inverse-closed setting. It is unclear whether this pattern persists for larger k. For example, for the full Cayley graph with inverse closed generators the “Riemann conjecture” holds up to k = 5, but already fails for k = 6. FIGURE 6. Roots of the H -polynomials in the complex plane. In the left panel, all non-trivial roots have modulus equal to one, so the “Riemann conjecture” holds, whereas in the right panel it is violated. Both panels correspond to k- consecutive cycle generators with k = 6: the left uses an inverse-closed gener- ating set, while the right uses generators without inverses. These correspond to Schreier coset graphs for the subgroup S ⌊n/2⌋ × S n−⌊n/2⌋ . Whether the property that all roots have modulus one persists for larger k remains unclear. Conjectures on diameters of the Cayley graph. Consider consecutive k-cycle generators including inverses and consider the Cayley graph. (Not the Schreier coset graph as above). We conjecture: the diameters are given by D k (n) = j n(n−1) 2(k−1) k + Q 0 (n), where Q 0 (n) is a periodic function of n, that is, a quasi-polynomial of degree zero. It should be understood as a small correction to the leading term j n(n−1) 2(k−1) k . The period is (k− 1) for even k and 2(k− 1) for odd k. The quasi-polynomial formulas are valid for n≥ 2k. Explicit quasi-polynomials for k ≤ 7 are presented in the Supplementary Material. It is worth looking at the corresponding H-polynomial, e.g. for k = 6 it is: x 14 + 2x 13 + 2x 12 + 3x 11 + 4x 10 + 2x 9 +x 8 + 2x 7 +x 6 + 2x 4 + 2x 3 +x 2 +x + 1, one can observe it is not unimodal. 4.3. SL(2,Z) and the Farey graph. Here we reinterpret the results of by one the present authors (D.Melnikov et.al. [Camilo2019]) in a way closer to the present exposition and to the principles of holography and “complexity = area”. It was argued there that to each element of SL(2,Z) one may associate a curve in the hyperbolic plane whose enclosed area approximately corresponds to the complexity of the element with respect to the standard generators S and T . In this language, the map from a group element (i.e. a node of the Cayley graph) to such a curve may be viewed as a holography map, in agreement with our framework. The cited results therefore imply that a variant of the “complexity = area” principle holds, at least approximately. An illustration is shown in Figure 7. 24 CayleyPy-4: HolographyCayleyPy collaboration - 1012 3 2 4 3 5 3 FIGURE 7. Farey graph on the upper halfplane. From [Camilo2019]. The dashed curves are “holography images” of some elements of SL(2,Z) areas under them correspond to complexities of these elements. “Complexity = area” holds approx- imately. 25 CayleyPy-4: HolographyCayleyPy collaboration 5. NEIGHBOR TRANSPOSITIONS CAYLEY AND SCHREIER GRAPHS 5.1. Multiset0 n 0 1 n 1 ... m n m . In this section we show how to build polygons for vertices of graphs generated by mulitsets. FIGURE 8. Multiset 0 4 1 3 2 5 3 2 , i.e. (n 0 ,n 1 ,n 2 ,n 3 ) = (4, 3, 5, 2) ; N = 4 + 3 + 5 + 2 = 14. The rectangles from left to right are R 0 (size 2 × 12), R 1 (5 × 7), and R 2 (3 × 4). E = [0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3], P = [2, 1, 3, 2, 2, 0, 0, 2, 0, 1, 2, 1, 3, 0], and d(P,E) = A 0 +A 1 +A 2 = 11 + 25 + 6 = 42. An example for 0 4 1 3 2 5 3 2 . We illustrate the construction on a graph G whose vertices are permutations of the multiset0 4 1 3 2 5 3 2 , with edges given by adjacent transpositions (i,i + 1). Let E = [0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3]. The construction produces a polygon f (E) such that, for any vertex P in G, the polygon f (P ) is a subset of f (E). Moreover, the bubble-sort distance from P to E equals the area difference area(f (E))− area(f (P )). Let (n 0 ,n 1 ,n 2 ,n 3 ) = (4, 3, 5, 2). Consider the rectangles • R 0 of size n 3 × (n 0 + n 1 + n 2 ) = 2× (4 + 3 + 5) = 2× 12 (area 24), • R 1 of size n 2 × (n 0 + n 1 ) = 5× (4 + 3) = 5× 7 (area 35), • R 2 of size n 1 × n 0 = 3× 4 (area 12). Define f (E) as the union of these rectangles, so that area(f (E)) = 24 + 35 + 12 = 71; see Figure 8. Given P = [2, 1, 3, 2, 2, 0, 0, 2, 0, 1, 2, 1, 3, 0], define • P 0 = P = [2, 1, 3, 2, 2, 0, 0, 2, 0, 1, 2, 1, 3, 0], • P 1 = P 0 −3 = [2, 1, 2, 2, 0, 0, 2, 0, 1, 2, 1, 0], • P 2 = P 1 −2 = [1, 0, 0, 0, 1, 1, 0]. Next, in each rectangle R i we draw a green lattice path from the lower-left corner to the upper- right corner, moving one step either up or right at each move. The choice of step is determined by the symbols of P i , read from left to right, as follows: • In R 0 , read P 0 : if the symbol is 3, step right; otherwise step up. • In R 1 , read P 1 : if the symbol is 2, step right; otherwise step up. • In R 2 , read P 2 : if the symbol is 1, step right; otherwise step up. Let f (P ) be the union of the regions below these green paths. Then f (P ) ⊆ f (E). Fur- thermore, if A i denotes the area in R i between the corresponding red and green curves, then the bubble-sort distance satisfies d(P,E) = area(f (E))− area(f (P )) = A 0 + A 1 + A 2 = 11 + 25 + 6 = 42, as shown in Figure 8. 26 CayleyPy-4: HolographyCayleyPy collaboration Generalization for an arbitrary multiset. Given a multiset 0 n 0 1 n 1 ... m n m . Let N = n 0 + n 1 +· + n m , and let P = π(E) for some permutation π ∈ S N , where E = 0,..., 0 |z n 0 , 1,..., 1 | z n 1 , ..., m,...,m | z n m . Set P 0 = P = π(E) and, for k = 1,...,m− 1, define P k to be the sequence obtained from P k−1 by deleting all occurrences of the letter m− k + 1. Note that P k contains only letters from 0, 1, 2,...,m− k. For each k = 0,...,m− 1, consider rectangle R k of size n m−k × P m−k−1 i=0 n i . Inside R k we draw two lattice paths (red and green); both start at the lower-left corner. The red path goes straight up to the top edge and then straight right to the upper-right corner. The green path is constructed by reading P k from left to right: if the current letter is (m− k), move one step to the right; otherwise, move one step up. After |P k | steps, the path reaches the upper-right corner. Let A k denote the area between the red and green paths in R k . Then the total bubble-sort swap steps converting P to E is d(P,E) = m−1 X k=0 A k . An example of this construction for the multiset0 4 1 3 2 5 3 2 is shown in Figure 8. More details for the multi-set0 n 0 1 n 1 ... m n m . countn 0 n 1 ... n m letter01 ... m Let’s fix notation for vectors • P 0 = P = π(E) contains letters0, 1, 2,...,m− 2,m− 1,m • P 1 = P 0 −m contains letters0, 1, 2,...,m− 2,m− 1 • P 2 = P 1 −m− 1 contains letters0, 1, 2,...,m− 2 • ... • P m−2 = P m−1 −3 contains letters0, 1, 2 • P m−1 = P m−2 −2 contains letters0, 1 RectangleR 0 R 1 ... R m−1 Rectangle Sizen m × P m−1 i=0 n i n m−1 × P m−2 i=0 n i ... n 1 × n 0 VectorP 0 P 1 ... P m−1 Curvem vs0,..., (m− 1) (m− 1) vs0,..., (m− 2) ... 1 vs 0 • For rectangle R 0 , read the word P 0 = P from left to right. If you see m, take a step to the right; otherwise move up. • For rectangle R 1 , read the word P 1 from left to right. If you see m− 1, take a step to the right; otherwise move up. • ... • For rectangle R m−1 , read the word P m−1 from left to right. If you see 1, take a step to the right; otherwise move up. 5.2. Cosets with 0,1,2 components - alternative representation. Coset space as an orbit of words. Throughout this section, cosets refers to the S n –orbits of words with fixed symbol multiplici- ties, equivalently to the Schreier coset space S n /(S λ 0 × S λ 1 × S λ 2 ) under the action by adjacent 27 CayleyPy-4: HolographyCayleyPy collaboration transpositions. Let E = 0 λ 0 1 λ 1 2 λ 2 . The Schreier coset space can be identified with the S n –orbit of E, i.e. the set of all words of length n with exactly λ 0 0’s, λ 1 1’s, and λ 2 2’s: O =W ∈0, 1, 2 n : #0 = λ 0 , #1 = λ 1 , #2 = λ 2 , |O| = n! λ 0 !λ 1 !λ 2 ! . Let s i = (i i+1) ∈ S n be the adjacent transposition. It acts on a word W = w 1 ...w n by swapping adjacent letters: s i · W = s i · (w 1 ...w i w i+1 ...w n ) = (w 1 ...w i+1 w i ...w n ). Thus, in the Schreier graph picture, vertices are words in O and edges connect W to s i · W for each adjacent transposition s i (ignoring loops when the swap does not change the word). Permuting positions among equal symbols does not change the word, so the stabilizer in S n is H = S λ 0 ×S λ 1 ×S λ 2 . To a word W ∈O we associate a lattice path P (W ) starting at the point (0, 0), using the following encoding of steps: 0 : (x,y)7→ (x + 1,y),1 : (x,y)7→ (x,y + 1),2 : (x,y)7→ (x− 1,y + 1). The reference word E is encoded analogously, yielding a path P (E). The two paths share the same start and terminal points. Claim 1. Let S(W,E) denote the oriented area of the polygon enclosed by the paths P (W ) and P (E). Then the area S(W,E) is equal to the number of bubble sort operations required to transform the word W into the reference word E, that is, the minimal number of adjacent transpositions needed to sort the symbols of W . Sketch. Let W be a word over the alphabet0, 1, 2 with fixed multiplicities λ 0 ,λ 1 ,λ 2 , and let E = 0 λ 0 1 λ 1 2 λ 2 be the sorted reference word. It is classical that the minimal number of adjacent transpositions required to transform W into E equals the total number of inversions with respect to the order 0 < 1 < 2, Inv(W ) = #(i,j) : i < j, W i > W j . The inversions decompose into three disjoint types, Inv(W ) = Inv 10 (W ) + Inv 20 (W ) + Inv 21 (W ), where Inv ab (W ) counts occurrences of the pattern a preceding b with a > b. The lattice path encoding 0 : (1, 0),1 : (0, 1),2 : (−1, 1) admits a corresponding decomposition of the oriented area S(W,E) into three independent con- tributions. Indeed, projecting the path P (W ) onto the directions associated with the pairs (1, 0), (2, 0), and (2, 1) yields three binary lattice paths whose enclosed areas coincide with Inv 10 (W ), Inv 20 (W ), and Inv 21 (W ), respectively. By the binary case, each such area equals the correspond- ing inversion count. Summing over the three pairs, we obtain S(W,E) = Inv 10 (W ) + Inv 20 (W ) + Inv 21 (W ) = Inv(W ). Since bubble sort (or insertion sort) performs exactly one adjacent transposition per inversion, the minimal number of adjacent swaps required to transform W into E equals S(W,E), as claimed. □ Hence, the word metric induced by adjacent swaps admits a geometric interpretation as the area between the corresponding lattice paths. 28 CayleyPy-4: HolographyCayleyPy collaboration Example 1. Consider E = [0, 1, 2], P = [2, 1, 0]. In Fig. 9, the words 012 and 210 are encoded by lattice paths, shown in blue and red, respec- tively. The area enclosed by the polygon formed by these two paths equals S = 3. (0, 0) (0, 2) S = 3 FIGURE 9. Encoding the words 012 and 210 by the lattice paths. The distance between the words coincides with the oriented area S between the associated paths. In Fig. 10, the same encoding is shown for several other pairs of words: (a) for the words 012 and 102, the enclosed area equals S = 1; (b) for the words 012 and 021, the enclosed area equals S = 1; (c) for the words 012 and 120, the enclosed area equals S = 2; (d) for the words 012 and 201, the enclosed area equals S = 2. (a) 012 and 102(b) 012 and 021(c) 012 and 120(d) 012 and 201 FIGURE 10. Encoding the words by the lattice paths. S = 8 (0, 0) (1, 3) FIGURE 11. Encoding of the words 00112 (blue) and 21100 (red) by lattice paths. 5.3. Box-Ball System evolution as a deterministic walk on a Schreier/Cayley graph and the AUC–AAC duality. The box–ball system (BBS) is a simple discrete dynamical system on a row of boxes, each either empty (0) or containing a ball (1). The system evolves over discrete time steps according to a deterministic rule. At each time step, balls are processed left to right. Each ball moves to the nearest empty box to its right that hasn’t already been claimed by another ball in this step. If no such empty box exists, the ball stays put. This rule produces soliton-like behavior 29 CayleyPy-4: HolographyCayleyPy collaboration S = 12 (0, 0) (0, 4) FIGURE 12. Encoding of the words 001122 (blue) and 221100 (red) by lattice paths. FIGURE 13. Box-Ball System Illustration. Example for K = 4 balls and N = 20 boxes. – clusters of balls propagate to the right and interact like solitons (they pass through each other and preserve their sizes). Box-ball demo application illustrates this concept. Figure 13 shows an orbit of the box–ball update on the configuration space Ω N,K = x ∈ 0, 1 N : P i x i = K (here N = 20, K = 4), displayed as rows t = 0, 1,...,T . Observe that N,K,T are the system parameters configurable in the application. Viewing Ω N,K as the vertex set of the Schreier graph obtained from the Cayley graph of S N with generators s i = (i i+1), each time step is a deterministic walk: the global “MOVE” map sends the current vertex x to a new vertex T (x), and this map can be realized (conceptually) as a word in adjacent transpositions that performs the needed swaps. Young-diagram / lattice-path encoding. On the right, each row is encoded as a monotone lattice path in the K × (N − K) rectangle by interpreting entries as follows: 0 (empty box)→ move UP; 1 (ball)→ move RIGHT. After starting at (0, 0), reading the whole word, the endpoint is 30 CayleyPy-4: HolographyCayleyPy collaboration (K,N − K), so the path indeed fits the rectangle. The displayed statistics satisfy the duality AUC(x) + AAC(x) = K(N − K) = 64, so AUC and AAC are complementary potential functions. In the Cayley/Schreier picture, a single adjacent swap 01 ↔ 10 flips a local corner of the path and changes AUC by±1, making area a height function on the graph. In this particular run, AUC increases monotonically from 8 to 64 while AAC decreases to 0, and the final state is the extremal sorted word 0 N−K 1 K (all balls packed to the right), whose path maximizes area under the curve. Thus the dynamics is visibly flowing toward a distinguished vertex in the Schreier/Cayley geometry, while the AUC/AAC complementarity provides the dual (above/below) viewpoint on the same walk. 5.4. Schreier graphs for orbits of tuples. In this section, we determine the diameters of a certain class of Schreier graphs that generalize the Cayley graph of S n for Coxeter generators (i.e. the permutohedron) and the 0, 1, 2 coset graphs seen earlier. Unlike Cayley graphs, Schreier graphs need no longer be vertex-transitive, so it is not immediately clear the diameter is always the dis- tance to a fixed vertex. For our chosen class of graphs, we explicitly describe the pair of vertices realizing the diameter, as well as compute it based on the graph isomorphism type. Let X n := 0, 1,...,n− 1 n be the set of n-tuples with entries in a finite set of order n. The symmetric group S n acts on X n by permutation of the components. The stabilizer of each x∈ X n under this action has the form G x ∼ = G λ := S λ 0 ×·× S λ m , where λ = λ 0 ,...,λ m and λ 0 +· +λ m = n (i.e. λ is a partition of n). Let X n,λ ⊂ X n consist of all n-tuples with stabilizer isomorphic to G λ . Let S ⊂ S n be the set of Coxeter generators of S n . Multiplication by elements of G x (on the right) induces an action on Cay(S n ,S) by graph automorphisms. Note that the Schreier graph Sch(S n ,G x ,S) of (left) cosets is obtained as the quotient graph of Cay(S n ,S) with respect to this action. Moreover, the vertices of Sch(S n ,G x ,S) may be identified with elements of the S n -orbit of x, with two vertices connected by an edge if and only if there is a transposition (i,i + 1) sending one n-tuple to the other. It is not hard to see that all graphs Sch(S n ,G x ,S) for x ∈ X n,λ are isomorphic. Therefore, from now on, we will assume x = [0,..., 0, 1,..., 1,...,m,...,m] is an n-tuple consisting of λ 0 0s, followed by λ 1 1s, and so on, such that λ 0 ≤ λ 1 ≤·≤ λ m . Example 2. (a) If λ i = 1 for all 0≤ i≤ n− 1, then x = [0, 1,· ,n− 1]. In this case, the stabilizer of x is trivial, so Sch(S n ,G x ,S) ∼ = Cay(S n ,S). (b) If λ = (n− k,k), then x∈ X n,λ = X n,k is a binary word and Sch(S n ,G x ,S) = G n,k . Let y ∈ X n , and let L i (y) :=|i < j|y i > y j | for each 0≤ i≤ n− 1 (we index our n-tuple starting from 0 for the sake of consistency). This is the number of inversions of y for fixed i. We call L(y) := (L 0 ,...,L n−1 ) the Lehmer code of y. For any y,y ′ ∈ S n · x, set Inv(y,y ′ ) := n−1 X i=0 L i (y ′ )− L i (y). We have the following: Proposition 1. For any y = [y 1 ,...,y n ]∈ S n · x, we have Inv(y, (i,i + 1)y) =      0if y i = y i+1 , −1 if y i+1 < y i , 1if y i < y i+1 Proof. Note the transposition only affects L i (y) and L i+1 (y). The case when y i = y i+1 is clear. Assuming y i+1 < y i , transposing these two entries decreases L i (y) by 1 and does not change 31 CayleyPy-4: HolographyCayleyPy collaboration L i+1 (y). For any j > i + 1, we have the following three cases: y i+1 < y i ≤ y j , y j < y i+1 < y i , and y i+1 ≤ y j < y i . In the first two cases, the transposition does not change L i (y),L i+1 (y). In the third case, it decreases L i (y) by the number of such j and simultaneously increases L i+1 by the same number. This means Inv(y, (i,i + 1)y) =−1. A similar argument for y i+1 > y i shows that Inv(y, (i,i + 1)y) = 1.□ Remark 2. Actually, the proof of Proposition 1 implies slightly more, as it completely describes L((i,i + 1)y) in terms of L(y) and y. Indeed, if y i = y i+1 , then L((i,i + 1)y) = L(y). If y i+1 < y i , then L((i,i + 1)y) is obtained from L(y) by subtracting 1 from L i (y) and exchanging it with L i+1 (y). If y i < y i+1 , then L((i,i + 1)y) is obtained from L(y) by adding 1 to L i+1 (y) and exchanging it with L i (y). Corollary 2. For any two vertices y ′ ,y ∈ Sch(S n ,G x ,S), we have d(y,y ′ )≥|Inv(y,y ′ )|. Proof. By definition, d(y,y ′ ) = l, where l is the minimal nonnegative integer such that y ′ = σ l ·σ 1 y and σ i ∈ S. Therefore, d(y,y ′ ) = P l i=1 d(v i−1 ,v i ), where v i = σ i v i−1 and v 0 = y. The statement follows by Proposition 1 and the triangle inequality.□ Corollary 3. For any y ∈ Sch(S n ,G x ,S), we have d(x,y) = Inv(x,y) = Inv(y). The furthest vertex from x in the graph is x ′ = [m,...,m,..., 1,..., 1, 0,..., 0]. Proof. By Corollary 2 we have d(x,y) ≥ Inv(x,y) = Inv(y), since the Lehmer code of x is (0,..., 0). Starting with the smallest component of y, we can apply adjacent transpositions (i,i + 1) to move each component of y leftward to its place in x. By Proposition 1 each such action decreases Inv(y) by exactly 1. This process inductively constructs a path from y to x, so Inv(y)≥ d(x,y). Thus d(x,y) = Inv(y). The maximal possible number of inversions is achieved by reversing x, which yields x ′ = [m,...,m,..., 1,..., 1, 0,..., 0].□ Proposition 4. The diameter of Sch(S n ,G x ,S) is equal to d(x,x ′ ) = P j>i λ i λ j . Proof. To prove the diameter is achieved for x,x ′ ∈ Sch(S n ,G x ,S), we show that for all y,y ′ ∈ Sch(S n ,G x ,S) we have d(y,y ′ ) ≤ d(x,x ′ ) = Inv(x ′ ) (see Corollary 3). Let Inv ′ (y) = |i < j|y i < y j |. Note that by definition Inv(x ′ ) = Inv(y) + Inv ′ (y). Let σ ∈ S n be the longest element. It acts by involution on the graph Sch(S n ,G x ,S), reversing each y ∈ S n · x, so that d(y,y ′ ) = d(σy,σy ′ ). It is also easy to see Inv(y) = Inv ′ (σy). By Corollary 3, it follows that d(y,x ′ ) = Inv ′ (y). By the triangle inequality and Corollary 3, we have 2d(y,y ′ )≤ d(x,y) + d(x,y ′ ) + d(y,x ′ ) + d(y ′ ,x ′ ) = Inv(y) + Inv ′ (y) + Inv(y ′ ) + Inv ′ (y ′ ) = 2Inv(x ′ ). Thus, d(x,x ′ ) is the diameter, which we can compute as: d(x,x ′ ) = Inv(x ′ ) = m X i=1 λ i i−1 X j=0 λ j = X j>i λ i λ j . □ One can interpret the Lehmer code of y ∈ S n · x as a path P y along the lattice Z 2 ⊂ R 2 , starting at (0,λ m ) and ending at (n, 0), with only vertical moves and horizontal moves to the right allowed. Indeed, for each component of L i (y) of L(y), define P y by connecting the points (i,L i (y)) and (i + 1,L i (y)) with a horizontal line segment. Then, connect the endpoints of adjacent line segments vertically, as well as the start of the first line segment with the starting point and the end of the last line segment with the ending point. The area bounded by P y and the two axes is equal to Inv(y). TODO: Figure to illustrate? 32 CayleyPy-4: HolographyCayleyPy collaboration 5.5. Large size limits, towards ”macroscopic” descriptions. Let us discuss large-size limits ofr the graphs we considered above. The main point is that, in the dual description, the limit shapes (i.e., the “typical” or “random” configurations) of “strings” can be described rather explicitly by tractable formulas. This can be seen as a manifestation of the “strong-to-weak” principle of duality, since the dual description becomes simple when original description is not. Moreover, the limiting dynamics can be described by (partial) differential equations, which corresponds to the standard microscopic-to-macroscopic change of description. In common terms, one may compare this to liquids: on one hand, they consist of atoms, which provide a microscopic description, while on the other hand, we typically describe them macroscopically using partial differential equations, such as the Navier–Stokes equations. A similar picture arises in our setting: in the large-size limit, we can expect the dynamics to be described in terms of partial differential equations. In a sense, this picture is analogous to the AdS/CFT correspondence. On the CFT side, there is a parameter N — the size of the matrix group — which is exactly analogous to n in our setting, the size of permutation matrices. One considers the limit N →∞, and in this limit the dual theory simplifies: the string-theoretic description reduces to its limiting gravitational description. Thus, the observables in the CFT correspond to areas or volumes of certain surfaces in AdS space. In our case, instead of AdS and gravity, the S n duality picture is simpler: planar polygons instead of AdS and in the large-n limit dynamics is expected to be described by hydrodynamic equations, such as the inviscid Burgers equation, as discussed below. We recall classical results and present several conjectures in this direction in the subsequent subsections. Modern studies of limit shapes for Young diagrams originate from the seminal works of Vershik– Kerov [Vershik1977] and Logan–Shepp [Logan1977], and have been greatly extended with nu- merous applications; see, e.g., [Borodin2000; Olshanski2001; Okounkov2006; Kenyon2006; An- gel2006; Okounkov2006; Petrov2013; Corwin2012]. It is honor to mention the landmark paper [Hooft1974] which influence on modern mathemati- cal physics is difficult to overestimate. It was proposed that in large N limit gauge theories admit description similar to string theory. It is tempting to think that similar ideas can be applied in our setting, we hope to elaborate that in future. 5.6. Vershik’s (1996) limit shapes for rectangular Young diagrams, general q. Here we recall the classical analysis of limit shapes for rectangular diagrams, which are essentially ROC curves (or Dyck paths). This fits into the general line of questions outlined in the previous subsection: one expects that, in the large-size limit, dual descriptions of discrete systems converge to contin- uous ones. Moreover, these limiting continuous systems often admit explicit descriptions, again confirming the “strong-to-weak” transformation paradigm, since continuous models are typically more tractable. By a limit shape we mean, roughly speaking, the shape of a “random” or “typical” ROC curve. This can be made precise by considering an average over all possible ROC curves with a given weight; the average is taken pointwise over the family of curves. Before proceeding, let us em- phasize that the results recalled here are, in a sense, q-deformations: they depend on a parameter q, which has a natural “quantum” interpretation. Our primary interest is the case q = 1, which will be discussed in the next subsection. However, this case requires a certain modification of the classical setup, so we first review the classical results. The shape of these limit curves was determined by A. M. Vershik [Vershik1996], building on the celebrated earlier joint work with S. Kerov [Vershik1977] (see also [Logan1977]). An interactive simulation widget has been developed by one of the authors (L. Petrov): link. (Press “About this simulation” for a detailed description.) Limit Shape: As N → ∞ with q = e −γ/N for fixed γ > 0, the rescaled partition boundary converges to a deterministic curve given by the implicit equation: Ae −γy + Be −γx = 1 33 CayleyPy-4: HolographyCayleyPy collaboration where A = 1− e −γ 1− e −γ(1+a) and B = 1− e −γa 1− e −γ(1+a) . FIGURE 14. Limit shape for ROC curves (rectangular Young diagrams, or equiv- alently Dyck paths), defined as the average shape of the curve when all config- urations are weighted by q area . (Here q is a parameter; in the plot it is chosen to be 0.95.) The solid red line represents the theoretical limit curve obtained by A. M. Vershik [Vershik1996]. An interactive simulation widget developed by L. Petrov is available at link. 5.7. Vershik’s (1985) limit shapes for rectangular Young diagrams q = 1. Here we remind related results on limit shapes for q = 1. A Young diagram chosen uniformly at random from the set of all Young diagrams with n boxes exhibits a limit shape phenomenon. After rescaling the diagram by a factor of 1/ √ n (so that the total area is normalized to 1), the boundary of the diagram concentrates, as n→∞, around the Vershik curve γ given by e − √ ζ(2)x + e − √ ζ(2)y = 1, see [Vershik1985] (last formula in the paper, attributed to unpublished work by A.M.Vershik). Here ζ(2) = π 2 /6. The proofs and actually certain generalizations appeared in works of several mathematicians. Here we will rely on [Petrov2009], where the following generalization has been obtain: Proposition 1. The limit shape for young diagrams in rectangle (ROC-curves) with fixed number of boxes (fixed area under the curve is given by: e −c(x−x 0 ) + e −c(y−y 0 ) = 1, for suitable constants c, x 0 , and y 0 . Moreover they can be seen as segments of the full Vershik’s curve above. Sketch of proof: consider fixed positive a and b with ab > 1 the Young diagrams with n boxes that fit in the a √ n×b √ n rectangle (in other words, the length is at most a √ n and the height is at most b √ n). Scaling by 1/ √ n we get a random set Y n of area 1 inside the rectangle a× b. 34 CayleyPy-4: HolographyCayleyPy collaboration If ab = 2, the boundary of Y n approaches (by probability) the diagonal of our a× b. If 1 < ab < 2, passing to a complement of Y n to the a× b rectangle (and making a symmetry with respect to its center), we get a set of area (ab− 1) < ab/2, and another scaling reduces the question to the ab > 2 case. If ab > 2, one can find the unique point P = (x 0 ,y 0 ) below the Vershik curve γ, points A and B on γ so that the segment PB is vertical, PA is horizontal, and PB : PA : √ S = b : a : 1, where S is the area in the triangle BAP below γ. Then this piece of γ is (after the scaling by a factor b : PB = a : PA) is the limit shape of Y n . So, the equation of the limit curve is e −c(x−x 0 ) + e −c(y−y 0 ) = 1 for appropriate constants c, x 0 , y 0 . This phenomenon was rediscovered independently by several mathematicians in various forms. An analogous result also holds for Young diagrams confined to a strip, that is, under restrictions either on the length or on the height. 5.8. Limit shape for ROC-curves (Dyck paths). The first natural question from our perspective regarding the large-n limits of discrete systems is: what is the shape of the “random” or “typical” string with a fixed area under it? As discussed above, one can expect tractable formulas for such limit shapes, which may be interpreted as a manifestation of a “strong-to-weak” holographic du- ality. Moreover from Vershik’s results above one can expect the answer to be given by the simple equation on the exponential of coordinates. Based on computational experiments we propose such answer below. Here, we present the results of simulations and a conjectural answer to this question in the case of ROC curves, which, according to our picture, are the holographic dual strings corresponding to the nodes of S n /(S k ×S n−k ). The same can be described as averaging over Dyck paths the same result, since the dominant contribution comes from the curve’s neighborhood (where all monotonic paths are Dyck paths); thus, averaging over Dyck paths or over all monotonic paths from (0, 0) to (1, 1) yields essentially the same outcome. Conjecture 1 (Limit shape). Fix C ∈ (0, 1) and let k → ∞ with L = L(k) satisfying L/k 2 → C. Then the normalized average path in layer L converges uniformly to the curve y = y C (x), x∈ [0, 1], given by y C (x) = 1− 1 λ ln 1 + e λ − e λx , λ̸= 0,(1) and y 1/2 (x) = x when C = 1 2 (i.e., λ = 0).(2) Here λ = λ(C)∈ R is the unique solution of C = ln 2 (1 + e λ ) + 2 Li 2 1 1+e λ − π 2 6 λ 2 .(3) Equivalently, the limit curve satisfies the implicit equation e −λ(1−x) + e −λy = 1 + e −λ .(4) Equation (4) is a rescaled and translated segment of the universal exponential curve e −αX + e −αY = 1, which arises in the arctic-circle phenomenon for boxed plane partitions [Cohn1998] and in Ok- ounkov’s theory of limit shapes for random surfaces [Okounkov2016]. Our constraint to the box [0, 1] 2 selects the appropriate segment; the parameter λ is determined by the area (equivalently, the layer number). 35 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 15. Limit shape for the ROC curves (Dyck paths) with fixed area under curve. That is - average of all ROC curves with fixed area under them. Related results by A.M.Vershik and [Petrov2009] were discussed in the previous subsection. 5.9. Dynamics in large size limits. TASEP-Burgers correspondence. KPZ.. As we discussed before Young tableaux can be considered as solutions of discrete string equations of motion and define the evolution of the paths (discrete strings). In large size limit it is natural to think that ”typ- ical” Young tableaux would produce such kind of evolution which would be possible to describe by partial differential equations, such that their solutions would be monotonic curves going from left-bottom to right-top. In particular the family of ”typical” curves above should be their solution, but more generally evolution might evolve any such monotonic curve. Mathematically speaking we can describe evolution as follows take a discrete curve take all possible Young bounded by the curve, make evolution by each of these curves and consider their averages. That defines an evolution on curves. For the case of the ROC-curves, educated guess is that such dynamics can be described by the inviscid Burgers equation and with fine granularity by KPZ-equations ([Kardar1986]). Relying on the known results on TASEP-Burgers correspondence [Rost1981; Ferrari2018; Ferrari2009; Quastel2021; Quastel2020] and relation to KPZ [Bertini1997; Corwin2012]. Instead of the microscopic dynamics implemented by random adjacent updates, i.e. TASEP- type evolution on 0, 1 p it is better to consider macroscopic dynamics. In the large-size (hy- drodynamic) regime, with p → ∞, k p /p → α ∈ (0, 1), and with the empirical initial profile converging to a limiting density u 0 (x), the coarse-grained field is described by a deterministic conservation law. Denoting the macroscopic density by u(x,s), the expected limit is the entropy solution of the inviscid Burgers equation ∂ s u + ∂ x u(1− u) = 0, u(x, 0) = u 0 (x). Analogy: think of a one-lane road with cars, where each site can be either occupied (1 = car) or empty (0 = space). The microscopic motion of the cars — each moving forward if the spot ahead is free — corresponds to a TASEP process or the evolution of nodes on a discrete graph. Now imagine observing this road from high above in the cosmos: individual cars are invisible, and you only see the average car density along the road. At this macroscopic level, the density evolves smoothly in time, and its dynamics are governed by the Burgers equation. If you zoom in and improve the resolution, the random fluctuations of cars around the average density become visible, and their collective dynamics are described by the KPZ equation, capturing the wiggly, stochastic behavior of the system. Fluctuations around this limit under the KPZ scaling (x∼ ε −1 ,t∼ ε −3/2 ), h ε converges to the solution of the KPZ equation: ∂ t h = ν∂ x h + λ(∂ x h) 2 + ξ, 36 CayleyPy-4: HolographyCayleyPy collaboration where ξ is space-time white noise. Thus, TASEP connects to KPZ through the limit of properly rescaled fluctuations around the hydrodynamic density. Figure 16 shows the Monte Carlo tab after a completed run (‘2000/2000‘) for the initial path A = (0 k 1 p−k ) with smoothing r = 13: on the left, the heatmap p(position, time) displays mean occupancy u(x,s) (blue≈ 0, red≈ 1) together with an active green ‘DRAWBOUNDARY‘ over- lay given by the fitted curve s(x)≈ 0.259 + 2.846x− 2.574x 2 , which tracks the visible separation between the mixed interior and near-pure corner regions; on the right, the multi-slice profile panel compares exact Burgers curves and MC curves at several times (including highlighted s = 0.2, s = 0.4, and breaking-time slices), with per-slice L 2 errors indicating stronger agreement at earlier times and larger discrepancies near later, shock-influenced regimes, so the two panels together pro- vide both a spacetime picture of the evolution and a quantitative slice-by-slice validation against the PDE prediction. FIGURE 16. Monte Carlo simulation results for starting sequence 0 k 1 p−k ): left, the averaged occupancy heatmap u(x,s) with fitted green boundary s(x) ≈ 0.259 + 2.846x− 2.574x 2 ; right, multi-time comparison of exact Burgers profiles and MC data with per-slice L 2 errors. An interactive simulation is available at link. 37 CayleyPy-4: HolographyCayleyPy collaboration 5.10. Spanning trees for finite spin-up sector, 1/n expansion, Benjamini-Schramm limit to lattices, Mahler measures. In this subsection we analyze the limit S n /(S d 1 ×...×S d j ×S n− P d j ) for d j fixed and n→ infty. Overview: math part. The key message here is that in certain sense (Benjamini-Schramm) these graphs tend just to the standard lattice graph Z P d j . Hence all the key quantities for them in the limit tend to those for the standard lattice. In particular the number of spanning trees nor- malized by graph size converges to logarithmic multi-variable Mahler measures. These measures count spanning trees for lattices and are interesting objects related to values of L-functions, deep Beilinson’s conjectures, supersymmetric Landau-Ginzburg models used in mirror symmetry for Fano varieties, etc. Additionally, we observe that many quantities here admit 1/n expansions which are efficient enough to reproduce numerical data with good accuracy. Overview: duality and string part.In physical language the claim is that the model that describes the graph Laplacian (the X spin chain) in the large n limit can be described in terms of d = P d j non-interacting particles. The appearance of d particles has a clear and simple explanation via our duality. Indeed, let us consider for simplicity the case S n /(S d ×S n−d ), with d fixed and n→∞. Our discrete duality describes that system as a discrete string whose worldsheet is a d×(n−d) rectangle. In the limit it becomes of size d×∞, so we can think that one variable on the worldsheet becomes continuous, while the other one is discrete. The continuous variable leads to particle-like behaviour, and the discrete variable means that we have not just a single particle but d of them. As we described above the action of the discrete system is a discretization of the standard string action, so we expect that in the limit it becomes an action for standard particles. In the next subsection we will describe another limit and argue that actual bosonic string-like scalar models appear in the same way. Our claims are of course closely related to the well-known fact that magnons in X models are almost non-interacting at large length. At the same time we were unable to find in the literature detailed discussions of the specific quantites we consider below, which are rather unusual from the spin chain point of view. In particular, surprisingly, it is not clear how to derive many of the proposals below directly from the Bethe ansatz, which in principle describes the full spectrum. It is interesting to note that a Van Hove singularity can be clearly seen in the spectrum of S n /(S 2 × S n−2 ) for large n – see figure 21 (and notebook). That confirms that these systems can be described as two non-interacting particles since the Van Hove spectrum appears as for the sum of two cosines. (A single cosine is spectrum of line graph which is S n /S n−1 ). Let us now present our main statements and observations. Conjecture 2 (Spanning trees, X-determinants and Mahler measure). Consider Coxeter (neig- bour transpositions) generators of S n and consider Schreier coset graphs S n /(S d 1 × ...× S d j × S n− P d j ) for d j fixed and n → ∞. Denote the number of spanning trees by τ (Γ n ) and the graph size by V (Γ n ), and denote d = P d j . (Remark: τ (Γ n ) is related to the determinant of the Laplace operator which is the X-Heisenberg spin chain Hamiltonian in a specific subsector, up to constants). With notation as above, logτ (Γ n ) |V (Γ n )| n→∞ −→ m 2d− d X i=1 (z i + z −1 i ) ! ,(5) where the logarithmic Mahler measure of a Laurent polynomial P ∈ Z[z ±1 1 ,...,z ±1 d ] is m(P ) = Z T d log P (e 2πiθ 1 ,...,e 2πiθ d ) dθ 1 ·dθ d (2π) d . For d = 2 (i.e. r = 2), the right-hand side equals 4G/π, where G is Catalan’s constant. Numeri- cally this Mahler measure is around 1.166243, and for d = 3 it is around 1.673389. It is related to special values of L-functions, explicitly known for d = 2, 3 and non-explicitly related to Beilin- son’s conjectures for general d (see discussion below on C.Denininger’s results). For general d, the measure equals the same expression for the lattice, i.e. logarithm of number of spanning trees 38 CayleyPy-4: HolographyCayleyPy collaboration on the finite lattice again normalized by size, and it is sometimes called ”spanning-tree entropy” of the d-dimensional integer lattice Z d . Similarly the traces, characteristic polynomial and entire spectral measure converges to appro- priate expressions for the lattice: log det(Laplacian G n − E) |V (Γ n )| n→∞ −→ m 2d− d X i=1 (z i + z −1 i )− E ! ,(6) Tr(Laplacian G n − E) s |V (Γ n )| n→∞ −→ Z T d 2d− d X i=1 (e 2πiθ d + e −2πiθ d )− E ! s dθ 1 ·dθ d (2π) d .(7) Instead of Tr or det one can put any other function and a similar equality is expected. In other words, moments and entire spectral measure with appropriate normalization tend to the one of the lattice. Remark. The expressions above are well-known and easy to understand for lattices (grid- graphs), see e.g. [Lyons2005; Silver2016]. The number of spanning trees is of course different for lattices and finite graphs above, but with normalization by graph sizes (which are also different) they coincide in the limit. Conjecture 3 (Benjamini–Schramm limit). The sequence (Γ n ) n≥1 converges in the Benjamini– Schramm (local weak) sense [Benjamini2001; Lyons2005] to the d-dimensional integer lattice Z d ,with its standard nearest-neighbour structure. Consequently, the spectral measure of Γ n con- verges weakly to the spectral measure of the Laplacian on Z d . Conjecture 4 (1/n expansion and numerical check). The expressions logτ (Γ n ) |V (Γ n )| and similar expres- sions for traces and characteristic polynomial admit an asymptotic expansion as series in 1/n. The constant term is the Mahler measure (given by expressions in the conjecture above). Convergence is fast enough such that determining only few values in n (which is easy numerically) one can determine the leading and several subleading coefficients, thus obtaining numerical results for infinite n from several quite small n values. Informal arguments. It is clear that the key idea is to understand that the graph is close to the lattice in a certain sense. The Coxeter generators are very similar to commutative – all generators commute except neighbor pairs. Moreover, for the case S n /(S 2 × S n−2 ) is has been already pictured on several figures above that the graph is quarter of the lattice (”quarter Aztec diamond”). The same can be achieved for any d for S n /(S d ×S n−d ) . Indeed, nodes of the graph are sequences of 0’s and 1’s with exactly d zeros. The embedding of the graph is defined by the following rule: associate to a vector d integer numbers which are just the positions where these zeros appear. One can see that this gives an embedding of our graph into the lattice graph Z d . Figure 17 illustrates that embedding for d = 3. See also figures 19, 20 for S n /S n−2 and S n /S n−3 . 39 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 17. The Schreier graph S 8 /(S 3 ×S 5 ) with Coxeter generators, similarity to lattice can be clearly seen. Widget is available link. Other graph examples available by the link. Numerical checks. To check the conjecture on Mahler measures numerically we rely on the 1/n expansion. We compute the number of spanning trees for several values of n, and then make a numerical fit for logτ (Γ n )/|V (Γ n )| in the form c 0 + c 1 /n + c 2 /n 2 + .... The leading term c 0 should match the desired Mahler measure. We observe excellent correspondence between the two computations: c 0 from finite data and numerical computation of the Mahler measure. The Mahler measure itself is just an integral, so it can be computed e.g. in Mathematica with high precision. Technically, to compute the number of spanning trees by Kirchoff’s theorem we take the prin- cipal minor of the Laplace matrix (to avoid problem with zero eigenvalue) and compute the log- arithm of the determinant directly, e.g. via the numpy function np.linalg.slogdet. In this way we get values up to n around 30. When we fit the obtained data we exclude the first several values (for example, we start from n ∼ 10). We can also fit by polynomials of various degrees, and we observe that degrees from 5 to 10 give nearly the same result for the leading term. Figure 18 provides a screenshot from the notebook with fit showing numerical coincidence up to 5 digits and near-independence on the choice of the degree of the fitting polynomial. Computations of the numerical determinants themselves are available in notebooks: notebook, notebook. 40 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 18. Fitted numerical data shows quite good coincidence of c 0 with Mahler measure which is around 1.67338. Leading coefficient c 0 does not de- pend much on the degree of the polynomial approximation. Notebook Potential route to the proof. The two conjectures are related. The first one follows from the second by a result of [Lyons2005]. By the visualization and arguments above it is clear that our graphs resemble lattices, the difficult part is to control the difference and show that is disappears in large size limit – while clear in principle, it may be nontrivial to do it fully rigorously without also relying e.g. on unproven properties of the Bethe ansatz. Step 1 (Local structure). A vertex of Γ n is an ordered partition [n] = B 1 ⊔ · ⊔ B r . The Coxeter generator s i acts non-trivially at v if and only if i and i+1 belong to different blocks. For a uniformly random vertex and a fixed radius R, the R-ball in Γ n is determined by O(R) consecutive positions of [n], whose block-membership pattern converges (as n → ∞) to that of an i.i.d. colouring of Z with r colours and frequencies (k j /n). The resulting limit graph is the Cayley graph of Z d with the 2d standard generators – exactly Z d . Step 2 (Følner / BS convergence). The fraction of “boundary” vertices – those whoseR-neighbourhood is not isomorphic to a ball in Z d —is at most O(R/n)→ 0. This is precisely the Følner condition, which implies Benjamini–Schramm convergence (Conjecture 3). Step 3 (Lyons’ theorem). By [Lyons2005, Theorem 1.2], if a sequence of finite connected graphs with uniformly bounded degree converges in the Benjamini–Schramm sense to an infinite unimod- ular random graph G ∞ , then logτ (G n ) |V (G n )| −→ h(G ∞ ), where h(G ∞ ) denotes the tree entropy of G ∞ and which is logarithmic Mahler measure in case when G ∞ is Z d . Step 4 (Tree entropy = Mahler measure). For G ∞ = Z d , the tree entropy equals h(Z d ) = Z T d log 2d− 2 d X i=1 cos(2πθ i ) ! dθ 1 ·dθ d = m 2d− d X i=1 (z i + z −1 i ) ! , 41 CayleyPy-4: HolographyCayleyPy collaboration establishing Conjecture 2 conditionally on Conjecture 3. The equality of the last two expressions is the tautological unfolding of the Mahler measure as a torus integral [Silver2016]. Related works. Multi-variable Mahler measures are very similar to supersymmetric Landau- Ginzburg models used for mirror symmetry of Fano varieties [Golyshev2007; Coates2013; Galkin2016] and earlier works [Witten1993; Hori2000; Eguchi1991; Vafa1989; Candelas1991; Kontsevich1995; Givental1998; Batyrev1994]. From physical point of view counting spanning trees on these graphs is equivalent to compu- tation of the determinant of Hamiltonian of X spin chain (by Kirchhoff’s theorem and identi- fication of the Hamiltonian with graph Laplacian). The conjectures above are closely related to the Bethe ansatz, but establish them fully rigorously and in full generality is still a nontrivial task which seems to not have been addressed yet. For the most simple case, the subsector with just two spins-up (i.e. graph S n /(S 2 ×S n−2 which is quatter of the Aztec diamond), there is a non-trivial result by R.Stanley, D.Knuth, T.Chow, et.al. (oeis-A007726, observed and conjectured in [Stanley1994], resolved in [Knuth1997; Chow1997; Ciucu1997], developed [Kenyon2000; Ciucu2008] etc.). which provides the number of spanning trees exactly for each finite n, (not only asymptotics that we discussed above). Our conjecture can be shown to be true in this case. At the same time, it would be very interesting to derive that exact finite-n result from the Bethe ansatz – which should be possible to do, but seems rather nontrivial. In general, the interest in this set of questions during that time seems to have come from seminal works on related questions of domino tilings (dimer model) for Aztec diamond [Elkies1991], and more generally remarkable results on spanning trees [Burton1993], in particular with relation to conformal field theories [Duplantier1989]. For Abelian groups, the relation of spanning trees and Mahler measures is quite natural and well documented in the literature. Indeed, by Kirchhoff’s theorem, the number of spanning trees of a finite graph equals the determinant of its Laplacian (with one row and column removed). In the Abelian case, the group algebra is essentially a quotient of the lattice algebra Z d . Thus group elements can be identified with Laurent polynomials, and convolution (or the action of the Laplacian) corresponds to multiplication by a Laurent polynomial. Consequently, the determinant of the Laplacian can be expressed in terms of this Laurent polynomial, and using the formula log(det(M )) = ]rmTr(log(M )) one arrives at the logarithmic Mahler measure. On the one hand, it counts spanning trees; on the other hand, in the infinite or periodic limit it naturally leads to the Mahler measure of the corresponding polynomial, modulo some technical details [Silver2005; Grunwald2019; Grunwald2021]. The non-trivial situation is that our graphs come from non-abelian groups, but this non-abelianity is not that large, and in the appropriate limit one obtains similar results to the abelian case. Mahler measure and their generalizations play a key role in the number theory: [Deninger2006; Lind1992; Lind1990; Deninger1997b; Arzhakova2021]. Wonderful relations were found be- tween Mahler measures and values of L-functions in special points [Smyth2008; Boyd1981; Rodriguez-Villegas1999; Boyd2003] and partly derived by C.Deninger [Deninger1997a] condi- tioned to Beilinson conjectures, we refer to [Brunault2020; Trieu2023] for further information. The general expectations according to C.Deninger and previous works by Smith, Boyd et.al. are the following. The Mahler measure m(P d ) is commensurable with the first non-vanishing deriva- tive of the L-function associated to the cohomology H d−1 (V P d ) (smooth compactification of the zero locus of Laurent polynomial) evaluated at s = 0 (or at the central point of the critical strip, depending on the normalization). It might be that m(P d )∼ Q × c d ·L ′ (M d , 0) where M d is the mo- tive attached to the projective closure of V P d , c d is a rational normalization factor (often involving powers of π), and∼ Q × denotes equality up to a rational factor. Cases d = 2, 3 have well-known explicit description. In some cases Mahler measure of A-polynomial of the knot gives hyperbolic volume of the knot complement manifold e.g. [Boyd2003]. Quantization of A-polynomial is con- jecturally related to colored Jones polynomials (”AJ-conjecture”) [Le2006; Gukov2005; Fuji2014; Grassi2016], fascinating topic deeply connected to various questions in topological string theory. 42 CayleyPy-4: HolographyCayleyPy collaboration Related important conjecture is L ̈ uck’s Determinant Conjecture. Let G be a finitely gener- ated group and ∆ the Laplacian on its Cayley graph. Then the Fuglede–Kadison determinant of ∆ satisfies det G (∆)≥ 1. More relations to deep conjectures and questions by Connnes, Gromov et.al briefly sketched on page 5/1458 [Aldous2007]. For abelian groups this is equivalent to a lower bound on the growth rate of spanning trees in the graph [L ̈ uck2002]. The papers above have more analytic flavor, related purely algebraic considerations for free group can be found in [Kont- sevich2009; Bellissard2007; Haiman1993] although terminology is different but constructions are essentially related - the key results show that resolvent of the Laplacian is algebraic function, more- over M.Kontsevich extends it to ”det(1-tM)” which is non-commutative analogue of Mahler mea- sure. Surprisingly, results rely on [Chomsky1968] theory of context free languages by Chomsky and Schutzenberger from 1968, and Kontsevich’s generalization additionally on Grothendieck’s conjecture on algebraic solutions of holonomic systems (see also lecture by M.Kontsevich and his newer lectures on the subject). In the case S n /(S d ×S n−d the graphs themselves apparently are special cases of ”token” graphs (applied to just a linear graph) [Dalfo2020]. Use of Benjamini-Schramm limits is quite wide- spread technique nowadays e.g. [Bille2023] section 2.2. FIGURE 19. Schreier coset graphs - Coxeter generators S n /S n−2 . Similarity with lat- tice is evident.Notebook. FIGURE 20. Schreier coset graphs - Coxeter generators S n /S n−3 . Similarity with lat- tice is evident. Notebook 43 CayleyPy-4: HolographyCayleyPy collaboration FIGURE21. Eigenvalues histogram Coxeter Schreier graph S 200 /(S 2 × S 198 ). Distributions tends to Van Hove distribution (sum of two cosines). Notebook. FIGURE22. Eigenvalues histogram Coxeter Schreier graph S 70 /(S 3 × S 67 ) Distri- butions tends to sum of three cosines. Notebook. 44 CayleyPy-4: HolographyCayleyPy collaboration 5.11. Limit to field theory string-like model. Spanning trees for many spins up and down sector. In the present subsection analyze limit S n /(S k 1 × ...× S k j ) for k j /n = p j fixed, and n→∞. Overview - math part. We propose that eigenvalue distribution tends to Gaussian, propose formulas for mean, variance and hence conjecture the number of spanning trees (Laplacian de- terminant), and also spectral gap. Numeric simulations are provided to support. It is worth men- tioning why Benjamini–Schramm technique is not applicable in the setup of the limit considered here. It is because the degrees of the graph nodes tends to infinity, thus it is obvious that there is no possible limit to usual graph. In some sense what is going on is defining the ”renormalization” of the UV-divergent limit. Overview - string part. We also conjecture that limiting theories are conformal field theories, which are almost the standard 1-dimensional scalar theories which here play the role of the string- like dual model. What is non-standard that in general the worldsheet has a non-standard shape in general - the shape of the polygon which appears in our discrete duality. For example for S n itself (without quotients) it is triangle. Appearance of the 1d boson is not surprising – the limit of X is described by such QFT goes back at least to [Luttinger1963] and standard in modern literature. More surprising is appearance of non-standard worldsheets, relation to discrete duality and predictions of spectrum of conformal dimensions. In that sense we give an answer what CFT may be viewed in some sense as dual to the standard 1-dimensional scalar compactified on a radius R circle – in our setup it is the CFT arising from the graph S n /(S d × S n−d ), where the normalized area of the dual polygon plays role of R 2 , i.e. R 2 = d(n− d)/n 2 , where n → ∞, in the other words this is X Heisenberg spin chain on subsector with k spins up, out of n total spins. The fact again goes back to [Luttinger1963], but what is new is to put into the framework of discrete analogue of AdS/CFT duality. It it worth to compare the limit in the present section with the previous from string theory perspective - the picture is surprisingly simple. Consider S n /(S d ×S n−d ) the dual polygon in our duality is rectangle d× (n− d), the limit in the previous section is: d fixed n → ∞, think of the rectangle as a worldsheet, imagine that one of the coordinates disappear - so strings reduces to particle, now imagine it disappeared not completely but has d discrete possible values - so we get d independent particles - that is exactly what happened in the previous subsection. In the present section both d,n goes to infinity and we get usual worldsheet, so appearance of usual string is natural. Our duality conjecture implies the answer on the spectrum of conformal dimensions for CFT arising as limits of X spins chains with various spin configurations - it corresponds to the spectrum of the Laplace operators on the planar polygons which shape is defined by d i in particular for the full S n it is triangle, and S n /(S d × S n−d ) to rectangle, general d i correspond to cutting down small triangles of with edges d i from the diagonal of the big triangle. Conjecture 5 (Guassian spectrum). The spectrum of the Laplacian for the Schreier graphS n /(S k 1 × ...× S k j ) (X Hamiltonian) tends to the Gaussian distribution in the limit k i /n = p i fixed, and n→∞. Informal explanation of the Gaussianity is quite simple and the following: graph Laplacian coincides with H X which up to constants is P i σ i,i+1 . Where σ i are commuting |i− j| > 1, moreover they acts on independent sets of spins. Thus eigenvalues are sums of individual eigenvalues, thus leading to Gaussian in large n limit. The effect of the non-commutativity for j = i + 1 disappears at large n. That is closely parallel to classical central limit theorems, where independence basically appears from variables acting on different coordinates of the probability space. And weak dependence of variables do not spoil the Gaussian at the limit. The question has been studied a lot in related contexts ...todo-references... Proposition 2. Mean value equals to 2 n ( P i<j k i k j ), that means in grows linearly with n. In particular for the case of S n (without quotients) it is (n− 1), for S n /(S d ×S n−d ) it is 2 n d(n−d). 45 CayleyPy-4: HolographyCayleyPy collaboration Small miracle. Factor ( P i<j k i k j ) coincides with diameter of the corresponding Schreier graph which is the area of the dual polygon according to our duality. It seems that fact is specific to Coxeter group, it resembles the classical miracle: number of reflections coincide with length of the longest element for Coxeter groups. Proposition 3. Variance for the case of S n (without quotients) is again (n− 1) (same as mean), for S n /(S d × S n−d ) it is (2d(n−d)) 2 +2d(n−d)(n 2 −2n) n 2 (n−1) , which is linear in n in leading order. In particular for n even and d = n/2, variance is 3/4n− 1/4− 1 4(n−1) . In general we expect that variance always grows linearly with respect to n in leading order in n. Proofs are standard. Remark. For the case of S n (without quotients) distribution is symmetric that means skewness and higher analogs exactly vanishes for finite n, not just asymptotically. Conjecture 6 (Spectral gap). The spectral gap (i.e. value of the first non-zero eigenvalue of the Laplacian) tends to zero as c/n 2 , for some constant c. For example for S n (without quotients) the smallest eigenvalue is λ 1 = 2 1− cos π n be- cause it comes from the standard representation of S n , it corresponds to line graph, where spec- trum is easy to see: λ k = 2 1− cos πk n , k = 1,...,n− 1. Conjecture 7 (Determinant. Spanning trees). Logarithm of the number of spanning trees (which is related to determinant of X Hamiltonian by Kirchhoff ’s theorem) normalized by the graph size – behaves as log(n) at leading order. This corresponds to central charge 1 from the CFT perspective, confirming the free boson as the dual model. The subleading term (independent of n – ”universal”) corresponds to twice the area of the dual polygon log(2 P i<j (k i /n)(k j /n)). Which should be thought as logarithm of ”renormalized” number of spanning trees. For S n /(S d × S n−d ), it is expected: 1 ( n d ) −1 logτ n = log(n) + log(2a(1− a))− 1+2a(1−a) 4a(1−a) 1 n + O(n −2 ), where a = d/n For S n (no quotients), it is expected: 1 n!−1 logτ n = log(n− 1)− 1 2(n−1) − 3 4(n−1) 2 − 15 6(n−1) 3 − ... (2m−1)!! 2m(n−1) m − ... + O(exp(−n)). The constant terms disappears since 0 = log(1) = log(2× 1/2), here 1/2 is triangle area, taken twice gives 1, logarithm gives zero. The arguments are quite simple. Assuming Gaussian distribution logdet is approximately log(Mean), the next orders of the approximation are given by the standard formula 1 N log det ′ H n ≈ logμ− σ 2 2μ 2 − 3σ 4 4μ 4 − ... (2m−1)!!σ 2m 2mμ 2m ... , which simply comes just from the log(m + delta) = log(m(1 +delta/m)) = log(m) +delta/m−delta 2 /2m 2 ..., then taking mean (over delta which is N (0, 1)) - odd degrees disappear, even degrees give (2m− 1)!!. To the best of our knowledge number of spanning trees for such graphs was not known be- fore. For example for permutohedron graph (i.e. S n without quotients) it is explicitly mentioned [Ehrenborg2021]: ”The associated graph is known as the 1-skeleton of the permutahedron. An open problem is to determine the number of spanning trees for this graph.” For small values of n for permutohedron graph numbers are given at A337083. Conjecture 8 (Limiting 1d bosonic model (”string side”)). In the limit above the quantum theory on the graph (X chain with corresponding spins set) tends to CFT described by 1d bosonic free model compactified on R , where R 2 = P i<j k i k j – the area of the dual polygon. However the worldsheet has an unusual shape - exactly the dual polygon. For the case S n /(S d × S n−d ) the worldsheet is the standard rectangle, but for the example of S n itself (no quotients) it is a triangle. 46 CayleyPy-4: HolographyCayleyPy collaboration Conjecture 9 (Conformal dimensions of primary fields). In the setup above the spectrum of con- formal dimensions is equal to the spectrum of the Laplacian on the dual polygon. Motivation 1. It is exactly the relation in conventional AdS/CFT where spectrum of conformal dimensions on CFT side can be read of the Laplacian on AdS side, since SL(2) subgroup of CFT it mapped into isometries of AdS and thus conformal algebra Laplacian is mapped to AdS geometric Laplacian. Motivation 2. Saleur’s formula [Itzykson1998]. The Saleur (or Coulomb Gas) formula ex- presses the conformal dimensions as ∆ n,m = 1 4 (gn 2 + m 2 g ), which represents the energy spectrum of a free boson on a circle of radius R∝ √ g. This formula matches the formula for the spectrum of the Laplacian on rectangle, indeed the eigenvalues of the Laplacian on a rectangle with side lengths L x and L y are given by λ n x ,n y = π 2 n 2 x L 2 x + n 2 y L 2 y for n x ,n y ∈ Z + . Numeric simulations. Numeric simulations can be found in the notebook. Let us present few figures, more can be found there. FIGURE 23. Eigenvalues his- togram Coxeter Cayley graph S 8 (Permutohedron graph). Distributions tends to Gauss- ian. FIGURE24. Eigenvalues histogram for Laplacian of Schreier graph S 18 /(S 9 ×S 9 ). Distributions tends to Gauss- ian. FIGURE 25. Logarithm of spanning trees number divided by graph size. Coxeter Cayley graphs S n (Permutohedron graph). X-axis is n. FIGURE 26. Logarithm of spanning trees number di- vided by graph size. Schreier graph S n /(S n/2 × S n/2 ). X-axis is n. 47 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 27. Schreier coset graphs - Coxeter generators S n /(S n/2 × S n/2 ). Variance - exact match with proposed theoretical formula. FIGURE 28. Schreier coset graphs - Coxeter generators S n /(S n/2 × S n/2 ). Spectral gap inverse square root - be- haves like a line. So gap is c/ p (n) 48 CayleyPy-4: HolographyCayleyPy collaboration 6. NEIGHBOR TRANSPOSITIONS EXTENDED BY (0,n− 1) (“WRAPPED” OR “AFFINE” CASE) Background and related work. A small modification of the nearest-neighbor transposition generators—namely, adding the transposition (0,n− 1)—leads to a more nontrivial Cayley graph, which we refer to as the cyclic, wrapped, or affine case. In CayleyPy, these generators are denoted Cyclic Coxeter and form a special case of wrappedkcycles with k = 2. The simplest quotient S n /S n−1 yields a cycle graph, corresponding to the affine Dynkin diagram of type e A n−1 , and may thus be viewed as an affine analogue of the nearest-neighbor transposition graph. The Laplacian of this graph corresponds to a periodic or closed spin chain, which is well known to be related to affine Lie algebras. The diameter of the graph was finally established to be ⌊n 2 /4⌋ in [Zuylen2016], building on earlier works cited therein; see also an alternative argument by Ville Salo (MO359699). A related recent result [Adin2025] establishes the circular version, yielding⌊(n− 1) 2 /4⌋. The graph was first studied in depth in the foundational work [Jerrum1985]. Let us note that the corresponding H -polynomial is given by 2x 4 + 4x 3 + 2x 2 (1− x 2 ) 3 , which is symmetric, has positive coefficients, and whose roots all have modulus 1 (”Riemann conjecture holds”). 6.1. Duality for the Wrapped Case. In this section we consider a modification of the binary Cayley/Schreier graph studied above, obtained by adding an extra generator that swaps the first and the last positions. This produces a natural wrapped (or cyclic) version of the model and leads to a new geometric interpretation of shortest-path distances in terms of areas between lattice curves minimized over cyclic shifts. 6.1.1. The wrapped binary graph. As before, the set of nodes is defined by vectors with 0 and 1 components, such that there are l-zeros: Gr aff l,n =x∈0, 1 n : n−1 X i=0 x i = n− l, We will distinguish the sorted vector denoting it by e: e := 0 l 1 n−l . Two vertices x,y ∈ Gr aff l,n are connected by an edge if either • y is obtained from x by an adjacent swap 01↔ 10, or • y is obtained from x by applying the transposition t = (0,n− 1), which exchanges the first and the last entries: (t· x) 0 = x n−1 ,(t· x) n−1 = x 0 (t· x) i = x i (1≤ i≤ n− 2). We denote by d wrap (x,y) the shortest-path distance in Gr aff l,n . The additional generator (0,n− 1) allows symbols to move across the boundary, leading to a metric behavior that is more complicated than the non-wrapped case. 6.1.2. Lattice-path encoding and cyclic shifts. Each word x ∈ Gr aff l,n canonically determines a monotone lattice path P (x) : (0, 0)−→ (l,n− l), obtained by reading x from left to right and applying the rule 07→ (1, 0),17→ (0, 1). 49 CayleyPy-4: HolographyCayleyPy collaboration Let Area(x) denote the number of unit lattice squares below P (x) inside the rectangle [0,l]× [0,n− l]. As shown earlier, Area(x) = Inv(x), where Inv(x) is the inversion number of the binary word x. In the wrapped setting, cyclic shifts arise naturally. For r ∈0, 1,...,n− 1 define the cyclic shift (sh r (x)) i := x i+r (mod n) . Each shift determines a path P (sh r (x)) and an associated area A r (x) := Area(sh r (x)). 6.1.3. Wrapped duality hypothesis. The main conjectural statement for the wrapped case is the following. Hypothesis 1 (Wrapped duality). For every x∈ Gr aff l,n , the distance from the sorted vertex e in the wrapped graph satisfies d wrap (e,x) =min r∈0,...,n−1 A r (x). Equivalently, if we define the prefix-sum function f x (i) := i X j=0 x j , i = 0, 1,...,n− 1, and let f min (i) := min r f sh r (x) (i), then the conjecture asserts that d wrap (e,x) = n−1 X i=0 f x (i)− f min (i) , i.e. the distance equals the area between the step function associated with x and the lower envelope of all its cyclic shifts. We illustrate the wrapped construction and highlight a subtlety of the conjecture. Example 3. Let n = 6, k = 3, and e = 000111. Consider the word x = 101010∈ X 6,3 . Non-wrapped area. The inversion number of x is Inv(x) = 6, hence Area(x) = 6. Cyclic shifts. The cyclic shift sh 1 (x) = 010101 has inversion number Inv(010101) = 3. Since further shifts repeat these patterns, we obtain min r A r (x) = 3. Wrapped distance. Using the additional generator (0, 5), we have 000111 (0,5) −→ 100110 (2,3) −→ 101010 = x. Thus d wrap (e,x)≤ 2. So, we have d wrap (e,x) = 2whilemin r A r (x) = 3. 50 CayleyPy-4: HolographyCayleyPy collaboration This shows that the naive wrapped-duality formula does not hold verbatim with the linear notion of area. Instead, the wrapped generator introduces a boundary effect that allows shortcuts not captured by minimizing the usual area over cyclic shifts. From the duality perspective, adding the generator (0,n− 1) transforms the linear correspon- dence vertex ←→ lattice path ←→ area into a wrapped version in which a vertex corresponds to a family of cyclic paths and the graph distance reflects a modified area functional that incorporates boundary moves. Determining the correct geometric invariant for the wrapped case remains an interesting open problem. 6.2. Diameters for the Schreier coset graphs “few-coincide”: S n /S d . The central state and the initial state are given as (0, 1, 2,...,n− d− 1,n− d,n− d,...,n− d) and ?? respectively, where (n− d) appears d times. Here we present some partial results on diameters of the Schreier coset graphs of the form S n /S D , which can be equivalently described as graphs with nodes corresponding to vectors where D elements coincide. In CayleyPy we define them by setting ”central state” to be (0, 1, 2,...n− D− 1,n− D,n− D,...,n− D) (D coincide at the end). The generators are the same - cyclic Coxeter (equivalently ”wrapped 2 cycles”). Conjecture 10. For D ≥ 2, the conjecture was found to be: D(n,d) = d− 1 2 n− δ(n,d), where the offset term δ(n,d) is given by δ(n,d) =                  d(d− 2) 4 ,if d is even and n is even, d(d− 2) 4 + 1 2 , if d is even and n is odd, d− 1 2 2 ,if d is odd. TABLE 1. Experimental data (d different, for 2-cycles) nd=2 d=3 d=4 d=5 d=6 d=7 d=8 d=9 2 312 4234 52456 635789 7368101112 8471012141516 948111416181920 1059131619212324 11510141821242628 12611162024273032 13612172226303336 14713192429333740 15714202631364044 16815222834394448 Continued on next page 51 CayleyPy-4: HolographyCayleyPy collaboration nd=2 d=3 d=4 d=5 d=6 d=7 d=8 d=9 17816233036424752 189172532394551 199182634414854 2010192836445158 2110202938465461 2211213140495765 2311223242516068 2412233444546372 25122435465666 26132537485969 27132638506172 28142740526475 29142841546678 30152943566981 7. BETHE ANSATZ AND GRAPH SPECTRUM Consider the Schreier graph for S n with generators adjacent (wrapped) transpositions, i.e. (12), (23),..., (n− 1,n), (n1), and central state 1... 10... 0 with k ones and n− k zeros. Its vertices are labelled by all possible permutations of this central state. We will assume its adja- cency matrix A is defined such that the sum of all entries in each row is the same 1 and equal to the number of generators, i.e. n. Now let us view each string of 0’s and 1’s as labelling a basis of a subspace with k flipped spins (corresponding to 1’s) in a 2 n -dimensional space. Then we find that the matrix A exactly matches the X spin chain Hamiltonian restricted to this subspace, in the form H = n X i=1 P i,i+1 (8) where the permutation operators switch the entries of our strings. This means that the spectrum of A can be found via Bethe ansatz. In general the Bethe equations read u a + i/2 u a − i/2 L = M Y b̸=a u a − u b + i u a − u b − i , a = 1,...,M(9) where we identify the length L with our n, while M is the number of magnons. We need eigen- values corresponding to all states with k flipped spins, so we need to consider M = k but also all lower values M = 0, 1,...,k− 1 as they correspond to states whose descendants will have k flipped spins. Adapting the standard Bethe ansatz formula to the normalisation of our Hamiltonian, the eigenvalues are found from E = n− M X j=1 1 u 2 j + 1/4 (10) For instance, a descendant of the vacuum state will have no Bethe roots and give the eigenvalue equal to n which indeed we always observe for the graph. As an example, we checked this explicitly for the casen = 5,k = 2, i.e. central state 11000. We find that the Bethe ansatz reproduces all the distinct eigenvalues of A (with more care one should be able to match the eigenvalue multiplicities as well). Concretely, here we need to consider: 1 this means we add an appropriate number of self-loops at each vertex depending on how many generators leave invariant the corresponding permutation 52 CayleyPy-4: HolographyCayleyPy collaboration M = 0 (one eigenvalue), M = 1 (four solutions, two different eigenvalues) and M = 2 (many solutions, three different eigenvalues once we restrict as usual to no repeated roots etc). 7.1. Spectral gap. With Bethe ansatz we can compute the gap between the largest (equal to n) and next-to-largest eigenvalues of the adjacency matrix. We observe (at least for the central state with two 1’s and n− 2 zeroes) it corresponds to a state with 1 magnon whose Bethe root solves u + i/2 u− i/2 = e 2πi/n .(11) Computing the energy for it we find the gap (for large enough n, unclear starting from which n) 2 cos(2π/n)− 2(12) which at large n gives≃−4π 2 /n 2 . This gives (for large enough n) the 2nd eigenvalue of the Laplacian as λ 2 = 2− 2 cos(2π/n) which at large n gives an upper bound for the diameter∼ n 4 – a huge overestimate. The bound comes from d≤ N − 1 λ 2 (13) where N is the number of graph’s vertices, so for us N = n(n− 1)/2. 7.2. Relation to Laplacian spectrum. Let us relate the above discussion to the spectrum of the graph Laplacian. The Laplacian is defined as the matrix L = D− A 0 , where D is the diagonal matrix with degrees of the vertices on the diagonal, and A 0 is the adjacency matrix with the only off-diagonal nonzero elements that are equal to 1 if the corresponding vertices are connected. Note this adjacency matrix is defined differently from A in the above Bethe ansatz discussion, as A there also has diagonal elements equal to the number of self-loops at each vertex. However, nicely, the Laplacian L differs from A only by flipping the overall sign and adding a scalar matrix nI , namely L = nI − A. This is easy to see since for all i we have A i = n− D i , because all edges at each vertex are either self-loops or contribute to D i . Here we assume that in D we have the degrees of each vertex not counting any self-loops. We conclude that the spectrum of the Laplacian is also captured by the Bethe ansatz and are given by E Laplacian = M X j=1 1 u 2 j + 1/4 (14) 7.3. Schreier graph with non-wrapped 2-cycles. We can also consider the Schreier graph for S n with central state of k 1’s and n− k zeros but taking as generators only the transpositions (12), (23),..., (n− 1,n), without the last one (n1). In the spin chain language this corresponds to the open spin chain, with Hamiltonian H = n−1 X i=1 P i,i+1 (15) The Bethe equations read u a + i/2 u a − i/2 2L = M Y b̸=a u a − u b + i u a − u b − i u a + u b + i u a + u b − i , a = 1,...,M(16) where the length L = n. The energy (eigenvalue of the Hamiltonian) is given by E = n− 1− M X j=1 1 u 2 j + 1/4 (17) Experimentally (at least for length n = 4, 5) we find some selection rules for Bethe roots, namely we should not allow roots that are: • zero 53 CayleyPy-4: HolographyCayleyPy collaboration • equal • equal up to sign For the case of only two 1’s, this Schreier graph is in fact the same as the quarter Aztec diamond (i.e. roughly a collection of squares filling the region 1 ≤ x < y ≤ n) if we add to the latter two edges (and two vertices) at the corners. This is shown in figure 29. FIGURE 29. The Schreier graph as a quarter Aztec diamond for n = 5. Note that (x,y) coordinates are indices of 1s. E.g., for A = (00011) (x A ,y A ) = (4, 5) or B = (00110) (x B ,y B ) = (3, 4), etc. Notice that adding the extra two edges does not affect the number of spanning trees. We have checked explicitly that taking the product of nonzero eigenvalues for the Schreier graph Laplacian 2 reproduces the number of spanning trees on the quarter Aztec diamond, sequence https:// oeis.org/A007726 in OEIS (what is called the ’order’ of the Aztec diamond there is our n− 1). 7.4. Relation between Schreier and Johnson graphs. In the Johnson graph J (n,k) the vertices correspond to subsets of (1, 2,...,n) consisting of k distinct elements. Two vertices are connected by an edge iff they differ by exactly 1 element. The Johnson graph J (n,k) is in fact precisely the Schreier graph for S n with central state being a string of k 1’s and n− k 0’s, and generators being all transpositions. To see this, we can label each string of 0’s and 1’s by the positions of 1’s in it – this will give precisely a subset of k elements out of n numbers like in J (n,k). It’s clear that the rule for placing edges is also the same for both graphs 3 . Let us also note that the Schreier graph for the case of only adjacent transpositions will then be a subgraph of J (n,k) (i.e. it will have the same vertices but fewer edges). 2 defined as for the closed case by appropriately subtracting a diagonal matrix to deal with self-loops at vertices 3 since if one string maps to another by a transposition this means we have moved exactly one of the 1’s to a new location 54 CayleyPy-4: HolographyCayleyPy collaboration 8. CONSECUTIVE-(K)-CYCLES. QUASI-POLYNOMIALITY, ETC. 8.1. Section outline. Quasi-polynomiality. In this section, we present several quasi-polynomial formulas for the diameters and word metrics associated with the consecutive k-cycle generators defined below. We also compute the corresponding H -polynomials and study their properties. Both Cayley graphs and Schreier coset graphs are considered. We also provide some other results like theoretical lower and upper bounds on diameters which are in agreement with our experimen- tal studies. As discussed above, the conjectural quasi-polynomiality of diameters and word metrics is one of the main pieces of evidence supporting the holography hypothesis. Indeed, if holography holds, then diameters and word metrics arise as Ehrhart quasi-polynomials, which naturally explains their quasi-polynomial behavior. Consequently, working out explicit quasi-polynomial formulas is essential for a deeper understanding of the holography principle. Generators definition. Related works. Fix an integer k. For n > k, we consider elements of S n given by the cyclic permutations (i,i + 1,...,i + k− 1) for i = 0,...,n− k. For k = 2, these are the neighbor transpositions (Coxeter or bubble sort generators) considered previously. In CayleyPy these generators are denoted: consecutive kcycles. For odd k, they generate A n inside S n , for even k they give S n . There are two natural options: whether or not to include inverses in the generating set. We consider both cases. To the best of our knowledge, these generators were not studied systematically in the literature, e.g. formulas for the diameters are new for k > 3 (k = 4 briefly discussed in our previous work: [Chervov2025b]). From the physical point of view, the Laplacian of such graphs corresponds to the situation when k neighboring spins interact, spin chains of that sort (but not exactly) appear e.g. [V. A. Kazakov2004] in AdS/CFT. Let us note that cyclic permutations are used to characterize triples of prefix–reversals generating the whole group S n , see [Blanco2025], and such triples are also discussed in the previous work. 8.2. (k− 1)-Shrinkage heuristics: Results and Difficulties. Before going into details let us first give some informal heuristic principle summarizing the results: to obtain results on diameters, word-metrics and other characteristics one should take results for k = 2 case (which is standard neighbor transposition or Coxeter generators) and just divide them by k − 1. In all considered cases it gives correct leading terms and moreover in some rare cases simple correction like adding ceil-rounding would suffice to get the correct results. However exact results typically are more complicated. The only case where we were able to conjecture the result for all k is the case of the coset Schreier graph with n//2 zeros and n− n//2 ones with not inverse closed generators. For the other cases we present conjectures for k up to 5 to 8 depending on cases, and only leading terms for all k. Informal motivation of that heuristics is rather simple: the k-cycle (i,i + 1,...,i + k− 1) is equal to product of (k− 1) neighbor transpositions (i,i + 1)(i + 1,i + 2)..., thus the nodes which are on distance 1 in k-cycle case are on distance k− 1 in standard neighbor transposition graph - so is some sense k-cycle graph shrinks standard graph k− 1 times. 8.3. Theoretical lower and upper bounds on the diameters. Cayley graphs. Here we prove lower and upper bounds for the diameters of the Cayley graph (not Schreier coset) of the con- secutive k cycles inverse closed generators. They quite correspond to informal (k− 1)-shrinkage principle above: for the standard neighbor transposition graph the diameter is equal to n(n− 1)/2 so we expect that for consecutive k-cycles we get n(n− 1)/(2(k − 1)). Indeed we can prove bounds with such leading term. Theorem 5. The diameter satisfies the inequality D k (n)≥ l n(n− 1)− 2 2(k− 1) m . 55 CayleyPy-4: HolographyCayleyPy collaboration Proof. Let ℓ(·) denote the Coxeter length on S n with respect to adjacent transpositions of the form (i i + 1), i.e. ℓ(σ) = #(i,j) : 1≤ i < j ≤ n, σ(i) > σ(j). For each consecutive k-cycle τ ∈ S n , we have ℓ(τ ) = k− 1. On the other hand, ℓ is subadditive, i.e. ℓ(σ 1 σ 2 ) ≤ ℓ(σ 1 ) + ℓ(σ 2 ). Hence, if σ is the product of m consecutive k-cycles, we have ℓ(σ)≤ m(k− 1) which implies that d(σ) = m≥ ℓ(σ) k− 1 . We consider the inverse permutation (reversal permutation?) defined by σ 0 (i) = n + 1− i, which has ℓ(σ 0 ) = n 2 = n(n−1) 2 . If k is odd or both k and σ 0 are even, then σ 0 lies in the subgroup generated by S and we may take σ = σ 0 . Otherwise, the subgroup generated by S is contained in A n , while σ 0 is odd (which happens when k is even and n ≡ 2, 3 (mod 4)). Then we cannot reach σ 0 itself, so we can consider σ = σ 0 s i for some adjacent transposition s i = (i i + 1). In that case σ is even and hence lies in the subgroup, and ℓ(σ) = ℓ(w 0 s i ) = ℓ(w 0 )− 1 = n(n− 1) 2 − 1 = n(n− 1)− 2 2 . Since the diameter is at least the distance d(σ) from the identity e to σ, we have that D k (n) ≥ d(σ)≥ ℓ(σ) k−1 ≥ n(n−1)−2 2(k−1) .□ Theorem 6. For any n > 3k, given k-cyclic generators, the distance between any permutation σ of n elements and the identity is at most n(n−1) 2(k−1) + O(n). Proof. First of all, we deliver the elements 1,...,k− 1 into [k; 2k− 2] by repeatedly moving them k− 1 steps to the left or to the right. Then we have done n + O(k) operations or less. Next we need to order the elements correctly. For this purpose we rotate [k; 2k− 1]. When- ever an element i is k − 1 to the right of the correct position, we place the element in the correct position by using the permutation (i,i + k − 1,i + 2k − 2) −1 . Since (k,..., 2k − 1) −1 (1, 2,...,k) −1 (k,..., 2k− 1)(1, 2,...,k) = (1,k,..., 2k− 1), we find that we can place the elements from 1 to k− 1 onto their rightful places in O(k) moves. Then we can forget about the first k− 1 elements and reorder the others. Repeating these operations with smaller and smaller n, we obtain the required result. □ 8.4. Theoretical diameters estimate. Schreier coset graphs S n / S ⌊n/2⌋ ×S n−⌊n/2⌋ . Here we provide theoretical diameters estimate for consecutive k cycles inverse closed generators for the Schreier coset graph S n /(S ⌊n/2⌋ × S n−⌊n/2⌋ ), which can be alternatively described as graph with nodes corresponding to vectors with components 0 and 1 with⌊n/2⌋ zeros, and n−⌊n/2⌋ ones. Estimate is again consistent with (k− 1)-shrinkage principle. Theorem 7. The diameter of the coset with [n/2] zeros and n− [n/2] ones is equal to n 2 4(k−1) + O(n). Proof. Consider the sequence of zeros and ones where the zeros are located at placesa 1 ,a 2 ,...,a l . Create a modified sequence where the zeros are located at a 1 ,...,a 1 + k− 2,a k ,...,a k + k− 2,...,a m(k−1)+1 ,...,a m(k−1)+k−1 ,... This modified sequence has the zeros come in groups of k− 1. Since a group of size k− 1 or less can be moved to a distance of 1 by 1 turn of the cycle, the modified position is reachable in [n/2/(k−1)] X m=0 a m(k−1)+1 − m(k− 1) moves. Which is at most⌈n/2⌉([n/2/(k− 1)] + 1) moves. 56 CayleyPy-4: HolographyCayleyPy collaboration Next we reconstruct the original sequence by guiding every zero to its rightful place in O(n) turns.□ 8.5. Quasi-polynomials for diameters. Cayley graphs. Here we present conjectural quasi- polynomial expressions for the diameters of the Cayley graph (not Schreier coset) of the con- secutive k cycles inverse closed generators for k ≤ 6, and partial information for larger k. The leading terms of the formulas is n(n− 1)/(2(k − 1)) consistent with the theoretical estimates above and naive (k− 1)-shrinkage principle. For n ≤ 14 the conjectures checked by effective realization of the BFS algorithm provided by CayleyPy. For larger n the data obtained by AI-component of CayleyPy - according to methodol- ogy described above. Theorem 8. Case k = 3 from n = 6: n≡ 0, 1 (mod 4) :D 3 (n) = n(n− 1) 4 , n≡ 2, 3 (mod 4) :D 3 (n) = n(n− 1) 4 − 1 2 . Or in the other words D(n) =⌊ n(n−1) 4 ⌋ Conjecture 11. Case k = 4 from n = 6: n≡ 0, 1 (mod 3) :D 4 (n) = n(n− 1) 6 − 1, n≡ 2 (mod 3) :D 4 (n) = n(n− 1) 6 + 2 3 . OGF = −x 9 + x 8 + 4x 6 + x 5 + 4x 4 + x 2 − x (1− x 3 ) 3 . FIGURE 30. case k = 4 57 CayleyPy-4: HolographyCayleyPy collaboration Case k = 5 from n = 8: n≡ 0, 1 (mod 8) :D 5 (n) = n(n− 1) 8 , n≡ 2, 7 (mod 8) :D 5 (n) = n(n− 1) 8 − 1 4 = n(n− 1)− 2 8 , n≡ 3, 6 (mod 8) :D 5 (n) = n(n− 1) 8 + 1 4 = n(n− 1) + 2 8 , n≡ 4, 5 (mod 8) :D 5 (n) = n(n− 1) 8 + 1 2 = n(n− 1) + 4 8 . OGF = H(x) (1− x 8 ) 3 , where H(x) = x 22 + 2x 21 + 3x 20 + 4x 19 + 5x 18 + 7x 17 + + 9x 16 + 11x 15 + 11x 14 + 11x 13 + 11x 12 + 11x 11 + + 11x 10 + 9x 9 + 7x 8 + 5x 7 + 4x 6 + 3x 5 + 2x 4 + x 3 . Nonnegative Symmetric Unimodal Weakly log concave TrueFalseTrueFalse FIGURE 31. case k = 5 58 CayleyPy-4: HolographyCayleyPy collaboration Case k = 6 from n = 10: n≡ 0, 1 (mod 5) :D 6 (n) = n(n− 1) 10 + 1, n≡ 2, 4 (mod 5) :D 6 (n) = n(n− 1) 10 + 4 5 = n(n− 1) + 8 10 , n≡ 3 (mod 5) :D 6 (n) = n(n− 1) 10 + 7 5 = n(n− 1) + 14 10 . OGF = H(x) (1− x 5 ) 3 , where H(x) = x 14 + 2x 13 + 2x 12 + 3x 11 + 4x 10 + 2x 9 + + x 8 + 2x 7 + x 6 + 2x 4 + 2x 3 + x 2 + x + 1. Nonnegative Symmetric Unimodal Weakly log concave TrueFalseFalseFalse FIGURE 32. case k = 6 59 CayleyPy-4: HolographyCayleyPy collaboration Case k = 7 (period 12): n≡ 0, 1, 4, 9 (mod 12) :D 7 (n) = n(n− 1) 12 , n≡ 2, 11 (mod 12) :D 7 (n) = n(n− 1) 12 − 1 6 = n(n− 1)− 2 12 , n≡ 3, 6, 7, 10 (mod 12) :D 7 (n) = n(n− 1) 12 + 1 2 = n(n− 1) + 6 12 , n≡ 5, 8 (mod 12) :D 7 (n) = n(n− 1) 12 + 1 3 = n(n− 1) + 4 12 . OGF = H(x) (1− x 12 ) 3 , where H(x) = x 34 + x 33 + 2x 32 + 3x 31 + 4x 30 + 5x 29 + 6x 28 + 8x 27 + + 9x 26 + 11x 25 + 13x 24 + 15x 23 + 15x 22 + 17x 21 + 17x 20 + + 17x 19 + 17x 18 + 17x 17 + 17x 16 + 15x 15 + 15x 14 + 13x 13 + + 11x 12 + 9x 11 + 8x 10 + 6x 9 + 5x 8 + 4x 7 + 3x 6 + 2x 5 + x 4 + x 3 . Nonnegative Symmetric Unimodal Weakly log concave TrueFalseTrueFalse FIGURE 33. case k = 7 60 CayleyPy-4: HolographyCayleyPy collaboration More generally: Conjecture 12. Let D k (n) be the diameter of S n Cayley graphs generated by consecutive cycles of length k and their inverses. They are given by quadratic quasi-polynomials in n (for n large enough) and the following holds: (1) All the statements below are valid for n ≥ 2k. (The starting point for quasi-polynomials to be valid.) (2) The periods of quasi-polynomials are: τ = ( k− 1,if k is even, 2(k− 1), if k is odd. (3) The general form of the diameter: D k (n) = n(n− 1) 2(k− 1) + Q 0 (n), where Q 0 (n) is quasi-polynomial of degree zero, i.e. just the periodic function of n. More- over in many cases Q 0 (n) is just the correction to make the first term expression to be the integer (floor/ceil), however it is not always the case. (4) The last layer always contains either the full reversal R = (n−1, n−2, ..., 1, 0), or its modified form with one adjacent inversion: R ′ = R◦ (i, i+1). (5) The alternation between R and R ′ in the last layers has a period T = k − 1. I.e. the sequence S(n) = R or R ′ whether the last layer contains R or R ′ will be periodic with the period k− 1. For diameter k we have the following formulas for polynomials H 5 (x) = Φ 2 (x) 3 Φ 4 (x) 2 Φ 8 (x) 2 Φ 10 (x). H 7 (x) = Φ 2 (x) 3 Φ 3 (x) 2 Φ 4 (x) 2 Φ 6 (x) 3 Φ 12 (x) 2 Φ 14 (x). Conjecture (odd k ≥ 5). For odd k, H k (x) = (x 2k−2 − 1) 2 (x k + 1) (x k−4 + 1) (x− 1) 2 (x + 1) . Equivalently, in cyclotomic form, H k (x) = Φ 2 (x) Y d|(2k−2) d>1 Φ d (x) 2 Y d|k d>1 Φ 2d (x) Y d|(k−4) d>1 Φ 2d (x). 61 CayleyPy-4: HolographyCayleyPy collaboration 8.6. Schreier coset graph: S n / S ⌊n/2⌋ ×S n−⌊n/2⌋ (not inverse-closed). Here we present a con- jectural formula for the diameters of Schreier coset graphs of the form S n S ⌊n/2⌋ × S n−⌊n/2⌋ . In this setting, we were able to obtain an explicit formula for all values of k. The graph can be equivalently described as follows: its vertices correspond to binary vectors with components in 0, 1, containing exactly ⌊n/2⌋ zeros and n−⌊n/2⌋ ones. In CayleyPy, we define these graphs by setting the centralstate to be [0] n//2 + [1] n−n//2 . The generating set consists of consecutive k-cycles; we consider the case where the generating set is not inverse-closed. The leading term of the diameter formulas are n 2 /(4(k− 1)) as it is expected from the (k− 1)- shrinkage principle, since for the standard k = 2 (Coxeter or neighbor transposition) case the diameter is exactly⌊n/2⌋(n−⌊n/2⌋). Informally, one may think of this graph as a “k-shrunken” version of GrassmannianGr(⌊n/2⌋,n) over the field with one element. From the point of view of that analogy diameter corresponds to di- mension of the manifold, and Poincare polynomial correspond to growth polynomial of the graph. Conjecture 13. Diameters of the Schreier coset graphs S n /(S ⌊n/2⌋ ×S n−⌊n/2⌋ ) with consecutive k-cycles, not inverse closed case are given by : n≡ 0 (mod (2k− 2)) : D k (n) = n 2 + 4(k− 1) 4(k− 1) , n≡−1 (mod (2k− 2)) : D k (n) = n 2 + 4(k− 1)− 1 4(k− 1) , and else, denoting p = n (mod (2k− 2)), D k (n) = n(n + 2k− p− 2) 4(k− 1) ,p is even D k (n) = (n− 1)(n + 2k− p− 2) 4(k− 1) , p is odd . The formula is expected to be valid for n large enough, exact bound is not clear, for k = 2, 3, 4, 5, 6 it starts from 4, 4, 7, 16, 31, so in general it might be for n > 2 k , which would be quite surprising since in the other cases it is typical to have linear bound in terms of k. For general k the generating function reads OGF = H(x) (1− x 2k−2 ) 3 (18) where where the H -polynomial is H(x) = x− x k x k + x (x− 1) 2 x 7 (x + 1) (−x 4k + x 2k+2 − 2x 2k+3 + kx 2k+4 − x 2k+4 + x 2k+5 −kx 2k+6 + x 4k+1 + x 4k+2 − x 4k+3 + x 8 − x 6 + x 5 ) One can show that in fact H(x) is a polynomial. Below we present picture of roots the H -polynomials for small k, not all roots have modules equal to 1, so ”Riemann conjecture is not true” for these cases. Nevertheless it is natural to believe that there are some patterns which remains to be understood. 62 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 34. Consecutive 2- cycles [0..01..1] FIGURE 35. Consecutive 3- cycles [0..01..1] FIGURE 36. Consecutive 4- cycles [0..01..1] FIGURE 37. Consecutive 5- cycles [0..01..1] 63 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 38. Consecutive 6-cycles [0..01..1] 64 CayleyPy-4: HolographyCayleyPy collaboration 8.7. Schreier coset graph: S n / S ⌊n/2⌋ × S n−⌊n/2⌋ (inverse-closed). Almost the same setting as in the previous subsection - but now inverse-closed generators. I.e. we study the Schreier coset graphs of the form S n S ⌊n/2⌋ × S n−⌊n/2⌋ . The graph can be equivalently described as follows: its vertices correspond to binary vectors with components in0, 1, containing exactly⌊n/2⌋ zeros and n−⌊n/2⌋ ones. In CayleyPy, we define these graphs by setting the centralstate to be [0] n//2 + [1] n−n//2 . The generating set consists of consecutive k-cycles; we consider the case where the generating set is inverse-closed. Informally, one may think of this graph as a “k-shrunken” version of Grassmanian Gr(⌊n/2⌋,n) over the field with one element. From the point of view of that analogy diameter corresponds to dimension of the manifold, and Poincare polynomial correspond to growth polynomial of the graph. We found quasi-polynomials and H -polynomials for small k. Surprisingly the roots of H - polynomials have modules equal one (”Riemann conjecture holds”) in the considered cases. How- ever it is not clear whether it would be true for larger k, there are plenty examples when it holds true for the beginning of the series, but not in general. Conjecture 14. All the quasi-polynomials below have leading term n 2 /(4(k− 1)) and liner term is absent, it natural to expect that it holds true for all k. H k (x) = x 2k−2 − 1 2 x 2k−3 + 1 (x− 1) 2 (x + 1) , k ≥ 2. H k (x) = Φ 2 (x) 2 Y d|(2k−2) d>2 Φ d (x) 2 Y d|(2k−3) d>1 Φ 2d (x), k ≥ 2. Proposition 9. Let k ≥ 2. Define integers a k,n for 0≤ n≤ 6k− 10 by a k,n =            j n 2 k + 1,0≤ n≤ 2k− 5, k− 1 + (n− (2k− 4)) mod 2 , 2k− 4≤ n≤ 4k− 6, k− 2− n− (4k− 5) 2 ,4k− 5≤ n≤ 6k− 10. Let H k (x) := 6k−10 X n=0 a k,n x n ∈ Z[x]. Then H k (x) = (x 2k−2 − 1) 2 (x 2k−3 + 1) (x− 1) 2 (x + 1) ∈ Z[x]. Proof. Define U (x) := x 2k−2 − 1 x− 1 = 2k−3 X i=0 x i , V (x) := x 2k−3 + 1 x + 1 = 2k−4 X r=0 (−1) r x r . Since x 2k−2 − 1 is divisible by x− 1 and 2k− 3 is odd (hence x 2k−3 + 1 is divisible by x + 1), the rational expression H ⋆ k (x) := (x 2k−2 − 1) 2 (x 2k−3 + 1) (x− 1) 2 (x + 1) lies in Z[x]. Moreover, H ⋆ k (x) = U (x) 2 V (x). Set W (x) := U (x)V (x). We compute W (x) explicitly: W (x) = (x 2k−2 − 1)(x 2k−3 + 1) (x− 1)(x + 1) = (x 2k−2 − 1)(x 2k−3 + 1) x 2 − 1 . 65 CayleyPy-4: HolographyCayleyPy collaboration Using x 2k−2 − 1 x 2 − 1 = 1 + x 2 + x 4 +· + x 2k−4 = k−2 X j=0 x 2j , we obtain W (x) = (1 + x 2k−3 ) k−2 X j=0 x 2j = k−2 X j=0 x 2j + k−2 X j=0 x 2k−3+2j . Therefore H ⋆ k (x) = U (x)W (x) = 2k−3 X i=0 x i k−2 X j=0 x 2j + k−2 X j=0 x 2k−3+2j . Let S :=0, 2, 4,..., 2k− 4 ∪ 2k− 3, 2k− 1,..., 4k− 7. Then the coefficient of x m in W (x) equals 1 if m∈ S and 0 otherwise. Since the coefficient of x i in U (x) is 1 for 0≤ i≤ 2k− 3 (and 0 otherwise), the coefficient of x n in H ⋆ k (x) = U (x)W (x) is c k,n = #m∈ S : n− (2k− 3)≤ m≤ n = # S∩ [n− (2k− 3), n ] . We show that c k,n = a k,n for every 0≤ n≤ 6k− 10. This proves H ⋆ k (x) = H k (x). Range I: 0≤ n≤ 2k− 5. Here n < 2k− 3, so S∩ [n− (2k− 3), n ] contains no element from the odd block2k− 3, 2k− 1,..., and it contains precisely the even integers in [0,n]. Thus c k,n = #0, 2, 4,·≤ n = j n 2 k + 1 = a k,n . Range I: 2k− 4 ≤ n ≤ 4k− 6. Put L := n− (2k− 3) (so−1 ≤ L ≤ 2k− 3 in this range). Split S = E∪ O with E :=0, 2,..., 2k− 4, O :=2k− 3, 2k− 1,..., 4k− 7. We count E∩ [L,n] and O∩ [L,n] separately. First, since n≥ 2k− 4, the interval [L,n] contains all evens up to 2k− 4 except those < L. The set E has exactly k− 1 elements. The number of even integers < L (and≥ 0) equals L + 1 2 for every integer L≥−1. Hence #(E∩ [L,n]) = (k− 1)− L + 1 2 . Second, since L ≤ 2k − 3, the interval [L,n] meets the odd block O starting at 2k − 3, and it contains precisely those odds between 2k− 3 and n. The number of such odds is #(O∩ [L,n]) = n− (2k− 3) 2 + 1 = L 2 + 1. Adding, c k,n = (k− 1)− L + 1 2 + L 2 + 1 = k + L 2 − L + 1 2 . Now L 2 − L+1 2 = 0 if L is even and equals−1 if L is odd. Thus c k,n = k if L is even and c k,n = k− 1 if L is odd. Since L = n− (2k− 3) and 2k− 3 is odd, the parity condition “L even” is equivalent to “n odd”. Therefore c k,n = k− 1 + n mod 2 = k− 1 + (n− (2k− 4)) mod 2 = a k,n . Range I: 4k− 5 ≤ n ≤ 6k− 10. Again put L = n− (2k− 3). Then 2k− 2 ≤ L ≤ 4k− 7. The interval [L,n] lies entirely above 2k− 2, so it contains no element of E =0, 2,..., 2k− 4. 66 CayleyPy-4: HolographyCayleyPy collaboration Also, since n ≥ 4k− 5 > 4k− 7, intersecting with O truncates at the maximal element 4k− 7. Hence c k,n = #t∈ Z : L≤ t≤ 4k− 7, t odd. The odd integers in that interval form an arithmetic progression of step 2, ending at 4k− 7, so c k,n = (2k− 3)− L 2 = 2k− 3− n− (2k− 3) 2 . Write n = (4k− 5) +t with 0≤ t≤ 2k− 5. Then n− (2k− 3) = 2k− 2 +t with 2k− 2 even, so n− (2k− 3) 2 = 2k− 2 + t 2 = (k− 1) + t 2 . Substituting, c k,n = 2k− 3− (k− 1) + t 2 = k− 2− t 2 = k− 2− n− (4k− 5) 2 = a k,n . We have shown c k,n = a k,n for every 0≤ n≤ 6k−10, hence H ⋆ k (x) = H k (x), as claimed. □ Case k = 3 from n = 4 n≡ 0 (mod 4) : D 3 (n) = n 2 8 , n≡ 1 (mod 4) : D 3 (n) = n 2 − 1 8 , n≡ 2 (mod 4) : D 3 (n) = n 2 + 4 8 , n≡ 3 (mod 4) : D 3 (n) = n 2 − 1 8 . OGF = x 10 + x 9 + 2x 8 + 3x 7 + 2x 6 + 3x 5 + 2x 4 + x 3 + x 2 (1− x 4 ) 3 . Nonnegative Symmetric Unimodal Weakly log concave TrueFalseFalseFalse 67 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 39. Consecutive 3-cycles +inv [0..01..1] Case k = 4 from n = 11 n≡ 0 (mod 6) :D 4 (n) = n 2 12 , n≡ 1, 5 (mod 6) : D 4 (n) = n 2 − 1 12 , n≡ 2, 4 (mod 6) : D 4 (n) = n 2 + 8 12 , n≡ 3 (mod 6) :D 4 (n) = n 2 + 3 12 . OGF = H(x) (1− x 6 ) 3 , where H(x) = x 16 + x 15 + 2x 14 + 2x 13 + 3x 12 + 4x 11 + 3x 10 + 4x 9 + 3x 8 + 4x 7 + 3x 6 + 2x 5 + 2x 4 + x 3 + x 2 . Nonnegative Symmetric Unimodal Weakly log concave TrueFalseFalseFalse 68 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 40. Consecutive 4-cycles +inv [0..01..1] Case k = 5 from n = 10 n≡ 0 (mod 8) : D 5 (n) = n 2 16 , n≡ 1, 7 (mod 8) : D 5 (n) = n 2 − 1 16 , n≡ 2, 6 (mod 8) : D 5 (n) = n 2 + 12 16 , n≡ 3, 5 (mod 8) : D 5 (n) = n 2 + 7 16 , n≡ 4 (mod 8) : D 5 (n) = n 2 + 16 16 . OGF = H(x) (1− x 8 ) 3 , where H(x) = x 22 + x 21 + 2x 20 + 2x 19 + 3x 18 + 3x 17 + 4x 16 + 5x 15 + 4x 14 + 5x 13 + 4x 12 + 5x 11 + 4x 10 + 5x 9 + 4x 8 + 3x 7 + 3x 6 + 2x 5 + 2x 4 + x 3 + x 2 . 69 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 41. Consecutive 5-cycles +inv [0..01..1] Nonnegative Symmetric Unimodal Weakly log concave TrueFalseFalseFalse 70 CayleyPy-4: HolographyCayleyPy collaboration Case k = 6, inverse-closed (empirical fit from n≥ 12) n≡ 0 (mod 10) : D 6 (n) = n 2 20 , n≡ 1 (mod 10) : D 6 (n) = n 2 − 1 20 , n≡ 2 (mod 10) : D 6 (n) = n 2 + 16 20 , n≡ 3 (mod 10) : D 6 (n) = n 2 + 11 20 , n≡ 4 (mod 10) : D 6 (n) = n 2 + 24 20 , n≡ 5 (mod 10) : D 6 (n) = n 2 + 15 20 , n≡ 6 (mod 10) : D 6 (n) = n 2 + 24 20 , n≡ 7 (mod 10) : D 6 (n) = n 2 + 11 20 , n≡ 8 (mod 10) : D 6 (n) = n 2 + 16 20 , n≡ 9 (mod 10) : D 6 (n) = n 2 − 1 20 . OGF = H(x) (1− x 10 ) 3 , where H(x) = x 28 + x 27 + 2x 26 + 2x 25 + 3x 24 + 3x 23 + 4x 22 + 4x 21 + 5x 20 + 6x 19 + 5x 18 + 6x 17 + 5x 16 + 6x 15 + 5x 14 + 6x 13 + 5x 12 + 6x 11 + 5x 10 + 4x 9 + 4x 8 + 3x 7 + 3x 6 + 2x 5 + 2x 4 + x 3 + x 2 . Nonnegative Symmetric Unimodal Weakly log concave TrueFalseFalseFalse 71 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 42. Consecutive 6-cycles +inv [0..01..1] 72 CayleyPy-4: HolographyCayleyPy collaboration nk=2k=3k=4k=5 d ∆d ∆ 2 d d ∆d ∆ 2 d d ∆d ∆ 2 d d ∆d ∆ 2 d 4421211210 563032-131-1210 6931511402310 7124062042-141-1 81641821611502 92050103-172-152-1 1025511321911711 1130601530 102082-1 1236611831 1221 1011 134270214-1 143-1 112-1 1449712531 1721 1311 1556802840 193-1 1420 1664813241 2221 1621 177290365-1 2430 183-1 1881914141 2731 2121 19901004550 304-1 233-1 20 1001015051 3431 2621 21 110110556-1 374-1 283-1 22 1211116151 4131 3121 23 1321206660 4440 3330 24 1441217261 4841 3631 25 156130787-1 525-1 394-1 26 1691318561 5741 4331 27 1821409170 615-1 464-1 28 1961419871 6641 5031 29 210151058705534-1 30 225113755731 316040 32644 3368 TABLE 2. Diameters D k (n) and increments, consecutive k-cycle, inverse closed 73 CayleyPy-4: HolographyCayleyPy collaboration 8.8. Schreier coset graph: S n / S l × S n−l (“k-shrunken” Grassmannian Gr(l,n,k)). Here we present conjectural formula for the diameters of Schreier coset graphs of the form S n / S l × S n−l . The graph can be equivalently described as follows: its vertices correspond to binary vectors with components in0, 1, containing exactly l zeros and n− l ones. In CayleyPy, we define these graphs by setting the central state to be [0] l +[1] n−l . The generating set consists of consecutive k-cycles. We discuss both cases inverse-closed and not in the present section. The leading term of the formulas are l(n−l)/(k−1) as it is expected from the (k−1)-shrinkage principle, since for the standard k = 2 (Coxeter or neighbor transposition) case the diameter is exactly l(n− l). Informally, one may think of this graph as a “k-shrunken” version of Grassmannian Gr(l,n,k) over the field with one element, since for k = 2 by standard analogies it is indeed Grassmanian over field with one element. From the point of view of that analogy diameter corresponds to dimension of the manifold, and Poincare polynomial correspond to growth polynomial of the graph. Striking new phenomena - bi-variable quasi-polynomiality of the diameter formulas, i.e. formulas behave as quasi-polynomials in both variables n and n− l. It is expected to be true in larger generality e.g. for more general vector with repeats (”partial flag manifolds”) and more general families of generators. Conjecture 15. (Not inverse closed case). For consecutive k-cycle generators not inverse closed, the diameters of the Schreier coset graph: S n / S l ×S n−l (“k-shrunken” GrassmannianGr(l,n,k)) are given by quasi-polynomials in two variables n and t = n− (l + 1). For k = 3 with period 2: D l (n) = l t 2 + 1 + (t mod 2), t = n− (l + 1), n≥ l + 1. Under the condition: l≥ 3. For l = 2, the formula is: D 2 (n) = n− 1, n≥ 4. For k = 4 with period 3 (again t = n− (l + 1)): D l (n) =                          l,t = 0, l + 1,t = 1, l + 2,t = 2, l 2 + t− 3 3 , t≥ 3, t≡ 0, 1 (mod 3), l 2 + t− 3 3 + 1, t≥ 3, t≡ 2 (mod 3). For k = 5 with period 4 (again t = n − (l + 1), for all L ≥ 4, starting from t + 1 ≥ 9that is, N ≥ L + 9): D(L,N ) = L t + 1 4 + 1 (t+1)≡0 (mod 4) . Conjecture 16. (Inverse closed case). Same setup as above , but inverse closed case. For k = 3 from L≥ 2(?): D(L,N ) = L(N − L) 2 For k = 4 from L≥ 2: D(L,N ) =      L 3 (N − L),3| L, j L(N−L)+2 3 k , 3 ∤ L. 74 CayleyPy-4: HolographyCayleyPy collaboration For k = 5 for N,L,N − L large enough: D(L,N ) = LN + 2 4 − 2 L 2 8 − 1 L≡2 (mod 4) 1 N≡2 (mod 4) 75 CayleyPy-4: HolographyCayleyPy collaboration 8.9. Schreier coset graph: ”few coincide”. Here we present some partial results on diameters of the Schreier coset graphs of the form S n /S D which can be equivalently described as graphs with nodes corresponding to vectors where D elements coincide. In CayleyPy we define them by setting ”centralstate” to be (0, 1, 2,...n− D− 1,n− D,n− D,...,n− D) (D coincide at the end). The generators are the same - consecutive k cycles, as in the everywhere in this section. 8.9.1. Inverse closed case. Conjecture 17. For k = 3 and coincide 4 the next quasi polynomial formulas are correct from n = 5: n≡ 0, 1 (mod 4) : D 3 (n) = n 2 − n− 12 4 , n≡ 2, 3 (mod 4) : D 3 (n) = n 2 − n− 10 4 , For k = 3 and coincide 3 the next quasi polynomial formulas are correct from n = 2: n≡ 0, 1 (mod 4) : D 3 (n) = n 2 − n− 4 4 , n≡ 2, 3 (mod 4) : D 3 (n) = n 2 − n− 6 4 , For k = 3 and coincide 3 D: OGF = H(x) (1− x 4 ) 3 , where H(x) =−x 12 − x 11 + 2x 9 + 7x 8 + 9x 7 + 9x 6 + 7x 5 + 2x 4 − x 2 − x. For k = 3 and coincide 4 D: OGF = H(x) (1− x 4 ) 3 , where H(x) =−3x 12 − 2x 11 − x 10 + 11x 8 + 11x 7 + 11x 6 + 11x 5 − x 3 − 2x 2 − 3x For the case k = 4 and the consecutive cycles with inverses coset coincide, God’s number D 4 (n) for different coincidence conditions D is described by the following quasipolynomials: For coincide 3D (from n = 7): D 4 (n) = n(n− 1) 6 − 1 For coincide 4D (from n = 8): D 4 (n) = n(n− 1) 6 − 2 For k = 4 and coincide 3 D: OGF = H(x) (1− x 3 ) 3 , where H(x) =−x 9 + 4x 6 + 3x 5 + 4x 4 − x For k = 4 and coincide 4 D: OGF = H(x) (1− x 3 ) 3 , where H(x) =−2x 9 − x 8 − x 7 + 6x 6 + 5x 5 + 6x 4 − x 3 − x 2 − 2x 76 CayleyPy-4: HolographyCayleyPy collaboration where n is the length of the graph. FIGURE 43. Consecutive k=3 +inv (coincide 3) FIGURE 44. Consecutive k=3 +inv (coincide 4) 77 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 45. Consecutive k=4 +inv (coincide 3) FIGURE 46. Consecutive k=4 +inv (coincide 4) 78 CayleyPy-4: HolographyCayleyPy collaboration TABLE 3. God’s numbers for consecutive cycles with inverses coset: central state (0, 1, 2,...n− D− 1,n− D,n− D,...,n− D) (D coincide at the end) knno coincide D coincide 2 D coincide 3 D coincide 4 D 34332– 355542 367765 37101098 3814141311 3918181715 3 1022222120 3 1127272625 3 1233333230 457532 467554 478766 4810998 4913121110 4 1016151413 4 1119181817 4 1223222120 4 132524 569974 578875 588877 599998 5 1011111110 5 1114141313 5 1217171615 67131198 68121097 69121098 6 101210109 6 1113121110 6 1214131312 79 CayleyPy-4: HolographyCayleyPy collaboration 8.9.2. Not inverse closed case. TABLE 4. God’s numbers for consecutive cycles without inverses coset: central state (0, 1, 2,...,n− D− 1,n− D,n− D,...,n− D) (D coincide at the end) knno c. D D = 2 D = 3 D = 4 D = 5 D = 6 2332– 24653– 2510974– 2615141295– 2721201815116 28282725221813 29363533302621 2 10454442393530 2 11555452494540 2 12666563605651 2 13787775726863 2 14????88858176 34443– 356653– 3699864– 37121211974 3816161513118 39202019171512 3 10252524222017 3 11303029272522 3 12363635333128 3 13424241393734 3 14????48464441 459854– 4697654– 471098764 481312111087 4916151413119 4 10191817161412 4 11232221201816 4 12272625242220 4 13313029282624 4 14??????333129 561111965– 5710109765 58111110876 591313121098 5 10151514131210 5 11181817161513 5 12212120191816 5 13242423222119 5 14??????262523 671916131176 Continued on next page 80 CayleyPy-4: HolographyCayleyPy collaboration knno c. D D = 2 D = 3 D = 4 D = 5 D = 6 681413111087 691413111098 6 1015131211109 6 11171514131211 6 12191817161514 6 13222120191817 6 14??????222120 81 CayleyPy-4: HolographyCayleyPy collaboration 8.10. Schreier coset graph: ”L-Different” inverse closed. Here we present some partial results on diameters of the Schreier coset graphs of the form S n /S n−L which can be equivalently de- scribed as graphs with nodes corresponding to vectors where only L elements are different. In CayleyPy we define them by setting the central state to (0, 1, 2,...,L− 2,L− 1,...,L− 1) = 0 1 1 1 2 1 ... (L− 2) 1 (L− 1) n−L−1 . The generators are the same — consecutive k cycles, as everywhere in this section. The next two conjectures for 2- and 3-Different coset graphs are based on the data obtained in this Kaggle notebook. 8.10.1. 2-Different coset. Conjecture 18. The diameter for k-cycles 2-Different coset graph with central state of length n is d k (n) = j 2n + k(k− 4) 2k− 2 k + 1− I[k is odd, 2⩽ n mod (k− 1)⩽ (k− 1)/2] 8.10.2. 3-Different. Conjecture 19. We conjecture that the diameter for k-cycles 3-Different coset graph with central state of length n equals to d k (n) = 2n + c k k− 1 , k ⩾ 2, n⩾ N k ,(19) where c 2 = −3, c 3 = −1, and for k ⩾ 4 the offset c k is a quadratic function of k whose form depends on the parity of k: c k =      k(k− 1) 2 − 5 if k is even, (k− 1) 2 2 − 4if k is odd. (20) The indices N k , from which the formula (19) starts working, and c k are tabulated in Table 5. The values N k also eventually seem to obey the quadratic law: N k =        k 2 − 9k + 14 2 , k is even, (k− 3) 2 2 ,k is odd. Our calculations for L = 4 show that d k (n) ≈ j 3n k−1 + k−5 2 k if k ⩾ 5. Combining this observation with two previous conjectures, we propose the following: Conjecture 20. The diameter for k-cycles L-Different coset graph with central state of length n is d k (n) = j Ln 2(k− 1) + c k k + r k (n), where c k = O(k), r k (n) is a ”small” remainder (possibly O(1)). 8.10.3. Last layer size (3-Different). Let s k (n) denote the size of the last layer of the k-cycle 3-Different inverse-closed Schreier coset graph with central state of length n. We conjecture that, for fixed k, the sequence s k (n) is eventually periodic in n, with behavior depending on the parity of k. Conjecture 21 (Even k). For even k ≥ 6, we conjecture that, for all sufficiently large n, the sequence s k (n) follows a single periodic pattern of odd values. Let C k = k(k− 1) 2 − 5, m = (2n + C k ) mod (k− 1). 82 CayleyPy-4: HolographyCayleyPy collaboration TABLE 5. 3-Different diameter conjectures for 2⩽ k ⩽ 20 kN k Offset (c k )d k (n) 23−32n− 3 34−1n− 1 451 2n+1 3 564 2n+4 4 6810 2n+10 5 7814 2n+14 6 81023 2n+23 7 91828 2n+28 8 101340 2n+40 9 113246 2n+46 10 122561 2n+61 11 135068 2n+68 12 144286 2n+86 13 157294 2n+94 14 1663115 2n+115 15 1798124 2n+124 16 1888148 2n+148 17 19128158 2n+158 18 20117185 2n+185 19 Then s k (n) = m + 1− (m mod 2) + 1 |m− (k− 2)| + 1 + 2 1 |m− (k− 3)| + 1 . Conjecture 22 (Odd k). For odd k ≥ 7, we conjecture that, for all sufficiently large n, the sequence s k (n) follows a two-half alternating periodic pattern, where each half has length H = k− 1 2 . Let C k = (k− 1) 2 2 − 4, m = n + C k 2 + H mod (k− 1). Define the two correction locations by (m origin ,m spike ) = ( (H, 2H − 1), k ≡ 1 (mod 4), (0, H − 1), k ≡ 3 (mod 4). Then s k (n) = m + 1− (m mod 2) + 1 |m− m origin | + 1 + 2 1 |m− m spike | + 1 . 83 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 47. Last layer size for k=6...10. Empirical data coincides with the pro- posed formula for large enough n. 84 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 48. Last layer size for k=16...20. Empirical data coincides with the proposed formula for large enough n. 85 CayleyPy-4: HolographyCayleyPy collaboration 8.11. Word-metrics for [012]-repeated full flips, 3-cycles. Here we consider consecutive 3 cy- cles generators. The Schreir coset graph of the form S n /(S n//3 × S n//3 × S n−2n//3 ), which alternatively can be described as graph with nodes given by taking all vectors with only 0, 1, 2 components where 0 is repeated n//3 times, 1 is repeated n//3 times, and the rest is 2. In Cay- leyPy such coset Schreier graph is defined by setting ”centralstate” appropriately. Here we present a conjectural quasi-polynomial expression for the word-metric between two families vertices of such graphs. Let us denote by r residue of n modulo 3 and by m its integer division by 3: n = 3m + rwith r ∈0, 1, 2. Consider the first family of states given by: [0, 1, 2] m if r = 0 [0, 1, 2] m + [0] if r = 1 [0, 1, 2] m + [0, 1] if r = 2 And the other states: [2] m + [1] m + [0] m if r = 0 [2] m + [1] m + [0] m+1 if r = 1 [2] m + [1] m+1 + [0] m+1 if r = 2 (21) Conjecture 23.(1) The second states are farthest states from the first ones (2) Starting from n = 5 the distance between them is given by the following degree two quasipolynomial n≡ 0 (mod 12) : D(n) = 1 12 n 2 + 1 4 n, n≡ 1 (mod 12) : D(n) = 1 12 n 2 + 1 12 n− 1 6 , n≡ 2 (mod 12) : D(n) = 1 12 n 2 + 1 12 n + 1 2 , n≡ 3 (mod 12) : D(n) = 1 12 n 2 + 1 4 n + 1 2 , n≡ 4 (mod 12) : D(n) = 1 12 n 2 + 1 12 n + 1 3 , n≡ 5 (mod 12) : D(n) = 1 12 n 2 + 1 12 n + 1 2 , n≡ 6 (mod 12) : D(n) = 1 12 n 2 + 1 4 n + 1 2 , n≡ 7 (mod 12) : D(n) = 1 12 n 2 + 1 12 n + 1 3 , n≡ 8 (mod 12) : D(n) = 1 12 n 2 + 1 12 n, n≡ 9 (mod 12) : D(n) = 1 12 n 2 + 1 4 n, n≡ 10 (mod 12) : D(n) = 1 12 n 2 + 1 12 n− 1 6 , n≡ 11 (mod 12) : D(n) = 1 12 n 2 + 1 12 n, 86 CayleyPy-4: HolographyCayleyPy collaboration of period 12. Note that the leading term is always n 2 /12 and the period appears as its denominator. Also note that we have D(0) = D(1) = 0. (3) The corresponding generating function is given by H(x) (1− x 12 ) 3 ,(22) where H(x) = x 32 + 2x 31 + 2x 30 + 4x 29 + 5x 28 + 5x 27 + 8x 26 + 9x 25 + 9x 24 + 13x 23 + 15x 22 + 15x 21 + 17x 20 + 17x 19 + 17x 18 + 17x 17 + 17x 16 + 17x 15 + 15x 14 + 15x 13 + 15x 12 + 11x 11 + 9x 10 + 9x 9 + 6x 8 + 5x 7 + 5x 6 + 3x 5 + 2x 4 + 2x 3 + x 2 The polynomial H(x) has the following properties: • its coefficients are nonnegative and unimodal; • its coefficients are neither symmetric nor weakly log concave; • its roots are not necessarily on the unit circle (can be both inside and outside). (4) The H -polynomial factors as H(x) = x 2 (x + 1) 3 (x 2 + x + 1)(x 2 + 1) 2 (x 2 − x + 1) 3 (x 4 − x 2 + 1) 2 (x 7 + x 6 − x 3 + x + 1) Hence, the generating function (22) can be rewritten as x 2 (x 7 + x 6 − x 3 + x + 1) (1− x) 3 (x 2 + 1)(x 2 + x + 1) 2 (x 4 − x 2 + 1) , The conjecture is checked up to n = 42. The following data was obtained on Kaggle using CayleyPy in this notebook. It is a dictionary of value pairs n : v, where n is as above and v is the (experimentally) shortest path from the initial state to the central state. 4 4 : 3, 5 : 3, 6 : 5, 7 : 5, 8 : 6, 9 : 9, 10 : 9, 11 : 11, 12 : 15, 13 : 15, 14 : 18, 15 : 23, 16 : 23, 17 : 26, 18 : 32, 19 : 32, 20 : 35, 21 : 42, 22 : 42, 23 : 46, 24 : 54, 25 : 54, 26 : 59, 27 : 68, 28 : 68, 29 : 73, 30 : 83, 31 : 83, 32 : 88, 33 : 99, 34 : 99, 35 : 105, 36 : 117, 37 : 117, 38 : 124, 39 : 137, 42 : 158 The computations regarding the properties of the H -polynomial are done in this notebook. 4 There might be some issue with the value for n=4, but we don’t need it below anyway. 87 CayleyPy-4: HolographyCayleyPy collaboration 9. WRAPPED (”AFFINE” OR ”PERIODIC”) (K)-CONSECUTIVE CYCLES. QUASI-POLYNOMIALITY, ETC. 9.1. Section outline. Quasi-polynomiality. In this section, we present several quasi-polynomial formulas for the diameters and word metrics associated with the ”wrapped” version of the consec- utive k-cycle generators (also can be called ”affine” or ”cyclic” or ”periodic”) defined below. We also compute the corresponding H -polynomials and study their properties. Both Cayley graphs and Schreier coset graphs are considered. We also provide some other results like theoretical lower and upper bounds on diameters which are in agreement with our experimental studies. The section is quite parallel to the previous one devoted to consecutive cycles case. Generators definition. Related works. Fix an integer k. For n > k, we consider elements of S n given by the cyclic permutations (i, (i+1) mod n,..., (i+k−1) mod n) for i = 0,...,n− 1. For k = 2, these are the neighbor transpositions (cyclic Coxeter) considered previously. In CayleyPy these generators are denoted: wrapped kcycles. For odd k, they generate A n inside S n , for even k they give S n . There are two natural options: whether or not to include inverses in the generating set. We consider both cases. To the best of our knowledge, these generators were not studied systematically in the literature, brief discussion is in our previous work: [Chervov2025b]). From the physical point of view, the Laplacian of such graphs corresponds to the situation when k neighboring spins interact. The simplest case k = 2 corresponds to ”periodic” or ”closed” or ”affine” spin chain. 9.2. 2(k − 1)-Shrinkage heuristics: Results and Difficulties. Before going into details let us first give some informal heuristic principle summarizing the results: to obtain results on diame- ters, word-metrics and other characteristics one should take results for the standard Coxeter (i.e. neighbor transpositions case) case and just divide them by 2(k− 1). In all considered cases it gives correct leading terms and moreover in some rare cases simple correction like adding ceil-rounding would suffice to get the correct results. However exact results typically are more complicated. 9.3. Theoretical diameters estimate. Schreier coset graphs S n / S ⌊n/2⌋ × S n−⌊n/2⌋ . Theorem 10. The diameter of the coset with⌊n/2⌋ zeros and n−⌊n/2⌋ ones is equal to n 2 8(k−1) + O(n). Proof. Since the k-cycle is wrapped, one can consider a modified setup where we are to obtain any permutation from the one where the zeros are located at the first ⌊n/4⌋ points and the last ⌊n/2⌋−⌊n/4⌋ points. Suppose that we want to achieve the sequence of zeros and ones where the zeros are located at places a 1 ,a 2 ,...,a l . Create a modified sequence where the zeros are located at (k− 1)⌊a 1 /(k− 1)⌋,..., (k− 1)⌊a 1 /(k− 1)⌋ + k− 2, (k− 1)⌊a k /(k− 1)⌋,..., (k− 1)⌊a k /(k− 1)⌋ + k− 2,..., (k− 1)⌊a m(k−1)+1 /(k− 1)⌋,..., (k− 1)⌊a m(k−1)+1 /(k− 1)⌋,... This modified sequence has the zeros and ones come in groups of k− 1. Since a group of size k− 1 or less can be moved to a distance of 1 to the right by 1 turn of the cycle, one can exchange two such groups in k − 1 moves. This time the modified position is reachable in at most (k − 1)⌈n/4/(k− 1)⌉(⌊n/2/(k− 1)⌋ + 1) moves, because one can choose whether any group will be delivered from the left or from the right. Next we reconstruct the original sequence by guiding every zero to its rightful place in O(n) turns.□ 88 CayleyPy-4: HolographyCayleyPy collaboration 9.4. Schreier coset graph: S n / S l ×S n−l (“k-shrunken” affine GrassmannianGr aff (l,n,k)). Here we present conjectural formula for the diameters of Schreier coset graphs of the formS n / S l × S n−l . The graph can be equivalently described as follows: its vertices correspond to binary vec- tors with components in 0, 1, containing exactly l zeros and n− l ones. In CayleyPy, we define these graphs by setting the central state to be [0] l + [1] n−l . The generating set con- sists of wrapped consecutive k-cycles. We discuss both cases inverse-closed and not in the present section. The leading term of the formulas are l(n− l)/2(k − 1) as it is expected from the 2(k − 1)- shrinkage principle, since for the standard k = 2 (Coxeter or neighbor transposition) case the diameter is exactly l(n− l). Informally, one may think of this graph as a “k-shrunken” version of affine Grassmannian Gr aff (l,n,k) over the field with one element, since for k = 2 by standard analogies it is indeed Grassmanian over field with one element. From the point of view of that analogy diameter corre- sponds to dimension of the manifold, and Poincare polynomial correspond to growth polynomial of the graph. Striking new phenomena - bi-variable quasi-polynomiality of the diameter formulas, i.e. formulas behave as quasi-polynomials in both variables n and n− l. It is expected to be true in larger generality e.g. for more general vector with repeats (”partial flag manifolds”) and more general families of generators. Conjecture 24. For k = 3, inverse closed, diameters are given by: d n = ℓ(n− ℓ) 4 + [n≡ 0 (mod 4), ℓ≡ 2 (mod 4)] FIGURE 49. Difference to naive estimation⌈l(n−l)/2(k− 1)⌉ wrapped 4 cycles inverse closed. Bi-variable quasi-polynomials pattern. 89 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 50. Difference to naive estimation⌈l(n−l)/2(k− 1)⌉ wrapped 3 cycles inverse closed. Bi-variable quasi-polynomials pattern. FIGURE 51. Difference to naive estimation⌈l(n−l)/2(k− 1)⌉ wrapped 5 cycles inverse closed. Bi-variable quasi-polynomials pattern. 90 CayleyPy-4: HolographyCayleyPy collaboration 9.5. Schreier coset graph: S n / S ⌊n/2⌋ ×S n−⌊n/2⌋ (not inverse-closed). Here we present a con- jectural formula for the diameters of Schreier coset graphs of the form S n S ⌊n/2⌋ × S n−⌊n/2⌋ . The graph can be equivalently described as follows: its vertices correspond to binary vectors with components in 0, 1, containing exactly ⌊n/2⌋ zeros and n−⌊n/2⌋ ones. In CayleyPy, we define these graphs by setting the centralstate to be [0] n//2 + [1] n−n//2 . The generating set consists of wrappped consecutive k-cycles; we consider the case where the generating set is not inverse-closed. The leading term of the diameter formulas are n 2 /(8(k− 1)) as it is expected from the 2(k− 1)- shrinkage principle, since for the standard k = 2 (Coxeter or neighbor transposition) case the diameter is exactly⌊n/2⌋(n−⌊n/2⌋). Informally, one may think of this graph as a “k-shrunken” version of affine Grassmannian Gr aff (⌊n/2⌋,n) over the field with one element. From the point of view of that analogy diam- eter corresponds to dimension of the manifold, and Poincare polynomial correspond to growth polynomial of the graph. We experimentally have found the coefficients and we conjecture that the formula is the follow- ing polynomial H k (x) = (x 4k−4 − 1) 2 (x 2k−1 + 1) (x− 1) 2 (x + 1) ∈ Z[x]. based on the following proposition. Proposition 11 (Closed form for H k ). Let k ≥ 2 and m := 3(k − 1). Define integers h i for 0≤ i≤ 10(k− 1) by h i =                                        1,i = 0, 1, 2 + i− 2 2 ,2≤ i≤ 2k− 3, i− (k− 2),2k− 2≤ i≤ 4k− 5, (m− 1) + (i− (4k− 4)) mod 2 , 4k− 4≤ i≤ 6k− 6, (m− 2)− (i− (6k− 5)),6k− 5≤ i≤ 8k− 10, (k− 1)− i− (8k− 9) 2 ,8k− 9≤ i≤ 10k− 14, 1,i = 10k− 13, 10k− 12, 0,i = 10k− 11, 10k− 10, where “mod2” denotes the remainder in0, 1. Let H k (x) := 10(k−1) X i=0 h i x i ∈ Z[x]. Then H k (x) = (x 4k−4 − 1) 2 (x 2k−1 + 1) (x− 1) 2 (x + 1) ∈ Z[x]. Before a proof we need two lemmas. Lemma 1 (Counting even/odd integers in an interval). Let A,B ∈ Z with A≤ B. Then #t∈ Z : A≤ t≤ B, t even = B 2 − A− 1 2 , and #t∈ Z : A≤ t≤ B, t odd = B + 1 2 − A 2 . 91 CayleyPy-4: HolographyCayleyPy collaboration Moreover, for every n∈ Z, j n 2 k + n− 1 2 = n− 1. Proof. For the even-count, the map t 7→ t/2 is a bijection between even integers t ∈ [A,B] and integers u ∈ [⌈A/2⌉, ⌊B/2⌋], hence the count equals ⌊B/2⌋ − ⌈A/2⌉ + 1. Using ⌈A/2⌉ = ⌊(A + 1)/2⌋ =⌊(A− 1)/2⌋ + 1, we obtain the stated formula. For the odd-count, odd t∈ [A,B] correspond bijectively to even t− 1∈ [A− 1,B− 1], so the first formula gives #t∈ [A,B]∩ Z : t odd = B− 1 2 − A− 2 2 = B + 1 2 − A 2 . Finally, write n = 2q or n = 2q + 1. If n = 2q, then⌊n/2⌋ +⌊(n− 1)/2⌋ = q + (q− 1) = 2q− 1 = n− 1. If n = 2q + 1, then⌊n/2⌋ +⌊(n− 1)/2⌋ = q + q = 2q = n− 1.□ Lemma 2 (A floor identity used in Range I). Let k ≥ 2 and i∈ Z. Then i 2 + i− (2k− 1) 2 = i− k. Proof. Write i = 2q or i = 2q + 1. If i = 2q, then i− (2k− 1) = 2q− 2k + 1 is odd, so i 2 + i− (2k− 1) 2 = q + q− k + 1 2 = q + (q− k) = 2q− k = i− k. If i = 2q + 1, then i− (2k− 1) = 2q + 2− 2k = 2(q− k + 1) is even, so i 2 + i− (2k− 1) 2 = q + (q− k + 1) = 2q− k + 1 = i− k. □ Proof. Define U (x) := x 4k−4 − 1 x− 1 = 4k−5 X a=0 x a , V (x) := x 2k−1 + 1 x + 1 = x 2k−2 − x 2k−3 +·− x + 1. Since 2k− 1 is odd, x 2k−1 + 1 is divisible by x + 1, hence V (x)∈ Z[x]; clearly also U (x)∈ Z[x]. Therefore H ⋆ k (x) := (x 4k−4 − 1) 2 (x 2k−1 + 1) (x− 1) 2 (x + 1) = U (x) 2 V (x)∈ Z[x]. We prove that H ⋆ k (x) = H k (x) by comparing coefficients. Set W (x) := U (x)V (x). Then W (x) = (x 4k−4 − 1)(x 2k−1 + 1) (x− 1)(x + 1) = (x 4k−4 − 1)(x 2k−1 + 1) x 2 − 1 . Because 4k− 4 is even, x 4k−4 − 1 x 2 − 1 = 1 + x 2 + x 4 +· + x 4k−6 = 2k−3 X j=0 x 2j . Hence W (x) = (x 2k−1 + 1) 2k−3 X j=0 x 2j = 2k−3 X j=0 x 2j + 2k−3 X j=0 x 2k−1+2j . The first sum has only even exponents, the second only odd exponents; thus the supports are disjoint and every nonzero coefficient of W equals 1. 92 CayleyPy-4: HolographyCayleyPy collaboration Let S :=0, 2, 4,..., 4k− 6 ∪ 2k− 1, 2k + 1,..., 6k− 7. Then the coefficient of x t in W (x) is 1 if t∈ S and 0 otherwise. Since H ⋆ k (x) = U (x)W (x) = 4k−5 X a=0 x a W (x), the coefficient of x i in H ⋆ k (x) is c i = #t∈ S : i− (4k− 5)≤ t≤ i = # S∩ [i− (4k− 5), i ] . We show c i = h i for all 0≤ i≤ 10(k− 1), by splitting into the index ranges defining h i . Range 0: i = 0, 1. Here i− (4k − 5) ≤ 0 and i < 2, so S ∩ [i− (4k − 5),i ] = 0. Thus c i = 1 = h i . Range I: 2 ≤ i ≤ 2k− 3. Then i < 2k− 1, so the odd block 2k− 1, 2k + 1,... does not contribute. Also i− (4k− 5) < 0, hence all even elements of S up to i are counted: c i = #0, 2, 4,·≤ i = i 2 + 1 = 2 + i− 2 2 = h i . Range I: 2k − 2 ≤ i ≤ 4k − 5. Again i− (4k − 5) ≤ 0, hence c i counts all elements of S in [0,i]. The even contribution equals ⌊i/2⌋ + 1. The odd contribution counts the odd numbers 2k− 1, 2k + 1,·≤ i, which is j i−(2k−1) 2 k + 1 (this is 0 when i = 2k− 2). Thus c i = i 2 + 1 + i− (2k− 1) 2 + 1. By Lemma 2,⌊i/2⌋ +⌊(i− (2k− 1))/2⌋ = i− k, so c i = i− k + 2 = i− (k− 2) = h i . Range I: 4k− 4≤ i≤ 6k− 6. Write i = 4k− 4 + s with 0≤ s≤ 2k− 2. Then the interval becomes [i− (4k− 5),i ] = [ 1 + s, 4k− 4 + s ]. Since 4k− 4 + s≤ 6k− 6 < 6k− 7, the odd block contributes odds from 2k− 1 up to 4k− 4 +s, while the even block contributes evens from 1 + s up to 4k− 6: # 0, 2,..., 4k−6∩[1+s, 4k−4+s] = #t∈ Z : 1+s≤ t≤ 4k−6, t even = 2k−3− j s 2 k , # 2k−1, 2k+1,...∩[1+s, 4k−4+s] = #t∈ Z : 2k−1≤ t≤ 4k−4+s, t odd = 4k− 3 + s 2 −(k−1), where we used Lemma 1 in each line. Adding, c i = 2k− 3− j s 2 k + 4k− 3 + s 2 − (k− 1) = k− 2 + 4k− 3 + s 2 − j s 2 k . Since 4k− 3 is odd, writing s = 2r or s = 2r + 1 gives 4k− 3 + s 2 − j s 2 k = ( 2k− 2, s even, 2k− 1, s odd. Hence c i = 3k− 4 for even s and c i = 3k− 3 for odd s. Since m = 3k− 3 and s = i− (4k− 4), c i = (m− 1) + s mod 2 = (m− 1) + (i− (4k− 4)) mod 2 = h i . Range IV: 6k − 5 ≤ i ≤ 8k − 10. Write i = 6k − 5 + t with 0 ≤ t ≤ 2k − 5. Then [i− (4k− 5),i ] = [ 2k + t, 6k− 5 + t ]. Since 6k− 5 + t ≥ 6k− 5 > 6k− 7, the odd block 93 CayleyPy-4: HolographyCayleyPy collaboration contributes odd integers from the moving lower bound up to the fixed top 6k − 7, and the even block contributes even integers from the same lower bound up to the fixed top 4k− 6: #t ′ ∈ Z : 2k + t≤ t ′ ≤ 6k− 7, t ′ odd = (3k− 3)− 2k + t 2 , #t ′ ∈ Z : 2k + t≤ t ′ ≤ 4k− 6, t ′ even = (2k− 3)− 2k + t− 1 2 , again by Lemma 1. Adding and using Lemma 1 with n = 2k + t, c i = (5k− 6)− 2k + t 2 + 2k + t− 1 2 = (5k− 6)− (2k + t− 1) = 3k− 5− t. Since m− 2 = 3k− 5 and t = i− (6k− 5), this gives c i = (m− 2)− (i− (6k− 5)) = h i . Range V: 8k − 9 ≤ i ≤ 10k − 14. Write i = 8k − 9 + t with 0 ≤ t ≤ 2k − 5. Then [i− (4k− 5),i ] = [ 4k− 4 + t, 8k− 9 + t ]. Because 4k− 4 + t > 4k− 6, the even block contributes nothing. The odd block contributes odd integers from the moving lower bound up to the fixed top 6k− 7, hence c i = #t ′ ∈ Z : 4k− 4 + t≤ t ′ ≤ 6k− 7, t ′ odd = (3k− 3)− 4k− 4 + t 2 . Since 4k− 4 is even, 4k−4+t 2 = 2k− 2 + t 2 , so c i = (3k− 3)− 2k− 2 + t 2 = (k− 1)− t 2 = (k− 1)− i− (8k− 9) 2 = h i . Range VI: i = 10k− 13, 10k− 12. We show S∩ [i− (4k− 5),i ] = 6k− 7, hence c i = 1. For i = 10k− 13, the interval is [i− (4k− 5),i ] = [ 10k− 13− (4k− 5), 10k− 13 ] = [ 6k− 8, 10k− 13 ], which contains 6k − 7 and is strictly above 4k − 6, so it meets S only at the last odd element 6k− 7. For i = 10k− 12, the interval is [i− (4k− 5),i ] = [ 6k− 7, 10k− 12 ], again containing 6k− 7 and lying above 4k− 6. Thus c i = 1 = h i in both cases. Range VII: i = 10k− 11, 10k− 10. For i = 10k− 11, the interval is [ 6k− 6, 10k− 11 ]; for i = 10k− 10, it is [ 6k− 5, 10k− 10 ]. In both cases the lower bound is > 6k− 7, the maximum element of S, so the intersection is empty and c i = 0 = h i . Thus c i = h i for all i, hence H ⋆ k (x) = H k (x), proving the closed form.□ Corollary 12 (Cyclotomic factorization). For k ≥ 2, H k (x) = Y d|(4k−4) d>1 Φ d (x) 2 Y d|(2k−1) d>1 Φ 2d (x), hence every irreducible factor of H k is cyclotomic. Proof. Use x n − 1 = Q d|n Φ d (x) and, for odd n, x n + 1 = Q d|n Φ 2d (x). Also (x 4k−4 − 1) 2 /(x− 1) 2 removes the Φ 1 (x) 2 factor, and (x 2k−1 + 1)/(x + 1) removes the Φ 2 (x) factor.□ 94 CayleyPy-4: HolographyCayleyPy collaboration Theorem 13. Case k = 2 from n = 4 n≡ 0 (mod 4) : D 2 (n) = n 2 8 , n≡ 1 (mod 4) : D 2 (n) = n 2 − 1 8 , n≡ 2 (mod 4) : D 2 (n) = n 2 + 4 8 , n≡ 3 (mod 4) : D 2 (n) = n 2 − 1 8 . OGF = x 10 + x 9 + 2x 8 + 3x 7 + 2x 6 + 3x 5 + 2x 4 + x 3 + x 2 (1− x 4 ) 3 . Conjecture 25. Case k = 3 from n = 8 n≡ 0, 6 (mod 8) : D 3 (n) = n 2 + 2n 16 = n(n + 2) 16 , n≡ 1, 5 (mod 8) : D 3 (n) = n 2 + 2n− 3 16 , n≡ 2, 4 (mod 8) : D 3 (n) = n 2 + 2n + 8 16 , n≡ 3, 7 (mod 8) : D 3 (n) = n 2 + 2n + 1 16 = (n + 1) 2 16 . OGF = H(x) (1− x 8 ) 3 , where H(x) = x 20 + x 19 + 2x 18 + 2x 17 + 3x 16 + 4x 15 + 5x 14 + 6x 13 + 5x 12 + 6x 11 + 5x 10 + 6x 9 + 5x 8 + 4x 7 + 3x 6 + 2x 5 + 2x 4 + x 3 + x 2 . Case k = 4 from n = 12 n≡ 0, 8 (mod 12) : D 4 (n) = n(n + 4) 24 , n≡ 1, 7 (mod 12) : D 4 (n) = n 2 + 4n− 5 24 , n≡ 2, 6 (mod 12) : D 4 (n) = n 2 + 4n + 12 24 , n≡ 3, 9, 11 (mod 12) : D 4 (n) = n 2 + 4n + 3 24 , n≡ 4 (mod 12) :D 4 (n) = n 2 + 4n + 16 24 , n≡ 5 (mod 12) :D 4 (n) = n 2 + 4n + 3 24 , n≡ 10 (mod 12) :D 4 (n) = n 2 + 4n + 4 24 . OGF = H(x) (1− x 12 ) 3 , 95 CayleyPy-4: HolographyCayleyPy collaboration where H(x) = x 30 + x 29 + 2x 28 + 2x 27 + 3x 26 + 3x 25 + 4x 24 + 5x 23 + 6x 22 + 7x 21 + 8x 20 + 9x 19 + 8x 18 + 9x 17 + 8x 16 + 9x 15 + 8x 14 + 9x 13 + 8x 12 + 7x 11 + 6x 10 + 5x 9 + 4x 8 + 3x 7 + 3x 6 + 2x 5 + 2x 4 + x 3 + x 2 . wrapped 2-cycles [0..01..1]wrapped 3-cycles [0..01..1]wrapped 4-cycles [0..01..1] FIGURE 52. Zeros of H -polynomials for wrapped k-cycle Schreier graphs S n / S ⌊n/2⌋ × S n−⌊n/2⌋ (not inverse-closed) for k = 2, 3, 4. 96 CayleyPy-4: HolographyCayleyPy collaboration nk=2k=3k=4k=5k=6 d ∆d ∆ 2 d ∆d ∆ 2 d ∆d ∆ 2 d ∆d ∆ 2 d ∆d ∆ 2 d 3110 4211201 532-1210300 6511310301400 7620410310400500 8821511410401500 9103-162-1510410500 10 1321811610510501 11153092-1710610510 1218311111811710610 13 214-1122092-1810710 14253114201111910810 1528401620122-11010910 16 32411821141111111010 17365-1203-1152-1122-11110 1841412321171114111210 19 4550253-11820152-11310 20 50512821202017111411 21 556-130302220182-1152-1 22 61513330242020111711 23666036302620212-1182-1 2472613931282123112011 25 787-1424-1303-12420212-1 2685614631332126202311 279170494-1353-12820242-1 2898715331382130202611 291058-15640403-13220272-1 30113716040432134202911 31 120806440453036203020 32128816841483038213220 331369-1725-15130403-13420 341458774543432362 3515381574538 TABLE 6. Diameters D k (n) and their increments and second increments for wrapped k-cycle coset S n / S ⌊n/2⌋ × S n−⌊n/2⌋ (not inverse-closed). In that example second increments allow to see periodic structure, i.e.fit quasi- polynomials. 97 CayleyPy-4: HolographyCayleyPy collaboration 9.6. Schreier coset graph: S n / S ⌊n/2⌋ ×S n−⌊n/2⌋ (inverse-closed). The setup is almost iden- tical to the previous subsection, but now consider inverse closed generators. Conjecture 26. Case k = 3 from n = 3 n≡ 0 (mod 8) : D 3 (n) = n 2 16 , n≡ 1, 7 (mod 8) : D 3 (n) = n 2 − 1 16 , n≡ 2, 6 (mod 8) : D 3 (n) = n 2 + 12 16 , n≡ 3, 5 (mod 8) : D 3 (n) = n 2 + 7 16 , n≡ 4 (mod 8) : D 3 (n) = n 2 + 16 16 . OGF = H(x) (1− x 8 ) 3 , where H(x) = x 22 + x 21 + 2x 20 + 2x 19 + 3x 18 + 3x 17 + 4x 16 + 5x 15 + 4x 14 + 5x 13 + 4x 12 + 5x 11 + 4x 10 + 5x 9 + 4x 8 + 3x 7 + 3x 6 + 2x 5 + 2x 4 + x 3 + x 2 . Case k = 4 from n = 12 n≡ 0 (mod 12) : D 6 (n) = n 2 − 2n + 48 24 = (n− 1) 2 + 47 24 , n≡ 1, 11 (mod 12) : D 6 (n) = n 2 − 1 24 , n≡ 2, 10 (mod 12) : D 6 (n) = n 2 + 20 24 , n≡ 3, 9 (mod 12) : D 6 (n) = n 2 + 15 24 , n≡ 4, 8 (mod 12) : D 6 (n) = n 2 + 32 24 , n≡ 5, 7 (mod 12) : D 6 (n) = n 2 + 23 24 , n≡ 6 (mod 12) : D 6 (n) = n 2 + 36 24 . OGF = H(x) (1− x 12 ) 3 , where H(x) = x 34 + x 33 + 2x 32 + 2x 31 + 3x 30 + 3x 29 + 4x 28 + 4x 27 + 5x 26 + 5x 25 + 9x 24 + 7x 23 + 6x 22 + 7x 21 + 6x 20 + 7x 19 + 6x 18 + 7x 17 + 6x 16 + 7x 15 + 6x 14 + 7x 13 + x 12 + 5x 11 + 5x 10 + 4x 9 + 4x 8 + 3x 7 + 3x 6 + 2x 5 + 2x 4 + x 3 + x 2 + 2. 98 CayleyPy-4: HolographyCayleyPy collaboration FIGURE53. wrapped3- cycles inverse closed [0..01..1] FIGURE54. wrapped4- cycles inverse closed [0..01..1] 99 CayleyPy-4: HolographyCayleyPy collaboration nk=2 invk=3 invk=4 invk=5 invk=6 inv d ∆d ∆ 2 d ∆d ∆ 2 d ∆d ∆ 2 d ∆d ∆ 2 d ∆d ∆ 2 d 3110 4211201 532-121-121-1 6511301301301 762031031-131-131-1 8821411401401401 9103-152-141041-141-1 10 1321711510501501 11153082-161-151-151-1 1218311011702601601 13 214-1112-172-161061-1 1425311311911710701 1528401420102-181-171-1 16 324116211211902801 17365-1183-1132-192-1810 184141212115111111910 19 4550233-1162-1122-1101-1 20 50512621181114111102 21 556-1283-1192-1152-1112-1 22 61513121211117111311 23666033302220182-1142-1 2472613631242120111611 25 787-1394-1263-1212-1172-1 2685614331292123111911 279170464-1313-1242-1202-1 2898715031342126112211 291058-1534-1363-1272-1232-1 30113715731392129112511 31 120806040413-13020262-1 32128816441442132212811 331369-1685-1463-1343-1292-1 341458734492372311 3515377513932 TABLE 7. Diameters D k (n) and their increments, wrapped k-cycle, inverse closed, S n / S ⌊n/2⌋ × S n−⌊n/2⌋ 100 CayleyPy-4: HolographyCayleyPy collaboration 9.7. Some eccentricities for Schreier coset graph: S n / S ⌊n/2⌋ × S n−⌊n/2⌋ . Here we consider eccentricities for elements [0, 1] ⌊n/2⌋ + [0] n−2⌊n/2⌋ ) for the Schreier coset graph: S n / S ⌊n/2⌋ × S n−⌊n/2⌋ , wrapped k cycles, both inverse-closed and not cases. And propose quasi-polynomial expressions for them, and study corresponding H -polynomials. Conjecture 27. Case k = 2 from n = 5 p r (n) = n 2 16 + a r n + b r . residue (r)a r b r polynomial p r (n) (0)00 n 2 16 (1) 1 8 − 3 16 n 2 16 + n 8 − 3 16 (2)0 12 16 = 3 4 n 2 16 + 3 4 (3) 1 8 1 16 n 2 16 + n 8 + 1 16 OGF = H(x) (1− x 4 ) 3 , H(x) = x 10 + x 8 + x 7 + 2x 5 + x 4 + x 3 + x 2 . Inverse close or not - coincide, case k = 3 from n = 9 p r (n) = n 2 32 + a r n + b r . residue ra r b r polynomial p r (n) 0 1 8 0 n 2 32 + n 8 1 3 16 − 7 32 n 2 32 + 3n 16 − 7 32 2 1 8 5 8 n 2 32 + n 8 + 5 8 3 3 16 5 32 n 2 32 + 3n 16 + 5 32 4 1 8 0 n 2 32 + n 8 5 3 16 9 32 n 2 32 + 3n 16 + 9 32 6 1 8 1 8 n 2 32 + n 8 + 1 8 7 3 16 5 32 n 2 32 + 3n 16 + 5 32 OGF = H(x) (1− x 8 ) 3 , where H(x) = x 18 + x 16 + x 15 + 2x 14 + 2x 13 + 3x 12 + 3x 11 + 2x 10 + 4x 9 + 3x 8 + 3x 7 + 2x 6 + 2x 5 + x 4 + x 3 + x 2 . 101 CayleyPy-4: HolographyCayleyPy collaboration Not inverse closed, case k = 4 from n = 7 p r (n) = n 2 48 + a r n + b r . residue (r)a r b r polynomial p r (n) 0 1 6 0 n 2 48 + n 6 1 5 24 − 11 48 n 2 48 + 5n 24 − 11 48 2 1 6 7 12 n 2 48 + n 6 + 7 12 3 5 24 3 16 n 2 48 + 5n 24 + 3 16 4 1 6 0 n 2 48 + n 6 5 5 24 7 16 n 2 48 + 5n 24 + 7 16 6 1 6 1 4 n 2 48 + n 6 + 1 4 7 5 24 25 48 n 2 48 + 5n 24 + 25 48 8 1 6 1 3 n 2 48 + n 6 + 1 3 9 5 24 7 16 n 2 48 + 5n 24 + 7 16 10 1 6 1 4 n 2 48 + n 6 + 1 4 11 5 24 3 16 n 2 48 + 5n 24 + 3 16 OGF = H(x) (1− x 12 ) 3 , where H(x) = x 26 + x 24 + x 23 + 2x 22 + 2x 21 + 3x 20 + 3x 19 + 4x 18 + 4x 17 + 5x 16 + 5x 15 + 4x 14 + 6x 13 + 5x 12 + 5x 11 + 4x 10 + 4x 9 + 3x 8 + 3x 7 + 2x 6 + 2x 5 + x 4 + x 3 + x 2 . Case k = 4 from n = 7 p r (n) = n 2 48 + a r n + b r . 102 CayleyPy-4: HolographyCayleyPy collaboration residue ra r b r polynomial p r (n) 002 n 2 48 + 2 1 1 24 31 16 n 2 48 + n 24 + 31 16 2 1 12 3 4 n 2 48 + n 12 + 3 4 3 1 8 7 16 n 2 48 + n 8 + 7 16 4 1 12 1 3 n 2 48 + n 12 + 1 3 5 1 8 41 48 n 2 48 + n 8 + 41 48 6 1 12 3 4 n 2 48 + n 12 + 3 4 7 1 8 5 48 n 2 48 + n 8 + 5 48 8 1 6 − 5 3 n 2 48 + n 6 − 5 3 9 5 24 − 25 16 n 2 48 + 5n 24 − 25 16 10 1 6 − 7 4 n 2 48 + n 6 − 7 4 11 5 24 − 29 16 n 2 48 + 5n 24 − 29 16 OGF = H(x) (1− x 12 ) 3 , where H(x) = − 2x 35 − 2x 34 − 2x 33 − 2x 32 + x 30 + x 29 + x 28 + x 27 + 2x 26 + 4x 25 + 5x 24 + 5x 23 + 6x 22 + 6x 21 + 7x 20 + 4x 19 + 3x 18 + 3x 17 + 4x 16 + 4x 15 + 3x 14 − x 12 + 3x 11 + 2x 10 + 2x 9 + x 8 + 2x 7 + 2x 6 + 2x 5 + x 4 + x 3 + x 2 + 2x + 2. 103 CayleyPy-4: HolographyCayleyPy collaboration FIGURE55. wrapped2- cycles [0101...] FIGURE56. wrapped3- cycles [0101...](inverse closed or not coincide) FIGURE57. wrapped4- cycles [0101...] FIGURE58. wrapped4- cycles inverse closed [0101...] 104 CayleyPy-4: HolographyCayleyPy collaboration nk=2k=3k=4k=5 d ∆d ∆ 2 d ∆d ∆ 2 d ∆d ∆ 2 d ∆d ∆ 2 d 311-1 4201201 521021-121-1 631-1301300300 7402310301301 842-141-131-131-1 9611501401401 10 72-251-141-141-1 11903602501501 1293-262-251051-1 13 1212802610601 14133-382-271-161-1 1516041002802701 16 164-3102-182-2710 17201312111002810 18214-4132-2102-291-1 19 2505150312021002 20 255-4153-3122-2102-2 21 3014180314021202 22 315-5183-3142-2122-2 233606210316021402 24366-5213-2162-1142-2 25 4215241218111602 26436-6253-3192-2162-2 274907280421031802 28497-6284-4213-3182-2 295616320424032002 30577-7324-4243-3202-2 31 6408360427032202 32648-7364-3273-3222-1 337217401330032411 34738414303252 3581453327 TABLE 8. eccentricities and increments for [0, 1] ⌊n/2⌋ + [0] n−2⌊n/2⌋ ) , wrapped k-cycle, not inverse-closed 105 CayleyPy-4: HolographyCayleyPy collaboration nk=2 invk=3 invk=4 invk=5 inv d ∆d ∆ 2 d ∆d ∆ 2 d ∆d ∆ 2 d ∆d ∆ 2 d 311-1 4201201 521021-121-2 631-13013-123-12 7402310210210 842-141-131-131-1 9611501401400 10 72-251-141-1401 11903602501410 1293-262-251-151-1 13 1212802601601 14133-382-261-161-1 1516041002702701 16 164-3102-172-271-1 1720131211901801 18214-4132-291-181-1 19 250515031002901 20 255-4153-3102-2910 21 3014180312021010 22 315-5183-3122-2111-1 233606210314021202 24366-5213-2142-1122-2 25 4215241216111401 26436-6253-3172-21410 274907280419031511 28497-6284-4193-3162-2 295616320422021802 30577-7324-4222-1182-2 31 6408360424122002 32648-7364-3253-3202-2 337217401328032202 34738414283222 3581453124 TABLE 9. eccentricities and increments for [0, 1] ⌊n/2⌋ + [0] n−2⌊n/2⌋ ) , wrapped k-cycle, inverse-closed 9.8. Word metrics to full flips for Cayley and Schreier (S n /S d ) graphs. Here we will consider inverse closed wrapped k-cycles generators. And will look on both Cayley and Schreier (S n /S d ) graphs. We will study the word metric to the ”full flipp”, i.e. the element [(n− d,n− d,...,n− d,n−d−1,..., 2, 1, 0), of in the other words lengths of the shortest paths between (0, 1, 2,...,n− d− 1,n− d,n− d,...,n− d) and (n− d,n− d,...,n− d,n− d− 1,..., 2, 1, 0). Computations are performed with CayleyPy for small n by BFS and for n > 14 by AI- compenent. TABLE 10. Experimental data (d coincide) knno c. d d = 2 d = 3 d = 4 d = 5 d = 6 453321 4644321 Continued on next page 106 CayleyPy-4: HolographyCayleyPy collaboration knno c. d d = 2 d = 3 d = 4 d = 5 d = 6 47655431 48766543 49776654 4 101088776 4 1110109998 4 12111010101010 4 13131212121211 4 14141414131313 4 15161616151515 4 16191919181717 4 17212121201919 4 18242424232222 4 19282726252424 4 20313029282727 4 21353433323129 4 22383736353432 4 23424140393836 4 24444443424139 4 25484847464543 4 26505252514948 4 27565656555352 4 28616161605857 4 296565646262 4 307070696767 4 317574737270 4 328079787775 4 3385848383 107 CayleyPy-4: HolographyCayleyPy collaboration Case k = 4, d = 2 from n = 12 n≡ 0 (mod 12) : D(n) = n(n− 2) 12 , n≡ 1 (mod 12) : D(n) = n(n− 2) 12 + 1 12 , n≡ 2 (mod 12) : D(n) = n(n− 2) 12 , n≡ 3 (mod 12) : D(n) = n(n− 2) 12 − 1 4 , n≡ 4 (mod 12) : D(n) = n(n− 2) 12 + 1 3 , n≡ 5 (mod 12) : D(n) = n(n− 2) 12 − 1 4 , n≡ 6 (mod 12) : D(n) = n(n− 2) 12 , n≡ 7 (mod 12) : D(n) = n(n− 2) 12 + 1 12 , n≡ 8 (mod 12) : D(n) = n(n− 2) 12 , n≡ 9 (mod 12) : D(n) = n(n− 2) 12 + 3 4 , n≡ 10 (mod 12) : D(n) = n(n− 2) 12 + 1 3 , n≡ 11 (mod 12) : D(n) = n(n− 2) 12 + 3 4 . The corresponding generating function is given by H(x) (1− x 12 ) 3 , where H(x) = x 35 + x 34 + 2x 33 + 2x 32 + 3x 31 + 4x 30 + 5x 29 + 7x 28 + + 8x 27 + 10x 26 + 12x 25 + 14x 24 + 14x 23 + 16x 22 + 16x 21 + 18x 20 + + 18x 19 + 18x 18 + 18x 17 + 16x 16 + 16x 15 + 14x 14 + 12x 13 + 10x 12 + + 9x 11 + 7x 10 + 6x 9 + 4x 8 + 3x 7 + 2x 6 + x 5 + x 4 108 CayleyPy-4: HolographyCayleyPy collaboration The polynomial H(x) has the following properties: • its coefficients are nonnegative and unimodal; • its coefficients are not symmetric; • its roots are not necessarily on the unit circle (can be both inside and outside). FIGURE 59. case k = 4, d = 2 The H -polynomial factors as H(x) = x 4 (x + 1) 2 (x 2 + 1) 3 (x 2 − x + 1) 2 (x 2 + x + 1) 2 (x 4 − x 2 + 1) 2 (x 7 − x 6 + x 3 − x + 1) Hence, the generating function can be rewritten as x 4 (x 7 − x 6 + x 3 − x + 1) (1 + x)(1− x) 3 (x 2 + x + 1)(x 2 − x + 1)(x 4 − x 2 + 1) 109 CayleyPy-4: HolographyCayleyPy collaboration Case k = 4, d = 3 from n = 14 n≡ 0 (mod 12) : D(n) = n(n− 2) 12 − 1, n≡ 1 (mod 12) : D(n) = n(n− 2) 12 − 11 12 , n≡ 2 (mod 12) : D(n) = n(n− 2) 12 , n≡ 3 (mod 12) : D(n) = n(n− 2) 12 − 1 4 , n≡ 4 (mod 12) : D(n) = n(n− 2) 12 + 1 3 , n≡ 5 (mod 12) : D(n) = n(n− 2) 12 − 1 4 , n≡ 6 (mod 12) : D(n) = n(n− 2) 12 , n≡ 7 (mod 12) : D(n) = n(n− 2) 12 − 11 12 , n≡ 8 (mod 12) : D(n) = n(n− 2) 12 − 1, n≡ 9 (mod 12) : D(n) = n(n− 2) 12 − 1 4 , n≡ 10 (mod 12) : D(n) = n(n− 2) 12 − 2 3 , n≡ 11 (mod 12) : D(n) = n(n− 2) 12 − 1 4 . The corresponding generating function is given by H(x) (1− x 12 ) 3 , where H(x) =−x 36 + x 33 + x 32 + 2x 31 + 4x 30 + 5x 29 + 7x 28 + + 8x 27 + 10x 26 + 11x 25 + 16x 24 + 16x 23 + 18x 22 + 18x 21 + 20x 20 + + 20x 19 + 18x 18 + 18x 17 + 16x 16 + 16x 15 + 14x 14 + 14x 13 + 9x 12 + + 8x 11 + 6x 10 + 5x 9 + 3x 8 + 2x 7 + 2x 6 + x 5 + x 4 − x 110 CayleyPy-4: HolographyCayleyPy collaboration The polynomial H(x) has the following properties: • its coefficients are neither nonnegative nor unimodal; • its coefficients are not symmetric; • its roots are not necessarily on the unit circle (can be both inside and outside). FIGURE 60. case k = 4, d = 3 The H -polynomial factors as H(x) =−x(x + 1) 2 (x 2 + 1) 2 (x 2 − x + 1) 2 (x 2 + x + 1) 2 (x 4 − x 2 + 1) 2 (x 13 − 2x 12 + x 11 − x 10 + x 9 − x 8 − x 7 + x 6 − x 5 + x 4 − x 3 + x 2 − 2x + 1) Hence, the generating function can be rewritten as −x(x 13 − 2x 12 + x 11 − x 10 + x 9 − x 8 − x 7 + x 6 − x 5 + x 4 − x 3 + x 2 − 2x + 1) (1 + x 2 )(1 + x)(1− x) 3 (x 2 + x + 1)(x 2 − x + 1)(x 4 − x 2 + 1) 111 CayleyPy-4: HolographyCayleyPy collaboration Case k = 4, d = 4 from n = 14 n≡ 0 (mod 12) : D(n) = n(n− 2) 12 − 2, n≡ 1 (mod 12) : D(n) = n(n− 2) 12 − 23 12 , n≡ 2 (mod 12) : D(n) = n(n− 2) 12 − 1, n≡ 3 (mod 12) : D(n) = n(n− 2) 12 − 5 4 , n≡ 4 (mod 12) : D(n) = n(n− 2) 12 − 2 3 , n≡ 5 (mod 12) : D(n) = n(n− 2) 12 − 5 4 , n≡ 6 (mod 12) : D(n) = n(n− 2) 12 − 1, n≡ 7 (mod 12) : D(n) = n(n− 2) 12 − 23 12 , n≡ 8 (mod 12) : D(n) = n(n− 2) 12 − 2, n≡ 9 (mod 12) : D(n) = n(n− 2) 12 − 5 4 , n≡ 10 (mod 12) : D(n) = n(n− 2) 12 − 5 3 , n≡ 11 (mod 12) : D(n) = n(n− 2) 12 − 5 4 . The corresponding generating function is given by H(x) (1− x 12 ) 3 , where H(x) =−2x 36 − x 35 − x 34 + x 31 + 3x 30 + 4x 29 + 6x 28 + + 7x 27 + 9x 26 + 10x 25 + 18x 24 + 18x 23 + 20x 22 + 20x 21 + 22x 20 + + 22x 19 + 20x 18 + 20x 17 + 18x 16 + 18x 15 + 16x 14 + 16x 13 + 8x 12 + + 7x 11 + 5x 10 + 4x 9 + 2x 8 + x 7 + x 6 − x 3 − x 2 − 2x 112 CayleyPy-4: HolographyCayleyPy collaboration The polynomial H(x) has the following properties: • its coefficients are neither nonnegative nor unimodal; • its coefficients are not symmetric; • its roots are not necessarily on the unit circle (can be both inside and outside). FIGURE 61. case k = 4, d = 4 The H -polynomial factors as H(x) =−x(x + 1) 2 (x 2 + 1) 2 (x 2 − x + 1) 2 (x 2 + x + 1) 2 (x 4 − x 2 + 1) 2 (2x 13 − 3x 12 + x 11 − x 10 + x 9 − x 8 − x 7 + x 6 − x 5 + x 4 − x 3 + x 2 − 3x + 2) Hence, the generating function can be rewritten as −x(2x 13 − 3x 12 + x 11 − x 10 + x 9 − x 8 − x 7 + x 6 − x 5 + x 4 − x 3 + x 2 − 3x + 2) (1 + x 2 )(1 + x)(1− x) 3 (x 2 + x + 1)(x 2 − x + 1)(x 4 − x 2 + 1) 113 CayleyPy-4: HolographyCayleyPy collaboration Case k = 4, d = 5 from n = 19 n≡ 0 (mod 12) : D(n) = n(n− 2) 12 − 3, n≡ 1 (mod 12) : D(n) = n(n− 2) 12 − 35 12 , n≡ 2 (mod 12) : D(n) = n(n− 2) 12 − 3, n≡ 3 (mod 12) : D(n) = n(n− 2) 12 − 13 4 , n≡ 4 (mod 12) : D(n) = n(n− 2) 12 − 8 3 , n≡ 5 (mod 12) : D(n) = n(n− 2) 12 − 13 4 , n≡ 6 (mod 12) : D(n) = n(n− 2) 12 − 3, n≡ 7 (mod 12) : D(n) = n(n− 2) 12 − 35 12 , n≡ 8 (mod 12) : D(n) = n(n− 2) 12 − 3, n≡ 9 (mod 12) : D(n) = n(n− 2) 12 − 9 4 , n≡ 10 (mod 12) : D(n) = n(n− 2) 12 − 8 3 , n≡ 11 (mod 12) : D(n) = n(n− 2) 12 − 9 4 . The corresponding generating function is given by H(x) (1− x 12 ) 3 , where H(x) =−3x 36 − 2x 35 − 2x 34 − x 33 − x 32 + x 30 + 2x 29 + 4x 28 + + 5x 27 + 7x 26 + 9x 25 + 20x 24 + 20x 23 + 22x 22 + 22x 21 + 24x 20 + + 24x 19 + 24x 18 + 24x 17 + 22x 16 + 22x 15 + 20x 14 + 18x 13 + 7x 12 + + 6x 11 + 4x 10 + 3x 9 + x 8 − x 6 − 2x 5 − 2x 4 − 3x 3 − 3x 2 − 3x 114 CayleyPy-4: HolographyCayleyPy collaboration The polynomial H(x) has the following properties: • its coefficients are neither nonnegative nor unimodal; • its coefficients are not symmetric; • its roots are not necessarily on the unit circle (can be both inside and outside). FIGURE 62. case k = 4, d = 5 The H -polynomial factors as H(x) =−x(x + 1) 2 (x 2 + 1) 3 (x 2 − x + 1) 2 (x 2 + x + 1) 2 (x 4 − x 2 + 1) 2 (3x 11 − 4x 10 − 2x 9 + 3x 8 + 3x 7 − 4x 6 − 3x 5 + 4x 4 + 2x 3 − 3x 2 − 3x + 3) Hence, the generating function can be rewritten as −x(3x 11 − 4x 10 − 2x 9 + 3x 8 + 3x 7 − 4x 6 − 3x 5 + 4x 4 + 2x 3 − 3x 2 − 3x + 3) (1 + x)(1− x) 3 (x 2 + x + 1)(x 2 − x + 1)(x 4 − x 2 + 1) 9.9. Coset 2-different. Consider inverse-closed coset with central state 011... 11 = 0 1 1 n−1 . According to calculations in the notebook, one can hypothesize that the diameter of the wrapped k-cycles coset is given by d k (n) = j n + (k− 1)(k− 2) 2k− 2 k + r k (n), where r k (n) is a ”small” remainder. This remainder term seems to be (2k− 2)-periodic function of n for n⩾ N k . See table 12 for details. 115 CayleyPy-4: HolographyCayleyPy collaboration TABLE 11. r k (n) period kN k periodr k (n) 344(0, 0, 0, 0) 456(0, 0, 0, 0, 0, 1) 568(0, 0, 0, 0, 0, 0, 0, 0) 6710(0, 0, 0, 0, 0, 0,−1, 0, 0, 1) 7912(−1,−1, 0, 0, 0, 0, 0, 0, 0, 0) 81014(0,−1, 0, 0, 0, 0, 0,−1,−1,−1, 0, 0, 1, 0) 91416(−1, 0, 0, 0, 0,−1,−1,−1, 0, 0, 0, 0, 0,−1,−1,−1) 101318(−1,−1,−1, 0, 0, 0, 0, 0,−1,−1,−2,−1,−1, 0, 0, 1, 0, 0) 111820(−1, 0, 0, 0, 0,−1,−1,−1,−1,−1, 0, 0, 0, 0, 0,−1,−1,−2,−2,−1) 122022(0, 0, 0, 0, 0,−2,−2,−2,−2,−2,−1,−1, 0, 0, 1, 0, 0,−1,−1,−2,−1,−1) 9.10. Coset 3-different. The following data correspond to the graph name part4 3Different 0122... 2222 line in the code. [t] :=⌊t⌋, p := 2k− 2, r := n mod p. For each k, the formula below is exact for all observed data points with n ≥ N k in the current spreadsheet, f k (n) = 2 n + x k 2k− 2 + e k (r),min r e k (r) = 0. • k = 3 (p = 4, x = 0, N k = 4, observed to n = 200; tail covers≈ 49.25 periods, well verified): f 3 (n) = 2 n + 0 4 + e 3 (r), r = n mod 4. e 3 = (0, 0, 1, 1). • k = 4 (p = 6, x = 2, N k = 7, observed to n = 200; tail covers≈ 32.33 periods, well verified): f 4 (n) = 2 n + 2 6 + e 4 (r), r = n mod 6. e 4 = (0, 1, 1, 1, 0, 0). • k = 5 (p = 8, x = 7, N k = 6, observed to n = 200; tail covers≈ 24.38 periods, well verified): f 5 (n) = 2 n + 7 8 + e 5 (r), r = n mod 8. e 5 = (1, 0, 0, 0, 0, 1, 1, 1). • k = 6 (p = 10, x = 6, N k = 8, observed to n = 200; tail covers≈ 19.30 periods, well verified): f 6 (n) = 2 n + 6 10 + e 6 (r), r = n mod 10. e 6 = (2, 2, 2, 2, 0, 1, 1, 2, 1, 2). • k = 7 (p = 12, x = 17, N k = 17, observed to n = 200; tail covers≈ 15.33 periods, well verified): f 7 (n) = 2 n + 17 12 + e 7 (r), r = n mod 12. e 7 = (0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0). 116 CayleyPy-4: HolographyCayleyPy collaboration • k = 8 (p = 14, x = 19, N k = 35, observed to n = 200; tail covers≈ 11.86 periods, well verified): f 8 (n) = 2 n + 19 14 + e 8 (r), r = n mod 14. e 8 = (1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 0, 0, 0, 0). • k = 9 (p = 16, x = 23, N k = 55, observed to n = 200; tail covers≈ 9.12 periods, well verified): f 9 (n) = 2 n + 23 16 + e 9 (r), r = n mod 16. e 9 = (2, 2, 2, 2, 1, 2, 1, 2, 2, 1, 1, 1, 0, 0, 1, 1). • k = 10 (p = 18, x = 35, N k = 69, observed to n = 200; tail covers≈ 7.33 periods, well verified): f 10 (n) = 2 n + 35 18 + e 10 (r), r = n mod 18. e 10 = (2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2). • k = 11 (p = 20, x = 40, N k = 88, observed to n = 200; tail covers≈ 5.65 periods, well verified): f 11 (n) = 2 n + 40 20 + e 11 (r), r = n mod 20. e 11 = (1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2). • k = 12 (p = 22, x = 43, N k = 129, observed to n = 200; tail covers≈ 3.27 periods, well verified): f 12 (n) = 2 n + 43 22 + e 12 (r), r = n mod 22. e 12 = (3, 2, 1, 2, 1, 0, 0, 1, 1, 1, 2, 2, 2, 3, 2, 2, 1, 2, 1, 2, 2, 3). • k = 13 (p = 24, x = 60, N k = 129, observed to n = 200; tail covers≈ 3.00 periods, well verified): f 13 (n) = 2 n + 60 24 + e 13 (r), r = n mod 24. e 13 = (2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1). • k = 14 (p = 26, x = 65, N k = 165, observed to n = 200; tail covers≈ 1.38 periods, moderate evidence): f 14 (n) = 2 n + 65 26 + e 14 (r), r = n mod 26. e 14 = (2, 3, 2, 3, 2, 2, 1, 1, 1, 2, 2, 2, 3, 2, 1, 2, 1, 1, 0, 0, 0, 1, 0, 1, 1, 2). • k = 15 (p = 28, x = 84, N k = 151, observed to n = 200; tail covers≈ 1.79 periods, moderate evidence): f 15 (n) = 2 n + 84 28 + e 15 (r), r = n mod 28. e 15 = (1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 1, 0, 0, 1, 1, 1, 1, 2, 2). • k = 16 (p = 30, x = 89, N k = 149, observed to n = 200; tail covers≈ 1.73 periods, moderate evidence): f 16 (n) = 2 n + 89 30 + e 16 (r), r = n mod 30. e 16 = (3, 2, 1, 2, 1, 1, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 2, 3, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 3). 117 CayleyPy-4: HolographyCayleyPy collaboration • k = 17 (p = 32, x = 97, N k = 127, observed to n = 200; tail covers≈ 2.31 periods, well verified): f 17 (n) = 2 n + 97 32 + e 17 (r), r = n mod 32. e 17 = (2, 2, 2, 2, 2, 1, 1, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 1, 1, 2, 2, 2, 3, 3, 3, 2). • k = 18 (p = 34, x = 119, N k = 167, observed to n = 200; tail covers≈ 1.00 periods, moderate evidence): f 18 (n) = 2 n + 119 34 + e 18 (r), r = n mod 34. e 18 = (2, 3, 3, 3, 2, 2, 1, 1, 0, 1, 1, 2, 2, 2, 2, 3, 3, 2, 1, 2, 1, 2, 1, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2). • k = 19 (p = 36, x = 127, N k = 230, observed to n = 300; tail covers≈ 1.97 periods, moderate evidence): f 19 (n) = 2 n + 127 36 + e 19 (r), r = n mod 36. e 19 = (3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3). • k = 20 (p = 38, x = 133, N k = 262, observed to n = 300; tail covers≈ 1.03 periods, moderate evidence): f 20 (n) = 2 n + 133 38 + e 20 (r), r = n mod 38. e 20 = (3, 4, 3, 4, 3, 3, 2, 2, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 3, 2, 3, 2, 3, 2, 2, 1, 0, 0, 1, 1, 2, 2, 1, 2, 2, 3, 4). • k = 21 (p = 40, x = 162, N k = 259, observed to n = 300; tail covers≈ 1.05 periods, moderate evidence): f 21 (n) = 2 n + 162 40 + e 21 (r), r = n mod 40. e 21 = (2, 2, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 2, 2). • k = 22 (p = 42, x = 169, N k = 253, observed to n = 300; tail covers≈ 1.14 periods, moderate evidence): f 22 (n) = 2 n + 169 42 + e 22 (r), r = n mod 42. e 22 = (2, 3, 3, 3, 2, 2, 2, 1, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 3, 4, 3, 3, 2, 2, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 3). Hypothesis (Last-Layer Periodicity, 3-Different Coset). Retain the notation of §11.11: let p := 2k− 2 and r := n mod p. The diameter of the 3-different wrapped coset graph is given by f k (n) = 2 n + x k 2k− 2 + e k (r),min r e k (r) = 0, where e k is p-periodic for n≥ N k . Let ℓ k (n) denote the size of the last BFS layer at depth n. We conjecture that ℓ k (n) is also eventually p-periodic (i.e., with the same period p = 2k− 2) for n≥ N k . Equivalently, the quasi-polynomial period of f k and the period of ℓ k coincide. This is supported empirically for k = 11, 12, 15, 17 (see Figures 63–66); in each case both e k (r) and ℓ k (n) exhibit synchronized periodicity with period p = 2k− 2. 118 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 63. k = 11: period p = 2k− 2 = 20 FIGURE 64. k = 12: period p = 2k− 2 = 22 FIGURE 65. k = 15: period p = 2k− 2 = 28 119 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 66. k = 17: period p = 2k− 2 = 32 120 CayleyPy-4: HolographyCayleyPy collaboration 9.11. Coset 4-different. Notebook link TABLE 12. Last layer size period for wrapped consecutive coset 4-different. kN k periodl k (n) 384(16, 4, 2, 30) 4116(2, 29, 2, 34, 114, 34) 5228(62, 124, 10, 84, 2, 6, 87, 6) 63710(258, 5, 26, 168, 10, 28, 48, 203, 18, 117) 74412(37, 14, 98, 278, 34, 96, 225, 218, 1, 34, 235, 2) 87614(1, 14, 61, 118, 137, 190, 3, 20, 225, 2, 4, 2, 9, 82) 99216(658, 2, 84, 276, 11, 20, 40, 34, 54, 170, 6, 22, 154, 252, 451, 392) 10− 11− 12− 121 CayleyPy-4: HolographyCayleyPy collaboration e (12)(123) (13) (132)(23) FIGURE 67. The Cayley graph Γ S 3 ,S where S =(12), (23) 10. REMINDER. BACKGROUND AND RELATED WORKS 10.1. Cayley and Schreier graphs, diameters, growth. Here we recollect basic definitions: Cayley and Schreier graphs, diameters, God’s numbers, growth, quasi-polynomials, discuss previ- ous and related works. There are not many works on Cayley graphs of consecutive cycles per se, and we focus on more general perspectives. Cayley graphs give a way of treating groups as geometric objects. In what follows, G denotes a (finite) group with a set of generators S. Definition 1. The Cayley graph of G with respect to S is a directed graph Cay(G,S) (or Γ G,S ) such that its set of vertices is precisely G, and its oriented edges are all pairs (g,gs) with g ∈ G and s∈ S. It is known that Cayley graphs are connected. The word metric on G with respect to S can be defined in two natural ways, depending on whether one considers the directed or the undirected Cayley graph. The directed word metric dist → S (g,h) is defined by as the length of the shortest word in letters from S representing the element g −1 h∈ G, if such a word exists. This metric corresponds to the path metric on the directed Cayley graph Cay(G,S). In contrast, the symmetric word metric dist S (g,h) is defined as the length of the shortest word in letters from S∪S −1 representing g −1 h∈ G. Geometrically, this is the standard path metric on the underlying undirected graph obtained from Cay(G,S) by forgetting edge orientations. Unless explicitly stated otherwise, all distances and diameters in this paper are understood with respect to the symmetric word metric induced by S∪ S −1 . Definition 2. Let Γ = (V,E) be a (directed or undirected) graph equipped with the path metric d(v,w), defined as the length of a shortest path from v to w (if such a path exists). The diameter of Γ is defined as diam(Γ) := sup v,w∈V d(v,w). If Γ is finite and strongly connected (or connected in the undirected case), the supremum is a maximum. Definition 3. Let H ≤ G be a subgroup. We define the Schreier coset graph (or simply the Schreier graph) Sch(G,H,S) as follows: • the vertices are the right cosets Hg of H in G, where g ∈ G; • for each vertex Hg and each generator s∈ S, there is a directed edge Hg s −→ Hgs. Notice that if H = e, the Schreier graph Sch(G,H,S) coincides with the Cayley graph Cay(G,S). 122 CayleyPy-4: HolographyCayleyPy collaboration He H(13)H(23) (12) (12) (23) (23) FIGURE 68. Schreier graph Sch(S 3 ,H,S) for H = ⟨(12)⟩ and S = (12), (23). Vertices are right cosets Hg (not group elements). In contrast to Cayley graphs, Schreier graphs are generally not vertex-transitive. As a conse- quence, metric properties such as growth and distances may depend on the choice of a basepoint. Definition 4. Let Γ = (V,E) be a finite graph endowed with a path metric d, and let v 0 ∈ V be a fixed basepoint. The God’s number of Γ relative to v 0 is defined as God(Γ,v 0 ) := max v∈V d(v 0 ,v). In graph theory, the quantity God(Γ,v 0 ) defined above is classically known as the eccentricity of the vertex v 0 , that is, ecc(v 0 ) := max v∈V d(v 0 ,v). We adopt the term God’s number to emphasize its interpretation as the maximal number of moves required to reach any state from a fixed initial state, following the terminology commonly used in the theory of combinatorial puzzles. Example 4. For the Rubik’s Cube group G with the standard generating set S of face turns, the diameter of the Cayley graph Cay(G,S) is 20. Since Cayley graphs are vertex-transitive, the God’s number (equivalently, the eccentricity of any vertex) coincides with the diameter. If Γ is vertex-transitive (in particular, if Γ is a Cayley graph), then God(Γ,v 0 ) does not depend on the choice of v 0 and coincides with the diameter diam(Γ). For Schreier graphs this need not be the case. For a Schreier graph Sch(G,H,S), the basepoint is typically chosen to be the trivial coset H . In this case, the God’s number measures the maximal distance from H to any coset Hg. Definition 5. Let Γ = (V,E) be a graph endowed with the path metric d, and let v 0 ∈ V be a fixed basepoint. The growth function of Γ relative to v 0 is defined as γ Γ,v 0 (n) := v ∈ V | d(v 0 ,v)≤ n . For Cayley graphs we write γ G,S , omitting the basepoint v 0 , since by vertex-transitivity the growth function is independent of the choice of basepoint. We say that the group G (or the Cayley graph Cay(G,S)) has • polynomial growth if there exist constants C,d > 0 such that γ G,S (n)≤ Cn d for all n, • exponential growth if there exists λ > 1 such that γ G,S (n)≥ λ n for all sufficiently large n. 123 CayleyPy-4: HolographyCayleyPy collaboration From this point on, when working with Cayley graphs, we denote the diameter by diam(G,S). For a finite group G, the diameter diam(G,S) is the minimal radius n such that the ball of radius n in the Cayley graph exhausts the group, i.e. γ G,S (n) =|G|. Thus, the diameter measures the extremal behavior of the growth function. 10.2. Quasi-polynomial functions. Definition 6. A function f : N→ Q is called a quasi-polynomial if there exists a positive integer m and polynomials P 0 ,P 1 ,...,P m−1 ∈ Q[n] such that f (n) = P r (n) for all n∈ N with n≡ r (mod m). The minimal such m is called the period of f . Example 5. Let S =(1, 2), (2, 3),..., (n− 1,n) be the set of Coxeter generators of S n . Then the diameter of the Cayley graph Cay(S n ,S) is given by diam(S n ,S) = n(n− 1) 2 , which is a quasi-polynomial of period 1 (i.e. an ordinary polynomial). Example 6. Let S =(1, 2,...,n− 1,n), (1, 2) be the set of LX generators of S n . In our previ- ous paper [Chervov2025b], we conjectured that the diameter of the LX Cayley graph Cay(S n ,S) is given by diam(S n ,S) = ( 3n 2 4 − 2n + 3, n≡ 0 (mod 2), 3n 2 4 − 2n + 9 4 , n≡ 1 (mod 2), which is a quasi-polynomial of period 2. 10.3. Ehrhart polynomials. Definition 7. LetL ⊂ R d be a lattice, and let P ⊂ R d be a d-dimensional convex polytope such that all vertices of P lie inL. For a positive integer t, let tP denote the t-fold dilation of P , that is, the polytope obtained by multiplying the coordinates of each vertex of P , with respect to a fixed basis ofL, by the factor t. Define L(P,t) := # tP ∩L , the number of lattice points contained in the polytope tP . Then L(P,t) is a polynomial in t of degree d with rational coefficients, called the Ehrhart polynomial of P . Example 7. Let P = conv(0, 0), (0, 1), (1, 0)⊂ R 2 . Then tP =(x,y)∈ R 2 : x≥ 0, y ≥ 0, x + y ≤ t. Hence the lattice points in tP are exactly the integer pairs (i,j) with i,j ≥ 0 and i + j ≤ t, so L P (t) = #(tP ∩ Z 2 ) = t X i=0 (t− i + 1) = (t + 1)(t + 2) 2 . Thus the Ehrhart polynomial of P equals L P (t) = t 2 + 3t + 2 2 . 124 CayleyPy-4: HolographyCayleyPy collaboration Definition 8 (Ehrhart counting function). LetL⊂ R d be a full-rank lattice, and let P ⊂ R d be a d-dimensional convex polytope. For t∈ Z >0 define the lattice-point counting function L L (P,t) := # tP ∩L , where tP =tx| x∈ P is the t-fold dilation of P . We say that P isL-integral if all vertices of P lie inL, andL-rational if all vertices of P lie in L⊗ Z Q. Then: • If P is L-integral, then L L (P,t) is a polynomial in t of degree d (with rational coeffi- cients), called the Ehrhart polynomial of P (with respect toL). • If P isL-rational, then L L (P,t) is a quasi-polynomial in t of degree d, called the Ehrhart quasi-polynomial of P (with respect toL). In the standard latticeL = Z d , if P =x∈ R d | Ax≤ b, A∈ Q k×d , b∈ Q k , then L Z d (P,t) = # tP ∩ Z d = #x∈ Z d | Ax≤ tb. In particular, when A,b are integral (equivalently, P is Z d -integral), this becomes an Ehrhart polynomial; otherwise it is an Ehrhart quasi-polynomial. WhenL = Z d , we write L(P,t) := L Z d (P,t). Example 8. Let P = conv(0, 0), (1, 0), (0, 1 2 )⊂ R 2 . For a positive integer t, the dilation tP is given by tP =(x,y)∈ R 2 | x≥ 0, y ≥ 0, x + 2y ≤ t. Hence the lattice points in tP are exactly the integer pairs (x,y) ∈ Z 2 ≥0 satisfying x + 2y ≤ t, and therefore L P (t) = #(tP ∩ Z 2 ) = ⌊t/2⌋ X y=0 (t− 2y + 1). A direct computation shows that L P (t) is a quasi-polynomial of period 2, given explicitly by L P (t) =        t 2 4 + t + 1, t≡ 0 (mod 2), t 2 + 4t + 3 4 , t≡ 1 (mod 2). We refer to [Stanley1997; Stanley2001] for more information. 10.4. ROC curves and AUC. Given a ranked list of examples with binary ground-truth labels, the induced label sequence x ∈ 0, 1 n defines the same monotone path P (x). In this encoding, the ROC curve is a scaled version of P (x), while the AUC is the corresponding normalized area. Theorem TODO shows that, in the binary orbit model, this area also measures the Cayley-graph distance to the sorted baseline, i.e., the number of adjacent misorderings (inversions) in the ranked sequence. Examples below illustrate that for x with n = 10 and k = 5, d(e,x) = AAC, area above ROC curve measured in unit squares. In the context of binary classification we can fix outputs R of a hypothetical binary classificaion model listed in ascending order, theshold values T , and predictions of classes PRED j corresponding to each selected threshold T j ∈ T . In this context x should be interpreted as ”ground truth”. True Positive Rate (TPR) and False Positive Rate (FPR) are used to build corresponding ROC curves. R = [0.05, 0.15, 0.25, 0.35, 0.45, 0.55, 0.65, 0.75, 0.85, 0.95], T = P + 0.01, PRED j = int(P > T j ), ∀j = 0,...,n− 1. 125 CayleyPy-4: HolographyCayleyPy collaboration 10.4.1. Example 1. x = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1], FPR = [1.0, 0.8, 0.6, 0.4, 0.2, 0.0, 0.0, 0.0, 0.0, 0.0, 0], TPR = [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 0.8, 0.6, 0.4, 0.2, 0]. In this example the area above the ROC curve is AAC = 0 (see Figure 69). Since e = x, we have d(x,e) = 0 = AAC. Now suppose x = [0, 0, 0, 0, 1, 0, 1, 1, 1, 1], then Inv(x) = 1 and d(x,e) = 1. Observe that exactly one unit square is above ROC curve (see Figure 70). So AAC = 1 = d(x,e). 10.4.2. Example 2. x = [0, 0, 0, 1, 0, 1, 1, 0, 1, 1], FPR = [1, 0.8, 0.6, 0.4, 0.4, 0.2, 0.2, 0.2, 0.0, 0.0, 0.0], TPR = [1, 1.0, 1.0, 1.0, 0.8, 0.8, 0.6, 0.4, 0.4, 0.2, 0.0]. FIGURE 69. ROC: TPR vs. FPR — perfect predictions. Here d(x,e) = 4. On the other hand, the ROC AUC is 0.84 (see Figure 71). The maximum possible area is (n− k)k = 25 (a 5× 5 grid), so the area above the curve is (1− 0.84)· 25 = 4. Thus AAC = 4 = d(x,e). 126 CayleyPy-4: HolographyCayleyPy collaboration FIGURE 70. ROC: TPR vs. FPR — one adjacent swap. FIGURE 71. ROC curve showing TPR vs. FPR — four adjacent swaps. 127 CayleyPy-4: HolographyCayleyPy collaboration ACKNOWLEDGMENTS A.C. is deeply grateful to M. Douglas, A. Hayt, C .Simpson, P.A. Melies, F. Charton, Y. Fregier, S. Nechaev, V. Rubtsov, G. Williamson, J. Ellenberg, for stimulating discussions, interest and encouragement, without whom the project may not exist in the present form. A.C. is grateful to J. Mitchel for involving into the Kaggle Santa 2023 challenge, from which this project originated, to M.Kontsevich, N.Nekrasov, T.Rokicki, M.Shapiro, A.Mironov, A.Gorski, V. Fock, V. Gorbunov, V.Golyshev, Y.Soibelman, S.Gukov, T. Smirnova-Nagnibeda, D.Osin, V. Kleptsyn, G.Olshanskii, A. Sutherland, I. Vlassopoulos, A.Zinovyev, M. Alekseyev, S.Klevtsov, A. Mellit, V.Dotsenko, L.Rybnikov, D.Grinberg for the discussions, interest and comments, to his wife A.Chervova and daugther K.Chervova for support, understanding and help with computa- tional experiments. We are deeply grateful to many colleagues who have contributed to the CayleyPy project at various stages of its development, including: D. Kamenetsky, S.Shakirov, N.Bukhal, J.Naghiev, K.Khoruzhii, A.Romanov, A. Naumov, A.Sychev, A.Lenin, E.Uryvanov, A. Abramov, M.Urakov, A.Kuchin, B.Bulatov, F.Faizullin, U.Kniaziuk, D.Naumov, S.Botman, A.Kostin, R.Vinogradov, N.Narynbaev, A.Korolkova, N. Rokotyan, S.Kovalev, A.Eliseev, A.Ogurtsov, G.Antiufeev, G.Verbii, A.Rozanov, V.Nelin, S.Ermilov, A. Trepetsky, A. Dolgorukova, N. Narynbaev, S. Nikolenko, R. Turtayev, K.Yakovlev, V.Shitov, E.Durymanov, R.Magdiev, M.Krinitskiy, P.Snopov, M. Evseev , A.Aparnev, A.Titarenko, M. Litvinov, N. Vilkin-Krom, A. Bidzhiev, A. Krasnyi, E. Geraseva, E. Koldunov, S. Diner, E. Kudasheva, A. Kravchenko, V. Zamkovoy, D. Kovalenko, O. Papulov, D. Mamayeva, M.Kazemina, et. al. The work of F. Levkovich-Maslyuk was supported by the STFC grant APP69281. 128 CayleyPy-4: HolographyCayleyPy collaboration REFERENCES [Adin2025] R. M. Adin, N. Alon, and Y. Roichman. Circular sorting. 2025. arXiv: 2502. 14398. [Alfarano2025] A. Alfarano, Franc ̧ois Charton, and A. Hayat. “Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers”. 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INSTITUT IMAGINE, PARIS, FRANCE Email address, A. Chervov: al.chervov@gmail.com CENTRE FOR MATHEMATICAL SCIENCE, CITY ST GEORGE’S, UNIVERSITY OF LONDON Email address, F. Levkovich-Maslyuk: fedor.levkovich-maslyuk@citystgeorges.ac.uk NEAPOLIS UNIVERSITY PAFOS, CYPRUS Email address, A. Smolensky: andrei.smolensky@gmail.com UNIVERSITY OF TEXAS AT DALLAS Email address, Farid Khafizov: farid.khafizov@utdallas.edu 138 CayleyPy-4: HolographyCayleyPy collaboration ACCENTURE Email address, I. Kiselev: igor.kiselev@gmail.com INTERNATIONAL INSTITUTE OF PHYSICS Email address, Dmitry Melnikov: dmitry.melnikov@iip.ufrn.br INDEPENDENT RESEARCHER Email address, I. Koltsov: ivankolt@gmail.com INDEPENDENT RESEARCHER Email address, S. Kudashev: sergey0474@gmail.com INDEPENDENT RESEARCHER Email address, D. Shiltsov: da.shiltsov@gmail.com RESEARCH CENTER OF THE ARTIFICIAL INTELLIGENCE INSTITUTE, INNOPOLIS UNIVERSITY Email address, M. Obozov: obozovmark9@gmail.com STANFORD UNIVERSITY Email address, Stanislav Krymskii: skrymskii@stanford.edu NRNU MEPHI (NATIONAL RESEARCH NUCLEAR UNIVERSITY) Email address, Valeriia Kirova: valeriia.kirova@mephi.ru THREE GORGES MATHEMATICAL RESEARCH CENTER, CHINA THREE GORGES UNIVERSITY, SOBOLEV IN- STITUTE OF MATHEMATICS, NOVOSIBIRSK STATE UNIVERSITY Email address, E. V. Konstantinova: e konsta@ctgu.edu.cn, ekonsta@math.nsc.ru IHES Email address, A. Soibelman: asoibelman@gmail.com PUC-RIO, DEPARTAMENTO DE MATEM ́ ATICA, RUA MARQU ˆ ES DE S ̃ AO VICENTE 225, G ́ AVEA, RIO DE JANEIRO, BRAZIL Email address, S. Galkin: sergey@puc-rio.br SOBOLEV INSTITUTE OF MATHEMATICS, THE MATHEMATICAL CENTER IN AKADEMGORODOK Email address, L. Grunwald: mathmanlily@gmail.com UNIVERSITY OF HRADEC KR ́ ALOV ́ E Email address, Alexei Kotov: alexei.kotov@uhk.cz IBS CENTER FOR GEOMETRY AND PHYSICS Email address, Alexander Alexandrov: alexander.alexandrov@ibs.re.kr KAZAKH-BRITISH TECHNICAL UNIVERSITY Email address, S. Lytkin: smlytkin@gmail.com UNIVERSITY OF WASHINGTON Email address, D. Fedoriaka: fedimser@cs.washington.edu INDEPENDENT RESEARCHER Email address, A. Chevychelov: heavy4evy@gmail.com INDEPENDENT RESEARCHER Email address, Z. Kogan: zahar1991@gmail.com INDEPENDENT RESEARCHER Email address, A. Natyrova: natyrovaaltana@gmail.com INDEPENDENT RESEARCHER Email address, L. Cheldieva: liuda.tarusina@gmail.com INDEPENDENT RESEARCHER Email address, O. Nikitina: ol.ya.nik.dev@gmail.com INDEPENDENT RESEARCHER Email address, Sergei Fironov: sergei.fironov@iai.spb.ru 139 CayleyPy-4: HolographyCayleyPy collaboration INDEPENDENT RESEARCHER Email address, Anton Vakhrushev: anton.vakhrushev.math@gmail.com INDEPENDENT RESEARCHER Email address, Andrey Lukyanenko: andrey.lukyanenko.math@gmail.com UNIVERSITY OF WASHINGTON Email address, Vasily Ilin: vasilyi@uw.edu INDEPENDENT RESEARCHER Email address, Denis Gorodkov: denis.gorodkov.math@gmail.com UNIVERSITY OF TORONTO Email address, Nikolay Bogachev: n.bogachev@utoronto.ca IHES (L’INSTITUT DES HAUTES ́ ETUDES SCIENTIFIQUES) Email address, Ilia Gaiur: ilia.gaiur@ihes.fr HIGHER SCHOOL OF ECONOMICS Email address, Mikhail Zaitsev: mrzaytsev@edu.hse.ru ST. PETERSBURG STATE UNIVERSITY Email address, Fedor Petrov: fedyapetrov@gmail.com UNIVERSITY OF VIRGINIA, CHARLOTTESVILLE Email address, Leonid Petrov: lenia.petrov@gmail.com QUEEN MARY UNIVERSITY OF LONDON Email address, Tatiana Gaintseva: t.gaintseva@qmul.ac.uk INDEPENDENT RESEARCHER Email address, Alina Gavrilova: alinagavrilova2024@gmail.com INDEPENDENT RESEARCHER Email address, Maxim N. Smirnov: maxim.n.smirnov@gmail.com GUANGDONG TECHNION-ISRAEL INSTITUTE OF TECHNOLOGY Email address, Nikita Kalinin: nikita.kalinin@gtiit.edu.cn INDEPENDENT RESEARCHER Email address, Anastasiia Khan: boykova.irk@yandex.ru INDEPENDENT RESEARCHER Email address, Kyuseok Jung: wjdrbtjr495@gmail.com CENTRALE LYON Email address, Hugo Mousset: hugo.mousset@etu.ec-lyon.fr INSTITUT CURIE, CNRS UMR168, PARIS, FRANCE Email address, H. Isambert: Herve.Isambert@curie.fr INSTITUT CURIE, CNRS UMR168, IMAGINE INSTITUTE, INSERM UMR 1163, PARIS, FRANCE Email address, O. Debeaupuis: orianne.debeaupuis@curie.fr, orianne.debeaupuis@institutimagine.org, orianne.debeaupuis@gmail.com 140

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