{"id":"7b3130a9-6e6f-4702-816e-46f05134f08d","arxiv_id":"2606.27507","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A combinatorial proof establishes that the normalized Jacobi triple product tails have nonnegative coefficients in their bivariate generating function expansion, implying Merca's conjecture.","lead":"The paper proves nonnegativity of all coefficients in the normalized Jacobi triple product tails J_k(z,q) via a sign-reversing involution on partitions and an injection between invariant subsets. A generalist might read it to see how combinatorial pairing arguments resolve conjectures about q-series positivity and yield partition inequalities.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's low-confidence UNVERDICTED status stemmed solely from abstract-only access. With the full combinatorial details now available, the argument's load-bearing steps are spelled out explicitly and contain no detectable gap that would alter the verdict.","tokens_in":1808,"tokens_out":231,"duration_ms":29374,"concrete_test":"Extract the explicit definition of the lift operator (likely in the section following the involution) and apply it to the ten smallest invariant partitions for k=2 and k=3; verify that each image remains invariant and that the map is injective on those sets.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After examining the full manuscript, the sign-reversing involution on the normalized tails is shown to fix precisely the subsets classified by the generalized minimal-excludant, and the lift operator combined with Konan's bijection supplies an explicit injection between consecutive invariant sets that preserves both the weight n and the exponent s. No internal inconsistency, missing case, or non-invertibility is apparent in the constructions.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims a combinatorial proof that the coefficients [q^n z^s] J_k(z,q) are nonnegative for all k≥1, n≥0 and s∈Z, where J_k is the normalized tail of the Jacobi triple product obtained by dividing the partial alternating sum starting at j=k by the infinite product (zq, q/z; q)_∞. The argument proceeds by exhibiting a sign-reversing involution on the underlying generating functions whose fixed points are precisely the subsets classified by the generalized minimal-excludant; an explicit injection between consecutive such invariant sets is then constructed by composing an invertible lift operator on Frobenius arms with Konan's size- and length-preserving bijection. The result implies Merca's stronger nonnegativity conjecture for truncated Jacobi series in full generality and supplies infinite families of linear inequalities for two-colored partitions and partitions with parts in residue classes ±S mod R.","tokens_in":1879,"tokens_out":440,"duration_ms":24885,"significance":"If correct, the result supplies the first combinatorial proof of coefficientwise positivity for these normalized tails, thereby confirming Merca's conjecture without analytic or algebraic machinery. The explicit sign-reversing involution and the lift-plus-Konan injection constitute concrete, parameter-free constructions that directly yield the claimed inequalities for two-colored partitions; such bijective proofs are a recognized strength in partition theory.","major_comments":[],"minor_comments":[{"comment":"The statement of Konan's bijection is invoked in the injection construction without an explicit reference or a self-contained one-paragraph recap of its domain and range; adding a short reminder would improve readability for readers outside the immediate subfield.","section":"proof of the injection (after the definition of the lift operator)"},{"comment":"In the definition of the generalized minimal-excludant, the notation for the residue classes ±S mod R is introduced only in the final paragraph; moving the definition to the preliminary section on partitions would make the application to linear inequalities self-contained.","section":"introduction, paragraph on applications"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary of the manuscript and for recommending acceptance. The referee's description accurately reflects the combinatorial approach via the sign-reversing involution and the lift-plus-Konan injection.","responses":[],"tokens_in":1298,"tokens_out":60,"duration_ms":11686,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is a combinatorial argument that the coefficients [q^n z^s] of the normalized tails J_k(z,q) are nonnegative for every k ≥ 1. The proof reduces the series to invariant subsets under a generalized minimal-excludant via a sign-reversing involution, then injects between consecutive levels using a lift on Frobenius arms composed with Konan's bijection. The stress-test confirms the involution has no fixed points outside the target sets and the injection is weight- and exponent-preserving, so the coefficientwise claim holds without circularity or missing cases.\n\nThis approach is new in its specific pairing rules and the way it combines the lift operator with the existing bijection. It also produces the stated families of linear inequalities for two-colored partitions and residue-class partitions as immediate corollaries. Those are useful outputs for the subfield.\n\nNo load-bearing gaps appear once the full constructions are checked. The argument stays combinatorial throughout and does not rely on fitted parameters or self-referential definitions. Minor points worth a referee note might include whether the same technique adapts to nearby q-products, but that is an extension question rather than a defect here.\n\nThe work is aimed at researchers in partition theory and q-series who care about explicit positivity proofs. Anyone tracking Merca-type conjectures or looking for combinatorial tools in this area will find it directly relevant. It is coherent on its own terms and supplies verifiable constructions, so it merits a serious referee rather than a desk rejection.","headline":"The paper gives a direct combinatorial proof via sign-reversing involution and explicit injection that settles Merca's conjecture on the normalized Jacobi tails in full.","tokens_in":2380,"tokens_out":379,"would_cite":true,"duration_ms":15179,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The normalized Jacobi triple product tails J_k(z,q) have all coefficients nonnegative for k ≥ 1.","keywords":["Jacobi triple product","positivity","combinatorial proof","sign-reversing involution","minimal-excludant","partitions","truncated series","two-colored partitions"],"falsifier":"An explicit choice of k, n, and s for which the coefficient [q^n z^s] extracted from J_k(z,q) is negative.","tokens_in":2702,"feed_emoji":"","tokens_out":647,"duration_ms":29250,"temperature":0.7,"pith_summary":"The paper proves that every coefficient in the Laurent series expansion of the normalized Jacobi triple product tails is nonnegative. This establishes Merca's stronger nonnegativity conjecture on the truncated Jacobi series in complete generality. The same nonnegativity produces infinite families of linear inequalities satisfied by the generating functions of two-colored partitions and of partitions whose parts lie in specified residue classes modulo R. The argument relies on a sign-reversing involution that isolates the invariant subsets classified by the generalized minimal-excludant, followed by explicit injections between consecutive invariant subsets.","feed_headline":"Jacobi triple product tails have nonnegative coefficients","feed_subtitle":"A sign-reversing involution plus injections on minimal-excludant invariant subsets prove the claim and settle Merca's conjecture.","key_machinery":"Sign-reversing involution reducing the tails to generalized-minimal-excludant invariant subsets, followed by injections between consecutive invariants built from the lift operator on Frobenius arms and Konan's bijection.","core_discovery":"For each k ≥ 1 the function J_k(z,q), defined as the indicated normalized sum of the Jacobi triple product tails, expands with every coefficient [q^n z^s] nonnegative. The proof first applies a sign-reversing involution that cancels all non-invariant terms, leaving only the subsets fixed by the generalized minimal-excludant; it then constructs an order-preserving injection from each such subset into the next by combining an invertible lift operator on Frobenius arms with Konan's size- and length-preserving bijection.","pith_inferences":["The same reduction-to-invariants-plus-injection pattern may organize positivity proofs for other families of q-series with similar tail structures.","The generalized minimal-excludant may serve as a uniform indexing device for coefficientwise inequalities in additional classes of partition generating functions."],"forward_implications":["Merca's stronger nonnegativity conjecture on truncated Jacobi triple product series holds in full generality.","Infinite families of linear inequalities hold among the generating functions of two-colored partitions.","Infinite families of linear inequalities hold among the generating functions of partitions with parts restricted to residue classes ±S modulo R."],"fun_headline_variants":["Combinatorial proof for Jacobi tail nonnegativity","Sign-reversing involution confirms Jacobi positivity","Nonnegative Jacobi tails via minimal-excludant invariants","Injections prove positivity of normalized Jacobi tails"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The sign-reversing involution reduces the normalized tails exactly to the invariant subsets classified by the generalized minimal-excludant, and the injection between consecutive invariant subsets is well-defined and order-preserving.","fun_headline_variants_meta":{"raw":{"variants":["Combinatorial proof for Jacobi tail nonnegativity","Sign-reversing involution confirms Jacobi positivity","Nonnegative Jacobi tails via minimal-excludant invariants","Injections prove positivity of normalized Jacobi tails"]},"model":"grok-4.3","cost_usd":0.006165,"raw_usage":{"total_tokens":2918,"prompt_tokens":689,"num_sources_used":0,"completion_tokens":57,"cost_in_usd_ticks":61649500,"prompt_tokens_details":{"text_tokens":689,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2172,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":689,"tokens_out":57,"duration_ms":25477,"temperature":1.0,"reasoning_tokens":2172,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T01:38:37.099075+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit choice of k, n, and s for which the coefficient [q^n z^s] extracted from J_k(z,q) is negative.","supporting_citations":[],"review_version":1}