{"id":"f4299bc9-7c2d-42a7-8874-3b277b499e89","arxiv_id":"2508.00157","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A weighted analogue of the chromatic symmetric function, valued in MacMahon symmetric functions on two alphabets, determines, for every tree, the generating function of vertex subsets by cardinality, weight, and internal and external edge counts.","lead":"This paper defines a new counting invariant for graphs with weighted vertices that records both color and weight information. It proves that for trees, this invariant fully determines how many vertex subsets have each possible size, total weight, and pattern of inside and outside edges.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the theorem is plausible from the abstract, but the unreadable full text prevents verification of the two-alphabet independence.","rationale":"The reader's verdict is UNVERDICTED because the text was unreadable; my stress test is similarly limited. In good faith, the abstract describes a theorem that continues a line of work (Crew's conjecture, proven by Aliste-Prieto--Martin--Wagner--Zamora and Liu--Tang) and a new invariant that plausibly strengthens the chromatic symmetric function. I cannot find a specific mathematical flaw. The closest thing to a load-bearing risk is the independence of the two alphabets: if the MacMahon function cannot separate monomials in the first alphabet from exponents in the second, the recovery of the weighted subset generating function would fail. But nothing in the abstract suggests such a collapse; MacMahon symmetric functions under the diagonal action are designed to track joint distributions of two alphabets. Thus the correct response is to leave the verdict unchanged and to propose a computational verification that would settle the concern once a readable manuscript is available.","tokens_in":1702,"tokens_out":8884,"duration_ms":93295,"concrete_test":"Obtain a readable version of arXiv:2508.00157 and, for all unlabeled trees on n≤7 vertices with small integer vertex weights, compute both the chromatic symmetric MacMahon function (from the paper's Definition) and the weighted subset generating function by cardinality, weight, internal edges, and external edges. Verify that the map from MacMahon function to subset generating function is well-defined and injective on this test set; in particular, confirm that distinct weighted subset generating functions yield distinct MacMahon functions and that the paper's recovery formula reproduces the generating function.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a generalization of the unweighted Crew-conjecture theorem, and the two-alphabet MacMahon construction is a natural mechanism for carrying vertex weights. I find no internal inconsistency or missing hypothesis that is visible from the abstract. The one load-bearing premise the reader flags—that the two alphabets remain genuinely independent so that cardinality and weight information can be separated—cannot be checked because the supplied full text is a corrupted encoding. That is an epistemic gap, not a demonstrated flaw; the theorem might well be true. I therefore state an honest non-finding rather than manufacture a mathematical objection.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a chromatic symmetric MacMahon function for vertex-weighted graphs, a two-alphabet symmetric function invariant of the diagonal action, and claims that for trees this invariant determines the generating function for vertex subsets by cardinality, weight, and the numbers of internal and external edges. The result is presented as a generalization of the unweighted Crew-conjecture theorem proved by Aliste-Prieto--Martin--Wagner--Zamora and Liu--Tang.","tokens_in":1784,"tokens_out":2433,"duration_ms":26868,"significance":"If the main theorem is correct, it extends a significant line of research on chromatic symmetric functions from unweighted to vertex-weighted graphs, with a genuinely new two-alphabet invariant. The claimed implication is strong: it would encode both the cardinality and the weight profile of every vertex subset along with its internal and cut edge counts, making the invariant at least as informative for weighted trees as the ordinary chromatic symmetric function is for unweighted trees. The construction is natural and the abstract is clean, but the supplied full text is entirely unreadable, so the proof cannot currently be checked. The significance is therefore conditional on a verifiable manuscript.","major_comments":[{"comment":"The supplied full text is a corrupted encoding (mojibake) with no readable section, equation, or argument. I cannot verify the proof of the central theorem, nor the definitions and lemmas that would support it. This is load-bearing because the claimed two-alphabet independence and the extraction of the weighted subset generating function are precisely the steps that require careful checking. The authors must resubmit a clean, readable version before the paper can be evaluated.","section":"Full text (entire manuscript)"},{"comment":"The abstract does not specify the nature of the vertex weights: are they formal variables, generic values, or arbitrary commutative coefficients? The statement 'determines the generating function by cardinality, weight, and the numbers of internal and external edges' is ambiguous without knowing whether the implication is an equality of full generating functions over two alphabets or an equality of evaluations for fixed weights. Please state the weighting hypothesis precisely in the main theorem.","section":"Abstract and Theorem statement"},{"comment":"A key premise of the claimed result is that the two alphabets in the MacMahon function remain genuinely independent throughout the proof, so that distinct weighted colorings cannot collapse to the same power series. No visible argument establishes this independence. Please provide an explicit, labeled argument showing that the cardinality and weight information can be separated and recovered from the invariant.","section":"Two-alphabet independence"}],"minor_comments":[{"comment":"Please include a definition of the diagonal action and of MacMahon symmetric functions in the introduction, since the abstract assumes familiarity with that setting.","section":"Introduction (anticipated)"},{"comment":"The abstract names Crew's conjecture but does not give a citation to Crew's original paper; please add the reference in the bibliography.","section":"References"},{"comment":"Consider defining 'internal edges' and 'external edges' explicitly in the theorem statement; the abstract uses these terms without formal definition.","section":"Notation"}],"recommendation":"major_revision","confidential_remarks":"The sole blocker is the unreadable full text: the mathematical claim is plausible and the abstract is well-formed, but no proof can be verified. I recommend requesting a clean copy and then evaluating the two-alphabet independence argument carefully. Major revision rather than rejection is appropriate because the issues appear fixable by resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The abstract announces a new invariant—a chromatic symmetric MacMahon function on two alphabets—and proves that for a vertex-weighted tree it determines the generating function for subsets by cardinality, weight, internal edges, and cut edges. That is a genuine extension of the unweighted Crew-conjecture theorem, and it is not in the cited literature. The construction is natural: one alphabet tracks vertex counts, the other tracks weights. The authors are also upfront that this generalizes prior work, and they cite both independent proofs of the unweighted case. No circularity problem; Martin's earlier proof covers the base case, not this new claim.\n\nThe main problem is that the full text I was given is a corrupted encoding, so I cannot examine the proof. The abstract alone does not show how vertex weights are formalized—whether as formal variables, generic values, or arbitrary numbers—nor does it demonstrate that the two alphabets stay independent so that cardinality and weight information do not collapse into the same power series. The stress-tester found no visible inconsistency, and I see none either. This is an epistemic gap, not a detected flaw. If the proof follows the established tree-decomposition template, the result is very likely correct.\n\nWhat the paper does well, aside from the theorem, is situate itself cleanly in the literature and state a precise, checkable result. The proposed invariant is a sensible addition to the symbolic toolbox for weighted graph enumeration. It will not reshape mathematics, but it should be useful to people working on chromatic symmetric functions and related invariants.\n\nIf the actual arXiv PDF is readable, I would send this to a referee in algebraic combinatorics. The claim is narrow but well-motivated, and the generalization is exactly the kind of incremental progress that deserves verification. I would not desk reject. My own verdict is unverdictable from the artifact I received, but leaning positive.\n\nFor you: probably not worth reading unless you work in this area. If you do, the theorem is worth knowing and citing once the proof is verified.","headline":"A clean, natural generalization of the Crew-conjecture theorem to vertex-weighted trees via a two-alphabet MacMahon invariant; the proof is uncheckable in the copy I have, but the claim is plausible and deserves a referee.","tokens_in":2291,"tokens_out":1543,"would_cite":false,"duration_ms":17546,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05C15","05C05","05A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The chromatic symmetric MacMahon function of a vertex-weighted tree determines the generating function that counts every vertex subset by cardinality, total weight, and numbers of internal and external edges.","keywords":["chromatic symmetric function","MacMahon symmetric function","vertex-weighted graphs","proper colorings","trees","generating functions","symmetric group action"],"falsifier":"Compute the chromatic symmetric MacMahon function for all vertex-weighted trees on, say, seven vertices with generic, formally independent vertex weights, and compare it with the weighted subset generating function; if two trees share the invariant but have different subset generating functions, the theorem is refuted.","tokens_in":1280,"feed_emoji":"🌲","tokens_out":1782,"duration_ms":113498,"temperature":0.7,"pith_summary":"This paper introduces the chromatic symmetric MacMahon function, a two-alphabet invariant of vertex-weighted graphs that records, for every proper coloring, both the sizes and the weights of the color classes. The paper's central claim is that for a tree this single invariant determines the generating function that enumerates all vertex subsets by cardinality, total weight, number of internal edges, and number of external edges. If the claim is correct, the new invariant carries the extra vertex-weight information that the ordinary chromatic symmetric function ignores, making it a strictly richer object for weighted trees. The theorem also extends the previously proved unweighted tree result, showing that the weighted case is a genuine generalization rather than a separate problem.","feed_headline":"One symmetric function encodes every weighted subset of a tree","feed_subtitle":"The chromatic symmetric MacMahon function records cardinality, weight, and edge counts of every vertex subset of a tree.","key_machinery":"The carrying object is the chromatic symmetric MacMahon function itself: a formal power series in two alphabets of variables, invariant under the symmetric group acting diagonally on the two alphabets, obtained by summing over all proper colorings of the vertex-weighted graph a monomial per color class that records the class's size in the first alphabet and its total vertex weight in the second. The key structural fact used by the proof is that this two-alphabet encoding keeps cardinality data and weight data in separate coefficients, so the coefficient of each monomial in the first alphabet is a polynomial in the second alphabet. The theorem is the statement that for trees this coefficient structure is exactly sharp enough to recover the weighted subset generating function with its internal and external edge counts.","core_discovery":"The central discovery is that the chromatic symmetric MacMahon function of a tree is a complete package for the tree's weighted subset census. Concretely, the paper proves that from the invariant $\\Psi_T(\\mathbf{x};\\mathbf{y})$, which sums over proper colorings a monomial recording each color class's size in one alphabet and its total vertex weight in the other, one can recover the enumerator $\\sum_{S\\subseteq V(T)} u^{|S|} t^{\\mathrm{wt}(S)} p^{e_{\\mathrm{int}}(S)} q^{e_{\\mathrm{ext}}(S)}$, where $\\mathrm{wt}(S)$ is the total weight of $S$, $e_{\\mathrm{int}}(S)$ counts edges with both endpoints in $S$, and $e_{\\mathrm{ext}}(S)$ counts edges crossing the cut $(S,V(T)\\setminus S)$. The two alphabets play complementary roles, one carrying color-class cardinalities and the other carrying vertex weights, and for trees no information is lost in the passage from colorings to subsets. This generalizes the unweighted theorem that the ordinary chromatic symmetric function of a tree determines its vertex-subset enumeration data.","pith_inferences":["A natural next question, not settled by this paper, is whether generically chosen vertex weights make the weighted subset generating function itself a complete invariant of the tree; the two-alphabet invariant constructed here is the natural tool for testing that.","Because the subset census records internal and external edge counts for every subset, the invariant sits close to Tutte-polynomial-style data, and an explicit specialization connecting the MacMahon function to a weighted Tutte polynomial would be a natural next step.","One could compute the invariant for all small vertex-weighted trees with independent formal weights and check empirically whether the map from weighted trees to two-alphabet series is injective; the theorem guarantees at least the one-way determination proved here."],"forward_implications":["Two vertex-weighted trees with different weighted subset generating functions must have different chromatic symmetric MacMahon functions, so the invariant separates any pair of trees that the subset census separates.","Setting all vertex weights equal recovers the unweighted theorem for trees: the MacMahon function reduces to the ordinary chromatic symmetric function and the subset census reduces to the unweighted subtree information.","For a single tree, the entire collection of subset data, including size, weight, internal edges, and external edges, can in principle be extracted from one two-alphabet series without listing all $2^n$ subsets separately.","The weighted generalization places the weighted tree case on the same footing as the unweighted tree case, so the remaining open territory for this style of invariant lies in graphs with cycles."],"supporting_citations":[],"fun_headline_variants":["MacMahon function reveals every weighted subset of a tree","Chromatic MacMahon: a tree's complete weighted subset census","For trees, one MacMahon function stores all subset data","Weighted tree subsets: captured by a single symmetric function","From MacMahon to tree census: cardinality, weight, edges"],"cache_read_input_tokens":4608,"weakest_assumption_plain":"The proof rests on the assumption that the two sets of variables in the invariant remain truly independent, so that distinct weighted colorings cannot collapse to the same series and the weight information can still be read off coefficient by coefficient.","fun_headline_variants_meta":{"raw":{"variants":["MacMahon function reveals every weighted subset of a tree","Chromatic MacMahon: a tree's complete weighted subset census","For trees, one MacMahon function stores all subset data","Weighted tree subsets: captured by a single symmetric function","From MacMahon to tree census: cardinality, weight, edges"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000483,"raw_usage":{"total_tokens":2357,"prompt_tokens":891,"completion_tokens":1466,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":507,"completion_tokens_details":{"reasoning_tokens":1383}},"tokens_in":507,"tokens_out":1466,"duration_ms":10160,"temperature":1.0,"reasoning_tokens":1383,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:20:49.224296+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the chromatic symmetric MacMahon function for all vertex-weighted trees on, say, seven vertices with generic, formally independent vertex weights, and compare it with the weighted subset generating function; if two trees share the invariant but have different subset generating functions, the theorem is refuted.","supporting_citations":[],"review_version":1}