{"id":"0c615da0-c4d4-4022-b345-56a426bb7452","arxiv_id":"2508.09107","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For fireworks permutations, the support of a Grothendieck polynomial is exactly the union, over Schubert monomials, of the componentwise intervals up to the top weight vector.","lead":"This paper proves an exact formula for which monomials appear in Grothendieck polynomials attached to fireworks permutations. For this class, the support is the full interval between a Schubert monomial and the maximal degree vector, so the Newton polytope is a generalized permutahedron.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3's proof begins with an unproved existence assertion about a row-a bump tile whose primary pipe is not a left-to-right maximum; the raising construction and hence Theorem 1.1 depend on it.","rationale":"The reader's weakest assumption identified the local pipe-dream topology in Theorem 1.3, especially the implicit counting claim about bump tiles. I agree with that general assessment, but the more precise stress point is the first sentence of the proof: the existence of the row-a bump tile with a non-left-to-right-maximum primary pipe is a substantial unproved existence statement, not just a detail of a figure. If it fails, the whole raising construction collapses. The same proof also uses an unproved assertion about P(ℓ) being primary at row a in Case 2. I do not see a clear counterexample in the text, and the construction may well be correct; the issue is that the reader cannot verify it from the written argument. The remaining parts of the paper—M-convexity in Corollary 1.2 and the layered containment in Corollary 1.4—are conditional on Theorem 1.1, so they inherit this risk. There is no formal verification and no code. Weighing all this, the concern is serious enough to justify the reader's CONDITIONAL verdict but not to reject the paper: the missing argument is a lemma about a finite, checkable combinatorial object, and the exhaustive small-n test I propose would settle whether the asserted existence actually holds. My read therefore leaves the verdict unchanged.","tokens_in":13375,"tokens_out":15993,"duration_ms":189854,"concrete_test":"Write an exhaustive checker over all pipe-dream tilings for n=6 (2^15 = 32768 tilings, or 2^21 for n=7) and, for every fireworks permutation w in S_n and every pipe dream P with wt(P)_a < wt(D(w))_a, test the two assertions: (1) some bump tile in row a has a primary pipe that is not a left-to-right maximum of w(1)...w(n); (2) in the Case 2 configuration, the secondary pipe at the highest real crossing below row a is indeed primary at a tile in row a. If any instance fails, Theorem 1.3 is false; if all pass through n=6, the missing counting lemma is strongly supported and the proof gap can be treated as expository.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the first sentence of the proof of Theorem 1.3 (Section 3): 'As wt(P)_a < wt(D(w))_a, there exists a primary pipe P(i) of a bump tile T in row a such that i is not a left-to-right maximum in w(1)...w(n).' This is asserted without proof, and it is not an immediate consequence of Proposition 2.10 alone. Proposition 2.10 counts, for each a, the number of non-initial-term positions j>a, but wt(P)_a counts cross tiles physically located in row a of the particular pipe dream P. A non-initial pipe may be primary in a different row, so the deficit in row a does not automatically place that pipe as a bump tile in that row. The subsequent case analysis starts from this T, so if the asserted bump tile does not exist, the raising construction cannot begin and Theorem 1.1 has no engine. A second unproved local assertion occurs in Case 2, where the proof states that the secondary pipe P(ℓ) of the highest crossing T' is primary for some tile S in row a; this is used to replace S with a cross tile, but it is not derived from Lemmas 3.1–3.3 in the text. Both are omitted counting/topology arguments rather than formal consequences of the stated lemmas. Since the paper has no machine-checked verification and the lemmas are justified by figures and prose, these gaps are the main risk to the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Newton polytopes of Grothendieck polynomials for fireworks permutations. Its main result, Theorem 1.1, asserts that the support of such a Grothendieck polynomial is as large as possible subject to known bounds: a monomial appears exactly when it divides x^{wt(\\overline{D(w)})} and is divisible by some monomial in the corresponding Schubert polynomial. The proof is built on a pipe-dream raising operation (Theorem 1.3) that increases the weight of a fixed row without changing the permutation. From the support formula the authors derive two corollaries: the homogenized Grothendieck polynomial has M-convex support, and for every fireworks permutation w there is a layered permutation whose Grothendieck polynomial has support containing that of w.","tokens_in":13805,"tokens_out":23929,"duration_ms":236169,"significance":"If the central theorem is correct, it gives a complete support description for a substantial family of Grothendieck polynomials, extending the known SNP and M-convexity results for Schubert polynomials and for vexillary Grothendieck polynomials. The resulting M-convexity of the homogenized support and the layered-permutation containment are natural and interesting consequences. The paper works with explicit pipe dreams and elementary polytopal tools, so the intended audience is broad. The main risk is that the proof of the key raising lemma, Theorem 1.3, relies on unproved local assertions about pipe-dream topology; these assertions are load-bearing for the main theorem.","major_comments":[{"comment":"The proof begins with the assertion that if wt(P)_a < wt(\\overline{D(w)})_a, then there exists a primary pipe P(i) of a bump tile T in row a such that i is not a left-to-right maximum. This is load-bearing: the entire raising construction starts from such a T. The assertion is not an immediate consequence of Proposition 2.10, which counts non-initial-term positions j>a, not primary pipes of bump tiles in row a. A simple counting check shows the gap: for the fireworks permutation w=31542 and a=3, wt(\\overline{D(w)})_3=2, while the number of non-left-to-right-maximum pipes with exit row greater than 3 is 1. Thus the deficit in the bound does not automatically produce the claimed bump tile. The authors should supply a rigorous counting or geometric argument, or state and prove the precise lemma they are using.","section":"Section 3, proof of Theorem 1.3, first sentence"},{"comment":"The case analysis asserts that 'The pipe P(ℓ) is primary for some tile S in row a' and then, using Lemma 3.1(1), that S is a bump tile. The first assertion is plausible because P(ℓ) has exit row greater than a, but it is not stated or proved; the second should be a formal application of Lemma 3.1 rather than an implicit step. In addition, the later reference to P(ℓ') is undefined: the secondary pipe of S was earlier called P(m), and the text then uses P(ℓ') without introducing it. The authors should rewrite this case so that every pipe label is defined and every use of Lemmas 3.1-3.3 is explicit.","section":"Section 3, Case 2 of proof of Theorem 1.3"},{"comment":"The sentence 'By repeated application of Theorem 1.3, there exists a pipe dream Q with x^{wt(Q)} = x_i x^\\alpha' is too terse, because Theorem 1.3 allows weights in rows b>a to decrease. To reach the exact vector \\alpha+ e_i, one must apply the theorem first to coordinate i and then restore rows n, n-1, ..., i+1 from right to left, since increasing a row b>i does not affect any row to its left. This ordering should be stated. Without it, the reader cannot verify that the desired monomial is obtained without further changes.","section":"Section 1, proof of Theorem 1.1"}],"minor_comments":[{"comment":"The sentence 'Every monomial appearing in any Grothendieck polynomial G_w divides x^{wt(\\overline{D(w)})}' is used in the statement of Theorem 1.1 but is neither proved nor cited. Please add a reference to a standard source or a short proof.","section":"Section 1, Introduction"},{"comment":"The statement contains grammatical and omission errors: 'the two pipes involved tiles T and T\\'' should be 'involved in tiles T and T\\'', and 'let denote the region R' should be 'let R denote the region'.","section":"Section 3, Lemma 3.3"},{"comment":"The symbol P(ℓ') is used in the bulleted construction and in Figure 8 but is never defined. It should either be replaced by P(m), the secondary pipe of S, or defined explicitly before use.","section":"Section 3, Case 2 of proof of Theorem 1.3"},{"comment":"In the displayed formula, 'Psp(SMdj (Dj))' is a typo; it should read Psp(SM(D_j)).","section":"Section 4, Proposition 4.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct in its broad strategy, and the results are valuable if the gaps are filled. The main risk is the proof of Theorem 1.3, where the initial existence assertion and the Case 2 pipe identifications are not fully justified. I would be willing to see a revised version with those arguments made rigorous; if they cannot be supplied, the main theorem would be unsupported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this paper proves the exact support description for fireworks Grothendieck polynomials, confirming the MSS25 conjecture in that case; the engine is a new pipe-dream raising operation (Theorem 1.3). Second, the proof of that engine has a couple of places where the authors assert existence of certain tiles without proof. I think these are fillable gaps rather than fatal flaws, but they are exactly what a referee should push on.\n\nWhat is actually new: the support formula itself, and the raising construction. The corollaries—M-convexity of the homogenized support and the layered containment—follow cleanly once Theorem 1.1 is in hand. The M-convexity part is a nice application of known generalized permutahedron facts; the layered containment is a tidy polytopal argument. Credit is earned: the paper does not rely on the conjecture, and the main theorem is a genuine new result for a nontrivial class of permutations.\n\nWhere I worry: the first sentence of the proof of Theorem 1.3 asserts that a deficit in row a weight gives a bump tile in that row whose primary pipe is not a left-to-right maximum. That is not immediate from Proposition 2.10, and it is load-bearing: the whole raising construction starts from that tile. A second, similar assertion appears in Case 2, where the pipe P(l) is said to be primary for some tile S in row a; this is also not derived from the stated lemmas. The proofs of Lemmas 3.1–3.3 are figure-based and terse; they may be correct, but they do not yet constitute a complete argument. None of this undermines the plausibility of the main theorem, but the paper as written has a real gap in the core proof.\n\nBottom line: this deserves a serious referee, but the referee should be instructed to ask for a rigorous proof of the two existence assertions (or a rewrite of Section 3 with more detail). If those steps check out, the paper is a solid contribution to Schubert calculus and polytope combinatorics. I would not cite it until the proof is fully settled.","headline":"The support formula for fireworks Grothendieck polynomials is new and likely correct, but the proof of the key raising lemma has two unproved existence assertions that a referee should push on.","tokens_in":14184,"tokens_out":2789,"would_cite":false,"duration_ms":29923,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","14M15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For fireworks permutations, a Grothendieck monomial appears exactly when it lies between a Schubert monomial and the top-degree bound.","keywords":["Grothendieck polynomials","Newton polytopes","support of polynomials","pipe dreams","fireworks permutations","M-convex sets","generalized permutahedra","Schubert polynomials"],"falsifier":"Compute the Grothendieck polynomial $\\mathfrak{G}_w$ for every fireworks permutation of small size, for example $n \\le 6$, and compare $\\mathrm{supp}(\\mathfrak{G}_w)$ with the union over $\\alpha \\in \\mathrm{supp}(\\mathfrak{S}_w)$ of $[\\alpha, \\mathrm{wt}(\\overline{D(w)})]$. If any monomial that divides $\\mathbf{x}^{\\mathrm{wt}(\\overline{D(w)})}$ and is divisible by some Schubert monomial has coefficient zero in $\\mathfrak{G}_w$, the theorem is false; even one minimal counterexample would settle it.","tokens_in":13179,"feed_emoji":"🎆","tokens_out":8517,"duration_ms":77715,"temperature":0.7,"pith_summary":"Grothendieck polynomials refine Schubert polynomials in K-theory, but their monomial supports are harder to describe. This paper proves that for fireworks permutations the support is as large as possible: a monomial appears in the Grothendieck polynomial precisely when it divides the maximal weight vector attached to the upward closure of the Rothe diagram and is divisible by some monomial of the corresponding Schubert polynomial. That interval-union description identifies the Newton polytope as a Minkowski sum of Schubert spanning set polytopes, one per column of the diagram. A reader should take away a complete, computable polytopal model for this family, along with two structural corollaries: the homogenized polynomial has M-convex support, and for every fireworks permutation there is a layered permutation whose support contains it.","feed_headline":"Fireworks Grothendieck support is exactly the full interval","feed_subtitle":"For fireworks permutations, a monomial appears exactly when it lies between a Schubert monomial and the top-degree bound.","key_machinery":"The engine of the proof is a local surgery on pipe dreams, which are tilings of a staircase grid by cross tiles and bump tiles that encode the monomials of Schubert and Grothendieck polynomials. Theorem 1.3 states that for a fireworks permutation, whenever a pipe dream's weight vector lies strictly below the maximal weight $\\mathrm{wt}(\\overline{D(w)})$ in some coordinate $a$, there is another pipe dream for the same permutation whose first $a-1$ weights agree, whose $a$-th weight is one larger, and whose later weights are no larger. The surgery replaces a carefully chosen bump tile in row $a$ by a cross tile, then repairs the pipe network below by flipping certain crossings to bumps; Lemmas 3.1, 3.2, and 3.3 control which pipes can pass between two boundary tiles so that the permutation is unchanged. Iterating this raising operation from reduced pipe dreams, which give the Schubert monomials, fills every lattice point between a Schubert exponent and the maximal vector, which is exactly the support formula.","core_discovery":"On the paper's own terms, the central result is Theorem 1.1: if $w$ is fireworks, then $\\mathrm{supp}(\\mathfrak{G}_w)$ equals the union over all Schubert monomial exponents $\\alpha \\in \\mathrm{supp}(\\mathfrak{S}_w)$ of the componentwise intervals $[\\alpha, \\mathrm{wt}(\\overline{D(w)})]$, where $\\overline{D(w)}$ is the upward closure of the Rothe diagram. Equivalently, every monomial that divides $\\mathbf{x}^{\\mathrm{wt}(\\overline{D(w)})}$ and is divisible by some monomial of $\\mathfrak{S}_w$ occurs with nonzero coefficient in $\\mathfrak{G}_w$. The proof builds, for any pipe dream of weight below the maximum, another pipe dream of the same permutation that increases one chosen row's crossing count by one without increasing later rows; iterating this surgery fills out the whole interval. From this formula the paper derives that the homogenized Grothendieck polynomial has M-convex support, hence a saturated Newton polytope and a generalized-permutahedron Newton polytope, and that the support is contained in that of the layered permutation whose block sizes are the lengths of the descending runs of $w$.","pith_inferences":["If the same interval-union principle holds beyond fireworks, it would give a uniform explanation of the known M-convexity cases; the natural test case is a permutation whose top-degree Grothendieck part is a single monomial, where the inclusion in Remark 4.5 would become an equality.","The column-by-column Minkowski description suggests an efficient membership algorithm: check the Schubert support via matroid polytopes and then check the upper bound; the paper does not discuss complexity, but the structure is algorithmically friendly.","The layered containment may be strict in general, and comparing it with known extremal behavior of layered permutations in Schubert calculus suggests that fireworks Grothendieck polynomials are also extremal for support size, a statement the paper does not make.","One could test the sharpness of the fireworks hypothesis by searching for a non-fireworks permutation for which the union of intervals differs from the actual support; a positive example would delineate exactly why the pipe-dream surgery needs the 3-12 avoidance condition."],"forward_implications":["For fireworks $w$, membership in $\\mathrm{supp}(\\mathfrak{G}_w)$ is decidable by two divisibility checks: $\\mathbf{x}^\\alpha$ divides $\\mathbf{x}^{\\mathrm{wt}(\\overline{D(w)})}$ and some Schubert monomial divides $\\mathbf{x}^\\alpha$.","The Newton polytope of $\\mathfrak{G}_w$ equals the Minkowski sum of Schubert spanning set polytopes attached to the columns of the Rothe diagram, intersected with $\\mathbb{Z}^n$; hence it is computable column by column.","The homogenized Grothendieck polynomial of a fireworks permutation has M-convex support, so its Newton polytope is a generalized permutahedron and the polynomial has the saturated Newton polytope property.","For every fireworks $w$, the layered permutation $\\pi(w)$ with block sizes equal to the descending runs of $w$ satisfies $\\mathrm{supp}(\\mathfrak{G}_{\\pi(w)}) \\supseteq \\mathrm{supp}(\\mathfrak{G}_w)$.","The maximal monomial $\\mathbf{x}^{\\mathrm{wt}(\\overline{D(w)})}$ appears in $\\mathfrak{G}_w$ for fireworks $w$, recovering a known existence statement as a byproduct of the construction."],"supporting_citations":[{"why":"Supplies the pipe dream formula expressing Schubert and Grothendieck polynomials as sums over pipe dreams, which translates support questions into pipe dream weights.","marker":"[FK94]"},{"why":"Gives the Schubert polynomial support as a Minkowski sum of Schubert matroid polytopes, the baseline description Theorem 1.1 extends to Grothendieck polynomials.","marker":"[FMS18]"},{"why":"States the conjectural interval formula for Grothendieck support and the containment result the paper refines, plus the top-monomial existence result Theorem 1.3 reproves.","marker":"[MSS25]"},{"why":"Provides the crossing lemma about pipes enclosed between two tiles that underpins the local surgery in Theorem 1.3.","marker":"[CY25]"},{"why":"Supplies the Minkowski-sum and integer-decomposition facts for generalized permutahedra used to prove M-convexity.","marker":"[Sch03]"},{"why":"Shows homogenized Schubert spanning sets are M-convex, the ingredient that makes the column-by-column sum a generalized permutahedron.","marker":"[Fuj84]"}],"fun_headline_variants":["Fireworks Grothendieck supports fill the full interval","Exact support formula for fireworks Grothendieck polynomials","M-convex support for fireworks Grothendieck polynomials","Fireworks supports are Schubert-to-max intervals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole interval description rests on the geometric fact that in a fireworks pipe dream, whenever a row has fewer crossings than the maximum, a suitable bump tile exists in that row whose primary pipe is not a left-to-right maximum; if that local tile search ever fails, the weight-raising construction stops and Theorem 1.1 does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Fireworks Grothendieck supports fill the full interval","Exact support formula for fireworks Grothendieck polynomials","M-convex support for fireworks Grothendieck polynomials","Fireworks supports are Schubert-to-max intervals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000366,"raw_usage":{"total_tokens":1954,"prompt_tokens":917,"completion_tokens":1037,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":972}},"tokens_in":533,"tokens_out":1037,"duration_ms":8844,"temperature":1.0,"reasoning_tokens":972,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:32:59.271799+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Grothendieck polynomial $\\mathfrak{G}_w$ for every fireworks permutation of small size, for example $n \\le 6$, and compare $\\mathrm{supp}(\\mathfrak{G}_w)$ with the union over $\\alpha \\in \\mathrm{supp}(\\mathfrak{S}_w)$ of $[\\alpha, \\mathrm{wt}(\\overline{D(w)})]$. If any monomial that divides $\\mathbf{x}^{\\mathrm{wt}(\\overline{D(w)})}$ and is divisible by some Schubert monomial has coefficient zero in $\\mathfrak{G}_w$, the theorem is false; even one minimal counterexample would settle it.","supporting_citations":[],"review_version":2}