{"id":"d111721d-d1f6-477c-89dc-f4325d3b0b81","arxiv_id":"2610.32672","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Conditionally on the Conrey-Snaith zeta-ratios formula, the count of prime pairs p and p+4 has the predicted Hardy-Littlewood asymptotic with power-saving error bounds.","lead":"This paper shows that, assuming a strong conjecture about averages of the Riemann zeta function, prime pairs separated by four occur exactly as the Hardy-Littlewood conjecture predicts. It also derives explicit error bounds, which improve if the Riemann hypothesis is true.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Factor-of-T error in the diagonal term D_x (Eq. 3.9) invalidates the cancellation central to the proof of Theorem 1.2.","rationale":"The reader's weakest assumption was Hypothesis 2.1, the external unproved input. I find a more immediate internal inconsistency: the diagonal term in Lemma 3.1 and Proposition 5.3 appears to be off by a factor of T. If confirmed, this breaks the central cancellation that converts the two-sided bounds into the correlation estimate for J_W, so Theorem 1.2 does not follow from the displayed arguments as written. The error is concrete and checkable; if it is merely a typo, the paper could be corrected, but as submitted the proof is not sound. Hence the verdict should move from CONDITIONAL to REJECT pending correction.","tokens_in":15638,"tokens_out":22627,"duration_ms":149014,"concrete_test":"Re-derive the diagonal term in Lemma 3.1: substitute v = t/T in I±(x) and isolate n = m, obtaining ∫_0^∞ g±(t/T) dt ∑ Λ(n)^2/n W(n/x)^2 = T H(0) ∑ Λ(n)^2/n W^2. Then check whether the claimed identity I± = D_x ± T J_W holds with D_x as defined in (3.9); if it fails by a factor of T, recompute the constant in Proposition 5.3 and verify whether the subtraction in Proposition 6.2 still yields (6.6) without an uncancelled diagonal term.","verdict_should_be":"REJECT","load_bearing_attack":"The proof's key cancellation in Proposition 6.2 relies on I±(x) = D_x ± T J_W(x) (Lemma 3.1) and on the corresponding diagonal term in Proposition 5.3, both using D_x = T^2 H(0) ∑ Λ(n)^2/n W(n/x)^2 (Eq. 3.9). However, the diagonal contribution from n=m in the expansion of |P_x(t)|^2 is ∫_0^∞ g±(t/T) dt ∑ Λ(n)^2/n W(n/x)^2 = T φ±(0) ∑ ... = T H(0) ∑ ..., since substituting v = t/T yields exactly one factor of T. Thus the defined D_x is too large by a factor of T. Consequently, the identity I± = D_x ± T J_W in Lemma 3.1 is false as written; the correct relation is I± = T H(0) ∑ Λ(n)^2/n W^2 ± T J_W. The same factor-of-T error appears in Proposition 5.3, where the diagonal of ∫ g±(t/T)|Q_x|^2 dt is also claimed to be T^2 H(0) ∑. In the inequality (6.5) of Proposition 6.2, subtracting the (incorrect) D_x would not cancel the true diagonal, leaving an extra (T−1) H(0) ∑ Λ^2/n W^2 term, which is O(x log^2 x) and dominates the claimed error O(x^{−1/16+ε}). Even if this is a typographical slip, the written derivation of the correlation estimate (6.4) is invalid without correcting this constant.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper proves, conditionally on Hypothesis 2.1 (a restricted uniform two-over-two zeta-ratios formula of Conrey and Snaith), the Hardy-Littlewood asymptotic for prime pairs of difference four: (1.3), (1.4), and the quantitative smoothed correlation estimate (1.5) with error O(x^{-1/16+eps}), improved under RH to O(x^{-3/8+eps}). The method uses Mellin inversion retaining crossed zero residues, a differentiated ratios mean, and two nonnegative time weights whose equal diagonal terms and opposite coefficients on the difference-four cross-term yield a sandwich for J_W(x). From this it also derives the B_4 asymptotics, the pi_4 logarithmic asymptotic, and a second-difference identity for the truncated Chebyshev function.","tokens_in":16126,"tokens_out":47505,"duration_ms":592749,"significance":"If the proof were correct, this would be a notable conditional result: the full prime-pair asymptotic for a fixed gap is obtained from a ratios conjecture without assuming RH, the singular-series constant 2C_2 emerges from the computation rather than being inserted, and the error term is quantitative. The treatment of the crossed zero residues and the two-weight bracket is an interesting device, and the supporting lemmas (Mellin truncations, zero-count bounds, singular-series inversion) are plausible in outline. However, two load-bearing algebraic inconsistencies in the displayed formulas (the diagonal term in Sections 3-5 and the definition of Y in Section 2) mean that, as submitted, the arguments do not currently establish the theorems; the paper is not yet in publishable form.","major_comments":[{"comment":"The diagonal term contains a factor-of-T error. Since g_+ and g_- are even, int_0^infty g_+(v) dv = phi_+(0)/2 = H(0)/2, so the n=m contribution to I_+ is (T/2)H(0) sum_n Lambda(n)^2 n^{-1} W(n/x)^2, not T^2 H(0) sum_n ... as in (3.9). The same erroneous diagonal is asserted in (4.9) and (5.7), and in the proof of Proposition 4.2 the displayed equality T G(1) sum ... = T^2 phi(0) sum ... is algebraically false because G(1)=phi(0)/2. Consequently the cancellation in Proposition 6.2 subtracts a term that is not present: with the printed D_x, the two sides of the inequality differ by (T^2 - T/2)H(0) sum_n Lambda(n)^2 n^{-1} W(n/x)^2, which is O(x^2 log^2 x) and, after division by T, dominates the claimed error O(x^{-1/16+eps}). The proof structure can be repaired by replacing T^2 H(0) by T H(0)/2 consistently, but as written (6.4), Theorem 1.2, and Theorem 1.1 are not established.","section":"Section 3, Eq. (3.9); Lemma 3.1; Propositions 4.2 and 5.3","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The displayed definition of Y does not have the property used in the proof. With beta=gamma=delta=eta, direct substitution in (2.1)-(2.2) gives A(alpha,eta;eta,eta)=1 and Y(alpha,eta;eta,eta)=zeta(1+2eta)^2 zeta(1+alpha+eta)^2, not 1, and Y(-eta,-alpha;eta,eta) does not vanish but contains zeta(1) factors. Hence the asserted cancellation leading to int_0^U zeta(1/2+alpha+it)/zeta(sigma_0+it) dt = U + O(U^{1/2+eps}) does not follow from the displayed formula. Since this is the basis for the zero-free strip (2.6) and the zero-count bound (2.8), and those are used in Lemma 6.1, the printed proofs of Lemmas 2.2 and 2.3 are not valid as they stand. The intended Y is presumably the standard ratios-theory expression with zeta factors in the denominator; that definition must be corrected and the subsequent identities rechecked.","section":"Section 2, Eq. (2.2), proof of Lemma 2.2","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The notation H = hat-h^2 is ambiguous; from (3.4) H is the Fourier transform of h^2, not the square of the Fourier transform of h.","section":"Section 3, Eq. (3.2)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The Mellin convolution denoted by tilde-W_2(z) in (4.9) and (5.8) is never defined explicitly; it should be stated to be (tilde-W * tilde-W)(z) = (2 pi i)^{-1} int tilde-W(s) tilde-W(z-s) ds.","section":"Sections 4-5, Eq. (4.9)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The Nyman-Beurling material feels disconnected from the main theorem; it is not used in the proof of Theorems 1.1-1.2 and could be presented as a separate remark with a clear bridge to (8.8).","section":"Section 8","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The two errors identified above are algebraic rather than strategic, and both appear to be repairable. In light of the closing declaration that GPT was used in generating proofs, I recommend that the author be asked to produce a version in which the constants in Sections 2-5 are independently derived, rather than patching individual display equations; the factor-of-T and Y-definition errors repeat across multiple locations, so a systematic recalculation is necessary."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Read this one. The paper does something genuinely new: from a restricted two-over-two zeta-ratios formula it derives the Hardy–Littlewood asymptotic for primes p and p+4 with a quantitative error, without assuming GRH, and it improves the error under RH. The differentiated ratios mean and the retention of crossed zero residues are the real innovations, and the singular series S(4) emerges from a Dirichlet series rather than being inserted. The conditional theorems are honestly stated, and the citation pattern is clean.\n\nThe stress-test note gets part of the diagnosis right but misses half the arithmetic. In Lemma 3.1, the diagonal contribution to |P_x(t)|^2 is T/2 times H(0) times the sum, not T^2 H(0) times the sum: g is even, so the integral from 0 to infinity of g(t/T) is T/2 times its Fourier transform at zero. The stress-test note's proposed correction T H(0) is also off by a factor of two. The same factor error appears in Proposition 4.2, where the line immediately before the display gives TG(1) = T phi(0)/2, and then the next display says T^2 phi(0). So the written equations are false as printed, but the error is consistent and has the shape of a propagating typo. Redefine D_x as (T/2)H(0) times the sum, and the cancellation in Proposition 6.2 goes through unchanged; the final correlation estimate and the main theorem are not affected. This is a referee request, not a refutation.\n\nThe larger caveat is structural: the entire tower rests on Hypothesis 2.1, a very strong uniform ratios conjecture with error O(U^{1/2+eps}) for shifts whose imaginary parts are allowed to grow like U^{1/4}. The paper is honest that every theorem is conditional, but a reader should not underestimate how strong this input is. The closing declaration that GPT was used in generating proofs matters: the diagonal factor slip is exactly the kind of error that needs a human pass, so independent verification should be required before the technical chain is trusted. Section 8, on Nyman–Beurling approximants, is mostly independent and adds little to the main result.\n\nIf I were an editor, I would send this to a number theorist who knows the ratios conjectures. It deserves a serious referee rather than a desk rejection, and after the diagonal correction and a careful independent check, the conditional theorem may well stand. I would not cite the current version until the factor error is fixed and the GPT-generated parts have been verified.","headline":"Serious conditional result on prime pairs of gap four; the diagonal-term factor error flagged by the stress test is real but repairable, and the central mechanism survives.","tokens_in":16641,"tokens_out":10521,"would_cite":false,"duration_ms":86864,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["11M06","11N05","11M26","11N32"],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"This paper proves the Hardy-Littlewood asymptotic for prime pairs of difference four from one restricted zeta-ratios formula, without the Riemann hypothesis.","keywords":["Hardy-Littlewood conjecture","prime pairs","difference four","zeta-ratios conjecture","L-functions ratios","Riemann zeta function","Mellin inversion","Nyman-Beurling criterion"],"falsifier":"The hypothesis has a directly checkable consequence: Lemma 2.2 shows it forbids any zero of $\\zeta(s)$ with real part in the strip $[35/64,37/64]$, so a computation certifying a nontrivial zero inside that strip would refute Hypothesis 2.1 and with it the theorems, and existing zero-location algorithms can do this. Independently, one can evaluate the left-hand side of (2.3) numerically at large $U$ for representative shifts inside the box (2.4), and a deviation from the main terms growing faster than $U^{1/2+\\varepsilon}$ would refute the formula directly.","tokens_in":17175,"feed_emoji":"🔢","tokens_out":19044,"duration_ms":123450,"temperature":0.7,"pith_summary":"This paper shows that the Hardy-Littlewood prediction for prime pairs of distance four — the claim that the number of primes $p\\le x$ with $p+4$ also prime behaves like $2C_2x/(\\log x)^2$ — follows from a single restricted version of the L-functions ratios conjecture, an assumed formula for averages of ratios of zeta values. That formula, Hypothesis 2.1, is the paper's only unproved input: the Riemann hypothesis is not assumed, and even the zero-free strip and zero-count bounds used in the proof are extracted from the formula itself rather than from standard theorems. The argument keeps the crossed zero residues in a Mellin inversion and uses two nonnegative weights whose Fourier transforms agree at zero but are equal and opposite at the frequency that selects the shift four, so the same prime-pair correlation is squeezed between two bounds that agree to the stated error. If the paper is right, the case $k=4$ of Conjecture B is a corollary of one statistical hypothesis about $\\zeta$, with a smoothed correlation error $O_\\varepsilon(x^{-1/16+\\varepsilon})$ that improves to $O_\\varepsilon(x^{-3/8+\\varepsilon})$ under the Riemann hypothesis.","feed_headline":"One zeta-ratios assumption yields the p, p+4 prime-pair asymptotic","feed_subtitle":"The k=4 case of Hardy-Littlewood's Conjecture B drops out of a single hypothesis about averages of zeta ratios.","key_machinery":"The argument is carried by four objects. First, Hypothesis 2.1, the uniform two-over-two zeta-ratios formula (2.3): an integral average of a ratio of four zeta values over a segment of the critical line, asserted to equal an explicit main term plus a swapped-term integral with error $O_\\varepsilon(U^{1/2+\\varepsilon})$; from this one formula the paper derives the zero-free strip $35/64\\le\\Re s\\le37/64$, the bound $\\zeta'/\\zeta(9/16+iv)\\ll\\log^2(2+|v|)$, and the zero count $N_{>}(9/16,T)\\ll_\\varepsilon T^{1/2+\\varepsilon}$ (Lemma 2.2). Second, the two nonnegative time weights $g_\\pm(t)=h(t)^2(1\\pm\\cos(8\\pi t))$, whose Fourier transforms $\\phi_\\pm$ agree at zero and satisfy $\\phi_-=-\\phi_+$ on the side intervals that select the shift $j=4$; the two inequalities of (6.5), each bounding $I_\\pm(x)$ from below by the $|Q_x|^2$ moment minus crossed pole-zero error terms, pinch the difference-four correlation between matching upper and lower bounds. Third, the Mellin decomposition $P_x(t)=Q_x(t)+R_x(t)-Z_x(t)$, which keeps the crossed zero residues $Z_x$ in play and bounds their mixed term with $Q_x$ using the zero-count estimates. Fourth, the singular-series Dirichlet series $D_S(z)=\\sum_k S(k)k^{-z}=\\zeta(z)\\zeta(1+z)A(z)$, whose Fourier inversion collapses the main terms exactly to $S(4)=2C_2$. Section 8 adds the Nyman-Beurling approximant $a_N(d)=\\mu(d)(1-\\log d/\\log N)$; through it the truncated Chebyshev function is expressed and the exact second-difference identity (8.7) is obtained.","core_discovery":"On its own terms, the paper's central claim is Theorem 1.1: under Hypothesis 2.1 — the uniform two-over-two zeta-ratios formula (2.3) with error $O_\\varepsilon(U^{1/2+\\varepsilon})$ for shifts with real parts in $[3/64,5/64]$ and imaginary parts of size at most $U^{1/4}$ — one obtains $\\sum_{n\\le x}\\Lambda(n)\\Lambda(n+4)=2C_2x+o(x)$, the same asymptotic for $A_4(x)$, and $\\pi_4(x)\\sim 2C_2x/(\\log x)^2$, where $C_2$ is the twin-prime constant and $S(4)=2C_2$ is the singular series for difference four. The quantitative engine is Theorem 1.2: for a fixed real smooth weight $W$ supported in $(1,101/100)$, the correlation sum with weight $W(n/x)^2/n$ equals $2C_2\\int_0^\\infty W(u)^2\\,du/u$ with error $O_{W,\\varepsilon}(x^{-1/16+\\varepsilon})$, improved to $O_{W,\\varepsilon}(x^{-3/8+\\varepsilon})$ if the Riemann hypothesis is added. Two corollaries extend the reach: the logarithmically weighted prime-pair sum satisfies $B_4(x)=2C_2\\log x+o(\\log x)$, and the second finite difference of integrals of squared increments of the truncated Chebyshev function equals $4C_2x+o(x)$, tying the pair problem to a finite Nyman-Beurling approximant. Throughout, the zero-free strip (2.6), the zero-count bound (2.8), and the differentiated ratio mean (2.14) are derived from the hypothesis itself rather than assumed.","pith_inferences":["The pinching mechanism is not visibly tied to the number four: the same scheme would handle any fixed even difference $2k$ provided one can construct nonnegative weights whose Fourier transforms agree at zero and are opposite in sign on the side interval selecting that $k$. The paper does not claim this; it is the natural next test of the method.","The unconditional error exponent $-1/16$ and the RH exponent $-3/8$ are set by the zero-count bounds extracted from Hypothesis 2.1, so any future strengthening of those bounds would feed directly into the correlation error; the proof shows this coupling but does not state it as a general principle.","Lemma 2.2 turns a statistical averaging assumption into the concrete arithmetic fact that $\\zeta$ has no zeros with real part in $[35/64,37/64]$. Because this is checkable with existing zero-location algorithms, the hypothesis carries a sharp, falsifiable fingerprint that deserves direct computational scrutiny.","If the same pinching argument could be run for every even difference at once under one uniform ratios formula, the entire Hardy-Littlewood Conjecture B would follow by the same mechanism; this paper proves only the case $k=4$, so the general claim remains an extrapolation."],"forward_implications":["The full Hardy-Littlewood asymptotic for difference four — Conjecture B at $k=4$ — is a corollary of the restricted ratios formula (2.3), with the Riemann hypothesis playing no role in the main theorem.","For fixed smooth weights the correlation error is $O_\\varepsilon(x^{-1/16+\\varepsilon})$ unconditionally and $O_\\varepsilon(x^{-3/8+\\varepsilon})$ under RH, so a qualitative conjecture becomes quantitative control on weighted prime-pair sums.","The logarithmically weighted sum $B_4(x)=2C_2\\log x+o(\\log x)$ and the second-difference asymptotic $F_x(5)-2F_x(4)+F_x(3)=4C_2x+o(x)$ follow, the latter connecting the pair problem to the Nyman-Beurling criterion.","Because $S(4)=2C_2>0$, the results give eventual lower bounds: $B_4(x)\\ge c\\log x$ for every $c<2C_2$, and the relevant second difference is at least $cx$ for every $c<4C_2$, for all sufficiently large $x$."],"supporting_citations":[{"why":"It supplies the two-over-two ratios conjecture that Hypothesis 2.1 restricts to the stated shift box, together with the differentiated ratio mean used as Lemma 2.3.","marker":"[3]"},{"why":"It is the 1923 paper stating the Hardy-Littlewood predictions whose case $k=4$ (Conjecture B) is the asymptotic proved in Theorem 1.1.","marker":"[4]"},{"why":"It provides the partial-fraction estimate for $\\zeta'/\\zeta$, the zero count in unit intervals, the density estimate used at $\\sigma=7/8$, and the polynomial growth of $\\zeta$ on vertical strips.","marker":"[6]"},{"why":"It is the precedent deriving the same truncated-ratio mean from a uniform ratios conjecture, and its Theorem 1.1 serves as the arithmetic template the proof adapts.","marker":"[2]"},{"why":"It is the precedent formally recovering the Hardy-Littlewood prediction from a two-point correlation formula for zeta zeros that keeps its arithmetic terms.","marker":"[5]"},{"why":"It supplies the Nyman-Beurling Dirichlet polynomial used in Section 8 to express the truncated Chebyshev function and derive the second-difference identity.","marker":"[1]"}],"fun_headline_variants":["Zeta-ratios assumption gives Hardy-Littlewood for primes p, p+4","Uniform zeta-ratios formula yields the p, p+4 prime-pair asymptotic","One zeta-ratios hypothesis recovers the p and p+4 Hardy-Littlewood law","Zeta-ratios conjecture implies the twin-prime constant asymptotic for p+4","A single zeta-ratios formula delivers the p, p+4 asymptotic"],"cache_read_input_tokens":0,"weakest_assumption_plain":"Every theorem rests on Hypothesis 2.1, the unproved assertion that an average of a specific ratio of four zeta-function values over an interval of length $U$ equals its two explicit main terms with error $O_\\varepsilon(U^{1/2+\\varepsilon})$, uniformly over the stated box of shifts; if that formula fails, the zero-free strip, the zero count, the differentiated ratio mean, and the prime-pair asymptotics all collapse with it.","fun_headline_variants_meta":{"raw":{"variants":["Zeta-ratios assumption gives Hardy-Littlewood for primes p, p+4","Uniform zeta-ratios formula yields the p, p+4 prime-pair asymptotic","One zeta-ratios hypothesis recovers the p and p+4 Hardy-Littlewood law","Zeta-ratios conjecture implies the twin-prime constant asymptotic for p+4","A single zeta-ratios formula delivers the p, p+4 asymptotic"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000156,"raw_usage":{"cost":0.000156065,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000156065,"upstream_inference_prompt_cost":3.9585e-05,"upstream_inference_completions_cost":0.00011648},"total_tokens":1903,"prompt_tokens":1263,"completion_tokens":640,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":1035,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":538}},"tokens_in":1263,"tokens_out":640,"cost_source":"openrouter_usage","duration_ms":6128,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":538,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-26T06:55:59.893203+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"The hypothesis has a directly checkable consequence: Lemma 2.2 shows it forbids any zero of $\\zeta(s)$ with real part in the strip $[35/64,37/64]$, so a computation certifying a nontrivial zero inside that strip would refute Hypothesis 2.1 and with it the theorems, and existing zero-location algorithms can do this. Independently, one can evaluate the left-hand side of (2.3) numerically at large $U$ for representative shifts inside the box (2.4), and a deviation from the main terms growing faster than $U^{1/2+\\varepsilon}$ would refute the formula directly.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the two-over-two ratios conjecture that Hypothesis 2.1 restricts to the stated shift box, together with the differentiated ratio mean used as Lemma 2.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is the 1923 paper stating the Hardy-Littlewood predictions whose case $k=4$ (Conjecture B) is the asymptotic proved in Theorem 1.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the partial-fraction estimate for $\\zeta'/\\zeta$, the zero count in unit intervals, the density estimate used at $\\sigma=7/8$, and the polynomial growth of $\\zeta$ on vertical strips."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is the precedent deriving the same truncated-ratio mean from a uniform ratios conjecture, and its Theorem 1.1 serves as the arithmetic template the proof adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is the precedent formally recovering the Hardy-Littlewood prediction from a two-point correlation formula for zeta zeros that keeps its arithmetic terms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the Nyman-Beurling Dirichlet polynomial used in Section 8 to express the truncated Chebyshev function and derive the second-difference identity."}],"review_version":1}