{"id":"d8bd4d8e-5047-4284-9064-763c78d9a416","arxiv_id":"2507.01529","paper_version":4,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper derives infinite congruence families for several biregular overpartition functions, but the headline family (4,3^t) with t=1 is invalid because \\bar{B}_{4,3}(3)=6.","lead":"This number theory paper claims infinitely many congruence families for biregular overpartition counts, covering pairs such as (4,3^t), (5,2^t), and (3,2^t). A central theorem is false: for t=1, \\bar{B}_{4,3}(3)=6, so the claimed congruence modulo 8 fails.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 8.1 fails for t=1: B_{4,3}(3)=6, so the key reduction (8.10) is false and the advertised (4,3^t) family is unsupported as stated.","rationale":"The reader's verdict is REJECT, and my independent check agrees. The central advertised family is the (4,3^t) congruence in Theorem 8.1, whose proof hinges on equation (8.10). Substituting t=1 into (8.7) and expanding gives a q-coefficient of 6, so the claimed 1 mod 8 reduction is false. Direct combinatorial enumeration of B_{4,3}(3) gives 6, confirming the counterexample without relying on any possibly misprinted derivation. This invalidates the abstract's claim that congruences hold for (4,3^t) for all t>=1. Other sections may be salvageable, but the paper's signature new infinite family is unsupported as stated; restricting to t>=2 and rechecking the reduction would be the natural repair. No change to the reader's verdict is needed.","tokens_in":27926,"tokens_out":8373,"duration_ms":82604,"concrete_test":"Expand the generating function (8.1) with (l1,l2)=(4,3) to order q^4, either in Sage/Python or by hand, and inspect the coefficient of q^3; it should be 6. Cross-check by enumerating all overpartitions of 3 whose parts are not divisible by 3 or 4: there are 6. If this coefficient is 6, Theorem 8.1 is refuted for t=1, n=1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing assumption is the reduction (8.10), on which Theorem 8.1 and the advertised (4,3^t) family rest. For t=1, equation (8.7) is the stated generating function for sum_n B_{4,3}(3n) q^n. Expanding the first term, f_2^3 f_3^6 f_12^2 f_8 / (f_1^6 f_6^3 f_24 f_4^2), gives (1-3q^2+...)(1+6q+21q^2+...) = 1+6q+18q^2+...; the second term is -8q^2 times a series with constant term 1, so (8.7) begins 1+6q+10q^2+... . The coefficient of q is 6, not 0 mod 8, so the claimed congruence (8.10), and therefore (8.2) for n=1, fails. Direct enumeration confirms this: overpartitions of 3 into parts not divisible by 3 or 4 are exactly 2+1 (4 choices) and 1+1+1 (2 choices), giving B_{4,3}(3)=6. The error is not a harmless typo: Lemma 2.8 cannot reduce the t=1 expression to 1 mod 8, and the abstract explicitly promises (4,3^t) for all t>=1. A corrected version might restrict the family to t>=2 and re-verify (8.10), but the theorem as stated is false.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the arithmetic of (ℓ1,ℓ2)-biregular overpartitions, i.e. overpartitions with no part divisible by ℓ1 or ℓ2, denoted B̄_{ℓ1,ℓ2}(n). Using generating function manipulations, dissection formulas, Hecke eigenforms, and Newman's identity, the author claims infinitely many congruence families modulo 3 and powers of 2 for the pairs (2,9), (5,2), (5,4), (8,3), and general families (5,2^t) for t≥3, (3,2^t) for t≥1, and (4,3^t) for t≥1. The main advertised new results are Theorem 8.1, asserting congruences for B̄_{4,3^t}(n) modulo 8 and 4 for all t≥1, and Theorem 9.1, asserting congruences for B̄_{3,2^t}(n) modulo 8 for all t≥1.","tokens_in":28151,"tokens_out":10397,"duration_ms":95143,"significance":"If the claimed infinite families were correct, they would constitute a meaningful extension of recent results on biregular overpartitions. The paper is self-contained in its use of standard tools: no free parameters are fitted, and the proofs are derivations from cited external results such as Martin's classification of multiplicative eta-quotients, Newman's identity, and classical dissection formulas. However, the significance of the current version is severely undermined because two of the headline theorems are false as stated, and the abstract advertises precisely those families.","major_comments":[{"comment":"The congruence asserted in (8.10), namely ∑_{n≥0} B̄_{4,3^t}(3n)q^n ≡ 1 (mod 8) for all t≥1, is false. For t=1, the right-hand side of (8.7) expands as 1+6q+O(q^2), so B̄_{4,3}(3)=6, which is not congruent to 0 mod 8. Hence (8.2) fails for n=1, and Theorem 8.1 is false as stated. The proof's appeal to Lemma 2.8 does not justify (8.10), because the eta-quotient in (8.7) is not of the form appearing in (2.12); the reduction to 1 mod 8 is a non-sequitur. Since the abstract explicitly promises the (4,3^t) family for all t≥1, this is a load-bearing error in the paper's central claim.","section":"Section 8, Eq. (8.10)"},{"comment":"Theorem 9.1 is false for t=1. Direct enumeration of overpartitions of 6 with no part divisible by 2 or 3 gives B̄_{3,2}(6)=6: the partitions 5+1 and 1+1+1+1+1+1 contribute 4 and 2 overpartitions respectively. Thus the assertion (9.2) with n=0, namely B̄_{3,2}(6)≡0 (mod 8), fails. Moreover, the derivation leading to (9.9) involves f_{2^{t-2}}, which is undefined for t=1, so the proof does not cover the stated range t≥1. The advertised (3,2^t) family for all t≥1 is therefore unsupported and false as stated.","section":"Section 9, Theorem 9.1"}],"minor_comments":[{"comment":"The opening sentence says 'By setting (ℓ1, ℓ2) = (4,3) in (1.1)', but the generating function that follows is for B̄_{2,9}(n); the pair should be (2,9).","section":"Section 3, Proposition 1"},{"comment":"Equation (4.11) states the congruence for B̄_{5,2}(4n+1), but the theorem concerns B̄_{5,2^t}(4n+1); this appears to be a typographical error.","section":"Section 4, Eq. (4.11)"},{"comment":"The proof of Proposition 4 is labeled 'Proof of Proposition 5.3', which is an incorrect cross-reference.","section":"Section 6, Proposition 4"},{"comment":"The statement of Lemma 2.8 is typeset ambiguously; the intended congruence f_{p^{k-1}m}^p ≡ f_{p^k m} (mod p^k) should be written explicitly to avoid confusion.","section":"Section 2, Lemma 2.8"},{"comment":"There are numerous typographical and grammatical issues (e.g., 'partions', 'notaion', 'expnasion', 'Devolepments') that should be corrected in any revision.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The paper cannot be accepted in its present form because two of the main advertised infinite families—(4,3^t) and (3,2^t) for all t≥1—are false as stated. The errors are not minor: they invalidate Theorem 8.1 and Theorem 9.1, which are central to the abstract's claims. A possible salvage would be to restrict the families to t≥2 and re-verify the proofs for that range, but this would shrink the advertised scope and require substantial reworking; the current version is therefore not suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Chris,\n\nThe main thing you should know: the paper's headline family for (4,3^t) is false as stated. Theorem 8.1 claims \\bar{B}_{4,3^t}(3n) ≡ 0 (mod 8) for all t≥1 and all n≥1. For t=1, the paper's own generating function (8.7) gives \\bar{B}_{4,3}(3)=6, so the congruence fails at n=1. The invalid step is (8.10), where the right side of (8.7) is asserted to be 1 mod 8 via Lemma 2.8. That reduction is false for t=1 and looks unsupported for t≥2 as well, since the lemma does not turn f_{3t-1} into f_{3t} modulo 8. So the advertised (4,3^t) family is unproven.\n\nThat said, there is real content elsewhere. The infinite families for (2,9), (5,2^t) with t≥3, and (3,2^t) with t≥2 appear to be new and are derived in the standard Hecke-eigenform/Newman-identity style. If those proofs hold up, they are legitimate additions to the partition-congruence literature. No free parameters are fitted; the derivations rest on cited external results. I did not check every eigenform assertion against Martin's classification, but the citations are to the right source.\n\nThe soft spots are in proportion. The t-range in Section 9 is also sloppy: the proof uses f_{2t-2}, which is undefined for t=1, so the (3,2^t) theorem should at least be restricted to t≥2. The sections on (5,2) and (8,3) overlap with Alanazi et al. and Nadji et al., so the novelty there is thin. And the abstract overclaims by listing (4,3^t) for all t≥1.\n\nThis is not a desk-reject. The false theorem is isolated to Section 8, and the rest of the paper is plausibly correct. A referee should be asked to check Section 8 carefully and to verify the t-ranges in Sections 8 and 9. The authors need to fix the abstract and either repair Theorem 8.1 for t≥2 or delete it. The t=1 counterexample is immediate, so they should catch it.\n\nThe paper is for specialists in partition congruences and q-series. If they salvage the valid families, the result is a modest but acceptable contribution. I would not bring it to a reading group as is, but it deserves a serious referee.","headline":"Theorem 8.1 is false for t=1 (B_{4,3}(3)=6), so the paper's headline (4,3^t) family is unsupported, but the other new families deserve a careful referee.","tokens_in":28838,"tokens_out":6201,"would_cite":false,"duration_ms":61416,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A17","11P83","11F11"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims infinitely many mod-3 and mod-2^k congruence families for biregular overpartitions, proved by reducing generating functions to Hecke eigenforms.","keywords":["biregular overpartitions","congruences","eta-quotients","Hecke eigenforms","dissection formulas","modular forms","partition congruences","arithmetic progressions"],"falsifier":"Evaluate the $t=1$ case of the claimed $(4,3^t)$ family: expanding the generating function for $(\\ell_1,\\ell_2)=(4,3)$ gives $\\sum_{n\\geq 0}\\bar{B}_{4,3}(3n)q^n = 1+6q+16q^2+\\cdots$, so $\\bar{B}_{4,3}(3)=6$ and the asserted congruence $\\bar{B}_{4,3}(3n)\\equiv 0 \\pmod{8}$ fails at $n=1$. This coefficient calculation settles the $t=1$ case directly.","tokens_in":27637,"feed_emoji":"🔢","tokens_out":11600,"duration_ms":110672,"temperature":0.7,"pith_summary":"The paper studies $\\bar{B}_{\\ell_1,\\ell_2}(n)$, the number of overpartitions of $n$ with no part divisible by $\\ell_1$ or $\\ell_2$ for coprime $\\ell_1,\\ell_2>1$. It aims to prove that these counts vanish along infinitely many arithmetic progressions: modulo 8 for pairs $(2,9)$ and $(5,2^t)$, modulo 4 for $(5,2)$, $(5,4)$, and $(3,2^t)$, and modulo 3 for $(2,9)$ and $(8,3)$, with the principal advertised family being $\\bar{B}_{4,3^t}(3n)\\equiv 0 \\pmod{8}$ for every $t\\geq 1$. The proofs isolate residue-class generating functions, identify the resulting eta-quotients as Hecke eigenforms, and transfer coefficient vanishing under primes in suitable residue classes back to the original counts. The proof of the $(4,3^t)$ family rests on a reduction that is false at $t=1$, since the same generating function then begins $1+6q+16q^2+\\cdots$ rather than $1 \\pmod{8}$.","feed_headline":"Biregular overpartition counts vanish on infinite progressions","feed_subtitle":"The new families cover (2,9), (5,2^t), (8,3), (3,2^t), and (4,3^t) modulo 3 and powers of 2.","key_machinery":"The engine is the eta-quotient form of the generating function, combined with dissection formulas that isolate the coefficients in a fixed residue class. After extraction the paper obtains identities such as $\\sum_{n\\geq 0}\\bar{B}_{2,9}(6n+1)q^n \\equiv 2\\eta^4(6z) \\pmod{8}$ and $\\sum_{n\\geq 0}\\bar{B}_{5,2^t}(4n+1)q^n \\equiv 2\\eta(4z)\\eta(20z) \\pmod{8}$. The products $\\eta^4(6z)$ and $\\eta(4z)\\eta(20z)$ are cusp forms that the paper identifies as Hecke eigenforms, meaning eigenvalues of the Hecke operators; applying $T_p$ at primes $p$ with vanishing eigenvalue forces certain coefficients $a(pn)+\\cdots$ to vanish, which translates into zero congruence classes for the partition function. A second mechanism, a classical multiplicative identity for eta-products due to Newman [19], iterates coefficient relations at primes $p\\equiv 1 \\pmod{6}$ or $p\\equiv 1 \\pmod{4}$ to generate additional infinite families.","core_discovery":"On the paper's own terms, the central discovery is a collection of infinite congruence families for $\\bar{B}_{\\ell_1,\\ell_2}(n)$. For primes $p_i\\not\\equiv 1 \\pmod{6}$ it claims congruences of the form $\\bar{B}_{2,9}(6p_1^2\\cdots p_{k+1}^2 n + (6j+p_{k+1})p_1^2\\cdots p_k p_{k+1})\\equiv 0 \\pmod{8}$ with an analogous family modulo 3, and multiplicative recurrence formulas for the same sequence. For $p_i\\not\\equiv 1 \\pmod{4}$ it claims the same progression shape for $\\bar{B}_{5,2^t}(4p_1^2\\cdots p_{k+1}^2 n + (4j+p_{k+1})p_1^2\\cdots p_k p_{k+1})\\equiv 0 \\pmod{8}$ for $t\\geq 3$, with the modulo-4 analogue for $(5,2)$ and $(5,4)$. It also claims $\\bar{B}_{8,3}(36n+33)\\equiv 0 \\pmod{3}$, $\\bar{B}_{3,2^t}(16n+6)\\equiv \\bar{B}_{3,2^t}(16n+10)\\equiv \\bar{B}_{3,2^t}(16n+14)\\equiv 0 \\pmod{8}$, and, for $(4,3^t)$, $\\bar{B}_{4,3^t}(3n)\\equiv 0 \\pmod{8}$ for all $n\\geq 1$ together with several modulo-4 families.","pith_inferences":["Because the $t=1$ counterexample breaks only the reduction in equation (8.10), the natural repair is to restrict the $(4,3^t)$ family to $t\\geq 2$; the modulo-4 portions of Theorem 8.1 derive from a different generating function and may survive unchanged.","The same template of reducing a residue class to an eta-product, identifying it as a Hecke eigenform, and iterating the eigenvalue recursion should produce analogous families for other coprime pairs $(\\ell_1,\\ell_2)$ whenever the reduced product is a low-weight newform with only finitely many bad primes.","The multiplicative recurrences stated for $(2,9)$ and $(5,2^t)$ give a cheap numerical test of any corrected $(4,3^t)$ conjecture: compute $\\bar{B}_{4,3^t}(3p)$ for small primes $p$ and compare with the recurrence predicted by the would-be eigenform.","The paper's methods could be pushed further to seek congruences for the missing residue classes in the $(8,3)$ case, where only a single progression modulo 3 is established, rather than an infinite family."],"forward_implications":["For every prime $p\\equiv 5 \\pmod{6}$, the claimed families give $\\bar{B}_{2,9}(6p^{2k+2}n+6p^{2k+1}j+p^{2k+2})\\equiv 0 \\pmod{8}$ and the analogous modulo-3 family with $18$ in place of $6$, whenever $p\\nmid j$.","For every prime $p\\equiv 3 \\pmod{4}$, the claimed families give $\\bar{B}_{5,2^t}(4p^{2k+2}n+4p^{2k+1}j+p^{2k+2})\\equiv 0 \\pmod{8}$ for every $t\\geq 3$, and the same progression shape modulo 4 for $(5,2)$ and $(5,4)$.","The paper claims $\\bar{B}_{8,3}(36n+33)\\equiv 0 \\pmod{3}$ for all $n$, and $\\bar{B}_{3,2^t}(16n+r)\\equiv 0 \\pmod{8}$ for $r=6,10,14$ and every $t\\geq 1$.","The paper claims $\\bar{B}_{4,3^t}(3n)\\equiv 0 \\pmod{8}$ for all $n\\geq 1$ and all $t\\geq 1$, together with $\\bar{B}_{4,3^t}(3n+2)\\equiv \\bar{B}_{4,3^t}(6n+4)\\equiv \\bar{B}_{4,3^t}(12n+7)\\equiv 0 \\pmod{4}$.","If the claimed families hold, then each listed pair has positive arithmetic density of zero residues modulo the relevant power of 2 or modulo 3, a stronger property than isolated congruences."],"supporting_citations":[{"why":"Supplies the prior congruence results for biregular overpartitions that the paper extends to infinite families.","marker":"[18]"},{"why":"Provides the earlier finite congruence lists, including pairs such as (2,3), (4,3), and (2,5), that serve as the baseline for comparison.","marker":"[1]"},{"why":"Records the recent extension of those congruences that motivates the families treated here.","marker":"[21]"},{"why":"Supplies the classical identity for eta-products whose iterative coefficient relations generate the Newman-type congruence families.","marker":"[19]"},{"why":"Provides the eta-quotient holomorphy criteria and Hecke operator background used to place reduced generating functions in spaces of modular forms.","marker":"[20]"},{"why":"Identifies the eta-products $\\eta^4(6z)$ and $\\eta(4z)\\eta(20z)$ as Hecke eigenforms, the key transfer step in the argument.","marker":"[17]"},{"why":"Gives the dissection formula used to isolate the $4n+1$ residue class in the $(5,2^t)$, $(5,2)$, and $(5,4)$ sections.","marker":"[12]"},{"why":"Supplies the dissection formulas used in the $(8,3)$ and $(3,2^t)$ sections to extract the vanishing residue classes.","marker":"[36]"}],"fun_headline_variants":["Infinite congruence families for biregular overpartitions","New infinite congruences for (5,2^t) and (4,3^t) overpartitions","Biregular overpartition counts vanish modulo 8 and 3 on infinite progressions","Infinite families of congruences for biregular overpartitions","Congruence nets for biregular overpartitions modulo 3 and powers of 2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the main $(4,3^t)$ family assumes that the reduced generating function $\\sum_{n\\geq 0}\\bar{B}_{4,3^t}(3n)q^n$ is congruent to $1$ modulo $8$ for every $t\\geq 1$; for $t=1$ the same function is $1+6q+16q^2+\\cdots$, so that assumption fails already in the first case.","fun_headline_variants_meta":{"raw":{"variants":["Infinite congruence families for biregular overpartitions","New infinite congruences for (5,2^t) and (4,3^t) overpartitions","Biregular overpartition counts vanish modulo 8 and 3 on infinite progressions","Infinite families of congruences for biregular overpartitions","Congruence nets for biregular overpartitions modulo 3 and powers of 2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001085,"raw_usage":{"total_tokens":4716,"prompt_tokens":1309,"completion_tokens":3407,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":925,"completion_tokens_details":{"reasoning_tokens":3294}},"tokens_in":925,"tokens_out":3407,"duration_ms":127496,"temperature":1.0,"reasoning_tokens":3294,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:54:04.413738+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the $t=1$ case of the claimed $(4,3^t)$ family: expanding the generating function for $(\\ell_1,\\ell_2)=(4,3)$ gives $\\sum_{n\\geq 0}\\bar{B}_{4,3}(3n)q^n = 1+6q+16q^2+\\cdots$, so $\\bar{B}_{4,3}(3)=6$ and the asserted congruence $\\bar{B}_{4,3}(3n)\\equiv 0 \\pmod{8}$ fails at $n=1$. This coefficient calculation settles the $t=1$ case directly.","supporting_citations":[{"cited_title":"Nadji, M","cited_arxiv_id":null,"evidence_quote":"Supplies the prior congruence results for biregular overpartitions that the paper extends to infinite families."},{"cited_title":"Newman, Modular forms whose coefficients possess multiplicative properties , Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the classical identity for eta-products whose iterative coefficient relations generate the Newman-type congruence families."},{"cited_title":"Ono, The web of modularity: arithmetic of the coefficients of modular forms and q−series, CBMS Regional Conference Series in Mathematics, 102 , Amer","cited_arxiv_id":null,"evidence_quote":"Provides the eta-quotient holomorphy criteria and Hecke operator background used to place reduced generating functions in spaces of modular forms."},{"cited_title":"Martin, Multiplicative η-quotients, Trans","cited_arxiv_id":null,"evidence_quote":"Identifies the eta-products $\\eta^4(6z)$ and $\\eta(4z)\\eta(20z)$ as Hecke eigenforms, the key transfer step in the argument."},{"cited_title":"Hirschhorn and J","cited_arxiv_id":null,"evidence_quote":"Gives the dissection formula used to isolate the $4n+1$ residue class in the $(5,2^t)$, $(5,2)$, and $(5,4)$ sections."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the dissection formulas used in the $(8,3)$ and $(3,2^t)$ sections to extract the vanishing residue classes."}],"review_version":1}