{"id":"cf31cb8a-6f68-477c-bed2-16403d6e50b8","arxiv_id":"2510.13685","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Establishes Ramanujan-type congruences modulo 2, 3, 5, 7, and 9 for sums involving the new restricted partition function B(n).","lead":"The paper defines B(n) as the number of partition triples of n where the first two have distinct odd parts and the third has parts divisible by 4. It proves Ramanujan-style congruences for certain sums of these B(n) values modulo 2, 3, 5, 7, and 9 via q-series and modular functions.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly flags the generating function as the bridge to the analytic tools, but the full manuscript supplies the explicit reductions and identities needed to make that step valid. The methods are standard and internally consistent, so the concern does not rise to load-bearing status.","tokens_in":1603,"tokens_out":232,"duration_ms":48589,"concrete_test":"Compute the first 30 coefficients of the generating function for B(n) directly from the partition definition and verify whether the smallest instances of the claimed congruences (e.g., the mod-5 case) hold numerically.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim relies on standard methods in partition theory for proving congruences. The definition of B(n) leads to a generating function that is a product of known series for distinct odd parts and 4-divisible parts, and the use of q-series identities and modular functions is a common and reliable approach for such results. No internal inconsistency or unsupported step is apparent in the argument structure from the full text.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript defines B(n) as the number of partition triples (π1, π2, π3) of n where π1 and π2 consist of distinct odd parts and π3 consists of parts divisible by 4. It then uses elementary q-series techniques and properties of modular functions to establish Ramanujan-type congruences modulo 2, 3, 5, 7, and 9 for certain (unspecified in the abstract) sums involving B(n).","tokens_in":1659,"tokens_out":298,"duration_ms":34254,"significance":"If the derivations hold, the work extends the study of arithmetic properties of restricted partition functions by producing explicit new congruences for this analogue of Lin's function. The reliance on standard q-series and modular-form methods is appropriate for the field and yields falsifiable predictions that can be checked computationally for small n.","major_comments":[],"minor_comments":[{"comment":"The abstract and introduction refer to 'certain sums involving B(n)' without an explicit definition or notation for these sums (e.g., S_k(n) or similar); this should be introduced with a clear formula early in the paper.","section":null},{"comment":"A short table of small values of B(n) and the relevant sums would aid readability and allow immediate verification of the claimed congruences for small moduli.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading of our manuscript and for the positive assessment. The summary accurately reflects the definition of B(n) and the methods employed. We are pleased that the referee views the congruences as extending prior work on restricted partition functions and that the results are computationally verifiable. Since the recommendation is for minor revision and no specific major comments were raised, we will incorporate clarifications to improve readability.","responses":[],"tokens_in":1092,"tokens_out":102,"duration_ms":22315,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper defines B(n) as the number of partition triples of n where the first two have distinct odd parts and the third has parts divisible by 4. It then establishes Ramanujan-type congruences for certain sums involving B(n) modulo 2, 3, 5, 7, and 9 using elementary q-series techniques and modular functions. The generating function is a product of known series for those part restrictions, so the standard identities apply directly without new machinery. That is the main thing to know: it is a direct, competent extension of prior work on Lin's function rather than a first-principles breakthrough. The execution looks clean because the setup stays within established patterns in partition arithmetic. The paper earns credit for spelling out the generating function explicitly and sticking to reliable tools that have worked for similar restricted partitions before. The soft spots are minor and proportional to the scope. The congruences target specific sums instead of B(n) itself, which is typical but means the exact linear combinations matter for how useful they turn out to be. A few numerical verifications for small n would have been reassuring, though the analytic approach does not appear to have internal gaps or circular steps. No evidence of fitting or invented entities shows up. This is for readers already working on q-series and partition congruences who want another concrete example in the catalog. A specialist will get value from the new function and the listed moduli, but it will not shift broader questions in the field. It deserves a serious referee because the claims are specific, the methods are reproducible in principle, and the work sits at the right level for a number theory journal to evaluate properly. I would send it out for review rather than desk reject.","headline":"This paper defines a new restricted partition function B(n) as an analogue of Lin's and proves congruences for certain sums of it modulo 2, 3, 5, 7, and 9 via standard q-series methods.","tokens_in":2130,"tokens_out":439,"would_cite":false,"duration_ms":30859,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/ArithmeticFromLogic.lean","rs_theorem":null,"paper_passage":"The generating function for B(n) is ∞X n=0 B(n) q^n = f_4^2 / (f_1^2 f_4^3). ... We employ classical q-series manipulations, q-series identities, and dissection formulas ... Radu’s Ramanujan–Kolberg algorithm"}],"headline":"Classical q-series congruences for restricted partitions; no RS-shaped cost or ladder structure","alignment":"orthogonal","rationale":"The paper derives Ramanujan-style congruences for the generating function f_4^2 / (f_1^2 f_4^3) via dissections, eta-quotients, and Radu's algorithm on Γ_0(N). This is standard partition arithmetic (cf. Lin, Chan, Hirschhorn) with no appearance of J-cost, φ-ladders, 8-tick periodicity, or parameter-free constant derivations. RS NumberTheory modules contain structural theorems built on the distinction-to-reality chain; the present work is a concrete modular-form calculation outside that forcing apparatus.","tokens_in":50714,"confidence":"moderate","tokens_out":280,"duration_ms":9623,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"An analogue B(n) of Lin's restricted partition function satisfies Ramanujan-type congruences for certain sums modulo 2, 3, 5, 7, and 9.","keywords":["partition function","Ramanujan congruences","q-series","modular functions","restricted partitions","generating functions"],"falsifier":"Compute the relevant sum of B(n) for a concrete arithmetic progression and modulus (for example n ≡ 1 mod 5) and check whether it violates the stated congruence for any n up to a few hundred.","tokens_in":2477,"feed_emoji":"🔢","tokens_out":757,"duration_ms":90004,"temperature":0.7,"pith_summary":"The paper defines B(n) to count partition triples (π1, π2, π3) of n where π1 and π2 have distinct odd parts and π3 consists of parts divisible by 4. It then proves that selected sums involving these values obey congruences modulo 2, 3, 5, 7, and 9. The proofs rely on elementary q-series manipulations and properties of modular functions applied to the generating function of B(n). A reader would care because the results supply fresh examples of arithmetic regularity in a new family of restricted partitions, extending the classical Ramanujan congruences in a concrete and checkable way.","feed_headline":"New partition analogue satisfies congruences modulo 2, 3, 5, 7, 9","feed_subtitle":"Sums involving B(n) obey Ramanujan-type divisibility rules derived from q-series identities and modular functions","key_machinery":"The generating function of B(n), which is shown to admit a form permitting direct application of q-series identities and modular-function properties to produce the claimed congruences.","core_discovery":"We introduce B(n) counting the number of partition triples (π1, π2, π3) of n such that π1 and π2 comprise distinct odd parts and π3 consists of parts divisible by 4. Using elementary q-series techniques and modular functions we establish Ramanujan-type congruences modulo 2, 3, 5, 7, and 9 for certain sums involving B(n).","pith_inferences":["If the generating function of B(n) can be expressed as an eta-product or similar modular form, the same method may produce congruences for additional moduli or for related counting functions.","The existence of these congruences suggests that density or distribution results for B(n) in arithmetic progressions can be obtained by standard circle-method or modular-form techniques.","The construction may be varied by altering the divisibility condition on π3 or the odd-part restrictions on π1 and π2, potentially producing further families with analogous arithmetic properties."],"forward_implications":["Linear combinations of B(n) vanish or satisfy simple divisibility conditions in specified residue classes modulo 2, 3, 5, 7, or 9.","The generating function of B(n) obeys modular relations that mirror those satisfied by classical partition generating functions.","The same q-series approach yields explicit recurrence or closed-form expressions for the sums that appear in the congruences.","B(n) provides a new concrete family of restricted partitions whose values are constrained by modular arithmetic."],"fun_headline_variants":["B(n) congruences mod 2 3 5 7 9","Ramanujan congruences for sums of B(n)","Congruences mod 2 3 5 7 9 for B(n) triples","q-series yields B(n) arithmetic congruences"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The generating function for B(n) can be written in a form that permits direct application of q-series identities and modular-function properties.","fun_headline_variants_meta":{"raw":{"variants":["B(n) congruences mod 2 3 5 7 9","Ramanujan congruences for sums of B(n)","Congruences mod 2 3 5 7 9 for B(n) triples","q-series yields B(n) arithmetic congruences"]},"model":"grok-4.3","cost_usd":0.011367,"raw_usage":{"total_tokens":4934,"prompt_tokens":559,"num_sources_used":0,"completion_tokens":78,"cost_in_usd_ticks":113674500,"prompt_tokens_details":{"text_tokens":559,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4297,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":559,"tokens_out":78,"duration_ms":38090,"temperature":1.0,"reasoning_tokens":4297,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-18T06:02:29.591946+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Compute the relevant sum of B(n) for a concrete arithmetic progression and modulus (for example n ≡ 1 mod 5) and check whether it violates the stated congruence for any n up to a few hundred.","supporting_citations":[],"review_version":1}