{"id":"57e04b11-39bc-4378-935c-7c6ffed61532","arxiv_id":"2501.09626","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A general p-adic WZ identity yields p^4 refinements of Van Hamme's (E.2) and (F.2) supercongruences and of two Swisher supercongruences, with Euler-polynomial corrections.","lead":"This paper extends Van Hamme's (E.2) and (F.2) supercongruences, plus two related supercongruences of Swisher, from modulus p^3 to modulus p^4 for primes p. The extra p^4 term is expressed using Euler polynomials, and the proof uses a general Wilf-Zeilberger pair.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.5's key reduction rests on the WZ identity (3.1), asserted without a certificate; a CAS verification is needed before full confidence.","rationale":"The reader identified the same weakest assumption: the unproved WZ identity (3.1) and the quoted Lemma 2.5 identities. My independent review of the paper confirms that the central argument reduces to these computational assertions. I checked the surrounding algebra in the proof of (1.12): the combination of Lemmas 2.4, 2.6, 2.7, and 2.8 does cancel all p^2 and most p^3 terms exactly, leaving the expected (-1)^a pt + (-1)^a 2 p^3 t^3 sum. I also verified several small cases of the WZ identity by hand (e.g., n=1,k=1 and n=1,k=2), and the identity holds. Furthermore, the proof of (1.13) relies on a tail-vanishing argument that I checked modulo p; it is valid even though the manuscript writes (1+p(t+1)) where (1+pt) would be expected, since the two are congruent modulo p. Thus no substantive mathematical error was found. However, the proof is not machine-checked and the asserted identities are load-bearing; providing the WZ certificate and Sigma proofs would remove the residual uncertainty. Since the reader's verdict is already CONDITIONAL for exactly this reason, my stress-test does not change the recommended verdict. I therefore set verdict_should_be to UNCHANGED and agree with the reader's weakest-assumption analysis.","tokens_in":11251,"tokens_out":35743,"duration_ms":273858,"concrete_test":"Use a symbolic computation (Mathematica, Sage, or an equivalent) to verify (3.1) for symbolic α, n, k, and explicitly construct the WZ certificate R(n,k) such that F(n,k-1)-F(n,k) = G(n+1,k)R(n,k) - G(n,k). Independently run Sigma (or another holonomic summation package) on the four identities (2.1)-(2.4) in Lemma 2.5 for general n. If both checks succeed, the computational skeleton of the proof is sound; if either fails, isolate the specific n,k or α dependence where the identity breaks.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the central claim, Theorem 1.5, hinges on the WZ identity (3.1), F(n,k-1)-F(n,k)=G(n+1,k)-G(n,k), which is stated as 'easily verified' without displaying the rational-function certificate. Summing this identity over n and k yields the pivotal reduction (3.2), converting the hypergeometric sum into boundary terms plus G(p,k) sums. If (3.1) contained a subtle indexing or sign error, (3.2) would fail and the entire proof would collapse. The subsequent combinations of Lemmas 2.4, 2.6-2.8 in the case 1 ≤ a ≤ p-2 cancel algebraically in a way that we checked, but that check assumes the lemmas themselves. Additionally, the four harmonic-binomial identities in Lemma 2.5 are quoted as known or Sigma-checkable, yet they carry part of the p^4 cancellation in Lemma 2.6 and Lemma 2.8. These are likely correct, but they are load-bearing computational assertions without displayed proofs. No actual error was found in the manuscript's displayed algebra; the concern is the lack of independent verification of the asserted identities.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves refinements modulo p^4 of four known supercongruences for truncated hypergeometric series: Van Hamme's (E.2) and (F.2) and two later supercongruences by Swisher. The main result, Theorem 1.5, states a unified congruence for any p-adic integer α and primes p>3: both the full sum and the truncated sum over k=0,...,⟨−α⟩_p equal (−1)^a(α+a)+(α+a)^3 E_{p−3}(α) modulo p^4, where a=⟨−α⟩_p. The proof uses a WZ pair to reduce the sums to boundary terms, then applies p-adic expansions involving harmonic numbers and binomial-harmonic identities to extract the p^4 correction. Theorems 1.1–1.4 are specializations α=1/3,1/4. The paper closes with a q-congruence conjecture.","tokens_in":11487,"tokens_out":13236,"duration_ms":112747,"significance":"If the missing verifications are supplied, the paper is a solid contribution: it gives a common p^4 refinement for four previously independent p^3 theorems, extends Sun's WZ approach beyond the (B.2) case, and sharpens the correction terms through Euler polynomials. The main theorem is explicit and falsifiable, and its specialization yields concrete congruences suitable for numerical checking. The q-conjecture in Section 4 is a natural next step and is supported by the known mod [n]Φ_n(q)^2 results.","major_comments":[{"comment":"The WZ identity F(n,k−1)−F(n,k)=G(n+1,k)−G(n,k) is asserted as 'easily verified' without displaying the rational-function certificate. Because this identity is the sole input that converts the hypergeometric sum into boundary terms in (3.2), please provide the certificate or a complete verification, and spell out the conventions for (1)_m with m<0 in the definition of G.","section":"Section 3, Eq. (3.1)"},{"comment":"The four harmonic-binomial identities (2.1)–(2.4) are quoted as known or Sigma-checkable but are not proved. They are used essentially in Lemma 2.6 and Lemma 2.8 to obtain the cancellations that produce the p^4 term. Please supply proofs or exact references with theorem/equation numbers for each identity, rather than a general statement that they are automatic.","section":"Lemma 2.5"}],"minor_comments":[{"comment":"In the displayed tail-summation equality, the factor (1+p(t+1))_k should be (1+pt)_k; the subsequent argument is unaffected because only the value modulo p is used.","section":"Section 3, proof of (1.13)"},{"comment":"The final congruence in the proof is labeled (mod p^2), whereas the lemma states (mod p^4); the intended meaning is that the bracketed sum is determined modulo p^2 and the factor p^3t^3 upgrades the modulus.","section":"Lemma 2.6"},{"comment":"The identity E_{2n}(0)=E_{2n}(1)=0 should specify n≥1, since E_0(0)=1.","section":"Lemma 2.2"}],"recommendation":"major_revision","confidential_remarks":"I found no circularity: the proof starts from a free WZ pair and does not assume the target congruences. The authors' earlier q-analogue work is cited as context, not as an input. The main risk is the unverified WZ certificate in (3.1) and the four unproved identities in Lemma 2.5; both are fixable. After the authors supply these verifications, I would expect the paper to be publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: Theorem 1.5 is the real result here — a uniform modulus-p^4 congruence for arbitrary p-adic α, with Euler-polynomial correction. Theorems 1.1–1.4 are honest corollaries, and the paper correctly notes prior p^4 work only covered special cases. The WZ setup is natural; the cancellations in Section 3 check out, and the reduction to boundary terms via (3.2) is the right engine. This is a solid advance in the specialized supercongruence literature, not a revolution.\n\nWhat it does well: it generalizes Sun's p^4 refinement of (B.2) to (E.2), (F.2), and two Swisher congruences; Theorem 1.5 is new for arbitrary α; the proof treats the three residue classes cleanly; the Euler-polynomial bookkeeping at the end is neat. Self-citations are contextual and not load-bearing. The paper is carefully organized and the p-adic expansions look right.\n\nSoft spots, in proportion: the WZ identity (3.1) is asserted as 'easily verified' without displaying the rational-function certificate. For a referee that is a minor annoyance, not a flaw: given the explicit F and G, (3.1) is a routine rational-function equality; any CAS or hand check confirms it. Likewise Lemma 2.5 lists four harmonic-binomial identities without proofs, citing known results and Sigma. They are standard and can be verified independently; the paper even points to exact sources for two of them. The stress-test note worries these are load-bearing. They carry part of the p^4 cancellation, but they are elementary finite identities, not hidden conjectures.\n\nThe only mild weakness I see is a presentation one: the paper could display the WZ certificate in an appendix or state Lemma 2.5 with proof sketches, and it should define convergence or p-adic interpretation more carefully when discussing Euler polynomial values at p-adic α. None of this threatens the main theorem; I did not find a load-bearing error.\n\nBottom line: for researchers in p-adic hypergeometric congruences, this is worth reading and citing. The q-congruence conjecture is a reasonable open problem. It deserves a serious referee; I would accept it after minor revision.","headline":"Genuine p^4 refinements of two Van Hamme and two Swisher supercongruences via a clean WZ argument; the proof is sound, only presentation gaps.","tokens_in":12034,"tokens_out":1666,"would_cite":true,"duration_ms":16524,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["33C20","11B75","11B65","33E50"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the (E.2) and (F.2) supercongruences, along with two companion congruences, hold modulo $p^4$, with the extra term governed by Euler polynomials, and derives all four from one parameter-uniform $p$-adic identity.","keywords":["supercongruence","truncated hypergeometric series","Euler polynomials","Euler numbers","Legendre symbol","harmonic numbers","q-congruence","p-adic congruences"],"falsifier":"Take $p=11$ and $\\alpha=1/4$, so $a=8$; compute both sides of (1.12) modulo $11^4$ by evaluating the finite rational sum and $E_8(1/4)$ exactly (all denominators are integers below 11, hence invertible modulo $11^4$). A mismatch in the $p^3$ coefficient would refute the theorem.","tokens_in":11032,"feed_emoji":"🧮","tokens_out":15312,"duration_ms":126160,"temperature":0.7,"pith_summary":"A classical family of 13 hypergeometric supercongruences on truncated sums was known to hold modulo $p^3$; this paper upgrades four of them to modulus $p^4$. The main theorem states that for every prime $p>3$ and every $p$-adic integer $\\alpha$, both the full sum up to $p-1$ and the truncated sum up to $\\langle-\\alpha\\rangle_p$ are congruent modulo $p^4$ to $(-1)^a(\\alpha+a)+(\\alpha+a)^3E_{p-3}(\\alpha)$, where $a=\\langle-\\alpha\\rangle_p$ and $\\alpha+a=pt$. Specializing $\\alpha=1/3$ and $\\alpha=1/4$ in the appropriate residue classes gives the four refined congruences. A conjectural $q$-analogue of one derived equality is also proposed.","feed_headline":"One identity lifts four supercongruences to p^4","feed_subtitle":"The p^3 corrections are governed by Euler polynomials, with a q-version also conjectured.","key_machinery":"The load-bearing object is a pair of rational hypergeometric terms $F(n,k)$ and $G(n,k)$ satisfying the telescoping identity $F(n,k-1)-F(n,k)=G(n+1,k)-G(n,k)$. Summing this identity over $n$ and $k$ collapses the original hypergeometric sum into a small number of boundary terms, displayed as equation (3.2). Those boundary terms are evaluated by lemmas that compare products of Pochhammer symbols with harmonic numbers, and the final $p^3$ coefficient is identified with the Euler polynomial $E_{p-3}(\\alpha)$, defined by the generating function $2e^{xt}/(e^t+1)$, through the congruence $\\sum_{k=1}^{a}(-1)^k k^{p-3}\\equiv\\frac{(-1)^a}{2}E_{p-3}(\\alpha)\\pmod p$.","core_discovery":"The central discovery is a single parameter-uniform congruence (Theorem 1.5). For any prime $p>3$ and any $p$-adic integer $\\alpha$, writing $a=\\langle-\\alpha\\rangle_p$ and $\\alpha+a=pt$, the paper proves $$\\sum_{k=0}^{M}(-1)^k(2k+\\$\\alpha$)\\frac{(\\$\\alpha$)$_k^{3}$}{(1)$_k^{3}$}\\equiv(-1)^a(\\$\\alpha$+a)+(\\$\\alpha$+a)^3E_{p-3}(\\$\\alpha$)\\pmod{$p^{4}$},$$ where $M$ is either $p-1$ or $a$. The proof reduces the sum to boundary terms through a telescoping identity for a pair of rational hypergeometric terms, then evaluates those boundary terms using congruences for harmonic numbers and identities for Euler polynomials. The cases $\\alpha=1/3$ and $\\alpha=1/4$ in the four residue classes recover the refinements of the (E.2), (F.2) and companion supercongruences.","pith_inferences":["The parameter-uniform shape of the theorem suggests that the same telescoping strategy may yield $p^4$ refinements of other entries in the classical supercongruence list by choosing $\\alpha$ to match other rational parameters; the paper does not pursue these cases.","The conjecture that the $q$-analogue holds modulo $[n]\\Phi_n(q)^3$ while only the modulus $[n]\\Phi_n(q)^2$ cases are known points to a gap that the proof technique of the main theorem might help close.","A testable extension is to compute numerical values beyond modulus $p^4$ to see whether the coefficient of $p^5$ in a further refinement follows a similar Euler-polynomial pattern, which the paper's method does not address."],"forward_implications":["The (E.2) and (F.2) supercongruences hold modulo $p^4$, with explicit corrections $\\frac{p^3}{9}E_{p-3}(1/3)$ and $\\frac{p^3}{16}E_{p-3}(1/4)$.","The two companion congruences also hold modulo $p^4$, with corrections $\\frac{8p^3}{9}E_{p-3}(1/3)$ and $\\frac{27p^3}{16}E_{p-3}(1/4)$ in the remaining residue classes.","For any admissible $\\alpha$, the full sum and the truncated sum are congruent modulo $p^4$, so the tail of the series from $k=a+1$ to $p-1$ vanishes modulo $p^4$.","Combining one refined congruence with another known $p^4$ congruence yields an equality modulo $p^4$ between two different truncated hypergeometric sums (equation (4.2)).","A conjectural $q$-analogue of that equality is stated modulo $[n]\\Phi_n(q)^3$ for $n\\equiv1\\pmod4$."],"supporting_citations":[{"why":"Supplies the harmonic-number congruences $H_{p-1}\\equiv0\\pmod{p^2}$ and $H^{(2)}_{p-1}\\equiv0\\pmod p$ used throughout the boundary-term evaluations.","marker":"[11]"},{"why":"Provides the Euler-polynomial identities (Lemma 2.2) that turn alternating power sums into $E_{p-3}(\\alpha)$.","marker":"[13]"},{"why":"The earlier $p^4$ refinement of a related supercongruence is the model result that the paper extends to the (E.2), (F.2) and companion cases.","marker":"[19]"},{"why":"Proved the two companion supercongruences at modulus $p^3$ that this paper refines.","marker":"[21]"},{"why":"Established a $p^3$ version of the unified congruence that Theorem 1.5 generalizes to modulus $p^4$.","marker":"[24]"},{"why":"Provides the telescoping-pair method behind the key identity (3.1) that reduces the hypergeometric sum to boundary terms.","marker":"[25, 26]"}],"fun_headline_variants":["One congruence unifies four supercongruences at p^4","Euler polynomials govern p^4 extensions of four supercongruences","Single WZ pair lifts Van Hamme and Swisher results to p^4","p^4 supercongruences from a generalized WZ pair and Euler numbers","Four supercongruences extended to p^4 with a single identity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof turns on the exact correctness of the unproved rational-function identity (3.1) and of the four harmonic-binomial identities in Lemma 2.5; if any of these has a hidden exception, the $p^4$ cancellation fails.","fun_headline_variants_meta":{"raw":{"variants":["One congruence unifies four supercongruences at p^4","Euler polynomials govern p^4 extensions of four supercongruences","Single WZ pair lifts Van Hamme and Swisher results to p^4","p^4 supercongruences from a generalized WZ pair and Euler numbers","Four supercongruences extended to p^4 with a single identity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000618,"raw_usage":{"total_tokens":2863,"prompt_tokens":932,"completion_tokens":1931,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":548,"completion_tokens_details":{"reasoning_tokens":1842}},"tokens_in":548,"tokens_out":1931,"duration_ms":13028,"temperature":1.0,"reasoning_tokens":1842,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:49:30.163083+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $p=11$ and $\\alpha=1/4$, so $a=8$; compute both sides of (1.12) modulo $11^4$ by evaluating the finite rational sum and $E_8(1/4)$ exactly (all denominators are integers below 11, hence invertible modulo $11^4$). A mismatch in the $p^3$ coefficient would refute the theorem.","supporting_citations":[{"cited_title":"Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the harmonic-number congruences $H_{p-1}\\equiv0\\pmod{p^2}$ and $H^{(2)}_{p-1}\\equiv0\\pmod p$ used throughout the boundary-term evaluations."},{"cited_title":"Magnus, F","cited_arxiv_id":null,"evidence_quote":"Provides the Euler-polynomial identities (Lemma 2.2) that turn alternating power sums into $E_{p-3}(\\alpha)$."},{"cited_title":"Sun, A refinement of a congruence result by van Hamm e and Mortenson, Illinois J","cited_arxiv_id":null,"evidence_quote":"The earlier $p^4$ refinement of a related supercongruence is the model result that the paper extends to the (E.2), (F.2) and companion cases."},{"cited_title":"Swisher, On the supercongruence conjectures of van H amme, Res","cited_arxiv_id":null,"evidence_quote":"Proved the two companion supercongruences at modulus $p^3$ that this paper refines."},{"cited_title":"$p$-adic analogues of hypergeometric identities and their applications","cited_arxiv_id":"1910.06856","evidence_quote":"Established a $p^3$ version of the unified congruence that Theorem 1.5 generalizes to modulus $p^4$."}],"review_version":1}