{"id":"f6839889-fe63-4f6d-b787-e8369f03d68d","arxiv_id":"1908.07955","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For well-behaved sequences of finite Coxeter groups, the descent-plus-inverse-descent statistic satisfies a central limit theorem exactly when its variance tends to infinity.","lead":"For finite Coxeter groups, this paper proves that the statistic counting descents plus inverse descents follows a normal distribution as the rank grows, provided the variance grows and a technical regular-growth condition holds. It conditionally answers an open question of Kahle and Stump and extends known results for permutation groups to all finite Coxeter groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Well-behaved hypothesis is non-vacuous; a natural product of A_{R/(i log R)} violates Eq. (6.1), so Theorem 6.5 is only a conditional answer to Kahle-Stump.","rationale":"The reader's CONDITIONAL verdict is confirmed. The well-behaved hypothesis is not vacuous: the constructed product of A_{R/(i log R)} fails Eq. (6.1) for every δ>0, so the theorem's scope is narrower than a full affirmative answer to Kahle-Stump. However, the conditional theorem itself appears internally coherent: the Lindeberg arguments, the fourth-moment bounds, and the decomposition into dihedral and non-dihedral parts are structurally sound, and the fourth-moment formula for A3 matches exact enumeration. There is a localized factor-4 inconsistency between Theorem 2.3(2) (V = Σ 1/m_i) and Lemma 5.2 (s^2 = 4/m_i); the correct dihedral variance is 4/m_i, and the error does not affect the divergence conditions in Theorem 6.5. These issues justify keeping the manuscript conditional rather than rejecting it; the authors should correct the factor-4 slip, acknowledge that non-well-behaved sequences exist, and soften the abstract's claim to answering Kahle-Stump's question unconditionally.","tokens_in":19155,"tokens_out":37894,"duration_ms":347855,"concrete_test":"For R ∈ {10^6, 10^8, 10^10}, build W_R = ∏_{i=1}^{R} A_{⌊R/(i log R)⌋} and compute, for δ ∈ {0.1, 0.2, …, 0.9} and k ∈ {1, 10, 100}, the tail ratio T_{k,R} = (Σ_{i=k}^{m_R} V(T_{n,i})) / (Σ_{i=1}^{m_R} V(T_{n,i})). Check that for fixed k, T_{k,R}→1 as R grows, so the sequence violates Eq. (6.1). Then compute the Lyapunov ratio L_R = Σ E(X_{n,i}^4) / (Σ V(T_{n,i}))^2 using the type-A four-moment formulas from Proposition 3.6; if L_R→0 and the max variance ratio tends to 0, the CLT holds for this non-well-behaved sequence, confirming that the well-behaved hypothesis is a genuine restriction rather than a necessary condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem is conditional on the 'well-behaved' property of Definition 6.2, Eq. (6.1). The authors state after Definition 6.2 that they failed to construct a non-well-behaved sequence, suggesting the condition is mild or automatic. This is not the case. Let R→∞ and set W_R = ∏_{i=1}^{R} A_{⌊R/(i log R)⌋}. Then rk(W_R)∼R and the largest component has rank ∼R/log R = o(R). For any fixed δ>0, a component is non-δ-small exactly when i ≲ R^δ/log R, so m_n ∼ R^δ/log R. The variance of the non-small part is V(T_{Mδ_n}) ∼ (1/6) Σ_{i≤m} r_i ∼ (δ/6)R. For any fixed k, the tail Σ_{i=k}^{m} V(T_{n,i}) is also ∼ (δ/6)R, because the first k terms contribute o(R). Hence the ratio in Eq. (6.1) tends to 1 as R→∞ for every fixed k; equivalently the supremum over n in Eq. (6.1) does not tend to 0 as k→∞. This sequence is not well-behaved for any δ>0, contradicting the authors' remark. Moreover this excluded sequence still satisfies the CLT: max_i s_{n,i}^2/s_n^2 ∼1/log R→0 and Σ E(X_{n,i}^4)/s_n^4 = O(1/log^2 R)→0, so the Lindeberg condition holds. Thus a natural family of Coxeter groups with growing rank satisfies the variance-divergence equivalence of Theorem 6.5 but is outside the theorem. This limits the paper's claim to answer Kahle-Stump's question for all sequences.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the statistic t(w)=des(w)+des(w^{-1}) on finite Coxeter groups and proves a central limit theorem for the associated random variable T_W under an additional 'well-behaved' hypothesis on the sequence of Coxeter groups. The main result, Theorem 6.5, states that for a well-behaved sequence of finite Coxeter groups with rank tending to infinity, the normalized T_n converges to N(0,1) if and only if its variance diverges, which is further equivalent to rk(G_n)+Σ 1/m_{n,i} → ∞, where G_n is the non-dihedral part and m_{n,i} are the dihedral parameters. The proof decomposes T_n into independent contributions from irreducible components, uses Lindeberg's theorem, and derives fourth-moment estimates for types A, B, and D via recursive conditional-expectation formulas solved with Sage and Mathematica.","tokens_in":19576,"tokens_out":10638,"duration_ms":164868,"significance":"If the result is correct, it provides a positive but conditional answer to a question of Kahle-Stump and generalizes prior CLTs for the two-sided descent statistic on symmetric groups and on irreducible types A, B, and D. The proof architecture is sensible, and the paper contains useful explicit mixed-moment computations in the appendix. However, the significance is limited by three issues: an internal factor-4 inconsistency in the variance formula for dihedral groups, a concrete natural sequence that violates the well-behaved hypothesis while still satisfying the CLT, and a reliance on unverified computer-algebra derivations for load-bearing fourth-moment bounds. These issues do not necessarily invalidate the theorem as stated, but they substantially weaken the claim that the paper answers Kahle-Stump's question in full generality.","major_comments":[{"comment":"Theorem 2.3(2) states that for a product of dihedral groups W=∏ I_2(m_i), the variance of T_W is Σ 1/m_i. However, Lemma 5.2 uses V(X_{n,i}) = 4/m_{n,i}, invoking Theorem 2.3 for this formula. A direct computation for I_2(m) gives V(T) = 4/m: the values of T are 0 for the identity, 2 for all non-identity elements except the longest element, and 4 for the longest element, yielding variance (4+4)/(2m) = 4/m. Thus the statement of Theorem 2.3(2) is incorrect and should be corrected to Σ 4/m_i. This factor affects the quantitative statement of Lemma 5.2 and the proof of Theorem 6.5, though the equivalence between variance divergence and rk(G_n)+Σ1/m_{n,i} → ∞ is preserved because of the constant factor 4.","section":"Section 2.3, Theorem 2.3(2) and Lemma 5.2"},{"comment":"The remark after Definition 6.2 asserts that the authors failed to construct a sequence that is not well-behaved. This is contradicted by an explicit example. Let R→∞ and set W_R = ∏_{i=1}^R A_{⌊R/(i log R)⌋}. Then rk(W_R) ∼ R. For any fixed δ>0, an irreducible component is non-δ-small precisely when i ≲ R^δ/log R, so m_R ∼ R^δ/log R. The variance of the non-small part satisfies V(T_{M_R^δ}) ∼ (1/6)Σ_{i≤m_R} r_i ∼ δ R/6. For any fixed k, the tail Σ_{i=k}^{m_R} V(T_{R,i}) is also asymptotic to δ R/6 because the first k terms contribute o(R). Hence the ratio in Eq. (6.1) tends to 1 as R→∞ for every fixed k, so the sequence is not well-behaved for any δ>0. This sequence nevertheless satisfies the CLT: max_i s_{R,i}^2/s_R^2 ∼ 1/log R → 0 and Σ E(X_{R,i}^4)/s_R^4 = O(1/log^2 R) → 0, so the Lindeberg condition holds. Thus a natural family of Coxeter groups with growing rank satisfies the variance-divergence equivalence of Theorem 6.5 but lies outside the theorem's hypothesis. The paper should either prove the CLT under a weaker condition that includes such examples or explicitly state that the answer to Kahle-Stump's question is only conditional on well-behavedness.","section":"Section 6, Definition 6.2 and the remark after it"},{"comment":"The proofs of the fourth-moment formulas are not self-contained. The text repeatedly states that starting values were 'computed with Sage' and the recursion was 'solved with the RSolve command of Mathematica', but no code, no session transcript, and no detailed derivation of the intermediate recursions are provided. These formulas are load-bearing: Lemma 5.3 depends on the estimate E((T−E(T))^4)=O(n^2) from Theorem 3.1, and the type D case is handled only by an upper bound. To make the proof verifiable, the authors should include the Sage/Mathematica code (e.g., as an ancillary file) or give analytic derivations of the recursions and closed forms. This is essential for a rigorous proof of a central result.","section":"Section 3, Lemmas 3.3, 3.4, Propositions 3.6 and 3.9"}],"minor_comments":[{"comment":"In the estimate for the first summand, the expression Σ_{i=k+1}^{∞} a_{n,i} X_{n,i} should be the finite sum to k_n; as written, the index range is formally undefined. The intended argument is clear, but the notation should be fixed.","section":"Section 4, proof of Lemma 4.4"},{"comment":"The definition of m_n := min {i ∈ N : W_{n,i+1} is δ-small} is unclear when all irreducible components are δ-small or when none are; a convention (e.g., m_n=0 or m_n=k_n) should be stated to make Eq. (6.1) well-defined in all cases.","section":"Section 6, Definition 6.2"},{"comment":"The sentence 'if every W_{n,i} is of non-dihedral type and for some δ, one has lim_{n→∞} m_n = 0' appears to be a typo; m_n is a positive integer, so it cannot tend to 0. This should likely read that m_n is bounded or that the maximum condition holds.","section":"Section 6, text before Definition 6.2"},{"comment":"There are several typographical errors, including 'random varibales' in the introduction, 'the the law of total expectation' in Section 3, and the rendered addresses 'F akult¨at' in the author affiliation lines. These should be corrected in a final revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper's central conditional theorem may be salvageable, but the three major issues should be addressed before publication. The factor-4 inconsistency in Theorem 2.3(2) is a clear error in a stated result. More importantly, the well-behaved counterexample shows that the authors' remark about the condition is incorrect and that the theorem excludes natural sequences satisfying the CLT; this limits the paper's claim to answer Kahle-Stump's question. The CAS-based moment computations also need to be made reproducible. I recommend major revision rather than rejection because the proof strategy appears coherent and the issues seem fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my take on arXiv:1908.07955. The new content is real: the weighted-sum CLT lemmas in Section 4 and the decomposition into dihedral/non-dihedral parts, plus the fourth-moment computations, go beyond Chatterjee–Diaconis, Özdemir, and Röttger. Theorem 6.5's equivalence between CLT, variance divergence, and rk(G_n)+Σ1/m_i diverging is a substantial step toward Kahle–Stump's question. The proof structure—split into irreducible components, check Lindeberg and fourth-moment bounds—is appropriate, and the honesty about using external CLTs from the literature is fine.\n\nThe soft spots are real but localized. First, the 'well-behaved' hypothesis is not as benign as the authors think. The stress-test construction W_R = ∏_{i=1}^R A_{⌊R/(i log R)⌋} violates Eq. (6.1) for every δ>0: the non-δ-small part has variance ∼δR/6, and the tail past any fixed k still ∼δR/6, so the ratio in (6.1) tends to 1, not 0. Yet this sequence satisfies the CLT. So the remark after Definition 6.2 claiming no non-well-behaved sequence exists is simply wrong, and Theorem 6.5 is only a conditional answer to Kahle–Stump. This doesn't invalidate the theorem, but it should be thoroughly qualified in revision.\n\nSecond, there is a factor-4 inconsistency between Theorem 2.3(2), which gives V(T)=Σ1/m_i for dihedral products, and Lemma 5.2's proof, which uses V(X_{n,i})=4/m_{n,i}. I suspect a typo, and the proof would survive with either constant, but it needs fixing.\n\nThird, the fourth-moment formulas in Section 3 rest on unshown Sage/Mathematica output. Not a fatal flaw—the recursions are there—but a referee should ask for the scripts or at least the starting values to be documented.\n\nCitations look appropriate; the self-citation to Röttger is a black-box use of a prior result, not self-serving.\n\nBottom line: this deserves a serious referee. I would accept it conditionally, asking for the well-behaved claim to be corrected, the variance typo fixed, and the moment computations made reproducible. The core architecture is sound and the result is a genuine advance.","headline":"Conditional CLT with a real hole: the well-behaved hypothesis is non-vacuous, and a factor-4 variance typo needs fixing, but the core proof architecture is sound and worth refereeing.","tokens_in":20092,"tokens_out":5000,"would_cite":true,"duration_ms":44871,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E15","20F55","60F05","05A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For well-behaved finite Coxeter groups, the two-sided descent statistic satisfies the central limit theorem exactly when its variance diverges.","keywords":["Coxeter groups","descent statistic","central limit theorem","two-sided Eulerian polynomial","dihedral groups","Lindeberg condition","well-behaved sequences"],"falsifier":"Build a sequence of finite Coxeter groups with $\\operatorname{rk}(W_n)\\to\\infty$ and $\\operatorname{V}(T_n)\\to\\infty$ that fails the well-behaved tail condition (6.1)—for instance, by letting the ranks of small components grow just slowly enough to defeat uniform tail decay—and check whether the normalized $T_n$ still converges to $N(0,1)$; if it does not, Theorem 6.5 fails without the hypothesis.","tokens_in":18894,"feed_emoji":"🎲","tokens_out":8115,"duration_ms":241856,"temperature":0.7,"pith_summary":"This paper establishes conditions under which the two-sided descent statistic—the number of descents of a random element of a finite Coxeter group plus the number of descents of its inverse—is asymptotically normal. For any \"well-behaved\" sequence of finite Coxeter groups whose ranks grow, the paper proves that the centred and scaled statistic converges to the standard Gaussian distribution exactly when its variance tends to infinity. Equivalently, normality holds precisely when the total rank of the non-dihedral irreducible components together with the sum of reciprocal dihedral parameters diverges. The result answers a question left open in earlier work on descents and extends previously known cases for symmetric and signed-permutation groups to arbitrary products of irreducible Coxeter groups.","feed_headline":"Two-sided descents go Gaussian as Coxeter groups grow","feed_subtitle":"For well-behaved Coxeter families, normality kicks in exactly when the statistic's variance blows up.","key_machinery":"One central object is the two-sided Eulerian polynomial, the generating function $\\sum_{w\\in W} x^{\\operatorname{des}(w)+\\operatorname{des}(w^{-1})}$, which lacks the real-rooted factorization that makes the one-sided descent statistic easy to handle. To replace that structure, the paper derives recursive conditional-expectation formulas for the joint distribution of $(\\operatorname{des}(w),\\operatorname{des}(w^{-1}))$ in types $A_n$ and $B_n$, solves the recursions for fourth moments, and bounds the type-$D_n$ moment by comparison with $B_n$. Around this, the paper builds a triangular-array argument: the variance of $T_n$ adds over irreducible components, $\\delta$-small components are separated from the rest, and a uniform tail-decay condition, called well-behaved, makes the Lindeberg condition hold.","core_discovery":"The central claim is Theorem 6.5: for a well-behaved sequence $(W_n)$ of finite Coxeter groups with $\\operatorname{rk}(W_n)\\to\\infty$, letting $T_n$ be the statistic $\\operatorname{des}(w)+\\operatorname{des}(w^{-1})$ on a uniform random element $w\\in W_n$, the following are equivalent: $(T_n)$ satisfies the CLT; $\\operatorname{V}(T_n)\\to\\infty$; and $\\operatorname{rk}(G_n)+\\sum_{i=1}^{\\ell_n}1/m_{n,i}\\to\\infty$, where $G_n$ collects the non-dihedral irreducible components and $I_2(m_{n,i})$ are the dihedral factors. The paper proves the equivalence by splitting $T_n$ into independent per-component contributions, applying Lindeberg's theorem for triangular arrays through a weighted-sum criterion, and using fourth-moment bounds of order $n^2$ for the irreducible types $A_n$, $B_n$, and $D_n$.","pith_inferences":["The authors' inability to construct a non-well-behaved sequence suggests the \"well-behaved\" hypothesis may be redundant; if so, Theorem 6.5 would hold for every sequence of finite Coxeter groups with growing rank, making the CLT criterion fully general.","The recursive conditional-expectation method for fourth moments is not tied to type $A$ or $B$; it could supply moment bounds for other Coxeter-invariant statistics or for higher moments, turning the CLT criterion into a finite-moment check.","Because $t$ counts geodesic-neighbour facets in the two-sided Coxeter complex, the theorem can be read probabilistically as a statement about the asymptotic size of the ball of radius one around a random chamber; the same variance condition might govern other local statistics on the complex.","The equivalence between CLT and variance divergence may hold more broadly for descent-like statistics on combinatorial groups, with the well-behaved condition serving as a technical bridge rather than a genuine restriction."],"forward_implications":["For any well-behaved sequence of finite Coxeter groups with growing rank, the two-sided descent statistic satisfies the CLT if and only if its variance diverges.","Equivalently, normality is controlled by the explicit quantity $\\operatorname{rk}(G_n)+\\sum_i 1/m_{n,i}$: once either the non-dihedral rank grows or the reciprocal dihedral parameters accumulate, the law of $T_n$ is Gaussian in the large-$n$ limit.","Sequences of products of dihedral groups obey the CLT exactly when $\\sum_i 1/m_{n,i}\\to\\infty$, as in the harmonic-series example where the $n$-th group is the product of the first $n$ dihedral groups.","If the dihedral contributions do not accumulate, the variance stays bounded and no Gaussian limit is possible, so the CLT fails."],"supporting_citations":[{"why":"introduces the two-sided descent statistic on permutations and proves the base CLT that the present theorem generalises.","marker":"[7]"},{"why":"computes descent statistic variances for all irreducible Coxeter types and poses the question answered here.","marker":"[8]"},{"why":"supplies the conditional-expectation recursion for the joint distribution used to compute fourth moments.","marker":"[10]"},{"why":"extends the CLT to the non-dihedral types $B_n$ and $D_n$, entering the main proof as the known cases.","marker":"[13]"},{"why":"gives the recursion for the type-$B_n$ two-sided Eulerian polynomial that yields Lemma 3.8.","marker":"[17]"},{"why":"defines the two-sided Coxeter complex and shows its $h$-polynomial is the generating function of the statistic.","marker":"[12]"}],"fun_headline_variants":["Coxeter groups: two-sided descents hit Gaussian limit","Two-sided descents obey CLT for Coxeter groups","Gaussian limit for two-sided descent statistic on Coxeter groups","Normality for two-sided descents as Coxeter groups grow"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The equivalence is conditional on the sequence being \"well-behaved\"—a uniform decay condition on the tail variances of small irreducible components—and the paper does not prove that every sequence of Coxeter groups satisfies this condition.","fun_headline_variants_meta":{"raw":{"variants":["Coxeter groups: two-sided descents hit Gaussian limit","Two-sided descents obey CLT for Coxeter groups","Gaussian limit for two-sided descent statistic on Coxeter groups","Normality for two-sided descents as Coxeter groups grow"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000985,"raw_usage":{"total_tokens":4113,"prompt_tokens":816,"completion_tokens":3297,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":432,"completion_tokens_details":{"reasoning_tokens":3227}},"tokens_in":432,"tokens_out":3297,"duration_ms":25777,"temperature":1.0,"reasoning_tokens":3227,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:54:27.039037+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build a sequence of finite Coxeter groups with $\\operatorname{rk}(W_n)\\to\\infty$ and $\\operatorname{V}(T_n)\\to\\infty$ that fails the well-behaved tail condition (6.1)—for instance, by letting the ranks of small components grow just slowly enough to defeat uniform tail decay—and check whether the normalized $T_n$ still converges to $N(0,1)$; if it does not, Theorem 6.5 fails without the hypothesis.","supporting_citations":[{"cited_title":"Chatterjee and P","cited_arxiv_id":null,"evidence_quote":"introduces the two-sided descent statistic on permutations and proves the base CLT that the present theorem generalises."},{"cited_title":"Counting inversions and descents of random elements in finite Coxeter groups","cited_arxiv_id":"1802.01389","evidence_quote":"computes descent statistic variances for all irreducible Coxeter types and poses the question answered here."},{"cited_title":"Martingales and descent statistics","cited_arxiv_id":"1901.01719","evidence_quote":"supplies the conditional-expectation recursion for the joint distribution used to compute fourth moments."},{"cited_title":"Asymptotics of a locally dependent statistic on finite reflection groups","cited_arxiv_id":"1812.00372","evidence_quote":"extends the CLT to the non-dihedral types $B_n$ and $D_n$, entering the main proof as the known cases."},{"cited_title":"Visontai","cited_arxiv_id":null,"evidence_quote":"gives the recursion for the type-$B_n$ two-sided Eulerian polynomial that yields Lemma 3.8."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the two-sided Coxeter complex and shows its $h$-polynomial is the generating function of the statistic."}],"review_version":1}