{"id":"74270986-f312-45cf-ba88-1243209673c2","arxiv_id":"2508.20200","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new chromatic quasisymmetric invariant for directed signed graphs and an algebra SQSym of signed quasisymmetric functions are defined and studied.","lead":"This paper introduces a new invariant for signed graphs with directed edges, refining the chromatic symmetric function to a quasisymmetric version. It defines a new algebra SQSym generalizing quasisymmetric functions, with a monomial basis, a fundamental family, and applications to hyperplane arrangements.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.2's proof leaves the 0-descent case of DES_Σ undefined; the final equality DES(ωτπ)=DES_Σ(π) is unverified exactly where F has its middle strict edge.","rationale":"The reader's weakest_assumption points to the bijection between SS_d and linear extensions of acyclic orientations. That bijection is not the real problem: every signed permutation induces an acyclic orientation τπ by the total order π, and π is a linear extension of exactly the poset P_{τπ}, so the union over acyclic orientations of their linear-extension sets is a disjoint partition of SS_d. The actual soft spot is the descent indexing, especially the 0-descent. Theorem 4.2's formula depends on DES_Σ(π) matching the strict edges of the F-chain, and the proof's rank-based argument is undefined or at best implicit at i=0. This is a concrete, local gap in the central proof, but it is plausibly fixable by an explicit convention, so the correct disposition remains conditional acceptance rather than rejection.","tokens_in":19419,"tokens_out":42284,"duration_ms":470797,"concrete_test":"Enumerate all signed permutations for the two-vertex signed graph of Example 2.15 (or a 3-vertex graph with a negative loop) and compute both sides of Theorem 4.2 with an explicit convention for the 0-descent, e.g., 0∈DES_Σ(π) iff ω(π(1))<0 for the chosen order-reversing labeling. Verify that the t-weighted F-sum equals the direct chamber sum (2.5) truncated to |colors|≤N. If the 0-descent convention cannot be made to match the direct sum, the expansion needs correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central deliverable is Theorem 4.2, the fundamental-family expansion of X_{−→Σ}. Its proof's final identification of the descent set is incomplete at i=0. Definition 4.1 defines rank(Σ,π)(m) only for m∈{±1,…,±d}, but the Σ-descent condition for i=0 applies rank to π_0, which under the paper's convention π(0)=0 is outside that domain. The indexing F^ϵ_S places a strict edge at 0 (the edge joining the two halves of the signed chain); if the 0-descent is computed by rank of 0, the middle strict edge in the formula can be wrong exactly when colorings cross zero. The proof also asserts a 'bijection between SS_d and the linear extensions of acyclic orientations of Σ' without proof; that assertion is in fact a true disjoint-union partition (each π induces τπ, and π∈L(P_{τπ})), so it is not the main concern. The unaddressed 0-descent is a genuine gap: the equivalence m∈DES(ωτπ)⇔… is only argued for m=1,…,d−1, and the chosen rank-labeling's behavior on the middle edge is not specified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a chromatic quasisymmetric invariant X_{−→Σ}(x;t) for directed signed graphs, refining the signed chromatic symmetric function of Zaslavsky and giving a signed analogue of Ellzey's directed chromatic quasisymmetric function. The invariant is defined via proper signed colorings weighted by ascents, with a hyperplane-arrangement interpretation. The authors then define and study the algebra SQSym of signed quasisymmetric functions, construct a monomial basis, introduce a fundamental family F indexed by signed chains, and prove a product formula in that family. The central deliverable is Theorem 4.2, which expands X_{−→Σ}(x;t) in the fundamental family with coefficients t^{inv_{−→Σ}(π)} and descent sets DES_Σ(π). Section 5 studies when the invariant lies in SSym[t], proving symmetry for a switched family of circular indifference signed graphs and giving counterexamples to broader symmetry.","tokens_in":19781,"tokens_out":11194,"duration_ms":121286,"significance":"If the main theorem is correct, this is a substantial contribution to the algebraig combinatorics of signed graphs. The invariant naturally specializes to the known signed chromatic symmetric function at t=1 and to the zero-free variant by adding negative loops. The paper provides a new algebra SQSym with a monomial basis, a dimension formula and generating function (3.5), and a fundamental family that interacts well with the expansion of the new invariant. The hyperplane-arrangement viewpoint is well motivated, and the definitions are for the most part clear and self-consistent. The paper gives concrete examples and tables that make the constructions accessible. The main concern is a gap in the proof of the central expansion, which is load-bearing for the paper's main claim.","major_comments":[{"comment":"The Σ-descent at i=0 is not defined: rank(Σ,π)(m) is defined only for m∈{±1,...,±d}, but the i=0 case compares rank(Σ,π)(π_0)=rank(Σ,π)(0). In the proof of Theorem 4.2, the equality DES(ωτπ)=DESΣ(π) is argued only for m=1,...,d-1. For i=0, the middle strict edge of F^{sgn(π)}_{DESΣ(π)} is governed by whether ω(π(1))<0 (see Eq. (3.9)), which depends on the position of π(1) in the order-reversing labeling, not on a rank of 0. The final identification DES(ωτπ)=DESΣ(π) is therefore unverified exactly where F has its critical middle edge. This gap affects the central expansion and needs to be repaired by a correct definition of the 0-descent or by a direct proof of the m=0 equivalence.","section":"Definition 4.1 and Theorem 4.2"},{"comment":"The proof of the symmetry theorem is incomplete. The involution Φ_i is described in three cases, but the rotation argument in the even-length path containing v' is not precisely specified, and the statement 'It is to check that Φ_i is an involution and that the number of ascents is preserved' is not a proof. Moreover, the generation statement 'SS is generated by π_i := (i, i+1)(i, i+1)' is evidently incorrect as written (that product is the identity), and π0 is later written as (1,-1). Since Theorem 5.4 is a stated result of the paper, these details need to be completed and corrected.","section":"Theorem 5.4"}],"minor_comments":[{"comment":"The product formula for monomial signed quasisymmetric functions is stated with 'The proof is omitted'. Since this is a structural property of SQSym, either a proof or a detailed reference to the cited quasi-shuffle argument should be included.","section":"Proposition 3.7"},{"comment":"The condition 'ij < ij+1 if i∈S' should presumably be 'if j∈S'. Please clarify the indexing of strict edges in the formula for F^ϵ_S.","section":"Eq. (3.9)"},{"comment":"The equality asc_{−→Σ}(τπ)=inv_{−→Σ}(π) is asserted as 'not difficult to see' without proof. A short argument should be supplied, since it is part of the main theorem's statement.","section":"Theorem 4.2 proof"},{"comment":"The asserted 'bijection between SS_d and the linear extensions of acyclic orientations of Σ' is stated without justification. It is true (via the disjoint union over τ' of L(P_{τ'},e)), but a sentence explaining the disjoint-union argument would remove ambiguity.","section":"Theorem 4.2 proof"},{"comment":"The table in Example 4.3 is hard to read: the same strings '2112' and '1221' appear repeatedly, but the statistics and sign vectors differ. The signed permutations should be displayed unambiguously (e.g., with explicit overlines or sign vectors) to avoid confusion.","section":"Table 1"},{"comment":"There are several small typos, e.g., the definition of π_i in Section 5 and SQQym/SQqym in Corollary 3.5. A careful proofreading pass is recommended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the journal's scope and the central construction is promising. The main issue is the 0-descent gap in Theorem 4.2; it is local and likely fixable, but it is load-bearing and must be addressed before publication. The symmetry section also needs a complete proof. I do not see signs of circularity or hidden parameter fitting; the invariant is defined independently and the external theorems are used appropriately."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: Aval and Melgar fill an obvious gap in the table by defining a signed chromatic quasisymmetric function and a signed analogue of QSym. The constructions look right, but the proof of the fundamental expansion (Theorem 4.2) has a gap at the i=0 position that needs to be addressed.\n\nWhat's actually new: the invariant X_{\\vec{\\Sigma}}(x;t), its hyperplane chamber interpretation, the algebra SQSym with monomial basis M_{k,\\lambda}, the fundamental family F^\\epsilon_S, the positive product rule, and the symmetry results for a specific family of directed signed graphs. The monomial basis dimension and generating function are concrete and check out. The specialization at t=1 to known invariants is verified.\n\nThe main soft spot is Theorem 4.2. Definition 4.1 defines \\Sigma-descents for positions i=0,...,d-1, but the rank function rank_{(\\Sigma,\\pi)} is only defined for m in {\\pm1,...,\\pm d}. Since \\pi(0)=0, the condition for i=0 uses rank of 0, which is outside the stated domain. The proof's final identification DES(\\omega\\tau_\\pi)=DES_\\Sigma(\\pi) is argued only for positions 1,...,d-1; the middle strict edge in F^\\epsilon_S is exactly what the 0-descent controls. This is a genuine gap, not a manufactured one. It may be fixable by an explicit convention for rank(0), but as written the derivation does not go through. The asserted 'bijection' between SS_d and linear extensions of acyclic orientations is actually fine as a disjoint union; that part is not the problem.\n\nTwo smaller issues: Proposition 3.7's monomial product formula is stated without proof (the authors say it follows from Hoffman/ABB), and Theorem 5.4's involution is sketched. Both are probably true but need details.\n\nBottom line: this is a real contribution to the QSym/signed graph literature. The definitions and the algebra are sound enough to be worth refereeing. But a serious referee should insist on a repaired proof of Theorem 4.2, with the 0-descent case handled explicitly. If that patch works, the paper is publishable. As it stands, my verdict is conditional.","headline":"New signed quasisymmetric invariant and algebra are solid, but the key expansion theorem has a gap at the middle descent.","tokens_in":20162,"tokens_out":4022,"would_cite":true,"duration_ms":43152,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05C22","05C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Signed graphs get a chromatic quasisymmetric invariant that expands over signed permutations in the fundamental family of SQSym.","keywords":["signed graphs","chromatic quasisymmetric functions","signed quasisymmetric functions","hyperplane arrangements","acyclic orientations","(P,ω)-partitions","signed permutations","fundamental basis"],"falsifier":"For a small signed graph, compute both sides of Theorem 4.2: the chamber sum (2.5) and the signed-permutation sum. If they differ for any graph, for example the two-vertex graph with one positive and one negative edge, the expansion fails. A direct count of $|SS_d|$ against the number of pairs (acyclic orientation $\\tau'$, linear extension of $P_{\\tau'}$) used in the proof would also expose the assumed bijection.","tokens_in":19383,"feed_emoji":"🎨","tokens_out":7301,"duration_ms":75209,"temperature":0.7,"texified_at":"2026-08-05T20:16:43.040628+00:00","pith_summary":"The paper sets out to extend the classical chromatic symmetric function and its directed-graph refinement to signed graphs, where edges carry a + or − sign and colorings assign integers with the condition that adjacent vertices cannot have equal or opposite colors according to the edge sign. Its new object is $X_{\\vec{\\Sigma}}(x;t)$, a polynomial in $t$ whose coefficients record, for each proper signed coloring, the number of edges where the coloring disagrees with the chosen orientation; at $t = 1$ it recovers the known signed chromatic symmetric function. The authors show that this invariant lives in $\\mathrm{SQSym}$, the algebra of signed quasisymmetric functions, which they construct with a monomial basis indexed by bicompositions and a fundamental family built from signed $(P,\\omega)$-partitions. The main result is an expansion of $X_{\\vec{\\Sigma}}(x;t)$ over signed permutations in that fundamental family, with the $t$-exponent given by a signing-weighted inversion statistic and the descent set coming from ranks in the associated signed poset. This gives a signed analogue of the directed chromatic quasisymmetric theory and connects signed graph coloring to hyperplane arrangements through the chamber–orientation bijection.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":5826,"prompt_tokens":922,"completion_tokens":4904,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":922,"completion_tokens_details":{"reasoning_tokens":3948}},"feed_headline":"Signed graphs get a chromatic quasisymmetric invariant","feed_subtitle":"A t-weighted sum over signed colorings lands in the new algebra SQSym and expands over signed permutations.","key_machinery":"The load-bearing object is the invariant $X_{\\vec{\\Sigma}}(x;t)$ itself, defined by summing $t^{\\mathrm{asc}(\\kappa)} x^\\kappa$ over proper signed colorings, where $\\mathrm{asc}(\\kappa)$ counts edges $e=\\{u,v\\}$ on which $\\tau(u,e)\\kappa(u)+\\tau(v,e)\\kappa(v)>0$. It is a signed quasisymmetric function, an element of the algebra $\\mathrm{SQSym}$ of bounded-degree series in variables $(\\ldots, x_{-1}, x_0, x_1, \\ldots)$ that are invariant under the quasisymmetrizing action of the symmetric group. The expansion is carried by the fundamental family $F^\\varepsilon_S$, the $(P,\\omega)$-partition enumerators of signed labeled chains: each $F^\\varepsilon_S$ is a sum over weakly increasing index sequences with strict steps prescribed by $S$ and sign pattern $\\varepsilon$. The proof machinery is the hyperplane arrangement $H_\\Sigma$ who","core_discovery":"On the paper's own terms, the central claim is that every directed signed graph $\\vec{\\Sigma} = (\\Sigma, \\tau)$ on $d$ vertices has a canonical $t$-refinement $X_{\\vec{\\Sigma}}(x;t) = \\sum_{\\kappa \\text{ proper}} t^{\\mathrm{asc}(\\kappa)} x^\\kappa$ of the signed chromatic symmetric function, and that this refinement expands as $X_{\\vec{\\Sigma}}(x;t) = \\sum_{\\pi \\in SS_d} t^{\\mathrm{inv}_{\\vec{\\Sigma}}(\\pi)} F^{\\mathrm{sgn}(\\pi)}_{\\mathrm{DES}_\\Sigma(\\pi)}$ in the fundamental family of $\\mathrm{SQSym}$, where $\\mathrm{inv}$ counts directed inversions of $\\pi$ in the symmetric double cover of $\\Sigma$ and $\\mathrm{DES}_\\Sigma(\\pi)$ is a descent set defined by ranks in the associated signed poset. The proof interprets the invariant chamber-by-chamber through the hyperplane arrangement $H_\\Sigma$, expresses each chamber's contribution as a signed $(P,\\omega)$-partiti","pith_inferences":["Editorial: the proof of the main expansion assumes a bijection between signed permutations and linear extensions of acyclic orientations of the signed graph; the paper does not prove this set-theoretic step, so the expansion's correctness rests on that missing verification.","If the fundamental expansion survives scrutiny, SQSym is likely to fit into a type-B or free-quasisymmetric-function framework, giving a signed analogue of the quasi-shuffle Hopf algebra that the paper only sketches.","A natural testable extension is to ask whether X_{\\vec\\Sigma}(x;t) distinguishes signed graphs that the t=1 specialization cannot, mirroring the open distinguishing-power questions for the unsigned invariant.","Setting x_0=0 recovers the zero-free signed coloring invariant; the expansion then suggests a zero-free analogue of the (Σ,π)-descent statistics, which could be studied combinatorially on its own."],"forward_implications":["Specializing t=1 turns X_{\\vec\\Sigma}(x;t) into the known signed chromatic symmetric function, so the new invariant is a genuine refinement of signed graph coloring.","The fundamental-family expansion gives a positive formula for X_{\\vec\\Sigma} in terms of signed permutations, making the invariant algorithmically computable from orientation data alone.","SQSym is a graded algebra with monomial basis indexed by bicompositions, and its Hilbert series (1−t)/(1−4t+2t^2) gives dimensions 1, 3, 10, 34, ...; the product of fundamental elements is governed by a shuffle rule.","For acyclic orientations containing at least one negative edge, X_{\\vec\\Sigma}(x;t) is never signed-symmetric, so full symmetry is restricted to special families such as the switched circular indifference digraphs studied in the paper."],"supporting_citations":[{"why":"defines the chromatic symmetric function that the new invariant refines.","marker":"[S95]"},{"why":"introduces chromatic quasisymmetric functions whose t-refinement is the unsigned model for the construction.","marker":"[SW16]"},{"why":"supplies the directed-graph version of the invariant and the fundamental-expansion proof that Theorem 4.2 adapts.","marker":"[E17]"},{"why":"defines proper colorings of signed graphs and the orientation-coloring compatibility convention used in Definition 2.14.","marker":"[Z82]"},{"why":"gives the bijection between chambers of the signed hyperplane arrangement and acyclic orientations, the geometric basis for the chamber sum.","marker":"[Z91]"},{"why":"studies the algebra SSym and the signed chromatic symmetric function X_Σ(x) that X_{\\vec\\Sigma}(x;1) recovers.","marker":"[KT21]"},{"why":"provides the classical chamber–acyclic orientation bijection used for the unsigned case and as model.","marker":"[G77]"}],"fun_headline_variants":["Signed graphs gain a t-refined chromatic invariant","New invariant for signed graphs lands in SQSym","t-weighted signed colorings define a quasisymmetric invariant","Signed graphs get a chromatic invariant in new algebra SQSym"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proof of the main expansion moves from a sum over acyclic orientations to a sum over all signed permutations by relying on an unproved bijection between signed permutations and linear extensions of acyclic orientations of the signed graph; if that bijection is wrong, the formula double-counts or misses terms.","fun_headline_variants_meta":{"raw":{"variants":["Signed graphs gain a t-refined chromatic invariant","New invariant for signed graphs lands in SQSym","t-weighted signed colorings define a quasisymmetric invariant","Signed graphs get a chromatic invariant in new algebra SQSym"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000215,"raw_usage":{"total_tokens":1245,"prompt_tokens":701,"completion_tokens":544,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":445,"completion_tokens_details":{"reasoning_tokens":479}},"tokens_in":445,"tokens_out":544,"duration_ms":5880,"temperature":1.0,"reasoning_tokens":479,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T15:13:42.853062+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a small signed graph, compute both sides of Theorem 4.2: the chamber sum (2.5) and the signed-permutation sum. If they differ for any graph, for example the two-vertex graph with one positive and one negative edge, the expansion fails. A direct count of $|SS_d|$ against the number of pairs (acyclic orientation $\\tau'$, linear extension of $P_{\\tau'}$) used in the proof would also expose the assumed bijection.","supporting_citations":[],"review_version":1}