{"id":"23704eb2-9881-4ca6-b9ec-358a8dd105e6","arxiv_id":"2507.23222","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Weighted K-k-Schur functions interpolate between two known Katalan function families, and their recursive expansion proves the K-k-Schur alternating conjecture for partitions whose first b_lambda parts are strictly decreasing.","lead":"The paper introduces a new family of symmetric functions, weighted K-k-Schur functions, and uses them to prove a long-standing alternating positivity conjecture for a large set of partitions. If correct, this gives a new structural result in K-theoretic Schubert calculus, the study of geometric invariants of flag varieties.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.6 applies its induction hypothesis (21) to a partition γ without proving that d^{c'-(c-1)}_γ(z) lies on γ's own bounce path; this can fail, leaving an unhandled undefined lowering in the proof of Theorem 1.2/1.3.","rationale":"The reader's weakest assumption (strictness in \\hat P) is about scope, not this gap. I agree the main theorem is likely true, but the proof of the recursive engine (Proposition 3.6) has a concrete unhandled case: the induction hypothesis (21) is applied to γ without verifying the relevant iterate d^{c'−(c−1)}_γ(z) exists on γ's bounce path. The explicit example lies inside \\hat P and is used in the induction for Theorem 1.3, so this is not merely a cosmetic issue for a peripheral case. Because the displayed proof of (24) is unjustified, the paper should be accepted only after the undefined case is either covered by the stated convention or ruled out under the hypotheses of Theorem 1.2/1.3. This motivates CONDITIONAL rather than REJECT.","tokens_in":22169,"tokens_out":42569,"duration_ms":435551,"concrete_test":"Directly compute, from Definition 2.12 and Lemmas 2.4/2.5, the Katalan expansion for k=5, ℓ=6, λ=(5,2,1,1,1,1), z=1 without using display (24): check whether g^{(5)}_{λ,2}=g^{(5)}_{λ}−g^{(5)}_{(5,1,1,1,1,1)}−g^{(5)}_{(5,2,1,1,1,0)} holds exactly. If it holds with L_{d^2_γ(1)} treated as 0, the gap is an omitted convention and the proof can be patched; if not, Proposition 3.6 fails and Theorem 1.2/1.3 needs a different argument in this case.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Proposition 3.6, after defining γ=λ−ε_{d^{c'−c}_λ(z)}, the text asserts “By the inductive hypothesis, L_{d^{c'−(c−1)}_γ(z)}g^{(k)}_{γ,z}=...” (display (24)). But (21) is only a statement about operators d^{c'_γ−t}_γ(z) along γ's own bounce path, and the paper never verifies that d^{c'−(c−1)}_γ(z) is defined or lies in this range. The case can fail: take k=5, ℓ=6, λ=(5,2,1,1,1,1)∈\\hat P^5_6, z=1. Then dλ(1)=2, dλ(2)=6, so c'=2; for c=1, γ=λ−ε_2=(5,1,1,1,1,1). In γ, row 2 has no root (k−γ_2+2=6), so d^2_γ(1) is undefined, while (24) is used to expand L_{d^2_γ(1)}g_{γ,1}. This same λ and z=1 are needed in the induction proving Theorem 1.2 for the final step z=bλ=2 of Theorem 1.3. Proposition 3.4 has a convention setting such an L to 0, but Proposition 3.6 does not carry this convention into (24), so the proof as written is incomplete. The expansion itself appears correct in this example, so the gap is likely patchable, but it is the load-bearing recursive step and must be fixed.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a new family of Katalan functions, the weighted K-k-Schur functions g^{(k)}_{λ,z}, which interpolate between K-k-Schur functions (z=1) and closed k-Schur Katalan functions (z=b_λ+1). The central result is Proposition 3.6, a recursive expansion of g^{(k)}_{λ,z+1} in terms of g^{(k)}_{μ,z} with alternating positive coefficients, and its consequences: Theorems 1.2 and 1.3, which prove the K-k-Schur alternating conjecture (Conjecture 1.1(e) of Blasiak–Morse–Seelinger) for the class \\hat P^k_ℓ of partitions satisfying strict decrease up to the bounce bottom b_λ, including all strictly decreasing k-bounded partitions. The proofs use the Katalan formula, the Mirror Lemma, and a delicate induction on the bounce path of λ.","tokens_in":22577,"tokens_out":20935,"duration_ms":229806,"significance":"If the proof is completed, the result is a substantial advance: it resolves a named conjecture from the 2022 work of Blasiak, Morse, and Seelinger for a natural and fairly broad class of partitions, and it introduces an interpolating family that is likely to be useful for the remaining conjectures in that program. The paper is careful and mostly self-contained: it gives explicit definitions, worked examples, and detailed verifications of the Mirror Lemma hypotheses in the key propositions. It relies on the externally established Katalan formula and Mirror Lemma, and the new weighted functions are defined rather than fitted to the conclusion, so there is no evident circularity. The main concern is a genuine gap in the central induction, described below; because the gap is local and the surrounding argument is plausible, the manuscript merits revision rather than rejection.","major_comments":[{"comment":"The induction hypothesis (21) is applied to the auxiliary partition γ without verifying that the lowering operator L_{d^{c'-(c-1)}_γ(z)} is defined and lies on γ's own bounce path. This is not automatic: for k=5, ℓ=6, λ=(5,2,1,1,1,1), z=1, one has c'=2 and, at c=1, γ=λ−ε_{d^1_λ(1)}=(5,1,1,1,1,1) has b_γ=1, so d^2_γ(1) is undefined. Proposition 3.4 has a convention setting such a lowering to 0, but Proposition 3.6 does not state that this convention applies in display (24); as written, (24) asserts an expansion for an operator that is not defined. This occurs at the final inductive step (c=1, a=c'-1) and is exactly the step on which the proofs of Theorems 1.2 and 1.3 depend. The expansion in the example appears to be correct, so the gap is likely repairable, but the proof must be amended: either the undefined case must be treated separately using the convention from Proposition 3.4, or one must prove that when the operator is defined, its exponent lies in the range of γ's own bounce path before applying (21).","section":"§3.2, Proposition 3.6, display (24)"}],"minor_comments":[{"comment":"The claim that d^{c'-c}_λ(z) ≠ z, z+1 is not generally true; for instance, with k=5, λ=(5,4,3,3,3,3), z=1, one has d^1_λ(1)=2=z+1. The needed conclusion γ_z>γ_{z+1} still follows from λ_z>λ_{z+1} in such cases, but the stated reason is inaccurate and should be corrected.","section":"§3.2, Proposition 3.6 proof"},{"comment":"The reference \"Remark 2.8 (b)\" should be \"Remark 2.8 (2)\".","section":"§3.1, Proposition 3.4"},{"comment":"Display (24) omits the membership \"∈ Z_{≥0}\" after the alternating coefficient; the text uses this membership immediately afterward and should include it.","section":"§3.2, display (24)"},{"comment":"The example writes the partition (7,6,5,4,4,4,4,3,3,3,2,2,0) with a trailing zero part; standard partition notation would omit the zero, which could confuse readers.","section":"Example 3.7"}],"recommendation":"major_revision","confidential_remarks":"The gap in Proposition 3.6 is load-bearing but appears local and likely patchable, so I recommend major revision rather than rejection. I see no evidence of circularity or inappropriate reliance on the authors' prior work; the use of [8,9] is contextual. The manuscript would benefit from a careful rewriting of the induction in Proposition 3.6 that explicitly tracks the bounce path of the auxiliary partition γ."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper introduces weighted K-k-Schur functions, which interpolate between the usual K-k-Schur functions and the closed k-Schur Katalan functions. That is a genuinely useful construction. The authors use it to prove the K-k-Schur alternating conjecture for the class of partitions defined by the strict inequalities in (1), which includes all strictly decreasing k-bounded partitions. The main theorems are new, and the overall strategy—recursion via the Mirror Lemma and an alternating-positivity induction—is sound in spirit. The paper is honest about what it does not prove: the full conjecture remains open for partitions with repeated initial parts.\n\nThe exposition is mostly clear. The definitions in Section 2 are standard Katalan machinery, and the examples (especially Example 4.1) help a lot. The proof of Theorem 1.2 from Proposition 3.6 is clean once Proposition 3.6 is granted.\n\nNow the soft spot. In the proof of Proposition 3.6, the induction hypothesis (21) is applied to the partition gamma = lambda - eps_{d^{c'-c}_lambda(z)} without checking that the lowering operator d^{c'-(c-1)}_gamma(z) is defined on gamma's own bounce path. This is not a minor pedantic point: it can fail. Take k=5, ell=6, lambda=(5,2,1,1,1,1), z=1. Then b_lambda=2, c'=2, and for c=1, gamma=(5,1,1,1,1,1). In Delta_k(gamma), row 2 has no root, so d^2_gamma(1) is undefined. Yet display (24) uses L_{d^2_gamma(1)}. Proposition 3.4 has a convention that treats such an L as zero, but Proposition 3.6 does not carry that convention into (24), and the induction hypothesis does not apply for an undefined operator. In this example the resulting expansion works anyway, so the gap is probably patchable, but it sits at the load-bearing recursive step and needs a fix. Either add a separate case for when the lowering operator is undefined, or prove that the relevant d exponent always lies on gamma's bounce path under the stated hypotheses.\n\nThere is also a small presentational note: the paper says \"we use the induction on c\" but the induction is on c for a fixed lambda, and then it reuses the same c range for gamma. That is the source of the confusion.\n\nWho is this for? Anyone working on K-theoretic Schubert calculus or Catalan/Katalan functions. The construction is worth knowing regardless of the gap. I would send it to a competent referee and ask for a revision that fixes Proposition 3.6. The main theorem is very likely true, but the proof as written is not complete.\n\nBest.","headline":"A genuinely new interpolation family and a partial resolution of a real conjecture, but the proof of the key Proposition 3.6 has an unverified step that must be patched.","tokens_in":23094,"tokens_out":11261,"would_cite":true,"duration_ms":111809,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05E10","14N15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces weighted K-k-Schur functions — a family interpolating between K-k-Schur functions and closed k-Schur Katalan functions — and uses them to prove the K-k-Schur alternating conjecture for all partitions in a class that…","keywords":["Katalan function","weighted K-k-Schur function","closed k-Schur Katalan function","K-k-Schur alternating conjecture","positivity","K-theoretic Schubert calculus","k-bounded partitions","Mirror Lemma"],"falsifier":"Compute the expansion $\\tilde g^{(k)}_\\lambda=\\sum_\\mu b_{\\lambda\\mu}g^{(k)}_\\mu$ for all $\\lambda\\in\\hat P^k_\\ell$ with small $k$ and $\\ell$ (say $k\\le 7$, $\\ell\\le 8$) by applying the defining Katalan relations symbolically; the theorem is false if any $(-1)^{|\\lambda|-|\\mu|}b_{\\lambda\\mu}$ comes out negative or non-integral, and the paper's own Example 4.1 with $k=5$, $\\lambda=(5,4,3,3,2,2)$ is one point where the signs all match.","tokens_in":21983,"feed_emoji":"🧮","tokens_out":8394,"duration_ms":84780,"temperature":0.7,"pith_summary":"This paper is trying to prove that a specific alternating sign pattern holds when closed k-Schur Katalan functions are expanded in the basis of K-k-Schur functions. The authors introduce a one-parameter interpolation between those two families, the weighted K-k-Schur functions, and show by induction on the weight that the expansion coefficients alternate in sign according to the size difference of the partitions. This resolves the K-k-Schur alternating conjecture for a large class of k-bounded partitions, including all strictly decreasing ones. The relevance is that such alternating positivity is the combinatorial backbone of K-theoretic Schubert calculus, where these symmetric functions represent geometric classes.","feed_headline":"Weighted K-k-Schur functions resolve alternating positivity conjecture","feed_subtitle":"New proof covers all strictly decreasing k-bounded partitions, linking K-theory to symmetric functions.","key_machinery":"The key object is the weighted K-k-Schur function $g^{(k)}_{\\lambda,z}$, defined as the Katalan function $K(\\Delta_k(\\lambda); L(\\Delta_k(\\lambda))\\setminus\\{d_\\lambda(x): x\\in [z,b_\\lambda]\\}; \\lambda)$. It is a Katalan function: an inhomogeneous symmetric function built from the dual stable Grothendieck determinant $g_\\gamma$ by applying lowering operators $L_z$ and raising operators $R_{ij}$ encoded in a root ideal. For $z=1$ this is the K-k-Schur function $g^{(k)}_\\lambda$; for $z=b_\\lambda+1$ it is the closed k-Schur Katalan function $\\tilde g^{(k)}_\\lambda$. The argument runs on two levers: the recursion $g^{(k)}_{\\lambda,z+1}=g^{(k)}_{\\lambda,z}-L_{d_\\lambda(z)}g^{(k)}_{\\lambda,z}$, and the Mirror Lemma of Katalan-function theory, which makes many lowering-operator terms vanish or collapse. Proposition 3.6 converts these local cancellations into an expansion of weight $z+1$ functions in weight $z$ functions with alternating-positive coefficients, and induction on $z$ carries the positivity to the endpoint.","core_discovery":"The central discovery is Theorem 1.3: for every partition $\\lambda\\in \\hat P^k_\\ell$, the closed k-Schur Katalan function $\\tilde g^{(k)}_\\lambda$ expands as $\\tilde g^{(k)}_\\lambda = \\sum_{\\mu\\in P} b_{\\lambda\\mu} g^{(k)}_\\mu$, and the coefficients satisfy $(-1)^{|\\lambda|-|\\mu|} b_{\\lambda\\mu}\\in \\mathbb{Z}_{\\ge 0}$. Here $\\hat P^k_\\ell$ is the class of k-bounded partitions with $\\lambda_{x-1}>\\lambda_x$ whenever $k-\\lambda_x+x<\\ell$, which includes all strictly decreasing k-bounded partitions. The proof goes through the stronger Theorem 1.2: under the strictness hypothesis $\\lambda_1>\\cdots>\\lambda_z$ (and $\\lambda_z>\\lambda_{z+1}$ when $z\\neq b_\\lambda$), the weighted function $g^{(k)}_{\\lambda,z+1}$ has an alternating-positive expansion in the weight-$z$ functions $g^{(k)}_{\\mu,z}$. Since $z=1$ recovers $g^{(k)}_\\lambda$ and $z=b_\\lambda+1$ recovers $\\tilde g^{(k)}_\\lambda$, the theorem interpolates from one side of the conjecture to the other.","pith_inferences":["Editorial inference: The weight parameter $z$ can be read as a filtration interpolating between two geometric bases; a $q$-analogue weighting each step by $q^z$ would produce a $q$-K-k-Schur function whose specialization at $q=0$ and $q=1$ gives the two extremes. The paper does not define such an object.","Editorial inference: If alternating positivity holds outside $\\hat P^k_\\ell$ as well, the obstruction is not the positivity itself but the Mirror Lemma's strictness hypotheses; a search for the smallest partition with a repeated part in the initial segment would test whether the theorem's class is sharp.","Editorial inference: The same induction, applied to other Katalan-function identities, may transfer these methods to the related alternating dual Pieri and k-branching conjectures, since those conjectures are logically linked in the same framework."],"forward_implications":["Conjecture 1.1(e) is now a theorem for every partition in $\\hat P^k_\\ell$, so the closed k-Schur Katalan function of any such partition is alternating-positive in the K-k-Schur basis.","Every strictly decreasing k-bounded partition lies in $\\hat P^k_\\ell$, so the conjecture holds for all strictly decreasing k-bounded partitions.","The weighted family $g^{(k)}_{\\lambda,z}$ gives a chain of alternating-positive expansions interpolating between $g^{(k)}_\\lambda$ and $\\tilde g^{(k)}_\\lambda$; passing from weight $z$ to weight $z+1$ never destroys the sign pattern.","Theorem 1.2 supplies a recursive algorithm for computing the coefficients $b_{\\lambda\\mu}$: expand $g^{(k)}_{\\lambda,z+1}$ via Proposition 3.6, then expand each weight-$z$ term by induction down to $z=1$.","The sign twist $(-1)^{|\\lambda|-|\\mu|}b_{\\lambda\\mu}$ is a nonnegative integer, so each signed coefficient is a whole number with a definite sign."],"supporting_citations":[{"why":"Supplies the Katalan-function framework, the Mirror Lemma, the Katalan formula for $g^{(k)}_\\lambda$, and the conjecture this paper resolves.","marker":"[4]"},{"why":"Introduced the K-k-Schur functions that form the target basis of the expansion.","marker":"[19]"},{"why":"Developed Catalan functions and k-Schur positivity methods that the proof adapts.","marker":"[2]"},{"why":"Gave Pieri rules characterizing K-k-Schur functions, used in their Katalan description.","marker":"[26]"},{"why":"Prior work of the authors on the alternating dual Pieri and k-branching conjectures, whose method is extended here.","marker":"[8]"},{"why":"Prior work of the authors on lowering operators for K-k-Schur functions that underpins the recursive step.","marker":"[9]"},{"why":"Motivated closed K-k-Schur functions in quantum K-theory, the endpoint of the interpolation.","marker":"[14]"}],"fun_headline_variants":["Weighted K-k-Schur functions settle alternating conjecture","New weighted functions prove K-k-Schur alternating conjecture","Alternating conjecture resolved by weighted K-k-Schur functions","Weighted K-k-Schur functions widen proof of alternating conjecture","K-k-Schur alternation cracked via weighted functions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the partition's first entries being strictly decreasing at each step of the recursion, and when that strictness fails the Mirror Lemma cannot be applied and the expansion is not proved.","fun_headline_variants_meta":{"raw":{"variants":["Weighted K-k-Schur functions settle alternating conjecture","New weighted functions prove K-k-Schur alternating conjecture","Alternating conjecture resolved by weighted K-k-Schur functions","Weighted K-k-Schur functions widen proof of alternating conjecture","K-k-Schur alternation cracked via weighted functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000265,"raw_usage":{"total_tokens":1612,"prompt_tokens":955,"completion_tokens":657,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":571,"completion_tokens_details":{"reasoning_tokens":578}},"tokens_in":571,"tokens_out":657,"duration_ms":7814,"temperature":1.0,"reasoning_tokens":578,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:55:44.715816+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the expansion $\\tilde g^{(k)}_\\lambda=\\sum_\\mu b_{\\lambda\\mu}g^{(k)}_\\mu$ for all $\\lambda\\in\\hat P^k_\\ell$ with small $k$ and $\\ell$ (say $k\\le 7$, $\\ell\\le 8$) by applying the defining Katalan relations symbolically; the theorem is false if any $(-1)^{|\\lambda|-|\\mu|}b_{\\lambda\\mu}$ comes out negative or non-integral, and the paper's own Example 4.1 with $k=5$, $\\lambda=(5,4,3,3,2,2)$ is one point where the signs all match.","supporting_citations":[{"cited_title":"Blasiak, J","cited_arxiv_id":null,"evidence_quote":"Supplies the Katalan-function framework, the Mirror Lemma, the Katalan formula for $g^{(k)}_\\lambda$, and the conjecture this paper resolves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the K-k-Schur functions that form the target basis of the expansion."},{"cited_title":"Blasiak, J","cited_arxiv_id":null,"evidence_quote":"Developed Catalan functions and k-Schur positivity methods that the proof adapts."},{"cited_title":"Morse, Combinatorics of the k-theory of affine grassmannians, Adv","cited_arxiv_id":null,"evidence_quote":"Gave Pieri rules characterizing K-k-Schur functions, used in their Katalan description."},{"cited_title":"Alternating dual Pieri rule conjecture and $k$-branching conjecture of closed $k$-Schur Katalan functions","cited_arxiv_id":"2501.04200","evidence_quote":"Prior work of the authors on the alternating dual Pieri and k-branching conjectures, whose method is extended here."},{"cited_title":"Lowering operators on $K$-$k$-Schur functions and a lowering operator formula for closed $K$-$k$-Schur functions","cited_arxiv_id":"2502.05618","evidence_quote":"Prior work of the authors on lowering operators for K-k-Schur functions that underpins the recursive step."},{"cited_title":"Ikeda, S","cited_arxiv_id":null,"evidence_quote":"Motivated closed K-k-Schur functions in quantum K-theory, the endpoint of the interpolation."}],"review_version":1}