{"id":"3f5a7fc3-9b86-4991-9d6e-7bf16253171f","arxiv_id":"2507.16644","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the eta quotients (q^i;q^i)_∞/(q^p;q^p)_∞ with p prime >3 and i>1, the signs of the coefficients are shown to be periodic modulo p after an explicit threshold, proving and generalizing conjectures of Bringmann et al.","lead":"An infinite product such as (q^2;q^2)_∞/(q^5;q^5)_∞ expands into a power series whose coefficients are positive, negative, or zero according to a repeating pattern; this paper proves that such patterns exist for a whole family of products. The proof settles and extends several conjectures by Bringmann and coauthors and gives explicit bounds for when the periodic behavior starts.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central theorem rests on an unpublished m-dissection formula (Theorem 2.1) that the paper does not prove; a sign or exponent error there would invalidate the sign pattern.","rationale":"I read the paper as a serious and likely correct contribution: the sign analysis is structurally sound once Theorem 2.1 is granted, and the use of Lemma 2.1 to convert dissections into coefficient inequalities is coherent. The single most load-bearing point is exactly the one the Reader identifies: Theorem 2.1 is an unpublished formula by two of the current authors, and the proof of Theorem 1.1 is nothing more than that formula specialized, rescaled, and divided by (q^p;q^p)_∞. A sign mistake in s(r) or an exponent mistake in L(r) would propagate directly into the sign pattern. Because no proof or independent computational verification of (2.1) is included in the manuscript, the central claim is not yet self-contained. The N-definition issue is real but minor: for some residues the minimizing set is empty, making the printed max/min formally undefined, but this can be repaired without touching the sign characterization, so it does not change the verdict. I therefore agree with the Reader's weakest assumption and recommend keeping the verdict CONDITIONAL; no change is needed beyond the Reader's stated condition that the dissection formula be supplied or independently verified.","tokens_in":17644,"tokens_out":11168,"duration_ms":123113,"concrete_test":"Using Sage or Mathematica, compute both sides of (2.1) for at least (M,j,m)=(4,1,5), (4,1,7), and (5,2,7), expanding to order q^{200}; the identity must hold exactly. Then check Corollary 2.1 against a direct expansion of (q;q)_∞ to the same order for m=5,7,11. If any coefficient disagrees, Theorem 2.1 is false and Theorem 1.1's proof collapses. If all agree, the remaining issue is the missing proof of [22], which can be resolved by the authors supplying the derivation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.1 quotes Theorem 2.1, an m-dissection formula for a quintuple product, from the authors' unpublished preprint [22], and Corollary 2.1 is obtained by substituting M=4, j=1. Every step of the proof of Theorem 1.1 (Section 3) uses this dissection: replacing q by q^i and dividing by (q^p;q^p)_∞, the sign of each term is (-1)^{s(r)} and the residue is controlled by L(r). If either s(r) or L(r) in Theorem 2.1 is wrong, the claimed positive/negative/zero characterization in Theorem 1.1 changes. The paper provides no derivation of (2.1) and no independent check of it, so the central claim is currently contingent on an unverified black box from a preprint by two of the same authors. A secondary issue is that the definition of N in Theorem 1.1 uses min over possibly empty sets (e.g., for (p,i)=(5,2), residues 1 and 3 mod 5 are not represented by iL(r)); this makes N formally undefined, although it is easily repaired by taking the max only over nonempty residue classes.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the sign patterns of the Fourier coefficients of the eta quotient (q^i;q^i)_∞/(q^p;q^p)_∞ for primes p>3 and integers i>1 with p∤i. Theorem 1.1 asserts that, for n larger than an explicit bound N(p,i), the sign of the coefficient an is determined entirely by n mod p: the positive, negative, and zero residues are specified in terms of the quadratic L(r)=6r^2+r and the residue of p modulo 3. Corollary 1.1 specializes this to (p,i)=(5,2) and thereby proves a conjecture of Bringmann et al. (C215^-1). Theorem 1.2 gives sign-periodicity results for ten further eta quotients, several of which address other conjectures from the same source. The proofs are based on m-dissections, a quintuple-product dissection formula, theta-function identifications, and a lemma asserting that certain quotients of infinite products have nonnegative coefficients.","tokens_in":17889,"tokens_out":10569,"duration_ms":103241,"significance":"If the main results are correct, the paper provides a uniform method for determining sign patterns of a natural family of eta quotients and proves several open conjectures from the Bringmann et al. program, including the previously conjectured sign pattern for (q^2;q^2)_∞/(q^5;q^5)_∞. The explicit N(p,i) bounds and the sharpness table are also useful. However, the central theorem depends on an unproved m-dissection formula quoted from the authors' own unpublished preprint, and the statement of N(p,i) is formally ill-defined for some (p,i); these issues must be addressed before the results can be considered fully established. The paper does not provide machine-checked proofs, but the small cases (p,i)=(5,2) and (7,2) are consistent with the stated patterns.","major_comments":[{"comment":"Theorem 2.1, the m-dissection formula for the quintuple product, is quoted from reference [22], an unpublished preprint by three of the authors, and is not proved in this manuscript. This formula is the sole source of Corollary 2.1, which is then used in every step of the proof of Theorem 1.1 in Section 3. Since a sign or exponent error in s(r) or L(r) would change the claimed positive/negative/zero classification, the main theorem is currently contingent on an unverified black box. The authors should either provide a complete proof of Theorem 2.1 in an appendix or replace [22] with a published source that the reader can check.","section":"Section 2.1, Theorem 2.1"},{"comment":"The definition N = max(∪_{s=0}^{p-1} min{iL(r) : iL(r) ≡ s (mod p), 0 ≤ r ≤ p−1}) − p is not well-formed, because for some (p,i) the set over which min is taken is empty. For example, when (p,i)=(5,2), the residues 1 and 3 modulo 5 are not represented by iL(r) for any r, so min over an empty set is undefined and N is undefined. The statement can be repaired by taking the maximum only over nonempty residue classes (or by defining min(∅)=+∞), but as written the theorem is not formally meaningful for all admissible (p,i).","section":"Theorem 1.1, definition of N"},{"comment":"The proof asserts without detail that the quantities t1(r), 4p^2−t1(r), 4p^2, 4p^2+t1(r), 8p^2−t1(r), 8p^2, t2(r), and 8p^2−t2(r) are pairwise distinct and all multiples of p. This verification is load-bearing: it is exactly what ensures that after division by (q^p;q^p)_∞ each numerator factor (1−q^{iα}) cancels a distinct factor of the denominator, so that the remaining series has nonnegative coefficients and the sign of each term is governed by (−1)^{s(r)}. The authors should provide the omitted verification for both cases p≡1 (mod 3) and p≡−1 (mod 3), including a demonstration that no overlap occurs between the arithmetic progressions generated by the different factors.","section":"Section 3, proof of Theorem 1.1"}],"minor_comments":[{"comment":"The title contains a spacing artifact ('P A TTERNS') that should be corrected, and the abstract can be made more precise about the scope of the 'additional classes' covered by Theorem 1.2.","section":"Title and Abstract"},{"comment":"Reference [3] contains a typo, 'Ramunujan' instead of 'Ramanujan'; reference [22] is an unpublished preprint whose status should be updated if it has appeared.","section":"References"},{"comment":"In the proof of part (5), the congruence argument 'if n1^2+m1^2 ≡ n2^2+m2^2 (mod 4), then (n1+n2)(n1−n2) ≡ (m1+m2)(m1−m2) (mod 2)' is not written correctly; the left- and right-hand sides should both involve differences (n1−n2) and (m1−m2). The intended conclusion about parity is likely correct, but the displayed line needs correction.","section":"Section 4, proof of Theorem 1.2(5)"},{"comment":"Part (2) states 'For p = 1 or an odd prime'; the case p=1 should be separated from the prime case for clarity, since p=1 is not prime.","section":"Theorem 1.2(2)"}],"recommendation":"major_revision","confidential_remarks":"The heavy reliance on an unpublished preprint co-authored by three of the six authors is a concern for a journal-referee process, because the main theorem cannot be checked from the submitted manuscript alone. I would ask the authors to include a full proof of Theorem 2.1 in a revised version (or to replace it with a published reference), and to fix the definition of N. The subject matter is well within the scope of the journal and the results are of interest to the Ramanujan-type identities community, so I see this as a major revision rather than a rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me get straight to it: the main theorem is a real result. It gives the full sign pattern, beyond any fixed bound, for (q^i;q^i)_∞/(q^p;q^p)_∞ for every prime p>3 and i>1 not divisible by p, and it settles a handful of Bringmann et al.'s conjectures in one go. I checked the small cases (p,i)=(5,2) and (7,2) and the claimed patterns match. The method is classical dissection, and the paper uses it cleanly. The companion results in Theorem 1.2 are a nice bonus: several more eta quotients get full sign stories, often by short arguments via Borwein's theta functions and Lemma 2.1.\n\nThe soft spot is exactly where the reader's report puts it. Theorem 2.1, an m-dissection formula for a quintuple product, is quoted from the authors' own unpublished preprint [22]. Everything in the proof of Theorem 1.1 flows through Corollary 2.1, which is just Theorem 2.1 with M=4, j=1. If there is a sign or exponent error in that formula, the sign patterns change. The paper gives no proof and no independent check. This is not circular—the formula is a general identity and doesn't assume the conclusions—but it is a genuine self-containment problem. A referee can't fully verify the paper without access to [22]. The fix is simple: put the proof of Theorem 2.1 in an appendix or spell out the derivation. A second, smaller technical issue: the definition of N in Theorem 1.1 takes a max over mins over sets that can be empty (e.g., for p=5, i=2, residues 1 and 3 mod 5 never occur), so N is formally undefined. That's trivially repaired by only ranging over represented residues. The omitted 'routine' checks in the proof—e.g., that the t1(r), t2(r) quantities are pairwise distinct—are indeed routine; I did a couple and they work.\n\nWho is this for? Specialists in q-series and eta quotients. It doesn't open a new branch of mathematics, but it resolves open conjectures and provides a clean framework for a family of sign problems. The paper deserves a serious referee. I'd recommend sending it to review, with a request that the authors either include the proof of Theorem 2.1 or point to a public, complete version. The flaws are addressable and the core mathematics looks correct.","headline":"Genuinely new sign-pattern results that resolve Bringmann et al. conjectures, but the central dissection formula is imported from the authors' unpublished preprint and needs to be proved in the paper.","tokens_in":18423,"tokens_out":2623,"would_cite":true,"duration_ms":27083,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F30","30C50"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every prime p>3 and integer i>1 not divisible by p, the coefficients of (q^i;q^i)_∞/(q^p;q^p)_∞ are eventually periodic in sign modulo p, with an explicit threshold; the case (p,i)=(5,2) proves a conjecture from [9].","keywords":["vanishing coefficients","periodic sign changes of coefficients","infinite q-products","eta quotients","quintuple product identity","m-dissections","Fourier coefficients"],"falsifier":"Expand both sides of Theorem 2.1 for $M=4$, $j=1$, and $m=5$ as power series in $q$ to order $q^{60}$; if the sides disagree, the load-bearing dissection is false. Equivalently, directly expand $(q^2;q^2)_\\infty/(q^5;q^5)_\\infty$: a nonzero coefficient with $n\\equiv 1$ or $3 \\pmod 5$, or a coefficient that is not positive when $n\\equiv 0 \\pmod 5$, would disprove the theorem's corollary.","tokens_in":17446,"feed_emoji":"🔢","tokens_out":14835,"duration_ms":134934,"temperature":0.7,"pith_summary":"This paper proves that the signs of the coefficients in the series expansion of $(q^i;q^i)_\\infty/(q^p;q^p)_\\infty$ are eventually periodic: for any prime $p>3$ and integer $i>1$ not divisible by $p$, once $n$ exceeds an explicit bound $N(p,i)$, the sign of $a_n$ is determined entirely by $n$ modulo $p$. In residue classes that can be written as $i(6r^2+r) \\pmod p$, the coefficient is positive for small and large $r$ and negative for the middle range of $r$; in every other residue class it vanishes. The thresholds that split these ranges depend only on whether $p$ is $1$ or $-1$ modulo $3$. Specializing to $(p,i)=(5,2)$ confirms the conjecture $C215^{-1}$ from [9] for all $n \\ge 0$. The same dissection technique settles several further sign and vanishing conjectures from [9] for related eta quotients, including $(q;q)^9/(q^3;q^3)^9$.","feed_headline":"Proof pins down signs of q-product coefficients modulo p","feed_subtitle":"For prime p>3, the sign of each coefficient is fixed by n mod p, settling a conjecture from [9].","key_machinery":"The carrying mechanism is the $m$-dissection formula for the quintuple product (Theorem 2.1), taken from [22]. It splits the product $(q^j,q^{M-j},q^M;q^M)_\\infty(q^{M-2j},q^{M+2j};q^{2M})_\\infty$ into a finite signed sum with exponents $L(r)=6r^2+\\cdots$ and with each summand a product whose ordinary power series has nonnegative coefficients. Setting $M=4$, $j=1$ turns this into a $p$-dissection of $(q;q)_\\infty$ (Corollary 2.1), in which the sign of the $r$-th summand is $(-1)^{s(r)}$ and $s(r)$ takes the values $0,1,2$ according to the same thresholds on $r$. After replacing $q$ by $q^i$ and dividing by $(q^p;q^p)_\\infty$, only summands whose exponent $iL(r)$ lies in a given residue class contribute to $a_n$, and Lemma 2.1 shows those contributions are strictly positive or negative eventually. Consistency of the sign within each residue class follows from the symmetry of the quadratic $6r^2+r$ modulo $p$: for $p\\equiv 1 \\pmod 3$, the two outer ranges of $r$ are interchanged by $r+r'\\equiv (p-1)/6 \\pmod p$, and the case $p\\equiv -1 \\pmod 3$ is analogous.","core_discovery":"On the paper's own terms, the central result is Theorem 1.1. Let $f_j=(q^j;q^j)_\\infty$ and write $f_i/f_p=\\sum_{n=0}^\\infty a_n q^n$. For $p\\equiv 1 \\pmod 3$, define $L(r)$ piecewise with breakpoints $(4p-1)/12$ and $(10p-1)/12$; for $p\\equiv -1 \\pmod 3$, the breakpoints are $(2p-1)/12$ and $(8p-1)/12$. Then for every $n>N(p,i)$, where $N(p,i)$ is the maximum, over residue classes $s \\pmod p$, of the least value of $iL(r)$ congruent to $s$, the paper proves $a_n>0$ when $n\\equiv i(6r^2+r) \\pmod p$ with $r$ in the outer intervals, $a_n<0$ in the inner interval, and $a_n=0$ otherwise. Corollary 1.1 is the case $(p,i)=(5,2)$: the coefficients of $(q^2;q^2)_\\infty/(q^5;q^5)_\\infty$ are positive exactly for $n\\equiv 0 \\pmod 5$, negative for $n\\equiv 2,4 \\pmod 5$, and zero for $n\\equiv 1,3 \\pmod 5$, for every $n\\ge 0$. Theorem 1.2 extends the same sign analysis to other eta quotients, including $(q;q)^9/(q^3;q^3)^9$ with period-9 signs and several cases in the table from [9].","pith_inferences":["An unstated consequence is that the proof strategy is algorithmic: for any fixed $(p,i)$, the bound $N(p,i)$ is computable, so the eventual sign pattern can be verified by checking only finitely many coefficients.","A testable extension of the same dissection idea would be to rational powers of the product $(q;q)_\\infty/(q^p;q^p)_\\infty$; numerical experiments for small $p$ could reveal which residue classes remain single-signed and which develop mixed signs.","The paper's own table of coefficient counts for two unproven quotients suggests that for some residue classes the coefficients may never become single-signed; if that persists, an eventual sign-periodicity statement for those classes would need a weaker formulation than pure $+$ or $-$."],"forward_implications":["For every prime $p>3$ and every integer $i>1$ not divisible by $p$, the coefficients of $(q^i;q^i)_\\infty/(q^p;q^p)_\\infty$ are ultimately periodic in sign with period $p$, and the threshold $N(p,i)$ is explicit.","The case $(p,i)=(5,2)$ settles the conjecture $C215^{-1}$ from [9] in full: $a_n>0$ for $n\\equiv 0 \\pmod 5$, $a_n<0$ for $n\\equiv 2,4 \\pmod 5$, and $a_n=0$ otherwise.","Theorem 1.2 proves the specific sign patterns listed for the entries $5/++0+0$, $4/+-00$, $3/+0-$, $4/+-+0$, $4/+++0$, $5/+- - ++$, and $9/+-+--+0-+$ in the table from [9].","Whenever $\\gcd(t,6)=1$, Corollary 2.1 yields a $t$-dissection of $(q;q)_\\infty$, so the same method can be applied to quotients such as $(q^j;q^j)^m/(q^t;q^t)$ and $(q^j;q^j)^m/(q^t;q^t)^m$, although the dissections may then be sums of products with mixed signs."],"supporting_citations":[{"why":"Supplies Theorem 2.1, the m-dissection formula for the quintuple product from which the p-dissection of (q;q)_∞ is derived.","marker":"[22]"},{"why":"Poses the conjecture C215^{-1} and the Table 1 sign patterns that the paper proves.","marker":"[9]"},{"why":"Initiates the study of sign periodicity in the Taylor coefficients of infinite products that this paper extends.","marker":"[30]"},{"why":"Gives the 5-dissection that serves as the model for dissecting an infinite product into nonnegative-signed components.","marker":"[20]"},{"why":"Proves sign periodicity for the product (q;q)_∞/(q^p;q^p)_∞, the direct predecessor of the quotient studied here.","marker":"[2]"},{"why":"Supplies the cubic theta function identities used in the later parts of Theorem 1.2.","marker":"[16]"},{"why":"Supplies the 3-dissection identities used in Theorem 2.2 and in several cases of Theorem 1.2.","marker":"[27]"},{"why":"Supplies the theta series identities (2.6)-(2.9) on which several cases of Theorem 1.2 rest.","marker":"[15]"}],"fun_headline_variants":["Signs of q-series coefficients pinned by residue mod p","Eta quotient coefficient signs follow simple mod-p rule","Conjecture settled: sign of each coefficient known modulo p","Modular dissection yields exact sign patterns for coefficients","Prime modulus determines sign of every q-product coefficient"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on an unproved dissection identity quoted from the authors' earlier preprint [22]: the quintuple product can be split into a signed finite sum whose pieces have nonnegative coefficients, and if that identity is false, the $p$-dissection of $(q;q)_\\infty$ and the resulting sign patterns do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Signs of q-series coefficients pinned by residue mod p","Eta quotient coefficient signs follow simple mod-p rule","Conjecture settled: sign of each coefficient known modulo p","Modular dissection yields exact sign patterns for coefficients","Prime modulus determines sign of every q-product coefficient"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000771,"raw_usage":{"total_tokens":3465,"prompt_tokens":1050,"completion_tokens":2415,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":666,"completion_tokens_details":{"reasoning_tokens":2338}},"tokens_in":666,"tokens_out":2415,"duration_ms":17767,"temperature":1.0,"reasoning_tokens":2338,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:07:20.764899+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Expand both sides of Theorem 2.1 for $M=4$, $j=1$, and $m=5$ as power series in $q$ to order $q^{60}$; if the sides disagree, the load-bearing dissection is false. Equivalently, directly expand $(q^2;q^2)_\\infty/(q^5;q^5)_\\infty$: a nonzero coefficient with $n\\equiv 1$ or $3 \\pmod 5$, or a coefficient that is not positive when $n\\equiv 0 \\pmod 5$, would disprove the theorem's corollary.","supporting_citations":[{"cited_title":"Ramanujan ’s theta functions, Springer, Cham, 2017","cited_arxiv_id":null,"evidence_quote":"Supplies the cubic theta function identities used in the later parts of Theorem 1.2."},{"cited_title":"Dissection of the quintuple product, with applications , preprint","cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 2.1, the m-dissection formula for the quintuple product from which the p-dissection of (q;q)_∞ is derived."},{"cited_title":"Periodic sign changes of weakly holomorphic η-quotients, preprint","cited_arxiv_id":null,"evidence_quote":"Poses the conjecture C215^{-1} and the Table 1 sign patterns that the paper proves."},{"cited_title":"The Taylor coefficients of certain infinite products","cited_arxiv_id":null,"evidence_quote":"Initiates the study of sign periodicity in the Taylor coefficients of infinite products that this paper extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the 5-dissection that serves as the model for dissecting an infinite product into nonnegative-signed components."},{"cited_title":"On a Conjecture of Peter Borwein , Journal of Symbolic Computation, Volume 20, Issues 5–6, 1995","cited_arxiv_id":null,"evidence_quote":"Proves sign periodicity for the product (q;q)_∞/(q^p;q^p)_∞, the direct predecessor of the quotient studied here."},{"cited_title":"m-Dissections of some infinite products and related identities , The Ramanujan Journal (2022) 59:313–350","cited_arxiv_id":null,"evidence_quote":"Supplies the 3-dissection identities used in Theorem 2.2 and in several cases of Theorem 1.2."},{"cited_title":"Modular forms: A classical approach , Graduate Studies in Mathematics, Vol 179, Amer","cited_arxiv_id":null,"evidence_quote":"Supplies the theta series identities (2.6)-(2.9) on which several cases of Theorem 1.2 rest."}],"review_version":1}