{"id":"b1424c89-763a-4ce3-8a06-e1e89aa855ea","arxiv_id":"2501.04200","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The alternating dual Pieri rule and k-branching conjectures for closed k-Schur Katalan functions are proven for strictly decreasing partitions, and the Pieri rule also for all partitions with k large.","lead":"Closed k-Schur Katalan functions are symmetric functions tied to quantum K-theory. This paper proves two conjectures about their positivity in special cases: the alternating dual Pieri rule for large k and for strictly decreasing partitions, and the k-branching rule for strictly decreasing partitions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of straightening Lemma 3.4(a) relies on a ceiling claim that is false in general; Theorems 1.3-1.4 are not established as written.","rationale":"The reader's weakest assumption identified Lemma 3.4 and the bounce-path combinatorics as the sensitive point. My stress-test confirms this and pinpoints a concrete error: the proof of Lemma 3.4(a) asserts a ceiling in columns y,y+1 where y=topΔk(µ)(z+1), but this fails for explicit µ. Since Lemma 3.4 feeds directly into Theorem 3.5, Proposition 4.3, and finally Theorems 1.3 and 1.4, the main results are not rigorously proven by the current manuscript. However, the lemma itself may be true; the identity appears to hold in the large-k examples tested, and the error may be repairable by choosing a different y (e.g., topΔk(µ)(z)) or by handling the boundary case y=ℓ separately. Therefore the appropriate verdict remains CONDITIONAL, as the reader recommended, but the required revision is more substantial than a routine check: the straightening lemma's proof must be corrected or replaced. I do not allege any bad faith; the issue is a technical gap in the combinatorial arguments. The proposed test would settle whether the lemma is actually false or merely unproven.","tokens_in":25174,"tokens_out":38797,"duration_ms":319897,"concrete_test":"Implement the Katalan function K(Ψ;M;γ) for small k,ℓ (e.g., all ℓ≤5, k≤6) using Definition 2.1, and enumerate all µ∈\\tilde{P}^k_ℓ with a single ascent µ_z+1=µ_{z+1} and all other descents. For each such µ, compute \\tilde{g}_µ and the claimed straightened expression from Lemma 3.4 (µ−ε_{z+1} in case (a), µ+ε_{up(y+1)}−ε_{z+1} in case (b)), and test equality. In particular, check cases where top(z+1)>top(z) but columns top(z+1) and top(z+1)+1 are not a ceiling, such as µ=(4,3,4,2,1), k=6. If any equality fails, Lemma 3.4 is false and the proof of Theorems 1.3–1.4 collapses; if all pass, the lemma is true but the proof needs a corrected argument that does not rely on the false ceiling assertion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The key straightening Lemma 3.4(a) contains a false structural assertion. It states: 'Suppose y := topΔk(µ)(z+1) > topΔk(µ)(z). Then the root ideal Δk(µ) has a ceiling in columns y, y+1.' This is not true. For example, take k=6, ℓ=5, µ=(4,3,4,2,1), z=2. Then µ∈\\tilde{P}^6_5, µ_2+1=µ_3, and all other adjacent inequalities hold. Δ_6(µ)={(1,4),(1,5)}. Bounce paths: 1→4, with 2,3,5 isolated, so top(3)=3 > top(2)=2. Column lengths: col3 has 0 roots, col4 has 1 root, so columns 3 and 4 do not have equal length; no ceiling in columns y,y+1. (If y=ℓ, the ceiling is not even defined.) Consequently, the application of Lemma 2.8 to deduce K(Δ;Δ;µ)=K(Δ;Δ;µ−ε_{z+1}) is unjustified for such µ. This straightening step is used in Theorem 3.5 to express lowering operators, and via Proposition 4.3 underpins Theorems 1.3 and 1.4. The lemma may still be true—e.g., in the large-k limit the identity reduces to a plausible determinant straightening g_(2,2,3)=g_(2,2,2)—but the present proof is incomplete and the central theorems are not established as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies two conjectures of Blasiak–Morse–Seelinger on closed k-Schur Katalan functions: the alternating dual Pieri rule and the k-branching conjecture. It proves the alternating dual Pieri rule in the large-k/stable limit (Theorem 1.2) and, for strictly decreasing k-bounded partitions, both the alternating dual Pieri rule (Theorem 1.3) and the k-branching expansion (Theorem 1.4). The proofs introduce a new Mirror Lemma (Lemma 2.9), analyze lowering operators on generalized closed k-Schur Katalan functions (Theorem 3.5), and reduce the positivity statements to a sign-pattern statement in Proposition 4.3.","tokens_in":25408,"tokens_out":8934,"duration_ms":78846,"significance":"If correct, the results would establish two open conjectures for a natural infinite family of partitions and give a simple, self-contained proof of the stable-limit Pieri statement. Theorem 1.2 is clean and convincing, and the lowering-operator expansion in Theorem 3.5 is a useful structural contribution. However, the proofs of Theorems 1.3 and 1.4 rest on Lemma 3.4(a), and the proof of that lemma contains a false structural assertion. Until this is repaired, the main positive results are not established as written. The paper does not provide machine-checked proofs or computational certificates for the intricate combinatorial steps.","major_comments":[{"comment":"The proof's claim that the root ideal has a ceiling in columns y,y+1 is false. Take k=6, ℓ=5, µ=(4,3,4,2,1), and z=2. Then µ∈\\tilde P^6_5 and satisfies µ_z+1=µ_{z+1} and µ_x≥µ_{x+1} for all x≠z. The root ideal is Δ_6(µ)={(1,4),(1,5)}, and the bounce paths are the singletons {2} and {3}, so top_{Δ_6(µ)}(3)=3>2=top_{Δ_6(µ)}(2). But column 3 of Δ_6(µ) has length 0 and column 4 has length 1, so there is no ceiling in columns 3,4. Thus the invocation of Lemma 2.8 to deduce K(Δ_k(µ);Δ_k(µ);µ)=K(Δ_k(µ);Δ_k(µ);µ−ε_{z+1}) is unjustified. Since Lemma 3.4(a) is used in the induction proving Theorem 3.5, which feeds Proposition 4.3 and then Theorems 1.3 and 1.4, the central results are not established as written.","section":"Section 3, Lemma 3.4(a)"},{"comment":"The iterative vanishing argument concluding X=0 via Proposition 2.10 requires hypotheses that are not verified. After proving X=L_d X for X=\\tilde g^{(k)}_µ−\\tilde g^{(k)}_{µ−ε_{z+1}}, the proof applies L_d^m to both summands and invokes Proposition 2.10. This requires that µ−ε_{z+1} lies in \\tilde P^k_ℓ, or at least that Δ_k(µ−ε_{z+1}) is a root ideal, which is not automatic: if µ_{z+1}=µ_{z+2}, the decrement at position z+1 destroys the inequality µ_{z+1}−1≥µ_{z+2}. The vanishing bound for the second summand should be justified separately, or the argument should be restructured.","section":"Section 3, proof of Lemma 3.4(a), Case 2"}],"minor_comments":[{"comment":"The phrase 'Hopf algbra' should read 'Hopf algebra'.","section":"Page 2"},{"comment":"The notation '(-1)a_{µµ(1)}' is ambiguous; it should be written with an explicit multiplication sign or parentheses, e.g. '(-1)a_{\\mu\\mu^{(1)}}', to avoid reading as a power.","section":"Equation (44)"},{"comment":"The figure captions refer to orange and green bounce paths, but the figures themselves are not labeled; the paths described in the captions should be marked directly in the figures.","section":"Remark 3.1(d) and Figures 1–2"},{"comment":"Reference [24] is cited as an arXiv preprint; if a published version exists, it should be cited instead or in addition.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"I verified the counterexample in the stress-test note; it is correct and directly invalidates the proof of Lemma 3.4(a) as written. The paper has a plausible chance of being repairable—perhaps the ceiling should be in columns top(z), top(z)+1 rather than y,y+1—so I recommend major revision rather than rejection. I would also ask the authors to check Lemma 3.4(a) in the case y=ℓ, where a ceiling in columns y,y+1 is not defined."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The paper does two things well: it proves the alternating dual Pieri rule in the stable limit (Theorem 1.2) with a clean argument from known tools, and it develops a new Mirror Lemma II (Lemma 2.9) plus a lowering-operator analysis that is the real technical core. If the core lemmas held, Theorems 1.3 and 1.4 would be plausible partial results toward the Blasiak–Morse–Seelinger conjectures.\n\nBut the core has a hole. Lemma 3.4(a) asserts that if top(µ)(z+1) > top(µ)(z), then Δ(µ) has a ceiling in columns y,y+1 with y = top(z+1). That is false. Take k=6, ℓ=5, µ=(4,3,4,2,1), z=2. This lies in ̃P^6_5 and satisfies the lemma's hypotheses: µ_2+1=µ_3 and all other adjacent inequalities are strict. The root ideal is Δ={(1,4),(1,5)}. Bounce paths: 1→4, and 2,3,5 isolated, so top(3)=3 > top(2)=2. But columns 3 and 4 have lengths 0 and 1, so no ceiling there. The subsequent application of Lemma 2.8 to conclude K(Δ;Δ;µ)=K(Δ;Δ;µ−ε_3) lacks the needed hypotheses; the natural candidate y=2 fails the mirror condition. So the straightening step is unjustified.\n\nThis is load-bearing: Lemma 3.4 is used in Theorem 3.5, then Lemma 4.2, Proposition 4.3, and finally Theorems 1.3 and 1.4. The error appears in the proof as written, not just in the statement; the proof claims the ceiling without proof. I have not checked whether the equality itself is true for the counterexample—it may be—but the paper does not give a valid argument.\n\nOn proportionality: the rest of the paper is careful and the new Mirror Lemma II is interesting. Theorem 1.2 is solid. But the main results as written are not established. This warrants extensive revision, not desk rejection; the ideas are worth salvaging. Send to peer review, but the referee should focus on Lemma 3.4 and its applications.","headline":"A well-intentioned but currently broken straightening lemma leaves Theorems 1.3–1.4 unproved as written.","tokens_in":26018,"tokens_out":8974,"would_cite":false,"duration_ms":67436,"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 proves the alternating dual Pieri rule for closed k-Schur Katalan functions in the large-k limit and for strictly decreasing partitions, and proves the k-branching conjecture for strictly decreasing partitions.","keywords":["k-Schur functions","Katalan functions","closed k-Schur Katalan functions","alternating dual Pieri rule","k-branching","lowering operators","symmetric function positivity"],"falsifier":"Take any strictly decreasing $k$-bounded partition $\\lambda$ and compute the expansion (38) for a single lowering operator $L_z$; if any coefficient outside $\\{0, \\pm 1\\}$ appears, Proposition 4.3 and hence Theorem 1.3 collapse. Alternatively, test Lemma 3.4 directly on a root ideal with $\\mathrm{top}_{\\Delta_k(\\mu)}(z+1) > \\mathrm{top}_{\\Delta_k(\\mu)}(z)$ and check whether the asserted equality $\\tilde{g}^{(k)}_\\mu = \\tilde{g}^{(k)}_{\\mu - \\epsilon_{z+1}}$ holds by direct computation; a single failure would break the proof at its first step.","tokens_in":24889,"feed_emoji":"➕","tokens_out":20787,"duration_ms":178411,"temperature":0.7,"pith_summary":"Closed $k$-Schur Katalan functions are a basis of a natural filtration of the symmetric functions, and their structure constants were conjectured to alternate in sign. This paper establishes the alternating dual Pieri rule for them in two regimes: in the large-$k$ limit, and for strictly decreasing $k$-bounded partitions for every $k$. It also establishes the $k$-branching expansion for strictly decreasing $k$-bounded partitions. The reason to care is that the alternating sign is the K-theoretic form of positivity: after multiplying by $(-1)^{|\\lambda|-|\\mu|-m}$, all coefficients are nonnegative integers, so the expansion is a genuine positive statement rather than a formal identity.","feed_headline":"Two k-Schur sign conjectures proven for decreasing shapes","feed_subtitle":"Every lowering operator contributes 0 or ±1, so the conjectured alternating signs hold.","key_machinery":"The mechanism is the lowering-operator calculus on the root ideals $\\Delta_k(\\lambda) = \\{ (i,j) : k - \\lambda_i + i < j \\}$. Lowering operators $L_z$ act on a generalized closed $k$-Schur Katalan function $\\tilde{g}^{(k)}_\\mu = K(\\Delta_k(\\mu); \\Delta_k(\\mu); \\mu)$ by shifting the subscript; Theorem 3.5 expands $L_z \\tilde{g}^{(k)}_\\lambda$ into at most three such functions. The crucial 'straightening' Lemma 3.4, proved with the new Mirror Lemma II, uses the bounce-path structure of $\\Delta_k(\\mu)$ to replace a subscript by an equal function whose entries are closer to being a partition. Iterating the straightening along the bounce path yields Proposition 4.3, the $0$-or-$\\pm 1$ expansion of a single lowering operator, which is the engine for the sign-alternation theorems.","core_discovery":"The central claim is that two sign-alternation conjectures for closed $k$-Schur Katalan functions are true on the stated families. For a strictly decreasing $k$-bounded partition $\\lambda$, every coefficient $c_{\\lambda\\mu}$ in $G_{1m}^\\perp \\tilde{g}^{(k)}_\\lambda = \\sum_{\\mu} c_{\\lambda\\mu} \\tilde{g}^{(k)}_\\mu$ satisfies $(-1)^{|\\lambda|-|\\mu|-m} c_{\\lambda\\mu} \\in \\mathbb{Z}_{\\ge 0}$, and every coefficient $a_{\\lambda\\mu}$ in the $k$-branching expansion $\\tilde{g}^{(k)}_\\lambda = \\sum_{\\mu} a_{\\lambda\\mu} \\tilde{g}^{(k+1)}_\\mu$ satisfies $(-1)^{|\\lambda|-|\\mu|} a_{\\lambda\\mu} \\in \\mathbb{Z}_{\\ge 0}$. In the large-$k$ limit, the first expansion is represented by the exact identity $G_{1\\ell}^\\perp g_\\lambda = g_{\\lambda - 1^\\ell}$. The proof shows that a single lowering operator $L_z$ acts on a strictly decreasing shape with coefficients equal to $0$ or $\\pm 1$, and that products of such operators, together with the binomial weights in $G_{1m}$, preserve the prescribed alternating sign.","pith_inferences":["Beyond the paper's stated $m=\\ell$ large-$k$ identity, the same $e_d^\\perp$-expansion calculation implies the full alternating dual Pieri rule in the stable limit for every $m$; the ingredients are already in Lemma 4.1 and the definition of $G_{1m}$.","The failure of the $\\pm 1$ structure for a single lowering operator when $\\lambda$ has equal parts (Example 3.7) suggests the full conjecture reduces to controlling equal-part descents; a direct test is whether the binomial-weighted sum over subsets $S$ in (45) repairs the signs even when individual $L_z$ do not.","A signed combinatorial model for the straightening lemma, in which each nonzero coefficient corresponds to a bounce-path configuration, could turn these proofs into a bijective proof and likely extend them to all $k$-bounded partitions.","Under the K-theoretic identification that motivated the conjectures, these results imply alternating-sign properties for the polynomial images, a translation the paper leaves implicit."],"forward_implications":["The alternating dual Pieri rule holds for every strictly decreasing $k$-bounded partition, for every $k$ and every $m \\ge 0$.","The $k$-branching conjecture holds for every strictly decreasing $k$-bounded partition, so the filtration steps $\\Lambda^{(k)} \\subset \\Lambda^{(k+1)}$ have the predicted alternating-sign expansion on this family.","In the large-$k$ regime, the operator $G_{1\\ell}^\\perp$ sends $g_\\lambda$ to the single function $g_{\\lambda - 1^\\ell}$, giving a clean identity that realizes the first conjecture's alternating sign as exactly one term.","Because Theorem 1.4 is deduced from Theorem 1.3 through shift invariance, any future extension of Theorem 1.3 to a larger class of partitions automatically yields a matching $k$-branching result.","The coefficient structure is rigid: individual lowering operators contribute $0$ or $\\pm 1$, and the sign is fixed solely by the difference in sizes, so the conjectures' alternating signs are not accidental cancellations."],"supporting_citations":[{"why":"The article that introduced Katalan functions and closed k-Schur Katalan functions, posed the two conjectures, and supplied the Mirror Lemma, the removal/addability identities, the e-perp expansion lemma, and the shift-invariance property used in the proofs.","marker":"[3]"},{"why":"The source of the generalized partition set and the root-ideal fact used for it, and of the Catalan-function method the straightening argument adopts.","marker":"[1]"},{"why":"The predecessor that proved two sibling conjectures of the same list and identified closed k-Schur Katalan functions with K-homology representatives, setting the comparison case this paper extends.","marker":"[12]"}],"fun_headline_variants":["Sign conjectures proven for decreasing k-Schur shapes","Alternating signs hold for k-Schur decreasing partitions","Two k-Schur sign rules proven on decreasing shapes","Decreasing shapes verify alternating signs in k-Schur"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof needs the straightening step in Lemma 3.4 to always terminate at zero: repeatedly applying a lowering operator along a bounce path eventually annihilates the difference of the two generalized functions, and the vanishing bound of Proposition 2.10 must apply to both summands of every difference.","fun_headline_variants_meta":{"raw":{"variants":["Sign conjectures proven for decreasing k-Schur shapes","Alternating signs hold for k-Schur decreasing partitions","Two k-Schur sign rules proven on decreasing shapes","Decreasing shapes verify alternating signs in k-Schur"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000201,"raw_usage":{"total_tokens":1358,"prompt_tokens":905,"completion_tokens":453,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":521,"completion_tokens_details":{"reasoning_tokens":390}},"tokens_in":521,"tokens_out":453,"duration_ms":4481,"temperature":1.0,"reasoning_tokens":390,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:40:09.940328+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any strictly decreasing $k$-bounded partition $\\lambda$ and compute the expansion (38) for a single lowering operator $L_z$; if any coefficient outside $\\{0, \\pm 1\\}$ appears, Proposition 4.3 and hence Theorem 1.3 collapse. Alternatively, test Lemma 3.4 directly on a root ideal with $\\mathrm{top}_{\\Delta_k(\\mu)}(z+1) > \\mathrm{top}_{\\Delta_k(\\mu)}(z)$ and check whether the asserted equality $\\tilde{g}^{(k)}_\\mu = \\tilde{g}^{(k)}_{\\mu - \\epsilon_{z+1}}$ holds by direct computation; a single failure would break the proof at its first step.","supporting_citations":[{"cited_title":"Blasiak, J","cited_arxiv_id":null,"evidence_quote":"The article that introduced Katalan functions and closed k-Schur Katalan functions, posed the two conjectures, and supplied the Mirror Lemma, the removal/addability identities, the e-perp expansion lemma, and the shift-invariance property used in the proofs."},{"cited_title":"Blasiak, J","cited_arxiv_id":null,"evidence_quote":"The source of the generalized partition set and the root-ideal fact used for it, and of the Catalan-function method the straightening argument adopts."},{"cited_title":"Ikeda, S","cited_arxiv_id":null,"evidence_quote":"The predecessor that proved two sibling conjectures of the same list and identified closed k-Schur Katalan functions with K-homology representatives, setting the comparison case this paper extends."}],"review_version":1}