{"id":"2383caae-c07c-4c87-be90-1c33c5de187c","arxiv_id":"2506.16789","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Cloning and unfolding, two new graph operations that preserve quasi-isometry type, plus new geometric obstructions, give a complete decision procedure for RAAGediness of triangle-free CFS graphs with at most 10 vertices.","lead":"Some infinite groups can be built from simple graphs, and this paper asks when two such graph-built groups are the same from far away. It gives new graph moves and tests that decide this question for all graphs with up to ten vertices that satisfy the relevant triangle-free condition.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The ≤10-vertex completeness claim rests on an unpinned computer enumeration; a single implementation bug would invalidate the headline.","rationale":"The reader's stated weakest assumption is the quasiisometry invariance of maximal product region graphs from Oh (Theorem 2.22/Corollary 2.23). That is a real dependency, but it is a cited published theorem and the paper explicitly checks that its hypotheses (compact weakly special* square complexes) hold for the Salvetti and commutator complexes. I do not see a paper-specific gap there beyond the usual risk of misquoting an external result. By contrast, the completeness claim for graphs with at most 10 vertices is unique to this paper, is central to the abstract, and is backed only by an unpinned computer program. The reader did flag this as a second dependency, but treated it as less fragile than the Oh theorem; I would rank it as the most load-bearing concern because it directly supports the headline claim and is not independently verifiable in the current version. The mathematical criteria may well be correct, but the complete classification cannot be certified without reproducible computational artifacts. Therefore the reader's CONDITIONAL verdict remains appropriate: the criteria are promising and largely proven, but the completeness assertion needs either a pinned, reproducible computation or a formal proof that the enumeration and criterion-checking are exhaustive.","tokens_in":61582,"tokens_out":21091,"duration_ms":221762,"concrete_test":"Pin the GitHub repository to a specific commit (or provide a container with a recorded checksum), run the published code, and regenerate: (1) Table 1's counts of triangle-free CFS graphs by (V,E); (2) the classification of all 3938 graphs with at most 11 vertices into the region labels of Figure 1; (3) the explicit list of the 8 unresolved 11-vertex graphs. Compare all outputs exactly with the paper's tables and figure. In addition, for a random sample of graphs in each region, independently verify the classification by hand or with a second, independent implementation of the criteria. If every output matches and the sample checks pass, the completeness concern is settled.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract's central applied claim is that the paper 'completely answer[s] the motivating question when the graph has at most 10 vertices'. This is not a theorem with an independent proof; it is the output of the authors' code at github.com/cashenchris/RACG, with no commit hash, no archived input/output, and no independent verifier. The criteria being implemented are genuinely intricate: iterated link doubling (up to three steps per the paper's own description), cloning, unfolding, near-double recognition, ladder detection via Theorem 6.16, and compliant-cycle detection via Theorem 7.5. Each of these has boundary cases where a small implementation error could flip the classification of a single graph. The paper provides Table 1 with counts of triangle-free CFS graphs by vertex and edge numbers, but it does not provide the per-graph classification data, nor a way to map Figure 1's region labels back to individual graphs without running the code. If even one of the graphs with at most 10 vertices is mislabelled, the advertised 'complete answer' is false. This is load-bearing because the completeness claim is the paper's main deliverable; the proofs of individual criteria can be sound and still leave the headline claim unsupported without a reproducible computation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies which right-angled Coxeter groups (RACGs) are quasiisometric to some right-angled Artin group (RAAG), focusing on triangle-free CFS presentation graphs. It introduces two new graph operations, cloning and unfolding, which change the presentation graph without changing the quasiisometry type of the corresponding RACG, and combines them with the existing link doubling operation. On the positive side, it gives criteria, such as coarse near doubles (Theorem 4.16) and transformations into known RAAGedy types (criterion (ΞR)), for deciding that a graph is RAAGedy and for producing a RAAG presentation graph. On the negative side, it develops obstructions from Morse boundaries, JSJ decompositions, maximal product region graphs (ladders), and compliant cycles. The paper reports a computer enumeration of triangle-free CFS graphs and claims that the criteria completely answer the motivating question for graphs with at most 10 vertices, with 8 unresolved 11-vertex graphs.","tokens_in":61766,"tokens_out":6925,"duration_ms":77176,"significance":"If the results hold, this is a substantial contribution to the quasiisometric classification of RACGs. The paper provides practical, graphically verifiable criteria for a problem that previously had only partial answers, and the new invariants (MPRG ladders, compliant cycles) go beyond existing RAAG rigidity tools. The constructive nature of the positive criteria and the large collection of worked examples are valuable. The paper is also unusually explicit: the main proofs are written out in detail, the operations are precisely defined, and the code is made available. The completeness claim for small graphs is a natural and useful deliverable, provided the computational component is made fully reproducible.","major_comments":[{"comment":"The headline claim that the motivating question is completely answered for triangle-free CFS graphs with at most 10 vertices is an output of the authors' code, but the manuscript gives only aggregate counts in Table 1 and region counts in Figure 1. No commit hash, archived input/output data, per-graph classification file, or independent verification procedure is provided. Because the implemented criteria include iterated link doubling (up to depth three), cloning, unfolding, near-double recognition, ladder detection via Theorem 6.16, and compliant-cycle detection via Theorem 7.5, a single coding error could flip the classification of one graph and invalidate the completeness claim. This is load-bearing; the paper should be accompanied by a fixed, versioned computational artifact with explicit inputs and outputs, or by an independent machine-checkable certificate.","section":"§3.3, Table 1, Figure 1, and Abstract"},{"comment":"The ladder and compliant-cycle obstructions both pass through Oh's maximal-product-region invariance (Theorem 2.22 and Corollary 2.23), which is quoted for compact weakly special* square complexes. The paper applies this to Davis and Salvetti complexes, noting that walls are 2-sided, but does not spell out why the finite-index cover passage in Definition 2.18 leaves the decorated maximal-product-region graph unchanged for exactly the complexes used later. This is not an observed error, but it is a correctness-risk point in a load-bearing dependency; please add an explicit verification or a precise citation covering the Davis and Salvetti cases.","section":"§2.6, §6, and §7"}],"minor_comments":[{"comment":"The word 'motiving' in the abstract should be 'motivating'.","section":"Abstract"},{"comment":"In the list of non-RAAGedy criteria, 'crietria' should be 'criteria'; also, the acronyms (CC') and (CC) are visually similar and should be cross-referenced explicitly where each is introduced.","section":"§1.2"},{"comment":"The caption says the region labels are list items from Section 1.2, but the figure has no legend; please add a legend or define the labels directly in the caption.","section":"Figure 1"},{"comment":"The text refers to red and blue vertices in Figure 5; if the figure appears in grayscale, add a shading or symbol key.","section":"Example 4.18 and Figure 5"},{"comment":"The statement about the horizontal line in Table 1 and the forced bipartiteness of graphs with more than (n-1)^2/4 + 1 edges would benefit from a brief explanation in the text, since the table alone does not make this clear.","section":"§3.3"}],"recommendation":"major_revision","confidential_remarks":"The central mathematical content appears sound and the paper fits the journal well. I am recommending major revision primarily because the advertised ≤10-vertex completeness theorem depends on an unpinned computer enumeration; this is fixable by archiving code and data. I would not want the paper accepted without that artifact."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here is my read. The headline: the paper nearly completes the small-graph classification of triangle-free CFS RACGs that are quasiisometric to a RAAG, and it introduces genuinely reusable graph operations and obstructions. That is real progress, not just a new example.\n\nWhat is new: cloning and unfolding change the presentation graph without changing the quasiisometry type of the Coxeter group, and they combine with link doubling to give the coarse-near-double criterion (Theorem 4.16). On the negative side, the paper goes beyond Behrstock's stable cycle and the Nguyen–Tran planar classification by deriving obstructions from JSJ cylinders (Corollary 5.14), wide ladders in the maximal product region graph (Theorem 6.16), and compliant cycles (Theorem 7.5). These are substantial tools. The paper is also honest: it leaves 8 graphs at 11 vertices unresolved.\n\nNow the soft spots, in proportion. First, the advertised complete answer for graphs with at most 10 vertices is the output of a computer enumeration with no commit hash, no per-graph data, and no independent verifier. The criteria are intricate, so a single boundary-case bug could flip the classification. This is load-bearing and fixable: post the data. Second, the MPRG invariance theorem of Oh (Theorem 2.22/2.23) carries most of the weight for the ladder and compliant-cycle obstructions. I have no reason to doubt it, but if it fails for Davis and Salvetti complexes, those non-RAAGedy criteria do not follow. Third, Theorem 5.16 is deferred to Theorem 7.5; that is manageable but needs checking.\n\nMy bottom line: the theoretical contribution is sound and the tools are worth having. The completeness claim should be treated as conditional until the code is pinned down. I would send this to a serious referee, with the explicit request to verify or at least demand reproducible enumeration artifacts. The paper is for people working on quasiisometric classification of Coxeter groups and RAAGs, and it deserves referee time.","headline":"Nearly settles the small-graph RAAGedy classification with genuinely reusable tools; the ≤10-vertex completeness claim depends on an unpinned computation.","tokens_in":62323,"tokens_out":2350,"would_cite":true,"duration_ms":24753,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","20F55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper gives graph-level criteria that decide, for every triangle-free CFS graph with at most 10 vertices, whether its right-angled Coxeter group is quasiisometric to a right-angled Artin group.","keywords":["right-angled Coxeter group","right-angled Artin group","quasiisometry","CFS graph","maximal product region graph","JSJ decomposition","graph modification operations","coarse near double"],"falsifier":"Re-run the paper's enumeration with independently versioned code on all triangle-free CFS graphs with at most 10 vertices; any graph for which neither the positive nor the negative criteria fire would refute the completeness claim. For the geometric engine, exhibit a quasiisometry between a Davis complex and a Salvetti complex whose induced map on maximal product regions is not an isomorphism of the decorated MPRG.","tokens_in":1881,"feed_emoji":"🧩","tokens_out":1834,"duration_ms":66804,"temperature":0.7,"pith_summary":"The paper asks which right-angled Coxeter groups (RACGs) are quasiisometric to some right-angled Artin group (RAAG), a property the authors call RAAGedy. It gives criteria on the presentation graph that decide this question, and when the graph is RAAGedy the criteria also produce a presentation graph for a quasiisometric RAAG. The positive direction uses three graph operations that change the presentation graph without changing the quasiisometry type of the group: link doubling, plus two new operations called cloning and unfolding. The negative direction gives several geometric obstructions, translated into graphical conditions, coming from Morse boundaries, JSJ decompositions, ladders in the maximal product region graph, and compliant cycles. A computer enumeration of all triangle-free CFS graphs with at most 10 vertices completely settles the motivating question in that range, leaving exactly 8 eleven-vertex graphs unresolved.","feed_headline":"All small Coxeter graphs classified: RAAG or not","feed_subtitle":"New graph moves preserve quasiisometry type, settling every triangle-free case with at most 10 vertices; only 8 eleven-vertex graphs remain.","key_machinery":"The load-bearing objects are three graph modification operations: link doubling, which replaces the graph by one whose Coxeter group is a finite-index subgroup of the original; cloning, which adds a twin of a vertex that is a satellite of at least two other vertices and leaves the quasiisometry type unchanged; and unfolding, which rewires along separating joins and also preserves the quasiisometry type. The twin graph, the quotient of the presentation graph by equal-link classes, carries the recognition algorithm for coarse near doubles. On the obstruction side, the maximal product region graph, whose vertices are maximal standard product regions and whose edges record intersections that are standard product regions, is the key object: a quasiisometry between universal covers induces an isomorphism of these decorated graphs, so the 1-bottleneck property of RAAG MPRGs and the contrary existence of wide ladders in RACG MPRGs become quasiisometry obstructions. Compliant subcomplexes, built inductively from maximal products by projections and line times bushy-tree factors, provide a second obstruction mechanism tailored to the projection-diameter dichotomy of RAAGs.","core_discovery":"The central claim is that for triangle-free CFS graphs without separating cliques, being RAAGedy is detectable at the level of the presentation graph. A graph is RAAGedy if it can be transformed, by link doubling, cloning, and unfolding, into a coarse near double or into a graph satisfying the Dani-Levcovitz conditions; the coarse near double condition is recognized from the twin graph and simplifies to: no unclonable singletons, or all unclonable singletons contained in the set of one vertex and its satellites, or in the set of two adjacent vertices and their satellites. In the converse direction, the paper shows several graph conditions force non-RAAGedy: presence of stable cycles or connected Morse boundary after iterated link doubling; rigid or hanging vertices in the JSJ graph of cylinders incompatible with RAAG JSJ decompositions; a ladder in the maximal product region graph that violates the 1-bottleneck property of RAAG MPRGs; and compliant cycles whose accumulated closest-point projections are too large. Together with the enumeration, these criteria completely answer the motivating question for all triangle-free CFS graphs with at most 10 vertices.","pith_inferences":["If the quoted invariance theorem for maximal product region graphs holds at the stated generality, the ladder and compliant-cycle obstructions should apply beyond 2-dimensional Davis and Salvetti complexes; testing them on higher-dimensional weakly special square complexes would be a natural extension.","The 8 unresolved 11-vertex graphs are the obvious next targets: either a new positive operation or a new obstruction will be needed, and those graphs are the minimal places to look for it.","The computer examples with deeply buried stable cycles suggest there may be no uniform bound on how many link doublings are needed to expose a stable cycle; if so, no finite search over link doubles can certify absence of the Morse-boundary obstruction, making the decomposition-sequence criterion potentially necessary.","Cloning and unfolding are only known to preserve quasiisometry type, not commensurability, so groups shown RAAGedy through them may be quasiisometric to a RAAG without admitting a finite-index RAAG subgroup; determining when these operations can be upgraded to commensurability is a testable next step."],"forward_implications":["For every triangle-free CFS graph with at most 10 vertices, RAAGediness is decided by the paper's criteria, and in the positive case the criteria produce an explicit presentation graph for a quasiisometric RAAG.","Cloning and unfolding are new quasiisometry-preserving graph operations, so any graph invariant invariant under all three operations is a quasiisometry invariant of the corresponding RACG.","The ladder obstruction shows that the maximal product region graph of a RACG can be a quasitree without the precise 1-bottleneck structure of a RAAG MPRG, so the MPRG carries information finer than its quasiisometry type.","The compliant-cycle obstruction subsumes the JSJ-based obstructions of no cycles of cuts and no virtually Z2 edge incident to a rigid non-Z2 vertex, giving a single mechanism for many non-RAAGedy examples.","The complete answer for up to 10 vertices and the list of 8 unresolved 11-vertex graphs give a concrete finite testbed for further questions, such as whether RAAGediness is constructible by coning from a square."],"supporting_citations":[{"why":"Supplies the maximal standard product region graph and proves that quasiisometries induce isomorphisms of decorated MPRGs, the engine of the ladder and compliant-cycle obstructions.","marker":"[74]"},{"why":"Establishes that a graph double is commensurable to a RAAG, the base positive criterion that near doubles and coarse near doubles feed into.","marker":"[34]"},{"why":"Gives the Dani-Levcovitz sufficient conditions for a RACG to have a finite-index RAAG subgroup, one of the target classes in the positive criteria.","marker":"[27]"},{"why":"Provides the CFS property as a necessary condition for a triangle-free RACG to be RAAGedy and relates divergence to graph structure.","marker":"[29]"},{"why":"Describes the JSJ graph of cylinders of a RACG in terms of cut pairs and cut 2-paths of the presentation graph, used for the JSJ obstructions.","marker":"[40]"},{"why":"Provides the theory of JSJ trees of cylinders and their quasiisometry invariance, on which the JSJ-based non-RAAGedy criteria depend.","marker":"[50]"},{"why":"Supplies the tree of quasiisometries framework used to patch together local quasiisometries in the proofs that cloning and unfolding preserve quasiisometry type.","marker":"[17]"},{"why":"Shows the Morse boundary of a RAAG is totally disconnected, which underlies the stable-cycle and Morse-boundary obstructions.","marker":"[19]"}],"fun_headline_variants":["Small Coxeter graphs: RAAG or not, now decided","All ≤10-vertex triangle-free Coxeter graphs classified","Cloning and unfolding crack Coxeter-RAAG detection","RAAGedy resolved: small Coxeter graphs fully classified","No more RAAGedy: complete criteria for small Coxeter graphs"],"cache_read_input_tokens":64512,"weakest_assumption_plain":"The negative criteria rest on Oh's theorem that a quasiisometry between universal covers of compact weakly special square complexes induces an isomorphism of their decorated maximal product region graphs; if that invariance fails for Davis and Salvetti complexes, the ladder and compliant-cycle obstructions do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Small Coxeter graphs: RAAG or not, now decided","All ≤10-vertex triangle-free Coxeter graphs classified","Cloning and unfolding crack Coxeter-RAAG detection","RAAGedy resolved: small Coxeter graphs fully classified","No more RAAGedy: complete criteria for small Coxeter graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001477,"raw_usage":{"total_tokens":5960,"prompt_tokens":992,"completion_tokens":4968,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":608,"completion_tokens_details":{"reasoning_tokens":4884}},"tokens_in":608,"tokens_out":4968,"duration_ms":39579,"temperature":1.0,"reasoning_tokens":4884,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:19:04.503760+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the paper's enumeration with independently versioned code on all triangle-free CFS graphs with at most 10 vertices; any graph for which neither the positive nor the negative criteria fire would refute the completeness claim. For the geometric engine, exhibit a quasiisometry between a Davis complex and a Salvetti complex whose induced map on maximal product regions is not an isomorphism of the decorated MPRG.","supporting_citations":[{"cited_title":"Oh, Quasi-isometry invariants of weakly special square complexes , Topology and its Applications 307 (2022), 107945","cited_arxiv_id":null,"evidence_quote":"Supplies the maximal standard product region graph and proves that quasiisometries induce isomorphisms of decorated MPRGs, the engine of the ladder and compliant-cycle obstructions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that a graph double is commensurable to a RAAG, the base positive criterion that near doubles and coarse near doubles feed into."},{"cited_title":"Dani and I","cited_arxiv_id":null,"evidence_quote":"Gives the Dani-Levcovitz sufficient conditions for a RACG to have a finite-index RAAG subgroup, one of the target classes in the positive criteria."},{"cited_title":"Dani and A","cited_arxiv_id":null,"evidence_quote":"Provides the CFS property as a necessary condition for a triangle-free RACG to be RAAGedy and relates divergence to graph structure."},{"cited_title":"Edletzberger, Quasi-isometries for certain right-angled Coxeter groups , Groups Geom","cited_arxiv_id":null,"evidence_quote":"Describes the JSJ graph of cylinders of a RACG in terms of cut pairs and cut 2-paths of the presentation graph, used for the JSJ obstructions."},{"cited_title":"Guirardel and G","cited_arxiv_id":null,"evidence_quote":"Provides the theory of JSJ trees of cylinders and their quasiisometry invariance, on which the JSJ-based non-RAAGedy criteria depend."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the tree of quasiisometries framework used to patch together local quasiisometries in the proofs that cloning and unfolding preserve quasiisometry type."},{"cited_title":"Charney, M","cited_arxiv_id":null,"evidence_quote":"Shows the Morse boundary of a RAAG is totally disconnected, which underlies the stable-cycle and Morse-boundary obstructions."}],"review_version":2}