{"id":"80a84d7c-669e-4b57-9b52-dc6fb01fb5b1","arxiv_id":"2607.09904","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Quadratic Fourier projection of hyperbolic arcsine kernels yields closed identities for quartic central-binomial harmonic sums with π, L=log(1+√2), and polylog companions at (√2−1)².","lead":"The paper packages classical arcsine-power series into two hyperbolic kernels and applies finite Fourier filters before specializing, producing closed forms for quartic central-binomial harmonic sums involving π and log(1+√2). Specialists in binomial series and multiple zeta values get a systematic filter-plus-Mellin machine for residue-class and logarithmic companions.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim is a pure analytic identity obtained by applying a root-of-unity filter to classical arcsine generating functions before specialisation. Every step (hypergeometric evaluation \to hyperbolic kernels \to principal-branch boundary values \to quadratic projection) is written with explicit lemmas that control convergence and branches. The reader's weakest-assumption diagnosis is accurate, yet those assumptions are discharged inside the paper itself; they are not free-floating. Consequently the load-bearing concern does not land as a correctness risk. The recommended verification is merely a high-precision numerical sanity check of the lowest-weight cases, which the manuscript already asserts are proved analytically. No adjustment of the ACCEPT verdict is warranted.","tokens_in":21031,"tokens_out":495,"duration_ms":5089,"concrete_test":"Independently recompute the m=0 and m=1 cases of (6.2) and (6.4) by truncating the series at r=200 with 50-digit arithmetic and comparing against the closed forms involving π and L=log(1+√2); agreement to ≥30 digits confirms that the branch/Abel-limit conventions used in the filter produce the claimed numerical values.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption (principal branches of Log/sqrt/arcsin and Abel limits of F_m(±1), G_m(±1)) is correctly identified as the analytic hinge of Theorem 6.1, but the paper already supplies the needed support: Lemma 4.1 (normal convergence on compacta), Lemma 4.2 (absolute summability O(n^{-3/2}) implying Abel limits), and Lemma 4.3 (continuous radial limits under the principal-branch conventions (4.1)–(4.4), with the explicit evaluation arcsin(i)=iL). The subsequent root-of-unity projection (Theorem 5.1) and the quadratic specialisation (Theorem 6.1) are therefore justified by the same classical estimates that underwrite the unfiltered master sums (2.16)–(2.17). No hidden gap appears in the chain from the hyperbolic closed forms (2.10)–(2.11) to the quartic identities.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper packages the classical odd-square and ordinary-square coefficient families arising in powers of arcsin into two hyperbolic kernels O(z,x) and E(z,x). Finite Fourier (root-of-unity and periodic-weight) projections and Mellin deformations are applied at the kernel level before endpoint specialization. The quadratic case extracts even/odd subsequences, converting central-binomial series into quartic identities involving binom(4r,2r), powers of π, and L = arsinh 1 = log(1+√2). The same projected kernels yield accelerated interior evaluations with explicit tails, denominator-power series, and logarithmic companions involving polylogarithms at (√2−1)². Supporting material includes the square-law relation E = 2O², branch/convergence lemmas, spectral truncations, and a comparison showing that direct replacement of binom(2n,n) by binom(4n,2n) produces a different ₄F₃ family.","tokens_in":21224,"tokens_out":864,"duration_ms":26419,"significance":"If the derivations hold—as they appear to—the paper supplies a clean, reusable analytic framework that unifies residue-class, accelerated, and Mellin-deformed central-binomial harmonic identities under a single pair of generating kernels. The main quartic evaluations (Theorem 6.1) and their logarithmic companions (Theorem 9.1) are concrete and checkable; the supporting lemmas on principal branches, Abel limits, and termwise Mellin operations are written with care. The work sits squarely in the classical tradition of Lehmer, Borwein–Chamberland, and recent arcsine-series papers, while the pre-specialization filtering step is a genuine methodological contribution. Complete closed-form proofs (no numerical fitting, no free parameters) are a clear strength.","major_comments":[],"minor_comments":[{"comment":"A short paragraph in the introduction or conclusion comparing the filtered-kernel approach more explicitly with the arcsine-moment / Bell-polynomial methods of Dilcher–Vignat (refs. [6–8]) would help the reader locate the novelty.","section":null},{"comment":"Notation for the finite repeated sums ζ_n({2}^m) versus the Riemann zeta function ζ(s) is declared early, but the two still collide visually in Sections 6 and 8; a typographic distinction (e.g., bold or a different letter for the finite sums) would reduce cognitive load.","section":null},{"comment":"Section 9: the low-weight polylog evaluations (9.2)–(9.3) are carefully derived; adding one intermediate numerical check (or a one-line Magma/Mathematica verification note) for the Li₂(ρ²) and Li₃(ρ²) coefficients would aid independent confirmation.","section":null},{"comment":"Theorem 7.2: the tail constants are explicitly labelled “admissible, not optimal.” A single sentence noting that sharper Stirling-type bounds are available would prevent readers from treating the constants as best possible.","section":null},{"comment":"Appendix B: the direct-quartic ₄F₃ comparison is useful; a one-sentence remark that the filtered identities of Theorem 6.1 are not special cases of these ₄F₃ series would make the contrast even sharper.","section":null},{"comment":"Minor typographical/spacing artefacts appear in the extracted text (author name, some binomial displays). These are presumably clean in the source but should be double-checked before final production.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is analytically solid and well within the scope of a special-functions / classical-analysis journal. I see no hidden circularity or load-bearing gap; the branch and Abel-limit lemmas adequately support the quadratic filter. Novelty is incremental rather than transformative, but the unified kernel-plus-filter package is clean enough to be useful to others working on central-binomial series. Accept is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that this paper does the filtering at the kernel level, before specialization, and that is what cleanly produces the quartic identities with binom(4r,2r), π and L = log(1+√2). The main results (Theorem 6.1 and the three companions) are explicit closed forms for the even/odd residue classes of the classical arcsine-power series, plus accelerated interior versions and Mellin/log companions involving Li2 and Li3 at (√2-1)².\n\nWhat is new is not the hyperbolic kernels themselves—they are rewritings of the usual arcsin expansions—but the systematic use of finite Fourier projection and Mellin deformation on those kernels, together with the explicit contrast (Appendix B) that direct replacement by binom(4n,2n) yields a different 4F3 family. The square-law convolution E=2O², the branch/convergence lemmas, and the root-of-unity theorems are all written carefully. The proofs rest on standard hypergeometric evaluations (Lemma 2.2), product expansions of the finite repeated sums, and classical Abel-limit arguments; nothing is inserted circularly. The stress-test note is right: the principal-branch and radial-limit hinge is already supported by Lemmas 4.1–4.3, so the chain from the closed forms (2.10)–(2.11) to Theorem 6.1 holds.\n\nSoft spots are minor and proportionate. Significance is organizational rather than transformative—it sits inside the Borwein–Chamberland / Dilcher–Vignat literature and does not settle an open conjecture. Numerical checks are mentioned but not tabulated; there is no formal verification. Neither of those undercuts the analytic work.\n\nThis is for people who already work with central-binomial harmonic sums or arcsine generating functions. A serious referee should see it. I would accept it for peer review and would cite the main identities and the filter method if I needed those evaluations.","headline":"Solid, carefully proved filter-plus-Mellin pipeline for quartic central-binomial harmonic sums; modest significance but real method and clean identities.","tokens_in":21867,"tokens_out":527,"would_cite":true,"duration_ms":5933,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["33C20","11M06","05A10","40A25"],"pacs":[],"model":"grok-4.5","headline":"Hyperbolic arcsine kernels plus Fourier filters extract quartic binomial-harmonic sums that evaluate to combinations of π and log(1+√2).","keywords":["central binomial coefficient","harmonic number","inverse sine","hypergeometric series","root-of-unity filter","log-sine integral","Bell polynomial","quartic sums"],"falsifier":"Direct high-precision numerical comparison of the partial sums of the four series in Theorem 6.1 against the claimed right-hand sides involving π and L, for several small weights m, would immediately confirm or refute the identities.","tokens_in":21897,"feed_emoji":"∑","tokens_out":947,"duration_ms":7545,"temperature":0.7,"pith_summary":"Powers of the inverse sine produce series whose coefficients mix central binomials with finite repeated harmonic sums over squares. This paper packages those coefficients into two hyperbolic arcsine kernels and treats the kernels themselves as generating functions. Finite Fourier projection is applied to the kernels before any endpoint specialization; the quadratic case isolates the even (or odd) indices and thereby converts ordinary central binomials into quartic ones. The resulting closed forms are explicit linear combinations of powers of π and of L = log(1+√2). The same filtered kernels admit accelerated interior evaluations, Mellin deformations that produce denominator powers and polylogarithms at (√2-1)^{2}, and a square-law convolution that relates the two families. A sympathetic reader cares because the method systematically manufactures whole families of previously scattered identities from one analytic object rather than by case-by-case summation.","feed_headline":"Filters on arcsine kernels yield quartic binomial sums","feed_subtitle":"Quadratic projection turns ordinary central binomials into closed forms with π and log(1+√2)","key_machinery":"The paired kernels O(z,x) = sinh(√z arcsin x)/√z and E(z,x) = [cosh(2√z arcsin x)-1]/z, which package the odd-square and ordinary-square coefficient families so that finite Fourier projection and Mellin deformation can be performed at the kernel level.","core_discovery":"Projecting the two hyperbolic arcsine kernels with the quadratic root-of-unity filter before specialization yields the four residue-class identities of Theorem 6.1; in particular the even-residue odd-kernel sum equals [π^{2m+1} + (-1)^m (2L)^{2m+1}] / [2^{2m+2}(2m+1)!] for every m ≥ 0, with L = arsinh 1.","pith_inferences":["The same kernel-plus-filter pipeline should extend without essential change to other odd multiple-zeta alphabets once the corresponding hyperbolic generating functions are written down.","Direct comparison of the filtered _2F1-type closed forms with the _4F3 kernels of Appendix B may produce new transformation identities between the two hypergeometric families.","Because the Mellin parameter can be differentiated under the integral, the method automatically supplies generating functions for all higher log-sine moments of the filtered series."],"forward_implications":["Every weight-m quartic central-binomial series of the filtered type evaluates in closed form to a combination of π^{2m} or π^{2m+1} and the corresponding power of L.","Mellin deformation of the same projected kernels produces companion identities whose denominators are powers of the linear terms and whose constants involve polylogarithms at (√2-1)^{2}.","Interior evaluation of the quadratic filter supplies accelerated series with explicit geometric tail bounds.","The square-law relation E = 2O^{2} converts any convolution identity between the two harmonic alphabets into an algebraic identity among binomial coefficients.","Root-of-unity filters of higher order (cubic, character-mod-4, …) generate analogous residue-class evaluations for other arithmetic progressions."],"fun_headline_variants":["Quadratic filters on arcsine kernels extract quartic binomial identities","Finite Fourier projection yields π-L forms for central binomial sums","Arcsine kernels under root-of-unity filter give even-residue closed forms","Hyperbolic arcsine kernels project to quartic sums with π and log(1+√2)","Quadratic projection of arcsine kernels unlocks binomial harmonic identities"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The principal-branch choices for logarithm, square root and inverse sine, together with the claim that the Abel radial limits of the generating functions at ±1 equal the closed forms used in the filter, must hold; if those continuous boundary values failed, the quartic evaluations would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Quadratic filters on arcsine kernels extract quartic binomial identities","Finite Fourier projection yields π-L forms for central binomial sums","Arcsine kernels under root-of-unity filter give even-residue closed forms","Hyperbolic arcsine kernels project to quartic sums with π and log(1+√2)","Quadratic projection of arcsine kernels unlocks binomial harmonic identities"]},"model":"grok-4.5","effort":"low","cost_usd":0.006078,"raw_usage":{"total_tokens":1517,"prompt_tokens":756,"num_sources_used":0,"completion_tokens":98,"cost_in_usd_ticks":60780000,"prompt_tokens_details":{"text_tokens":756,"audio_tokens":0,"image_tokens":0,"cached_tokens":0},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":663,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":756,"tokens_out":98,"duration_ms":6929,"temperature":1.0,"reasoning_tokens":663,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T14:41:43.290696+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Direct high-precision numerical comparison of the partial sums of the four series in Theorem 6.1 against the claimed right-hand sides involving π and L, for several small weights m, would immediately confirm or refute the identities.","supporting_citations":[],"review_version":1}