{"id":"7a1c235d-91e1-413e-83d9-113e77a41de1","arxiv_id":"2607.10968","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every k≥1 and n≥0 the coefficients J_{k,n}(m) of the Jacobi theta tails are positive whenever |m|≤k+n.","lead":"The paper proves that the coefficients in the generating function for tails of the Jacobi theta series are strictly positive in a natural range of degrees. This settles the infinite-product case of a positivity conjecture and implies Merca's conjecture on truncated Jacobi triple products.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The Reader correctly isolates the non-negativity of N_k as the sole non-formal step, yet that step is immediate from the definitions once the reciprocal q-Pochhammers are expanded. All preceding identities are standard q-series manipulations (Heine transformation, elementary telescoping of partial theta series) that introduce no hidden sign changes. The explicit lower bound J_{k,n}(m)≥1 on the stated range follows at once from the geometric-series expansion of the principal term. No circularity, free parameters or external conjectures remain. The paper therefore stands as a clean elementary proof of the d=∞ case of the author’s earlier conjecture, and the Reader’s ACCEPT verdict with high confidence requires no adjustment.","tokens_in":7953,"tokens_out":488,"duration_ms":4846,"concrete_test":"For the smallest case k=1 expand both sides of identity (1.4) as Laurent series in z and q up to total degree 20 (or higher) by truncated products for the Pochhammers and direct summation of f_1,f_2,g_2; verify that every coefficient of N_1(z,q) is non-negative and that the extracted J_{1,n}(m) equal 1 or greater precisely on the claimed range |m|≤1+n.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central positivity claim of Corollary 1.6 rests on the non-negativity of the Laurent coefficients of the remainder N_k(z,q) after the explicit identity of Theorem 1.5 is rewritten. That non-negativity follows immediately from the series definitions (1.3) of f_a and g_a: every summand is a monomial q^{i^2+...}x^i divided by products of (q)_i and (x)_j, and the reciprocal of each (·)_∞ expands as a generating function with non-negative coefficients (the ordinary partition generating function). The algebraic rearrangements in Lemmas 2.1–2.3 and Proposition 2.4 are elementary and free of sign-changing cancellations. Consequently the only potential soft spot identified by the Reader is in fact secure; no further load-bearing gap appears in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves a positivity property for the tails of the Jacobi theta series. For integers k≥1 and n≥0 it defines coefficients J_{k,n}(m) via the two-variable generating function in the abstract (equivalently the left-hand side of (1.4)), and shows that J_{k,n}(m)>0 for every integer m with -(k+n)≤m≤k+n. The argument proceeds by establishing an explicit identity (Theorem 1.5) that rewrites the truncated series in terms of the auxiliary series f_a and g_a of (1.3). After algebraic rearrangement the identity yields Corollary 1.6, in which the target generating function equals an elementary positive term plus a remainder N_k(z,q) whose Laurent coefficients are non-negative. The non-negativity of N_k follows at once from the series definitions of f_1, f_2 and g_2 together with the ordinary partition expansions of the reciprocal q-Pochhammer symbols. As a consequence the limiting case d=∞ of the author’s earlier Conjecture 1.3 is settled, and Merca’s Conjecture 1.2 is recovered.","tokens_in":8172,"tokens_out":805,"duration_ms":6927,"significance":"The result settles the d=∞ case of a natural two-variable refinement of Merca’s conjecture on truncated Jacobi triple products, and thereby recovers that conjecture itself. The proof is entirely elementary (Heine’s second transformation, a short recursion for f_a/g_a, and a partial-theta decomposition), self-contained, and free of circularity. The explicit identity of Theorem 1.5 supplies a concrete positive expansion that may be useful for further refinements or for combinatorial interpretations. The work therefore constitutes a clean and substantial advance in the theory of truncated theta series and positivity of q-series coefficients.","major_comments":[],"minor_comments":[{"comment":"In the abstract and in the definition preceding Corollary 1.6 the factor (q/z)_∞ appears, while the body of Theorem 1.5 and Lemma 2.3 write (z^{-1}q)_∞; the two are identical, but a uniform choice of notation would improve readability.","section":null},{"comment":"The final counting argument in the proof of Corollary 1.6 (the double sum over h and r that produces the lower bound 1) is correct but slightly compressed; a one-sentence reminder that every residue class t mod 2 is hit at least once for 0≤t≤2(n+k) would make the strict positivity completely transparent.","section":null},{"comment":"References [3] and [12] are cited as preprints with arXiv numbers; if they have since appeared or been updated, the bibliographic data should be refreshed before publication.","section":null},{"comment":"A short remark after Corollary 1.6 noting that the same argument yields the non-negativity claimed in Merca’s Conjecture 1.2 (by the specialisation indicated in Remark 1.4) would make the logical dependence fully explicit for the reader.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is short, self-contained and technically clean. The only soft spot flagged by the reader (non-negativity of the remainder N_k) is in fact immediate from the definitions of f_a and g_a; no load-bearing gap remains. I see no reason to delay acceptance."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper settles the infinite-d limit of the author’s earlier positivity conjecture on two-variable Jacobi tails and, as a corollary, Merca’s conjecture on truncated triple products. The new piece is the explicit identity in Theorem 1.5 that rewrites the signed tail as a sum of three series built from the auxiliary functions f_a and g_a; once that identity is in hand, non-negativity of the remainder N_k is immediate from the partition generating functions, and a short counting argument upgrades it to the strict positivity claimed in Corollary 1.6.\n\nWhat works well is the elementary character of the argument. Heine’s second transformation, the recursive relation between f_a and g_a (re-proved for completeness), and the partial-theta decomposition of Lemma 2.3 are all written out cleanly; there are no free parameters and no circular appeal to the target positivity. The self-citations are only for motivation and for a previously established recursion that is re-derived here, so the logical chain is self-contained.\n\nThe only soft spot the reader flagged—the claim that N_k has non-negative Laurent coefficients—is in fact secure: every summand of f_1, f_2, g_2 is a monomial over products of Pochhammers, and the reciprocal infinite products expand with non-negative coefficients. No sign-changing cancellations appear in the rearrangements. The paper does not treat finite d ≥ 2, but it never claims to; the limiting case is already enough to recover Merca.\n\nThis is for people who work on truncated q-series and partition inequalities. The proofs are short enough that a reading-group session could walk through them. I would send it to referees without hesitation; the result is new, the method is transparent, and the literature engagement is honest.","headline":"Solid elementary proof of the d=∞ case of Zhou’s own positivity conjecture for Jacobi tails; implies Merca’s conjecture and is ready for referees.","tokens_in":8698,"tokens_out":498,"would_cite":true,"duration_ms":4292,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11P82","11B83","05A16"],"pacs":[],"model":"grok-4.5","headline":"Tails of the Jacobi theta series have strictly positive coefficients in a symmetric band of width 2(k+n).","keywords":["positivity","Jacobi theta series","partial theta series","q-series","truncatedations","basic hypergeometric series"],"falsifier":"Expand the remainder N_k(z,q) for a fixed small k (say k=1 or 2) up to moderate total degree in z and q and check whether any coefficient is negative; a single negative coefficient would refute the claim.","tokens_in":8892,"feed_emoji":"∑","tokens_out":684,"duration_ms":5077,"temperature":0.7,"pith_summary":"The paper proves a positivity property for the infinite tails of Jacobi’s triple-product (theta) series. After normalizing by the usual infinite products, the remaining series, expanded as a double power series in z and q, has coefficients J_{k,n}(m) that are strictly positive whenever the exponent m of z lies between -(k+n) and k+n. The result settles the limiting case of a two-variable positivity conjecture previously stated by the author and, as a corollary, confirms Merca’s conjecture on non-negative coefficients of truncated Jacobi triple-product series. The argument is purely algebraic: elementary identities for basic hypergeometric series are used to rewrite the tail as an explicit positive main term plus a remainder series whose non-negativity is visible term-by-term. A sympathetic reader cares because positivity of such tails supplies uniform lower bounds on partial theta series and sharpens several classical truncation results that appear throughout partition theory and q-series.","feed_headline":"Jacobi theta tails stay positive in a wide band","feed_subtitle":"Elementary q-series identities prove every coefficient inside |m|≤k+n is strictly positive","key_machinery":"The explicit q-series identity of Theorem 1.5 that rewrites the Jacobi tail in terms of three families of basic hypergeometric series f_a and g_a; once these series are expanded, every coefficient is visibly non-negative and the main term already supplies the claimed positivity band.","core_discovery":"For every integer k≥1 and every n≥0 the Laurent coefficients J_{k,n}(m) of the normalized Jacobi tail are positive for all integers m with -(k+n)≤m≤k+n. Equivalently, the generating function of the tail equals z^{-k}(1+z+⋯+z^{2k})/((1-qz)(1-z^{-1}q))+z^{-k}N_k(z,q), where N_k has exclusively non-negative coefficients.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Jacobi theta tails positive for all m with |m|≤k+n","Central band |m|≤k+n holds only positive Jacobi tail coeffs","q-series show Jacobi theta tails positive throughout |m|≤k+n","Elementary proof: Jacobi tails stay positive inside |m|≤k+n","All J_{k,n}(m) positive precisely when |m|≤k+n"],"cache_read_input_tokens":128,"weakest_assumption_plain":"That the three auxiliary series f1, f2 and g2 expand with only non-negative coefficients once the reciprocal q-Pochhammer symbols are written as power series.","fun_headline_variants_meta":{"raw":{"variants":["Jacobi theta tails positive for all m with |m|≤k+n","Central band |m|≤k+n holds only positive Jacobi tail coeffs","q-series show Jacobi theta tails positive throughout |m|≤k+n","Elementary proof: Jacobi tails stay positive inside |m|≤k+n","All J_{k,n}(m) positive precisely when |m|≤k+n"]},"model":"grok-4.5","effort":"low","cost_usd":0.005708,"raw_usage":{"total_tokens":1534,"prompt_tokens":778,"num_sources_used":0,"completion_tokens":91,"cost_in_usd_ticks":57080000,"prompt_tokens_details":{"text_tokens":778,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":665,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":778,"tokens_out":91,"duration_ms":4839,"temperature":1.0,"reasoning_tokens":665,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T07:58:11.453857+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Expand the remainder N_k(z,q) for a fixed small k (say k=1 or 2) up to moderate total degree in z and q and check whether any coefficient is negative; a single negative coefficient would refute the claim.","supporting_citations":[],"review_version":1}