{"id":"235d6654-f321-43b6-8bb5-a644ece6e211","arxiv_id":"1908.06642","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New infinite families of congruences modulo 5, 11, and 17 are proved for four (s,t)-regular bipartition functions.","lead":"This paper proves new infinite families of divisibility formulas for counts of restricted bipartitions, adding four new parameter pairs to a catalogue of Ramanujan-style partition congruences. The value is for specialists in number theory, where each new congruence is a data point for deeper structure in partition generating functions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's weakest assumption was the unshown 'Similarly, we find' block in Section 4, and that is indeed the natural place to scrutinize Theorem 1.2. I therefore stress-tested exactly that block. The extraction step that advances from one line of (4.7) to the next can be reduced to the linear recurrence above; iterating it reproduces every printed coefficient, including the crucial final reduction to (0,0,8). This confirms that the congruences for B_{7,11} follow as claimed. The paper would be easier to verify if the recurrence or the intermediate dissections were printed, so the CONDITIONAL verdict is understandable, but no actual correctness defect emerges. I also spot-checked Lemma 2.4, Lemma 2.6, and the final steps of Sections 3, 5, and 6; all are internally consistent. Hence the reader's conditional assessment should stand unchanged rather than being strengthened or weakened.","tokens_in":10718,"tokens_out":47933,"duration_ms":405626,"concrete_test":"Implement the recurrence (A',B',C') = (9A+10B+10C, 9A+5B, 8A) mod 11, or equivalently run a computer-algebra expansion of f_7 f_1^9 modulo 11, iterating the q^{7n+4} dissection ten times; verify that the coefficient triples match all eleven lines of (4.6)-(4.7), especially the final (0,0,8). Independently expanding both sides of the final line (4.7) to O(q^20) settles the theorem's key computational input.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claims are supported by the printed computations; I found no load-bearing error. The only genuinely delicate spot is the unshown iterative dissection recorded as 'Similarly, we find' between (4.6) and (4.7). I checked this block by deriving the underlying linear recurrence on the coefficient triple (A,B,C) in R_k = A f_7 f_1^9 + B q f_7^5 f_1^5 + C q^2 f_7^9 f_1. Extracting the residue 4 modulo 7 gives (A',B',C') = (9A+10B+10C, 9A+5B, 8A) mod 11, i.e. (-2A-B-C, -2A+5B, -3A). Starting from (A_1,B_1,C_1) = (9,9,8), this recurrence reproduces every displayed line of (4.7) exactly: (9,5,6), (4,7,6), (1,5,10), (5,1,8), (3,6,7), (3,2,2), (1,4,2), (3,7,8), (1,7,2), and finally (0,0,8). The final step then follows: substituting (2.9) and extracting residue 4 yields coefficient -8 ≡ 3 mod 11, giving (4.10); residues 1,5,6 are absent, giving (4.9). The remaining inductions and Sections 3, 5, and 6 are consistent with the printed q-series identities. Thus the omitted block is an expositional gap, not a correctness defect.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies B_{s,t}(n), the number of (s,t)-regular bipartitions of n, whose generating function is f_s f_t / f_1^2. Using modular q-series dissections and known identities from Ramanujan, Berndt, Hirschhorn, and Sellers, the authors prove four theorems: Theorem 1.1 gives a two-parameter family of congruences modulo 5 for B_{2,15}; Theorem 1.2 gives infinite families modulo 11 for B_{7,11}, including a strong congruence with a power of the modulus in the argument; Theorem 1.3 gives a family modulo 11 for B_{27,11} for all m ≥ 4; and Theorem 1.4 gives two residue classes modulo 17 for B_{243,17}. The proofs proceed by extracting coefficients from generating functions, applying dissections, and iterating the resulting recurrences.","tokens_in":11002,"tokens_out":15964,"duration_ms":119887,"significance":"The paper provides new infinite families of congruences for four families of regular bipartition functions, extending a recent line of results by Lin, Dou, Xia and Yao, Adiga and Ranganatha, and Kathiravan. The methods are standard but the computations are explicit, checkable, and free of fitted parameters. The final congruences are concrete and falsifiable. The main families in Theorems 1.1 and 1.2 are particularly clean, and the paper would be a useful addition to the literature once the omitted derivation in Section 4 is supplied.","major_comments":[{"comment":"The ten iterative dissections displayed after the phrase 'Similarly, we find' are stated without any derivation. This block is essential: the last line of (4.7) is substituted into (2.9) to obtain (4.8) and hence (4.9)-(4.10), so Theorem 1.2 rests on the correctness of these ten lines. I have checked the block independently: writing R_k = A f_7 f_1^9 + B q f_7^5 f_1^5 + C q^2 f_7^9 f_1, the operation of extracting the residue class 4 modulo 7 gives (A',B',C') ≡ (9A+10B+10C, 9A+5B, 8A) modulo 11, and starting from (9,9,8) this reproduces exactly the coefficient triples (9,5,6), (4,7,6), (1,5,10), (5,1,8), (3,6,7), (3,2,2), (1,4,2), (3,7,8), (1,7,2), (0,0,8) in (4.7). The mathematics is therefore correct, but as written the proof is not self-contained in a load-bearing place. The authors should include the recurrence, or at least one full representative iteration with the remaining lines listed as the result of repeating it.","section":"Section 4, between (4.6) and (4.7)"}],"minor_comments":[{"comment":"The final sentence 'This completes the proof of Theorem 1.4 follow from (6.5)' omits the key observation that every term on the right-hand side of (6.5) is a power series in q^3 multiplied by q, so the coefficients of q^{3n} and q^{3n+2} vanish. This extraction should be stated explicitly.","section":"Section 6, after (6.5)"},{"comment":"The passage from (5.7) and (5.8) to the full infinite family (1.12) is not shown. A one-line induction, with base case m=4 given by (5.7) and the induction step following from (5.8), should be included.","section":"Section 5, proof of Theorem 1.3"},{"comment":"There are numerous typographical and grammatical errors, such as 'Ramanujan [18] give', 'th e identities', 'mo dulo', 'regu lar', and 'co efficient'. A careful proofreading pass is recommended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The omitted block in Section 4 is the only genuine gap; I verified that the stated dissections follow from a simple linear recurrence, so the gap is fillable without changing the results. If the authors supply that derivation, the paper is suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou should know this paper is a modest but solid contribution to the (s,t)-regular bipartition congruence literature. It proves four new infinite families of congruences modulo 5, 11, 11, and 17 for B_{2,15}, B_{7,11}, B_{27,11}, and B_{243,17}. The method is not new — it is the standard dissection technique using known q-series identities from Berndt, Hirschhorn-Sellers, and Berndt-Yee-Yi — but the specific congruences are new and the paper does not oversell them.\n\nWhat it does well: the proofs are mostly elementary and checkable. I sampled the expansions in Lemmas 2.4 and 2.6 and Sections 3, 5, and 6; they are internally consistent. The paper is honest about its scope and cites the relevant prior literature, including Dou, Adiga-Ranganatha, Xia-Yao, and the authors' own earlier work, only as context. The circularity burden is low: the target congruences are never assumed.\n\nThe soft spot is Section 4. Between (4.6) and (4.7) the authors write 'Similarly, we find' and then list ten iterative dissections without derivation. Theorem 1.2 rides on that block. This is a genuine expositional gap — a referee should ask for the recurrence or a lemma. But I verified the block by deriving the linear recurrence on the coefficient triple in R_k = A f_7 f_1^9 + B q f_7^5 f_1^5 + C q^2 f_7^9 f_1. Extracting residue 4 mod 7 gives (A',B',C') = (9A+10B+10C, 9A+5B, 8A) mod 11, which reproduces every displayed line in (4.7). So the omitted block is correct; it is a presentation problem, not a correctness defect. That matches the reader's conditional verdict but resolves the main uncertainty.\n\nOther soft spots are minor: the inductions in Theorems 1.1, 1.3, and 1.4 are terse, and the abstract has a typo ('bipartitions' for 'bipartitions'). Neither affects the results.\n\nWho this is for: partition-theory specialists working on congruences for regular bipartitions. It is a competent extension of a known method, not a breakthrough. I would send it to peer review; it deserves a serious referee who can fill in the omitted dissection details. If the authors add a short recurrence lemma for the Section 4 block and expand the inductions, it should be accepted.","headline":"A narrow but solid congruence paper; the unshown Section 4 block checks out, so it deserves a serious referee.","tokens_in":11573,"tokens_out":2894,"would_cite":true,"duration_ms":24894,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11P83","05A17"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves four infinite families of congruences modulo 5, 11, and 17 for the numbers of $(s,t)$-regular bipartitions, including scaled recurrences and vanishing along arithmetic progressions.","keywords":["Congruence","Regular bipartition","Partition function","Generating function","q-series identity","Dissection","Modulo 11","Infinite family"],"falsifier":"Evaluate the generating function $\\sum B_{7,11}(n)q^n=f_7f_{11}/f_1^2$ to the first few hundred terms and check the residue of $B_{7,11}((2\\cdot7^{12}-2)/3)$ against $3\\pmod{11}$, and check $B_{7,11}(7^{11}+(2\\cdot7^{11}-2)/3)\\equiv0\\pmod{11}$; a single mismatch would disprove Theorem 1.2.","tokens_in":10482,"feed_emoji":"🔢","tokens_out":11396,"duration_ms":95550,"temperature":0.7,"pith_summary":"The paper establishes four new infinite families of congruences for $(s,t)$-regular bipartitions, the number $B_{s,t}(n)$ of bipartitions of $n$ whose first part is $s$-regular and second part is $t$-regular. Theorem 1.1 gives congruences modulo $5$ for $B_{2,15}$, including the exact scaling law $B_{2,15}(3^{2m+1}n+(7\\cdot 3^{2m+1}-5)/8)\\equiv 2^m B_{2,15}(3n+2)\\pmod 5$. Theorem 1.2 gives a scaling law modulo $11$ for $B_{7,11}$ and shows that $B_{7,11}$ vanishes modulo $11$ along three families of arithmetic progressions. Theorem 1.3 shows that $B_{27,11}$ vanishes modulo $11$ for all $m\\ge 4$, and Theorem 1.4 shows that $B_{243,17}$ vanishes modulo $17$ in two residue classes modulo $81$. These results extend a line of congruence results for regular bipartitions and give infinitely many explicit values of $n$ for which each function is divisible by the relevant modulus.","feed_headline":"New congruence families for bipartition counts modulo 5, 11, 17","feed_subtitle":"The proofs give exact recurrences and vanishing laws for (2,15), (7,11), (27,11), and (243,17)-regular bipartitions.","key_machinery":"The argument runs on the generating function $\\sum_{n\\ge0}B_{s,t}(n)q^n=f_sf_t/f_1^2$, with $f_k=\\prod_{m\\ge1}(1-q^{mk})$. Since $f_t\\equiv f_1^t\\pmod t$ by the binomial theorem, the problem reduces to extracting coefficients from products of $f_1$'s and a single $f_t$. The dissections use classical identities: Berndt's identity (2.2), the Hirschhorn--Sellers identity (2.3), and Hirschhorn's cube and reciprocal-cube identities (2.4)--(2.6), together with Lemmas 2.4--2.6 for $p_5(7n+3)$, $p_7(7n)$, and $p_9(7n+4)$ built from $\\theta$-function relations from Ramanujan's lost notebook. Extracting terms in a fixed residue class modulo $3$ or $7$, then replacing $q^3$ or $q^7$ by $q$, produces the recurrences that iterate to Theorems 1.1--1.4.","core_discovery":"On the paper's own terms, the central discovery is that the generating function $\\sum_{n\\ge0}B_{s,t}(n)q^n=f_sf_t/f_1^2$ can be dissected, after reduction modulo $t$ via $f_t\\equiv f_1^t\\pmod t$, to yield exact recurrences between values of $B_{s,t}$ at arithmetic progressions whose step multiplies by $3$ or $7$ at each iteration. For $(2,15)$ and $(7,11)$ the recurrences are nonzero scaling laws: $B_{2,15}(3^{2m+1}n+(7\\cdot3^{2m+1}-5)/8)\\equiv2^mB_{2,15}(3n+2)\\pmod5$ and $B_{7,11}(7^{12m}n+(2\\cdot7^{12m}-2)/3)\\equiv3^mB_{7,11}(n)\\pmod{11}$. For $(7,11)$ with $k=1,5,6$, for $(27,11)$ with $m\\ge4$, and for $(243,17)$ in the residue classes $23$ and $77$ modulo $81$, the same method yields outright vanishing modulo the modulus. All four families are asserted for all $n\\ge0$ and all permitted $m$, so the paper claims infinitely many congruence identities, not merely finitely many checked cases.","pith_inferences":["The same binomial reduction and dissection recipe should apply to other pairs $(s,t)$ with $t$ prime once $f_sf_t/f_1^2$ collapses to a manageable product modulo $t$; the four pairs here display the pattern for $t=5,11,17$.","The $(243,17)$ congruences are proved only at the base scale, so a natural open question is whether the same two residue classes recur at higher powers of $3$, as the other theorems' base congruences do."],"forward_implications":["For $(2,15)$-regular bipartitions, the paper proves that $B_{2,15}$ vanishes modulo $5$ on the progressions $3^{2m+2}n+(23\\cdot3^{2m+1}-5)/8$ and $3^{2(m+1)+1}n+(13\\cdot3^{2(m+1)}-5)/8$ for every $m\\ge0$.","For $(7,11)$-regular bipartitions, $B_{7,11}(7^{12m}n+(2\\cdot7^{12m}-2)/3)\\equiv3^mB_{7,11}(n)\\pmod{11}$ for all $m\\ge0$, so the same residue recurs with a $3^m$ multiplier.","For $(7,11)$-regular bipartitions, $B_{7,11}(7^{12m+11}(7n+k)+(2\\cdot7^{12m+11}-2)/3)\\equiv0\\pmod{11}$ for $k=1,5,6$ and all $m,n\\ge0$.","For $(27,11)$-regular bipartitions, $B_{27,11}(3^m n+(5\\cdot3^{m-1}-3)/2)\\equiv0\\pmod{11}$ for all $m\\ge4$ and $n\\ge0$, and for $(243,17)$-regular bipartitions, $B_{243,17}(81n+23)\\equiv B_{243,17}(81n+77)\\equiv0\\pmod{17}$ for all $n\\ge0$."],"supporting_citations":[{"why":"Supplies the dissection identity (2.2) and Entry 17(v) in (2.9); the $(2,15)$ and $(7,11)$ proofs substitute both into their generating functions.","marker":"[2]"},{"why":"Supplies Entry 31 on page 174, which with [4] evaluates the A, B, C expressions used in Lemma 2.4.","marker":"[3]"},{"why":"Provides identities (2.12)--(2.15) and Theorem 3.2 (Lemma 2.5), which evaluate the p_5, p_7, and p_9 dissections in Lemmas 2.4--2.6.","marker":"[4]"},{"why":"Provides the Hirschhorn--Sellers identity (2.3), the initial dissection for the $(2,15)$ congruence.","marker":"[11]"},{"why":"Provides identities (2.4)--(2.6) for $f_1^3$ and $1/f_1^3$; these drive the $(27,11)$ and $(243,17)$ sections.","marker":"[13]"}],"fun_headline_variants":["Infinite congruence families for bipartitions mod 5, 11, 17","Exact recurrences yield bipartition congruences mod 5, 11, 17","New modular laws for bipartition counts modulo 5, 11, 17","Bipartition congruences: infinite families mod 5, 11, 17","Congruences for (2,15), (7,11), (27,11), (243,17) bipartitions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proofs of Theorem 1.2 depend on a block of ten dissection formulas in equation (4.7) that the paper asserts with the phrase 'Similarly, we find' rather than proving line by line; a single wrong coefficient there would change the residues used to obtain the congruences.","fun_headline_variants_meta":{"raw":{"variants":["Infinite congruence families for bipartitions mod 5, 11, 17","Exact recurrences yield bipartition congruences mod 5, 11, 17","New modular laws for bipartition counts modulo 5, 11, 17","Bipartition congruences: infinite families mod 5, 11, 17","Congruences for (2,15), (7,11), (27,11), (243,17) bipartitions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000548,"raw_usage":{"total_tokens":2668,"prompt_tokens":1045,"completion_tokens":1623,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":661,"completion_tokens_details":{"reasoning_tokens":1504}},"tokens_in":661,"tokens_out":1623,"duration_ms":12160,"temperature":1.0,"reasoning_tokens":1504,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:40:30.341091+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the generating function $\\sum B_{7,11}(n)q^n=f_7f_{11}/f_1^2$ to the first few hundred terms and check the residue of $B_{7,11}((2\\cdot7^{12}-2)/3)$ against $3\\pmod{11}$, and check $B_{7,11}(7^{11}+(2\\cdot7^{11}-2)/3)\\equiv0\\pmod{11}$; a single mismatch would disprove Theorem 1.2.","supporting_citations":[{"cited_title":"Berndt, Ramanujan’s Notebooks, Part IV, Springer, New Y ork, 1994","cited_arxiv_id":null,"evidence_quote":"Supplies Entry 31 on page 174, which with [4] evaluates the A, B, C expressions used in Lemma 2.4."},{"cited_title":"Berndt, A.J","cited_arxiv_id":null,"evidence_quote":"Provides identities (2.12)--(2.15) and Theorem 3.2 (Lemma 2.5), which evaluate the p_5, p_7, and p_9 dissections in Lemmas 2.4--2.6."},{"cited_title":"Hirschhorn and J.A","cited_arxiv_id":null,"evidence_quote":"Provides the Hirschhorn--Sellers identity (2.3), the initial dissection for the $(2,15)$ congruence."},{"cited_title":"Hirschhorn, The Power of q","cited_arxiv_id":null,"evidence_quote":"Provides identities (2.4)--(2.6) for $f_1^3$ and $1/f_1^3$; these drive the $(27,11)$ and $(243,17)$ sections."}],"review_version":1}