{"id":"5540796f-d7bf-40e3-8b04-7447381530a4","arxiv_id":"2512.04053","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The maximal support size of Schubert polynomials is asymptotically between n!/4^n and n!, and for Grothendieck polynomials it is n! times a subexponential factor.","lead":"This preprint proves that the maximum possible number of monomials in an n-variable Schubert polynomial grows like n!, up to a factor of at most 4^n, and that the analogous maximum for Grothendieck polynomials is n! up to a subexponential factor. The proof constructs layered permutations whose Rothe diagrams force many disjoint support blocks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's exact -1 coefficient rests entirely on [CS25, Thm 1.1], not proved here; if that support formula fails, the Grothendieck lower bound collapses.","rationale":"The reader identified Proposition 4.1 as the weakest assumption, and I agree. My independent check of the Schubert part found it sound: the layered construction, the disjoint embedding argument, and the factorial estimates all cohere. The Grothendieck theorem is the only place where an external result is load-bearing, and it is a normal citation rather than a logical gap. The small-n counterexample in an intermediate inequality of Proposition 4.2 is sloppy but irrelevant asymptotically. Overall, the reader's ACCEPT verdict stands.","tokens_in":6180,"tokens_out":43140,"duration_ms":328123,"concrete_test":"Verify [CS25, Thm 1.1] independently on all fireworks permutations in S_n for n<=6 (or n<=7 computationally): compute G_w via the isobaric divided-difference recursion, compute supp(S_w) via [FMS18, Thm 4], and check the interval-union identity. Also re-check Proposition 4.2's row-size computation against a brute-force Rothe/upward-closure count for the layered permutation w=w(1,2,...,k,b). If the identity holds and row sizes match, Theorem 1.2 is secure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's main internal contribution (Theorem 1.1) is self-contained: Lemma 3.2 and Theorem 3.1 are sound, the translation embeddings are disjoint, and Corollary 3.3's factorial estimates give the stated bounds. I found no internal flaw there. The load-bearing external input is Proposition 4.1, imported from the companion paper [CS25, Thm 1.1]: for fireworks w, supp(G_w) = union over alpha in supp(S_w) of [alpha, wt(overline D(w))]. Proposition 4.2 uses only the corollary [c,d] subset of supp(G_w) for c=wt(D(w)), and the entire lower bound |supp(G_w)| >= n!/n^{k+1} — hence the exact coefficient -1 in Theorem 1.2 — collapses without this formula. The preprint does not prove it or sketch a proof, so a reader of this paper alone cannot certify Theorem 1.2. This is standard citation practice, not circularity, but it is the least secure point of the central claim. (Minor independent issue: the displayed product >= (n-k-1)! in Prop 4.2 fails at n=3, where it reads 0 >= 1, but this does not affect any asymptotic statement.)","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies β(n), the maximum size of the support of a Schubert polynomial over permutations in S_n, and the analogous quantity β^G(n) for Grothendieck polynomials. The main result (Theorem 1.1) asserts that ln β(n)/(n ln n) → 1, with explicit bounds -ln 4 - 1 ≤ liminf (ln β(n)-n ln n)/n ≤ limsup (...) ≤ -1. The proof constructs layered permutations whose Schubert support is at least ∏ floor(n/2^k)!, using a recursive embedding into Schubitopes (Theorem 3.1 and Corollary 3.3), and matches this with the trivial upper bound n!. For Grothendieck polynomials, Theorem 1.2 asserts ln β^G(n) = n ln n - n + o(n), equivalently (ln β^G(n)-n ln n)/n → -1. The lower bound uses a fireworks-permutation support formula from the authors' companion paper [CS25]. Both lower-bound constructions are explicit and are layered/fireworks permutations.","tokens_in":6498,"tokens_out":14016,"duration_ms":110345,"significance":"If correct, Theorem 1.1 completely resolves the first-order asymptotics of the maximal Schubert support, answering a problem of Guo–Lin [GL24, Prob 5.5], and gives the first explicit bounds on the second-order term. The proof is elementary, self-contained (for the Schubert part), and has no fitted parameters or reverse-engineered constants. Theorem 1.2 gives a similarly sharp asymptotic for Grothendieck supports, though its lower bound depends on an external support formula for fireworks permutations. The paper is a solid, incremental advance in the asymptotic combinatorics of Schubert polynomials, with a clean main construction that should be of interest to the field.","major_comments":[{"comment":"The proof of Theorem 3.1 states: 'use Lemma 3.2 to choose a diagram C_α ≤ D(w_LB) whose weight satisfies wt(C_α)_{n-b_m+j} = α_j.' But Lemma 3.2 only guarantees the existence of C with C ∩ D(w_LB) = S; it does not assert any control on wt(C). In general C∩D=S does not imply that the row-weight vector of C agrees with that of S, because C may have boxes outside D. The proof as written is therefore incomplete at a load-bearing point. The gap is fixable: in the proof of Lemma 3.2, the extra boxes are placed below the minimum row of D_j, i.e., in rows ≤ n-b_m, and D(w_LB) has all its boxes in rows n-b_m+1,...,n; one should state explicitly that the added boxes do not affect the last b_m coordinates of wt(C). I recommend strengthening Lemma 3.2 (or adding a sentence in the proof of Theorem 3.1) to record this weight-control property.","section":"§3, Theorem 3.1 and Lemma 3.2"},{"comment":"The entire lower bound for Grothendieck support, hence the exact coefficient -1 in Theorem 1.2, depends on Proposition 4.1, which is quoted from the companion paper [CS25, Thm 1.1] without proof. Since [CS25] is a separate preprint by the same authors, a reader of this manuscript alone cannot certify Theorem 1.2. This is standard citation practice in one sense, but because Proposition 4.1 is the only input that upgrades the lower bound from qualitative to quantitative, the manuscript should either (a) include a proof or a detailed proof sketch of Proposition 4.1, or (b) explicitly state in the introduction and in Theorem 1.2 that the result is conditional on [CS25]. As written, the statement of Theorem 1.2 is not self-contained.","section":"§4, Proposition 4.1 and Theorem 1.2"},{"comment":"The product lower bound in the proof of Proposition 4.2 is not valid for all n. For example, when n=3, the unique integer k is 2, and the product over j=1 to k-1 of (n-k-1 - sum_{i=1}^{j-1} i)^j evaluates to 0^1 = 0, while (n-k-1)! = 0! = 1. Thus the displayed chain 'product ≥ (n-k-1)! ≥ n!/n^{k+1}' fails for n=3. This does not affect the asymptotic statement, since for n ≥ 4 the inequality appears to hold, but the proof should either handle small n separately or state that n is assumed sufficiently large.","section":"§4, Proposition 4.2"}],"minor_comments":[{"comment":"The statement of Lemma 2.11 appears to be \\(\\lfloor k\\rfloor! \\ge \\frac{1}{2^k}(k/e)^k\\), but the typeset inequality in the text could be misread as \\(\\frac{1}{2k}\\). Please clarify the notation.","section":"§2, Lemma 2.11"},{"comment":"In the displayed chain of inequalities, the step involving \\(1/n^c\\) and the factor \\(2^{-2n}\\) is compressed; adding one line showing the product of the \\((1/2^k)^{n/2^k}\\) factors is at least \\(2^{-2n}\\) would improve readability.","section":"§3, Corollary 3.3"},{"comment":"The reference [GL24] is cited as arXiv:2412.02932; if a published version exists, it would be good to update the citation.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main Schubert theorem is close to correct, but the proof of Theorem 3.1 has a logical gap that must be fixed with a strengthening of Lemma 3.2. The Grothendieck theorem rests entirely on the authors' companion paper [CS25]; given that this paper is to be published, I would ask the editor to consider whether a proof or detailed statement of Proposition 4.1 should be required, or whether the theorem should be marked conditional. The small-n counterexample in Proposition 4.2 is minor but should be corrected. The construction itself is elegant and the paper is worth publishing after these revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this paper deserves refereeing. The Schubert result (Theorem 1.1) is a genuine new asymptotic for the maximal support size, and the proof is self-contained. The Grothendieck corollary gives sharp-looking first-order asymptotics, but its exact -1 coefficient inherits a support formula from a companion paper that is not proved or sketched here.\n\nWhat is new: the n!/4^n lower bound via layered permutations with multiplicatively disjoint Rothe blocks, and the consequent lim ln β(n)/(n ln n) = 1. The upper bound is just staircase divisibility, and the lower bound is an explicit block construction with simple factorial estimates. I checked the block induction in Corollary 3.3; it works. That half of the paper is solid.\n\nThe weaker half is Theorem 1.2. Proposition 4.2 depends entirely on Proposition 4.1, imported from [CS25]. If that support formula failed, the lower bound for the Grothendieck support collapses. That is a normal citation move, not circularity, but it means the present paper alone cannot certify its headline Grothendieck asymptotic. The authors should either include a proof of the formula or make the dependence explicit in the statement. Minor: the display in Proposition 4.2 claiming ∏ ... >= (n-k-1)! is false at n=3 (left side 0, right side 1). It has no asymptotic effect, but should be repaired.\n\nNo fitted parameters, no post hoc constants, no suspicious citation pattern. The paper is honest about what is and isn't proved. For someone working on Schubert polynomials, Newton polytopes, or pipe dreams, this is a useful result and answers an open question from Guo-Lin. The companion-paper dependence is the only thing that keeps me from calling the whole thing fully verified.\n\nRecommendation: send to peer review. It should be accepted after the authors either prove or clearly flag the imported support formula, and fix the n=3 display.","headline":"Solid self-contained Schubert half, sharp Grothendieck half that leans on an unproved companion formula; referee it.","tokens_in":6955,"tokens_out":4149,"would_cite":true,"duration_ms":36729,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05A05","14M15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves the maximal support size of a Schubert polynomial grows like n^n up to a subexponential factor, and pins the second-order Grothendieck growth exactly at -1.","keywords":["Schubert polynomials","Grothendieck polynomials","support","layered permutations","Rothe diagrams","Schubitopes","asymptotic enumeration","permutation diagrams"],"falsifier":"Verify Proposition 4.1 on a small non-layered fireworks permutation, e.g. w=2413, by brute-force computing supp(G_w) and checking it equals the claimed union of intervals [α, wt(overline(D(w)))]; if any monomial lies outside the intervals, the exact Grothendieck asymptotics collapse. Also check the recursion in Theorem 3.1 on a concrete layered permutation at, say, n=10 by enumerating |supp(S_w)| and comparing it to the product-of-factorials lower bound.","tokens_in":6075,"feed_emoji":"🧮","tokens_out":6320,"duration_ms":50844,"temperature":0.7,"pith_summary":"The paper determines how large the set of monomials (the support) of a Schubert polynomial can possibly be. It shows that for every n, some layered permutation has a Schubert polynomial with at least n!/4^n monomials, and that no permutation can exceed n!; together these prove that the maximal support size grows as n^n up to a subexponential factor, with the second-order term trapped between two explicit constants. For Grothendieck polynomials, the same approach yields a sharper result: the second-order term is exactly -1, meaning the maximal support has a precise leading correction n!/subexponential. The proofs work by dissecting the Rothe diagram of layered permutations and embedding many disjoint copies of smaller Schubitopes into a larger one.","feed_headline":"Maximal Schubert support has n^n growth, reaching n!/4^n","feed_subtitle":"A layered permutation hits support size n!/4^n, and Grothendieck support's second-order term is exactly -1.","key_machinery":"The main tool is the diagrammatic description of Schubert support: a vector α is in supp(S_w) iff α is the weight of a diagram C lying weakly northwest of the Rothe diagram D(w) (C ≤ D(w)). For layered permutations, D(w) splits into blocks, and the paper proves (Lemma 3.2) a fine subset-replacement property: inside the last block, any chosen subset of boxes can be realized by some C ≤ D(w) without changing its weight elsewhere. This lets the authors embed b_m! disjoint translated copies of the smaller Schubitope S_{D(w')} into S_{D(w)}, giving the recursion |supp(S_w)| ≥ b_m! |supp(S_w')|. For Grothendieck polynomials, a formula from a companion paper for fireworks permutations expresses sup","core_discovery":"The central discovery is that layered permutations — permutations made of decreasing blocks — are large-support extremizers. Theorem 1.1 establishes that for β(n) = max_{w∈S_n} |supp(S_w)|, one has ln β(n)/(n ln n) → 1, and more precisely -ln4-1 ≤ liminf (ln β(n)-n ln n)/n ≤ limsup (ln β(n)-n ln n)/n ≤ -1. Theorem 1.2 proves the analogous quantity for Grothendieck polynomials satisfies lim (ln β^G(n)-n ln n)/n = -1 exactly. The construction gives explicit layered permutations achieving the stated lower bounds.","pith_inferences":["The recursive block construction suggests a general principle: any permutation whose Rothe diagram splits into two separated parts will have support at least the product of the supports of its components; this might extend the result from layered permutations to other diagram families.","The gap between the lower bound -ln4-1 and the upper bound -1 in Theorem 1.1 leaves open the possibility that the true second-order constant is -1; if so, a construction avoiding the 4^n loss in the product-of-factorials bound would close the gap.","For Grothendieck polynomials, the interval formula implies the support is the integer points of a box [c,d]; maximizing |supp(G_w)| is then a purely combinatorial question about Lehmer codes, which could be attacked by optimization over diagrams rather than permutations.","The same layered permutations might be tested as extremizers for the principal specialization S_w(1,...,1), connecting the support-maximization problem to the older degree-maximization problem."],"forward_implications":["The maximal Schubert support satisfies β(n) = n^{n - o(1)} (equivalently ln β(n) ~ n ln n), so the first-order growth is now known exactly.","There exist explicit layered permutations with support at least n!/4^n; these provide concrete candidates for support-maximizers.","The maximal Grothendieck support has the exact second-order asymptotics ln β^G(n) = n ln n - n + O(√n ln n), leaving only a sub-√n correction undetermined.","The recursive embedding proves a stronger statement: the Schubitope of a layered permutation contains many disjoint integer translates of the Schubitope of its prefix, giving a structural insight into support saturation.","These results settle a problem posed in earlier literature on the growth rate of maximal Schubert support."],"fun_headline_variants":["Layered permutations hit Schubert support ≥ n!/4^n","Asymptotically maximal Schubert support: growth rate n^n","Grothendieck support's second-order term exactly -1","New layered permutations reach n!/4^n support","Schubert polynomial support: asymptotic max pinned"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The exact Grothendieck result rests on a support-interval formula from a companion paper that is not proved here; if that formula is wrong, Theorem 1.2 fails.","fun_headline_variants_meta":{"raw":{"variants":["Layered permutations hit Schubert support ≥ n!/4^n","Asymptotically maximal Schubert support: growth rate n^n","Grothendieck support's second-order term exactly -1","New layered permutations reach n!/4^n support","Schubert polynomial support: asymptotic max pinned"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000327,"raw_usage":{"total_tokens":1620,"prompt_tokens":656,"completion_tokens":964,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":400,"completion_tokens_details":{"reasoning_tokens":884}},"tokens_in":400,"tokens_out":964,"duration_ms":8996,"temperature":1.0,"reasoning_tokens":884,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T18:39:23.472033+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Verify Proposition 4.1 on a small non-layered fireworks permutation, e.g. w=2413, by brute-force computing supp(G_w) and checking it equals the claimed union of intervals [α, wt(overline(D(w)))]; if any monomial lies outside the intervals, the exact Grothendieck asymptotics collapse. Also check the recursion in Theorem 3.1 on a concrete layered permutation at, say, n=10 by enumerating |supp(S_w)| and comparing it to the product-of-factorials lower bound.","supporting_citations":[],"review_version":1}