{"id":"aec535f9-a107-48ab-be1e-03544e0da264","arxiv_id":"1908.05123","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A new bilateral semi-terminating identity for Ramanujan-type 1/π^k series is proved, and a family of supercongruences is conjectured with proofs of several rank-3 and rank-5 cases.","lead":"Jesús Guillera constructs bilateral versions of Ramanujan-like series for 1/π^k and conjectures new supercongruences for them, proving several special cases. The paper is a contribution to the arithmetic theory of hypergeometric supercongruences, with a side observation linking certain discriminants to Asai L-functions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The α_k, β_k in identity (5) are obtained by numerical solving followed by an unverified 'easy' rational identification; the explicit semi-terminating identities and supercongruence tables in Section 4 inherit this uncertainty.","rationale":"The reader's weakest_assumption identifies the same load-bearing concern: the explicit α_k and β_k are obtained by numerical solving and unverified rational identification. This is indeed the most consequential gap for the paper's explicit and conjectural claims, because identity (5) is proved only with unspecified constants. The semi-terminating identities (6), the tabulated ε0, and the numerical evidence for the supercongruence conjecture (7) all depend on the exact values of these constants. If the numerical identification is incorrect, those explicit statements need revision even though the existence form of (5) could remain true. The reader's CONDITIONAL verdict already captures this: the paper should be accepted only if the rational identifications are verified or explicitly downgraded to conjecture. I found no reason to move the verdict further, since Section 5's proved examples are supported by independent terminating hypergeometric identities and classical congruences such as Morley's and Wolstenholme's. The concrete test above would settle whether the concern lands; if it passes, the central conditional claims become substantially more reliable.","tokens_in":16000,"tokens_out":11408,"duration_ms":123501,"concrete_test":"For Example 4.2, compute the left side of (5) to at least 50 significant digits at several values of x using the convergent series with z0 = 1/7^4 and a rigorous tail bound, then solve the linear system for α1, α2, β1, β2. Use rational reconstruction to see whether the values are exactly −79/2, 23/2, 5i/2, −3i/2, and additionally plug the proposed rational values back into (5) at new x values to check residuals. Independently, compute the Fourier coefficients c_k = ∫_0^1 f(x) e^{−2πikx} dx by high-precision numerical quadrature for k = 1, 2 and compare the derived α_k, β_k. This directly checks whether the coefficients used in (6) and Table 3 are exact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After proving that (5) holds for some constants α_k, β_k, Section 2 determines them by numerically solving the 2m×2m linear system at 2m values of x and then states: 'Once we get the approximated solutions, is easy to identify the exact rational values they are.' No exact verification is supplied. Section 4 then uses these values (e.g., α1 = −14/3, β1 = 5i/2, α4 = 1/6) to compute the semi-terminating constant r0 in (6), the discriminant ε0 in the tables, and the numerical checks of the supercongruence conjecture (7). If any of these identifications is merely a very good rational approximation rather than exact, then the displayed semi-terminating identities in Examples 4.1–4.4 and the tabulated ε0 are wrong, and the claimed numerical support for (7) collapses. This is load-bearing precisely because theorem (5) alone does not provide the explicit constants; the conditional formulas in (6) and (7) require the exact coefficients. Section 5's proven congruences are independent of this step, since they are proved directly from terminating hypergeometric identities, so the flaw is localized to the unproved examples and the conjectural supercongruence tables.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs bilateral series associated with rational Ramanujan-like series for negative powers of π. It introduces functions A(n,x) and B(n,x), forms a periodic holomorphic function f(x) with a cosine prefactor, and states identity (5) asserting that f(x) equals v0√((−1)^m χ0)/π^m times a finite trigonometric polynomial of degree m with constants α_k, β_k (conjectured rational). From the limit x → −p/2 the author derives a semi-terminating identity (6) and conjectures supercongruences (7) modulo p^{2m+1}. Section 4 gives unproved examples with explicit α_k, β_k, r0, and ε0. Section 5 proves several B-supercongruences for rank-3 and rank-5 series using terminating hypergeometric identities quoted from [6] together with Morley's and Wolstenholme's congruences. The paper ends with tables of known rational series and a speculative observation linking the ε0 values to Asai L-functions in [4].","tokens_in":16241,"tokens_out":11480,"duration_ms":107217,"significance":"If identity (5) is rigorously established, the paper would provide a uniform bilateral completion for rational Ramanujan-like series and a new mechanism for producing B-supercongruences with modulus p^{2m+1}. The proved supercongruences in Section 5 are concrete and appear correct; reducing them to terminating hypergeometric identities and classical congruences is a clean and effective strategy. The conjecture (7), with explicit ε0 values, is falsifiable and the tabulated data constitute a useful experimental resource. The link to Asai L-functions, though speculative, is intriguing. However, the proof of the central identity is sketchy, and the exact values of α_k and β_k used in Section 4 are obtained by numerical identification rather than proof, so the rigorous core of the paper is currently limited to Section 5.","major_comments":[{"comment":"The proof of (5) hinges on the unproved estimate f(x)=O(e^{(2m+1)π|Im(x)|}). This bound is stated without derivation, and a reader cannot check how the infinite sum A(n,x) and the prefactors behave near infinity or how the pole cancellations contribute to the growth. Since this estimate is what forces the Fourier expansion to contain only the modes up to k=m, identity (5) is not fully proved as written. Please supply the necessary estimates (or a precise reference) and spell out how the bound yields the exact trigonometric form in (5).","section":"Section 2, proof of identity (5)"},{"comment":"The constants α_k and β_k are determined by 'solving numerically the linear system' and then 'is easy to identify the exact rational values they are.' This is not a proof of exactness. These constants enter directly into the semi-terminating constant r0 in (6) and into the tabulated ε0 and the numerical checks of conjecture (7) in Examples 4.1–4.4 and Tables 1–5. The author should either prove the exact values by an exact method (for example, by differentiating (5) at x=0 and using exact arithmetic, or by evaluating at enough special points and solving the linear system exactly) or explicitly state that the listed α_k, β_k, r0, and ε0 are conjectural. As written, the sentence invites the reader to treat numerical approximations as exact identities.","section":"Section 2, paragraph after (5); Section 4"},{"comment":"The proofs of Examples 5.1–5.7 start from terminating hypergeometric identities quoted from [6] (for instance, (24), (30), (35), (40), (45), (50)). Although Remark 5.2 states that Zeilberger's algorithm can verify such identities, no recurrences, initial values, or certificates are shown, so the displayed identities cannot be checked by the reader. Since these identities are the foundation of the proofs in this section, please either reproduce the relevant derivation from [6] with the exact substitutions and limits, or include an appendix with the Zeilberger recurrences and initial values. Otherwise the 'proved' examples rest on unverifiable input.","section":"Section 5, use of terminating identities from [6]"}],"minor_comments":[{"comment":"The word 'semicongruences' appears where 'supercongruences' is clearly intended; please correct the terminology consistently.","section":"Section 3 and Introduction"},{"comment":"The right-hand side of (17) is typographically ambiguous: '= 8 8p (2p − 1)!/(p − 1)!' should probably read '= 8/8^p (2p − 1)!/(p − 1)!' or similar. Please clarify the intended denominator.","section":"Equation (17)"},{"comment":"The congruence (44) is missing the modulus at the end; it should read '(mod p^5)'.","section":"Example 5.6, congruence (44)"},{"comment":"The word 'identy' should be 'identity' in the proof of Example 5.7.","section":"Example 5.7"},{"comment":"The tables do not state how the ε0 values were obtained. A short note on the provenance (numerical evaluation of (6), conjectural pattern, or exact derivation) would improve reproducibility and clarify the status of each entry.","section":"Section 6, Tables"},{"comment":"The final observation would be more useful if the author explicitly identified which entries in [4, Table 5.2] match the ε0 values in Table 3; as written, the correspondence is not checkable from the text.","section":"Section 6, Intriguing observation"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The author works in a well-established area and builds on their own published work [6]. The main concern is the gap between numerical identification and exact proof in the derivation of (5) and its coefficients. This is a common methodological issue in experimental number theory, but for a rigorous journal the paper should either close that gap or be reframed as a conjectural/experimental contribution with a clearly separated rigorous Section 5. The Section 5 results appear sound, and the extensive numerical data are a strength. I would encourage a revision that makes the status of every unproved assertion explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is identity (5): a bilateral completion of rational Ramanujan-like series that, at x = -p/2, produces semi-terminating identities and a whole family of conjectured p^{2m+1} supercongruences (7). That is a real idea, and the connection to Asai L-functions via the epsilon_0 table is worth taking seriously even though it is only an observation. The paper is also honest about what is proved versus conjectured: Section 4 is explicitly flagged as conjectural, and Section 5 proves several rank-3 and rank-5 B-supercongruences using terminating hypergeometric identities plus Morley and Wolstenholme. Those proofs look correct to me on inspection, and Remark 5.2 gives a credible Zeilberger route for certifying the identities, so the proved examples are not floating on trust.\n\nThe soft spots are real but localized. The proof of (5) is sketchy: the growth bound O(e^{(2m+1)pi|Im x|}) is asserted without derivation, and the termination at k=m is the load-bearing step. More importantly, the explicit alpha_k and beta_k are obtained by numerically solving a linear system and then an unverified \"easy\" rational identification. The reader's stress-test note is right: if those numbers are merely good rational approximations, the semi-terminating formulas in Examples 4.1-4.4 and the tabulated epsilon_0 are wrong, and the numerical support for (7) collapses. That is a genuine gap, not a manufactured one. It is also fixable: one could verify the coefficients exactly by checking the Fourier expansion at enough rational points with rigorous interval arithmetic, or by proving the finite Fourier structure directly. The A-side congruence (12) is not new, being equivalent to Zudilin-type supercongruences, so the real contribution is the B-side framework.\n\nI disagree with any reading that calls this circular. The conjectures are not fitted to data; the proved examples are derived from independently known identities, and the numerical checks are presented as checks. The self-citation is fine.\n\nWho is this for? People working on Ramanujan-type series, supercongruences, and hypergeometric motives. A serious referee should engage with it, mainly because the bilateral identity is novel and likely to be useful even if some of the conjectured constants need adjustment. I would not desk-reject. My recommendation: send to a specialist in hypergeometric congruences, and push the author to supply exact verification of the alpha_k, beta_k values or to state clearly that they are conjectural throughout Section 4.","headline":"New bilateral completion of Ramanujan-like 1/pi^k series with conjectured supercongruences; the main theorem is plausible but the explicit constants are numerically guessed, and the proved cases rest on cited identities.","tokens_in":16807,"tokens_out":671,"would_cite":true,"duration_ms":8733,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B65","33C20","11A07","11F67"],"pacs":[],"model":"deepseek-v4-flash","headline":"A rational Ramanujan-like series for 1/π^m can be completed to a bilateral sum that is exactly a degree-m trigonometric polynomial, from which the paper derives supercongruences modulo p^{2m+1}.","keywords":["Ramanujan-like series","bilateral series","semi-terminating series","supercongruences","hypergeometric identities","binomial congruences","Hilbert modular forms"],"falsifier":"Take Example 4.4, the rank-9 series for $768/\\pi^4$. Evaluate both sides of (5) at ten values of $x$ to high precision, solve the resulting $8\\times8$ linear system for $\\alpha_1,\\alpha_2,\\alpha_3,\\alpha_4,\\beta_1,\\beta_2,\\beta_3,\\beta_4$, and check whether the outputs are exactly $-25/6, 8/3, -2/3, 1/6, 0,0,0,0$; any discrepancy beyond rounding error would refute the explicit constants behind the semi-terminating identities and the tabulated $\\varepsilon_0$ values. A direct test of the supercongruence conjecture would be to compute the difference in (7) for a tabulated example at a large prime $p$ and verify divisibility by $p^{2m+1}$.","tokens_in":15765,"feed_emoji":"🔢","tokens_out":16137,"duration_ms":133715,"temperature":0.7,"pith_summary":"Ramanujan-like series are hypergeometric sums—infinite sums built from rising factorials—that evaluate to a reciprocal power of π, for example $\\sum_{n\\ge0}\\frac{(\\frac12)_n^5}{(1)_n^5}(820n^2+180n+13)(-1/1024)^n=128/\\pi^2$. This paper proves that any rational series of this type can be completed to a bilateral sum over all integers, and that the completion is exactly $\\frac{v_0\\sqrt{(-1)^m\\chi_0}}{\\pi^m}$ times a trigonometric polynomial of degree $m$ with coefficients $\\alpha_k,\\beta_k$. Sending the completion parameter to $-p/2$ for odd integers $p$ turns the identity into a finite 'semi-terminating' sum equal to a $\\beta$ function times a quadratic-field constant; the paper conjectures that these finite sums obey supercongruences modulo $p^{2m+1}$ for almost all primes, and it verifies the conjecture numerically in all tabulated cases. A few of the supercongruences are proved by reducing them to terminating hypergeometric identities and classical binomial congruences. The paper also records that the quadratic discriminants in its rank-5 table coincide with values of certain L-functions of Hilbert modular forms, and leaves that coincidence unexplained.","feed_headline":"Bilateral twins turn 1/π^k series into finite Fourier sums","feed_subtitle":"Completing the sums over all integers yields new p^{2m+1} supercongruences, some proved.","key_machinery":"The load-bearing object is the function $$f(x)=$e^{{-i\\pi x}}$(\\cos\\pi x)^{2j+1}\\prod_{s_k\\ne1/2}\\frac{\\cos\\pi x-\\cos\\pi s_k}{1-\\cos\\pi s_k}\\sum_{n\\in\\mathbb Z}A(n,x).$$ It is constructed so that the zeros of $\\cos\\pi x-\\cos\\pi s_k$ cancel the poles of the shifted hypergeometric summands, making $f$ holomorphic; it is $1$-periodic because the fractions $s_k$ appear in pairs $s,1-s$; and it grows like $O(e^{(2m+1)\\pi|\\operatorname{Im}x|})$, which forces the Fourier expansion to have only finitely many terms, up to degree $m$. That finite Fourier identity is the step that converts a Ramanujan-like series into a bilateral identity, then into semi-terminating sums and supercongruences. For the proved cases, the same Fourier reduction is supplemented by terminating hypergeometric identities and classical binomial congruences for primes.","core_discovery":"The paper's central claim is identity (5): for the shifted summand $A(n,x)$ of a rational Ramanujan-like series for $1/\\pi^m$, the normalized bilateral sum $$$e^{{-i\\pi x}}$(\\cos\\pi x)^{2j+1}\\prod_{s_k\\ne1/2}\\frac{\\cos\\pi x-\\cos\\pi s_k}{1-\\cos\\pi s_k}\\sum_{n\\in\\mathbb Z}A(n,x) =\\frac{v_0\\sqrt{(-1)^m\\chi_0}}{\\pi^m}\\Bigl(1+\\sum_{k=1}^m(\\alpha_k(\\cos2\\pi kx-1)+\\beta_k\\sin2\\pi kx)\\Bigr),$$ with constants $\\alpha_k,\\beta_k$ conjectured rational. The proof shows the left-hand side is a holomorphic $1$-periodic function of exponential type $2m+1$, so its Fourier expansion terminates at degree $m$. Letting $x\\to-p/2$ with $p$ odd gives the finite semi-terminating identity (6), and the paper conjectures the $p^{2m+1}$ supercongruence (7) for the partial sums $0\\le n\\le(p-1)/2$; it proves several rank-3 and rank-5 instances by certifiable terminating hypergeometric identities.","pith_inferences":["If identity (5) is as general as the proof suggests, the same Fourier-termination argument should extend by analytic continuation to the divergent ($|z_0|>1$) cases in the tables, yielding supercongruence predictions there.","The recorded match between the rank-5 discriminants and Hilbert modular L-function values suggests the quadratic fields in the supercongruences are controlled by the same hypergeometric motives; proving this would require linking the finite Fourier completion to modular-form arithmetic.","The finite Fourier structure of (5) gives an algorithmic rationality test for candidate series: compute the first $2m$ Fourier coefficients numerically and check whether they are rational (or rational times $i$); a failure would rule out a Ramanujan-like series for $1/\\pi^m$.","Computing the coefficients $\\alpha_k,\\beta_k$ exactly for a rank-7 or rank-9 example beyond the tables would test the rationality conjecture and, if it failed, would force a revision of the explicit semi-terminating constants."],"forward_implications":["Every rational Ramanujan-like series for $1/\\pi^m$ has a bilateral completion whose Fourier polynomial has degree exactly $m$, producing new finite identities at half-integer shifts.","The conjectured supercongruences (7) would hold modulo $p^{2m+1}$ for all odd primes outside a finite exceptional set, with sign and power determined by the series' quadratic discriminant and rank.","For integer shifts the same mechanism yields integer-shift supercongruences modulo $p^{2m+1}$, and adding one further term gives strengthened versions modulo $p^{2m+2}$.","The proved rank-3 and rank-5 cases show the route is effective: terminating hypergeometric identities plus classical binomial congruences are enough to certify the pattern in concrete instances.","If the rationality conjecture for the Fourier coefficients $\\alpha_k,\\beta_k$ is correct, the semi-terminating constants in (6) are rational, so the supercongruence conjecture is a statement about genuinely arithmetic numbers."],"supporting_citations":[{"why":"Supplies the x-expansion of Ramanujan-type series used to prove rationality of the Fourier coefficients in the m=1 case.","marker":"[5]"},{"why":"Provides the terminating hypergeometric identities that turn Section 5's supercongruence proofs into finite checks.","marker":"[6]"},{"why":"Gives the classical binomial congruence used to complete the p-adic reductions in the proved examples.","marker":"[11]"},{"why":"Supplies the cubic binomial congruence used in one rank-5 proof.","marker":"[13]"},{"why":"Introduces the integer-shift supercongruence pattern and contributes a congruence used in the final proof.","marker":"[17]"},{"why":"Contains the table of L-function values that match the epsilon_0 column of the paper's rank-5 table.","marker":"[4]"}],"fun_headline_variants":["Bilateral Ramanujan series truncate to finite Fourier sums","Bilateral sums to finite Fourier, some p^{2m+1} congruences proven","Finite Fourier termination yields some proven supercongruences","Bilateral trick turns 1/π^k into finite sums, some congruences proven"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the constants $\\alpha_k$ and $\\beta_k$ in identity (5), which the paper determines by solving a finite linear system numerically, have been correctly identified as exact rational values (or exact values times $i$); the paper states this identification without giving an exact proof for the tabulated examples.","fun_headline_variants_meta":{"raw":{"variants":["Bilateral Ramanujan series truncate to finite Fourier sums","Bilateral sums to finite Fourier, some p^{2m+1} congruences proven","Finite Fourier termination yields some proven supercongruences","Bilateral trick turns 1/π^k into finite sums, some congruences proven"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001326,"raw_usage":{"total_tokens":5364,"prompt_tokens":877,"completion_tokens":4487,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":493,"completion_tokens_details":{"reasoning_tokens":4407}},"tokens_in":493,"tokens_out":4487,"duration_ms":34328,"temperature":1.0,"reasoning_tokens":4407,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:23:41.577566+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take Example 4.4, the rank-9 series for $768/\\pi^4$. Evaluate both sides of (5) at ten values of $x$ to high precision, solve the resulting $8\\times8$ linear system for $\\alpha_1,\\alpha_2,\\alpha_3,\\alpha_4,\\beta_1,\\beta_2,\\beta_3,\\beta_4$, and check whether the outputs are exactly $-25/6, 8/3, -2/3, 1/6, 0,0,0,0$; any discrepancy beyond rounding error would refute the explicit constants behind the semi-terminating identities and the tabulated $\\varepsilon_0$ values. A direct test of the supercongruence conjecture would be to compute the difference in (7) for a tabulated example at a large prime $p$ and verify divisibility by $p^{2m+1}$.","supporting_citations":[{"cited_title":"Guillera , A matrix form of Ramanujan-type series for 1 /π , Gems in Experimental Mathematics: Contemp","cited_arxiv_id":null,"evidence_quote":"Supplies the x-expansion of Ramanujan-type series used to prove rationality of the Fourier coefficients in the m=1 case."},{"cited_title":"Guillera , More hypergeometric identities related to Ramanujan-type serie s, The Ramanujan J","cited_arxiv_id":null,"evidence_quote":"Provides the terminating hypergeometric identities that turn Section 5's supercongruence proofs into finite checks."},{"cited_title":"Morley , Note on the congruence 2 4n ≡ (−1)n(2n)!/ (n!)2, where 2 n + 1 is prime, Ann","cited_arxiv_id":null,"evidence_quote":"Gives the classical binomial congruence used to complete the p-adic reductions in the proved examples."},{"cited_title":"Wolstenholme , On certain properties of prime numbers, Quart","cited_arxiv_id":null,"evidence_quote":"Supplies the cubic binomial congruence used in one rank-5 proof."},{"cited_title":"Zudilin , Ramanujan-type supercongruences, J","cited_arxiv_id":null,"evidence_quote":"Introduces the integer-shift supercongruence pattern and contributes a congruence used in the final proof."},{"cited_title":"Special hypergeometric motives and their $L$-functions: Asai recognition","cited_arxiv_id":"1906.07384","evidence_quote":"Contains the table of L-function values that match the epsilon_0 column of the paper's rank-5 table."}],"review_version":1}