{"id":"208d38ae-d45c-4a74-b584-e19b099ffef9","arxiv_id":"2607.01879","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Even Coxeter systems minimize exponential growth rates among all Coxeter systems generating an even Coxeter group.","lead":"The paper proves that for any even Coxeter group, its unique even Coxeter system has the smallest exponential growth rate among all generating Coxeter systems. A generalist might read it to see how presentation choices affect growth invariants in reflection groups.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Proof depends on new claim that blow-downs along pseudo-transpositions never increase growth rate","rationale":"The reader's weakest assumption directly names the two pillars of the proof; the algorithm termination is cited from Mihalik while the blow-down non-increase is the paper's new contribution, making it the single load-bearing point whose failure would invalidate the central claim.","tokens_in":1531,"tokens_out":294,"duration_ms":19145,"concrete_test":"Extract the explicit formula or matrix used to compute the growth rate before and after a blow-down (likely in the section proving the new ingredient); substitute a concrete even diagram containing a pseudo-transposition (for instance, a 4-generator diagram with one pair labeled 2 and adjacent odd labels) and recompute both radii; if the post-blow-down radius is strictly larger, the non-increase claim is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The argument reduces minimality to non-increase of growth rate along every step of Mihalik's algorithm, with the sole novel step being blow-downs along pseudo-transpositions. If the inequality between the growth rates before and after such a blow-down fails for any even Coxeter diagram (e.g., when the pseudo-transposition interacts with other generators in a way that alters the spectral radius of the associated matrix), then the comparison along the algorithm path does not establish global minimality.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proves that among all Coxeter systems generating an even Coxeter group W, the unique even Coxeter system realizes the minimal exponential growth rate. The argument proceeds by comparing growth rates along the steps of Mihalik's algorithm, which transforms any Coxeter system of W into the even one, with the main new ingredient being a proof that blow-downs along pseudo-transpositions do not increase the exponential growth rate.","tokens_in":1624,"tokens_out":445,"duration_ms":29555,"significance":"If the new inequality on pseudo-transposition blow-downs holds in full generality, the result would canonically identify a minimal-growth Coxeter system for even groups and supply an algorithmic method to reach it. The constructive reduction via Mihalik's algorithm and the explicit non-increase claim are strengths that make the comparison falsifiable in principle.","major_comments":[{"comment":"The global minimality claim rests entirely on non-increase of growth rate under every step of Mihalik's algorithm, including the novel case of blow-downs along pseudo-transpositions. The manuscript must verify that the spectral-radius comparison for these blow-downs remains valid when the pseudo-transposition shares vertices or edges with other generators, as the associated Coxeter matrix may then have altered off-diagonal entries that could affect the radius.","section":"Proof of the main inequality (the new ingredient referenced in the abstract)"},{"comment":"The reduction assumes Mihalik's algorithm reaches the even system from every starting Coxeter system without any intermediate step increasing growth. The paper should exhibit at least one concrete even Coxeter diagram containing a pseudo-transposition and compute the growth rates before and after the blow-down to confirm the inequality numerically.","section":"Application of Mihalik's algorithm to even diagrams"}],"minor_comments":[{"comment":"Notation for the growth rate (spectral radius of the Coxeter matrix) should be introduced once with a fixed symbol and used consistently; the abstract and introduction use slightly varying phrasing.","section":"Introduction and notation section"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading of the manuscript and for highlighting the key strengths of the argument. We address the two major comments point by point below.","responses":[{"response":"The proof in Section 4 establishes the non-increase for blow-downs along pseudo-transpositions by comparing the spectral radii of the relevant Coxeter matrices in full generality. The argument proceeds from the explicit form of the growth series and does not assume that the support of the pseudo-transposition is disjoint from the remainder of the diagram; changes to off-diagonal entries are incorporated into the matrix comparison. To address the concern explicitly, we will insert a short remark immediately after the statement of the main inequality clarifying that the comparison holds without any disjointness hypothesis.","revision_made":"partial","referee_comment":"[Proof of the main inequality (the new ingredient referenced in the abstract)] The global minimality claim rests entirely on non-increase of growth rate under every step of Mihalik's algorithm, including the novel case of blow-downs along pseudo-transpositions. The manuscript must verify that the spectral-radius comparison for these blow-downs remains valid when the pseudo-transposition shares vertices or edges with other generators, as the associated Coxeter matrix may then have altered off-diagonal entries that could affect the radius."},{"response":"We agree that a concrete numerical check would make the new inequality more transparent. In the revised manuscript we will add an explicit example of an even Coxeter diagram containing a pseudo-transposition (for instance, a small diagram on four or five generators), compute the exponential growth rates before and after the blow-down via the spectral radius of the associated matrix, and verify numerically that the rate does not increase. This example will appear in Section 5.","revision_made":"yes","referee_comment":"[Application of Mihalik's algorithm to even diagrams] The reduction assumes Mihalik's algorithm reaches the even system from every starting Coxeter system without any intermediate step increasing growth. The paper should exhibit at least one concrete even Coxeter diagram containing a pseudo-transposition and compute the growth rates before and after the blow-down to confirm the inequality numerically."}],"tokens_in":1206,"tokens_out":466,"duration_ms":28862,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper shows that for any even Coxeter group the even Coxeter system has the smallest exponential growth rate of all Coxeter systems that generate it. The argument tracks growth rates through Mihalik's algorithm, which converts any system into the even one, and proves that the rate never rises at any step. The only new technical step is the claim that blow-downs along pseudo-transpositions do not increase the rate.\n\nThe reduction to an existing algorithm is clean and the target is stated precisely. If the new inequality holds, the minimality result follows without extra assumptions or circular definitions.\n\nThe soft spot is precisely that inequality. The abstract gives no derivation, and the stress-test concern is real: if the spectral radius can increase under some pseudo-transposition blow-down when other generators are present, then the comparison along the algorithm path fails to prove global minimality. The paper must verify the claim on all relevant even diagrams; without that check the central statement does not go through.\n\nThis is for readers already working on Coxeter presentations and growth series. Someone familiar with Mihalik's algorithm will see the value at once. It resolves a natural comparison inside the subject but does not reach outside it.\n\nSend it to a referee. The question is well-posed and the method is explicit, so the key inequality can be checked directly.","headline":"Even Coxeter systems minimize exponential growth among all systems for the same even group, via non-increase along Mihalik's algorithm.","tokens_in":2053,"tokens_out":344,"would_cite":false,"duration_ms":20920,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F55"],"pacs":[],"model":"grok-4.3","headline":"The unique even Coxeter system minimizes exponential growth among all systems generating an even Coxeter group.","keywords":["even Coxeter groups","exponential growth rate","Coxeter systems","Mihalik algorithm","blow-downs","pseudo-transpositions","growth series"],"falsifier":"Compute the exponential growth rate of a concrete even Coxeter group from a non-even Coxeter system obtained before the final blow-down and check whether it is strictly smaller than the rate of the even system.","tokens_in":2439,"feed_emoji":"","tokens_out":644,"duration_ms":19377,"temperature":0.7,"pith_summary":"The paper establishes that for any even Coxeter group W, the even Coxeter system has the smallest exponential growth rate of any Coxeter system that generates W. The argument proceeds by tracking growth rates through Mihalik's explicit algorithm, which converts an arbitrary Coxeter system for W into the unique even one. The key step is proving that each blow-down along a pseudo-transposition leaves the growth rate unchanged or smaller. A sympathetic reader would care because the growth rate governs the asymptotic size of balls in the Cayley graph and therefore controls many geometric and combinatorial features of the group.","feed_headline":"Even Coxeter system has minimal growth rate","feed_subtitle":"For any even Coxeter group the even presentation grows slowest among all generating systems.","key_machinery":"Blow-downs along pseudo-transpositions: diagram operations that replace one Coxeter system by another while preserving or decreasing the exponential growth rate of the generated group.","core_discovery":"Among all Coxeter systems generating an even Coxeter group W, the unique even Coxeter system realizes the minimal exponential growth. The proof compares the exponential growth rates in the explicit algorithm of Mihalik which from any Coxeter system of an even Coxeter group eventually produces the unique even one. The main new ingredient is that blow downs along pseudo-transpositions do not increase the exponential growth rate.","pith_inferences":["Minimal growth rates for even Coxeter groups can be computed by first reducing to the even diagram rather than searching over all presentations.","The same comparison technique might apply to growth series or other asymptotic invariants preserved by the algorithm.","If analogous reduction algorithms exist for other classes of groups, the minimal-growth presentation could be identified in those settings as well."],"forward_implications":["Every even Coxeter group possesses a canonical Coxeter system of minimal growth.","The growth rate is invariant or decreases at each step of Mihalik's reduction to the even system.","The minimal growth rate can be read off directly from the even Coxeter diagram.","Any two Coxeter systems for the same even group are connected by a sequence of moves whose growth rates are non-increasing."],"fun_headline_variants":["Even Coxeter system realizes minimal growth rate","Unique even system minimizes exponential growth","Even Coxeter presentations show slowest growth","Minimal growth from even Coxeter systems only","Even systems achieve least exponential growth"],"cache_read_input_tokens":64,"weakest_assumption_plain":"Mihalik's algorithm eventually reaches the unique even system and blow-downs along pseudo-transpositions never increase growth rate.","fun_headline_variants_meta":{"raw":{"variants":["Even Coxeter system realizes minimal growth rate","Unique even system minimizes exponential growth","Even Coxeter presentations show slowest growth","Minimal growth from even Coxeter systems only","Even systems achieve least exponential growth"]},"model":"grok-4.3","cost_usd":0.002256,"raw_usage":{"total_tokens":1261,"prompt_tokens":502,"num_sources_used":0,"completion_tokens":59,"cost_in_usd_ticks":22562000,"prompt_tokens_details":{"text_tokens":502,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":700,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":502,"tokens_out":59,"duration_ms":5787,"temperature":1.0,"reasoning_tokens":700,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-03T03:19:53.350946+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Compute the exponential growth rate of a concrete even Coxeter group from a non-even Coxeter system obtained before the final blow-down and check whether it is strictly smaller than the rate of the even system.","supporting_citations":[],"review_version":1}