{"id":"9d96b5cf-5b6a-4a76-8a55-9ee5e2896ea7","arxiv_id":"2607.11113","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":4.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Odd-order reciprocal power sums satisfy a uniform Bernoulli-polynomial congruence modulo n, and Lehmer-type products admit truncated Bell-polynomial expansions modulo n^{K+1}.","lead":"The paper proves modular congruences for odd-power reciprocal sums over residues coprime to n, and expands generalized Lehmer binomial products to higher order via Bell polynomials. Specialists in arithmetic congruences can use the formulas for explicit computation and verification of higher moduli.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the Kummer-type lemma as the sole external dependency and correctly notes that it is a published result whose hypotheses match the paper’s range restrictions on m. The remainder of the derivation (prime-power Faulhaber reduction \to local congruence \to CRT lift for Theorem 1.3; logarithmic expansion + Bell generating function + Möbius inversion for Theorem 1.4) is elementary and self-contained. No stronger load-bearing gap is present, so the ACCEPT verdict stands without adjustment.","tokens_in":12345,"tokens_out":432,"duration_ms":5039,"concrete_test":"Pick a concrete admissible triple, e.g. n=35, e=2, m=3 (satisfies (n,6)=1, m odd, 3≤m≤min(φ(p^l)-l) and m≢1 mod(p-1)). Directly compute both sides of (5) in exact rational arithmetic modulo 35 and verify numerical equality; simultaneously expand the product of Theorem 1.4 to K=2 and check agreement with the known n^3 formula of Zhong–Chern–Cai.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims of Theorems 1.3 and 1.4 rest on standard tools (Euler theorem reductions of reciprocal powers, Faulhaber/Bernoulli expansions of power sums, Möbius inversion of the binomial product, and the exponential generating function for complete Bell polynomials). The only external input flagged by the reader—the Kummer-type congruence for Bernoulli polynomials (Lemma 2.2)—is a published result whose hypotheses (even indices >l, not divisible by p-1, denominator of the argument coprime to p) are exactly the range enforced by the paper’s conditions on m. No internal inconsistency, hidden free parameter, or circular reduction appears in the local-to-global lift or the truncated logarithmic expansion. The argument therefore holds under the stated arithmetic hypotheses.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper extends quadratic reciprocal-sum congruences of Zhong–Chern–Cai to odd powers: under (n,6)=1, e∈{2,3,4,6} and odd m in the range 3≤m≤min(φ(p^l)-l) with m≢1 (mod p-1) for all p|n, Theorem 1.3 asserts that the truncated sum S_m(n) is congruent modulo n to an explicit multiple (via a generalized Jordan totient) of the difference of two Bernoulli-polynomial values at 1/e. Independently, Theorem 1.4 supplies a truncated expansion of the Möbius product of binomial coefficients ∏_d|n binom(kd-1,⌊ d/e⌋)^μ(n/d) in complete exponential Bell polynomials, valid modulo n^{K+1} whenever (n,K!)=1; the K=2 case recovers the earlier cubic congruence. Both results are obtained by local prime-power analysis (Euler reduction of inverses, Faulhaber/Bernoulli expansions, Kummer-type index reduction) followed by CRT and Möbius inversion, with an explicit recursive form of the Bell coefficients provided for computation.","tokens_in":12489,"tokens_out":984,"duration_ms":29755,"significance":"If the statements hold, the work supplies a uniform, algorithmically usable framework that lifts the classical Morley–Lehmer–Cai line from squares and cubic products to odd reciprocal powers and arbitrary modulus order. The Bell-polynomial expansion and its recursive companion make higher-order verification concrete rather than merely existential; the local-to-global passage via CRT and the careful matching of hypotheses with the cited Kummer congruence for Bernoulli polynomials are clean applications of standard tools. The results therefore constitute a genuine, if incremental, advance in the arithmetic of truncated harmonic sums and binomial products.","major_comments":[],"minor_comments":[{"comment":"Throughout §§1–2 the displayed formulae for the generalized totient φ_f^{(k)}(n) and for the right-hand side of Theorem 1.3 are typographically broken (superscripts and subscripts collide, e.g. “n φ(n)-mφ(m-φ(n)) 1 (n)”). These must be restored to standard LaTeX so that the main statements are readable.","section":null},{"comment":"Lemma 2.2 (the Kummer-type congruence) is cited from Ma–Li arXiv:2211.15874; a one-sentence reminder of the precise range of indices and the coprimality condition on the argument would help the reader verify that the paper’s hypotheses on m exactly match the lemma’s requirements.","section":null},{"comment":"In the derivation of the lifting relation (33) the case c ≡ -1 (mod e) uses the identity 1/r^m ≡ (-1)^m / (pl-tr)^m; the subsequent sign (-1)^{m-1} is correct for odd m, but a parenthetical remark that the same identity fails for even m (hence the restriction) would prevent confusion.","section":null},{"comment":"The recursive definition (53) of the coefficients C_t is useful; it would be clearer if the authors explicitly noted that it is the standard Newton–Girard recurrence for the elementary-to-power-sum conversion underlying the Bell polynomials.","section":null},{"comment":"Several references appear only as arXiv preprints (e.g. [1],[4]–[10],[12]); where published versions exist they should be cited, and the arXiv numbers should be updated to the final versions if available.","section":null},{"comment":"Minor linguistic points: “high-order” \to “higher-order” in the title and abstract; “we already know the form” is informal for a journal abstract; the phrase “certain reciprocal sums of odd order” should specify the precise arithmetic constraints already at the abstract level.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript continues a well-defined series by the same author group; the novelty relative to [14] is genuine but modest. The technical content is solid and the presentation issues are purely cosmetic, so the paper is suitable for a solid number-theory journal after a light revision. No concerns about correctness or circularity."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is a clean, self-contained next step in the Cai–Zhong program on Lehmer-type products and reciprocal power sums. Two concrete new statements: Theorem 1.3 gives a uniform mod-n congruence for the truncated odd-order sum Sm(n) under the usual (n,6)=1 and e in {2,3,4,6} hypotheses, expressed as an explicit multiple of a difference of Bernoulli-polynomial values; Theorem 1.4 packages the Möbius product of binomials as a truncated expansion in complete exponential Bell polynomials modulo n^{K+1}. Both go beyond the square-sum and cubic-product results of their earlier arXiv:1604.00445 and Cai’s series.\n\nWhat they do well is keep the bookkeeping honest. Local prime-power congruences come from Euler reduction of inverses, Faulhaber plus the addition formula for Bernoulli polynomials, then a Kummer-type index drop (Lemma 2.2, cited from Ma–Li) whose hypotheses match exactly the range they enforce on m. The lift to composite n is by CRT after a short Möbius argument; the product side is just the truncated log-exp generating function that defines the Bell polynomials, with an optional recursive form for computation. No free parameters, no circular reductions, and the base cases from [14] are used only as motivation, not as black boxes that hide the new work.\n\nSoft spots are minor and proportional. The significance is specialized—useful computational scaffolding inside a narrow arithmetic-congruence niche, not a method that travels far. The Kummer input is external, but it is published and the paper’s conditions sit inside its hypotheses, so the dependence is clean rather than load-bearing risk. Presentation is dense but readable if you already know the earlier papers.\n\nThis is for people already working on Bernoulli-polynomial congruences or algorithmic verification of Lehmer products. A serious referee will find the derivations standard and carefully tracked; it deserves peer review and should clear a specialized number-theory journal after routine polishing. I would cite the Bell expansion if I needed higher-order product formulae, and I would bring the odd-power statement to a reading group only if we were already deep in this literature.","headline":"Solid incremental extension of the Cai–Zhong line: odd-order reciprocal sums via Bernoulli polynomials and a clean Bell packaging of higher Lehmer products.","tokens_in":13107,"tokens_out":554,"would_cite":true,"duration_ms":6046,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A07","11B68","11B73"],"pacs":[],"model":"grok-4.5","headline":"Odd-order reciprocal power sums over residues coprime to n are congruent modulo n to an explicit Bernoulli-polynomial expression, while generalized Lehmer products expand as truncated Bell polynomials modulo n^{K+1}.","keywords":["reciprocal power sums","Bernoulli polynomials","Lehmer-type products","complete exponential Bell polynomials","higher-order congruences","Möbius inversion","Euler totient"],"falsifier":"Pick a concrete prime power p^l (p≥5) and an admissible odd m, compute the truncated sum S_m(p^l) by direct enumeration and the right-hand side of Theorem 2.2 involving Bernoulli polynomials, and check whether they agree modulo p^l; any mismatch falsifies the local step on which the whole paper depends.","tokens_in":13230,"feed_emoji":"∑","tokens_out":831,"duration_ms":7723,"temperature":0.7,"pith_summary":"The paper extends classical Morley–Cai–Zhong congruences for reciprocal squares and binomial products to two broader settings. First, for n coprime to 6 and e in {2,3,4,6}, it proves that the truncated sum of 1/r^m over residues r ≤ floor(n/e) coprime to n is congruent modulo n to a simple multiple of a difference of Bernoulli polynomials, provided m is an odd integer lying in a range controlled by the prime-power factors of n. Second, the Möbius product of binomial coefficients binom(kd-1, floor(d/e)) is shown to admit an explicit power-series expansion in n whose coefficients are complete exponential Bell polynomials evaluated at the reciprocal power sums; the expansion holds modulo n^{K+1} whenever n is coprime to K!. Together the two theorems supply a uniform, computable framework that recovers all previously known low-order cases and makes higher-order verification algorithmic.","feed_headline":"Odd-power reciprocal sums get Bernoulli formulas mod n","feed_subtitle":"Bell polynomials then expand the associated binomial products to any order","key_machinery":"The complete exponential Bell polynomials that convert the truncated logarithm of the product ∏(1-kn/r) into an explicit power series in n, together with a Kummer-type congruence for Bernoulli polynomials that reduces the large local index M+1 to the global totient index φ(n)-m+1.","core_discovery":"Under the arithmetic hypotheses (n,6)=1, e∈{2,3,4,6} and m odd with 3≤m≤min(φ(p^l)-l) and m≢1 mod(p-1) for every p^l∥n, the sum S_m(n) satisfies the uniform congruence modulo n given in Theorem 1.3; independently, for any K with (n,K!)=1 the generalized Lehmer product equals (-1)^{φ_e(n)} times the partial sum of Bell polynomials ar B_m(-k S_1,…,-(m-1)!k^m S_m)/m! · n^m modulo n^{K+1} (Theorem 1.4).","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Uniform mod-n congruences for odd reciprocal power sums","Bell polynomials expand Lehmer products to higher orders","Higher-order forms for reciprocal sums and Lehmer products","Truncated Bell expansions for generalized Lehmer products","Odd-power reciprocal sums get unified congruences mod n"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The argument rests on a Kummer-type congruence that lets one replace a large Bernoulli-polynomial index by a smaller congruent index; if that replacement fails for the allowed range of m, both the local formula and its lift to composite n collapse.","fun_headline_variants_meta":{"raw":{"variants":["Uniform mod-n congruences for odd reciprocal power sums","Bell polynomials expand Lehmer products to higher orders","Higher-order forms for reciprocal sums and Lehmer products","Truncated Bell expansions for generalized Lehmer products","Odd-power reciprocal sums get unified congruences mod n"]},"model":"grok-4.5","effort":"low","cost_usd":0.004764,"raw_usage":{"total_tokens":1480,"prompt_tokens":933,"num_sources_used":0,"completion_tokens":84,"cost_in_usd_ticks":47640000,"prompt_tokens_details":{"text_tokens":933,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":463,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":933,"tokens_out":84,"duration_ms":4705,"temperature":1.0,"reasoning_tokens":463,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T06:58:03.471121+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Pick a concrete prime power p^l (p≥5) and an admissible odd m, compute the truncated sum S_m(p^l) by direct enumeration and the right-hand side of Theorem 2.2 involving Bernoulli polynomials, and check whether they agree modulo p^l; any mismatch falsifies the local step on which the whole paper depends.","supporting_citations":[],"review_version":1}