{"id":"4bc76fd9-a6ea-4db9-93fe-f17d56bf2b59","arxiv_id":"1908.08613","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A randomly sieved set that mimics the primes has largest gap almost surely equal to g((2e^{-γ}+o(1)) log log x), with g defined through an extremal interval sieve problem; the same framework turns Hardy-Littlewood conjectures into lower bounds on prime gaps.","lead":"The authors introduce a random sieve model of the primes and prove that, in this model, the largest prime gap up to x is governed by an extremal interval sieve quantity. The paper also shows that any set satisfying the Hardy-Littlewood conjectures must contain long gaps, connecting prime gaps to sieve theory and exceptional zeros.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem 1.1 is sound and Conjecture 1.2 is a clearly labeled heuristic, not a load-bearing correctness claim.","rationale":"The reader identified the R-to-primes transfer as the weakest assumption, and I agree that this is the main limitation of the paper's heuristic claim. However, I do not regard that transfer as a load-bearing correctness concern: it is explicitly separated as Conjecture 1.2, and the rigorous Theorem 1.1 stands independently. I checked the natural places where an internal flaw might hide: the dependence on the extremal sieve function W_y, the large-deviation estimates in Section 5, and the lower-bound construction in Section 7. The lower-bound step initially looks as if it requires the small-prime residue tuple to be an exact minimizer, which would have tiny probability, but translation transitivity over the product of residue class groups shows that for every realized tuple some interval shift realizes the global minimum W_y. This makes the proof coherent. The cited bound (1.12) is standard and even if its constants were improved, the theorem's g-based formulation is robust. The paper is honest about the conjectural nature of the prime-gap prediction, and its numerical comparison with existing prime-gap tables is presented as motivation rather than evidence. Accordingly, I do not see a reason to move the verdict away from ACCEPT.","tokens_in":29720,"tokens_out":26867,"duration_ms":295662,"concrete_test":"Independently re-derive Lemma 5.2, the Bennett-inequality estimate for w2 < p <= w3, and verify the arithmetic in (5.9) and (5.10), including the bound on Mt/sigma^2 used to justify the O(x^{-100}) probability. This is the most delicate large-deviation step supporting the Borel-Cantelli argument in Theorem 7.1; if the claimed probability fails by a polynomial factor, the upper-bound proof would need repair.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing internal concern in the paper's rigorous core. The proof of Theorem 1.1 is long, but the probabilistic estimates in Lemmas 5.1 through 6.5 are mutually consistent, and the conditional-minimizer step in Theorem 7.2 is not a hidden rare-event assumption: because the torus of residue tuples is acted on transitively by translations, every realization of the small-prime residues has some interval shift whose effective residue pattern realizes the global extremal value W_y. Thus the lower bound does not require an unlikely small-prime configuration. The cited interval-sieve bound (1.12) is standard Iwaniec linear-sieve input, and Theorem 1.1 is formulated through g so an improved W_y would refine, not invalidate, the statement. The only substantive leap is Conjecture 1.2, which transfers the same g to the primes; the paper explicitly labels this as a conjecture, and the model-to-primes transfer is a recognized limitation rather than an unstated assumption. The numerical tension between the conjectured lower bound and current prime-gap data is real but asymptotic, and it does not bear on the correctness of Theorem 1.1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a new probabilistic model R of the primes: an integer n>e^2 belongs to R if it survives sieving by one uniformly random residue class modulo each prime p≤z(n), where z(n) is chosen so that Θ_{z(n)}^{-1} is approximately log n, i.e. z(n)∼n^{1/e^γ}. The central result, Theorem 1.1, states that with probability one the largest gap G_R(x) in R up to x satisfies g((ξ−o(1))log_2 x)≤G_R(x)≤g((ξ+o(1))log_2 x), where ξ=2e^{-γ} and g is the inverse of the extremal interval-sieve function W_y log y. The proof passes through five checkpoint sieving ranges (Lemmas 5.1–6.5), Borel–Cantelli arguments in Section 7, and a deterministic translation argument for the lower bound. The paper also proves a uniform Hardy–Littlewood type asymptotic for R (Theorem 1.3), a Riemann-hypothesis analogue (Theorem 1.4), deterministic large-gap consequences from Hardy–Littlewood type assumptions (Theorems 1.5 and 1.6), and a conditional theorem showing that exceptional zeros force W_y to be small infinitely often (Theorem 2.2). Conjecture 1.2 transfers the random-model gap asymptotic to the actual primes.","tokens_in":29891,"tokens_out":5231,"duration_ms":56174,"significance":"The paper is significant for several reasons. The model R is conceptually new: it is a sieve-based random set that automatically reproduces the Hardy–Littlewood singular series while remaining amenable to rigorous moment analysis, unlike the earlier Cramér and Granville models. Theorem 1.1 is a genuine almost-sure statement about a natural random sifted set, proved with substantial first- and second-moment estimates rather than heuristics. The formulation through the deterministic quantity W_y makes the connection between prime gaps, Jacobsthal's function, and the interval sieve problem precise, and it identifies the exact scale ξ log_2 x predicted by Granville's heuristic. The paper is also admirably honest: Conjecture 1.2 is explicitly labeled a conjecture, the dependence on the unknown behavior of W_y is encoded in the function g, and the exceptional-zeros connection is stated conditionally. If the transfer conjecture is correct, the paper yields the prediction G_P(x)∼g(ξ log_2 x), refining Cramér's model and matching Granville's lower-bound heuristic.","major_comments":[],"minor_comments":[{"comment":"The abstract says a random residue class is selected for every prime modulus below a 'specific bound', but in (1.10) the cutoff z(n) depends on n. The wording should say 'a bound depending on n' to avoid confusion.","section":"Abstract and §1.3"},{"comment":"In the displayed statement of Theorem 7.1 the expression g((1+ε)ξ(log x 2)^2) appears to be a typesetting artifact; it should presumably read g((1+ε)ξ(log_2 x)^2). Please correct this in the final version.","section":"Theorem 7.1"},{"comment":"The table at the end of Section 1.6 lists the same asymptotic for Granville's model G as for R, but Section 2.7 gives only a short proof sketch for G. A footnote in the table stating that the G row is supported by a sketch rather than a full theorem would help readers distinguish rigorous results from heuristic ones.","section":"§1.6 table and §2.7"},{"comment":"In the proof of Theorem 1.4 the parameter m is used both for the dyadic decomposition and for the exponent in x=2^m; renaming one of the two indices would improve readability.","section":"Proof of Theorem 1.4"}],"recommendation":"accept","confidential_remarks":"I see no grounds for rejection. The theorem-level claims for the model R appear fully proved, and the transfer to the primes is clearly labeled as a conjecture. The only caution I would convey to the editor is that the headline prediction for actual prime gaps rests on Conjecture 1.2, which is a heuristic; the numerical tension with current data for x≤10^18 noted in Section 1.1 is real but asymptotic and does not undermine the rigorous content of the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe one thing to know: this paper introduces a genuinely new probabilistic model of the primes, and it rigorously solves the largest-gap problem for that model. R is not Cramer's independent set or Granville's block model. Each n is sieved by a random residue class modulo every p <= z(n), with z(n) ~ n^{1/e^gamma} chosen so the surviving density mimics the primes. The dependence structure is the whole point, and the authors handle it.\n\nWhat's new and what's well done. Theorem 1.1 is the centerpiece: almost surely, G_R(x) lies between g((xi-o(1)) log_2 x) and g((xi+o(1)) log_2 x), where xi = 2e^{-gamma} and g inverts W_y log y, W_y being the minimal survivor count in an interval sieve. This connects prime gaps (in the model) to the classical extremal interval-sieve problem, and the proof—via five checkpoint sieving lemmas, Azuma, Bennett, and a combinatorial large-prime argument—is detailed and holds together. I specifically checked the conditional-minimizer step in Theorem 7.2: because translations act transitively on the residue torus, the small-prime configuration can always be shifted to the extremal one, so no rare event is hiding there. Theorems 1.3 and 1.4 (HL and RH-analog for R) follow from first/second moments and are what the model was designed to satisfy, which is fine; the largest-gap theorem is where the real work lies. The deterministic results (Theorems 1.5, 1.6) and the exceptional-zeros corollary (Theorem 2.2) are honest conditional statements.\n\nSoft spots, in proportion. The headline for the primes is Conjecture 1.2, and the paper says so plainly: the transfer from R to P is a heuristic, not a theorem. Second, W_y is not known asymptotically; (1.12) only brackets it. So even the model's prediction is a range, and the folklore g(u) ~ u would give G_P(x) ~ xi log^2 x ≈ 1.12 log^2 x, while the data to 10^18 sit at 0.92. That gap between heuristic and data is real, but it is asymptotic and the paper does not pretend otherwise. Third, the hypotheses of Theorems 1.5/1.6 are strong—uniform or averaged HL over tuples of size up to log x or beyond—and, as the authors note, Elsholtz's examples make the pointwise version fail for some admissible tuples of that size. So those results are conditional on assumptions that may not hold for the actual primes. None of this touches the rigorous results about R.\n\nWho it's for: anyone working on prime gaps, sieves, or probabilistic models of primes. It would make a good reading-group paper. It deserves a serious referee; the core theorems look right, the conjectural parts are labeled, and the framework will be referenced.\n\nRecommendation: send to peer review.","headline":"A genuinely new random-sieve model whose largest-gap theorem is rigorously reduced to the interval-sieve problem; the prime-gap prediction is a clearly labeled conjecture, not a theorem.","tokens_in":30469,"tokens_out":5995,"would_cite":true,"duration_ms":56784,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N05","11B83"],"pacs":[],"model":"deepseek-v4-flash","headline":"A random residue-class sieve predicts that the largest prime gap below $x$ is governed by the interval-sieve inverse $g$ at the scale $2e^{-\\gamma}(\\log x)^2$.","keywords":["primes","prime gaps","probabilistic models","random sieve","interval sieve","Hardy-Littlewood conjectures","extremal sieve bounds","largest gaps"],"falsifier":"For some explicit large $x$, find a prime-free interval of length greater than $g((\\xi+\\varepsilon)(\\log x)^2)$, with $g$ computed from the extremal problem (1.11): Theorem 1.1 says the random model almost surely has no such gap, so such a prime gap would disprove Conjecture 1.2 in the upper direction; conversely, showing that for infinitely many $x$ no prime-free interval of length $g((\\xi-\\varepsilon)(\\log x)^2)$ exists would disprove it in the lower direction.","tokens_in":29477,"feed_emoji":"🔢","tokens_out":19371,"duration_ms":164291,"temperature":0.7,"pith_summary":"The paper proposes a new probabilistic model of the primes: an integer $n$ survives only if it avoids a randomly chosen residue class modulo each prime $p\\le z(n)$, where the cutoff $z(n)\\sim n^{1/e^\\gamma}$ is tuned so that the survival density is about $1/\\log n$. Working rigorously inside this model, the authors prove that the largest gap below $x$ is almost surely bracketed by $g((2e^{-\\gamma}-o(1))(\\log x)^2)$ and $g((2e^{-\\gamma}+o(1))(\\log x)^2)$, where $g$ inverts the extremal interval-sieve function $W_y\\log y$. They conjecture the same bracket for the actual primes, which would make the maximal prime gap asymptotic to $g(2e^{-\\gamma}(\\log x)^2)$, and to $2e^{-\\gamma}(\\log x)^2$ under a folklore bound on the interval sieve. The model also satisfies the singular-series asymptotics to near-square-root error, and the paper proves a partial converse: any set obeying such uniform asymptotics must contain large gaps. The payoff is that the largest-gap problem is reduced to a deterministic extremal sieve question while the model still reproduces the expected local prime statistics.","feed_headline":"Random sieve sets prime gaps near 1.1229 (log x)^2","feed_subtitle":"A residue-class sieve reproduces prime tuples and fixes the largest-gap scale; the same bracket is conjectured for actual primes.","key_machinery":"The load-bearing object is the random sieved set $R=\\{n>e^2:n\\in S_{z(n)}\\}$, where $S_z$ is what remains after deleting one random residue class modulo each prime $p\\le z$, and $z(t)$ is the largest prime with $\\prod_{p\\le z(t)}(1-1/p)^{-1}\\le\\log t$; the prime number theorem gives $z(t)\\sim t^{1/e^\\gamma}$. Large-gap statistics are controlled by the extremal interval-sieve quantity $W_y=\\min_{\\{a_p\\}}|[0,y]\\cap S_{(y/\\log y)^{1/2}}|$, the minimum number of survivors of an interval of length $y$ after optimally choosing one deleted residue class at each prime up to $(y/\\log y)^{1/2}$, and by its inverse $g(u)=\\max\\{y:W_y\\log y\\le u\\}$. The proof transfers probability estimates for $|[0,y]\\cap S_w|$ across five sieving ranges, using an upper-bound sieve, the large sieve with concentration inequalities, a martingale step, and a random-graph expansion for large primes.","core_discovery":"On the paper's own terms, the central discovery is that a prime model built by deleting one uniformly random residue class for every prime $p\\le z(n)$ admits a rigorous two-sided largest-gap law. For $g(u)=\\max\\{y:W_y\\log y\\le u\\}$ and $\\xi=2e^{-\\gamma}$, one has almost surely $$g((\\xi-o(1))(\\log x)^2)\\le G_R(x)\\le g((\\xi+o(1))(\\log x)^2)$$ as $x\\to\\infty$. The same random set obeys the strong form of the singular-series conjecture for admissible tuples of size up to $\\log^c x$ with near-square-root error, and the authors conjecture, as Conjecture 1.2, that the same gap bracket holds for the primes. They further prove deterministic conversions: uniform singular-series asymptotics for any set force large gaps, with averaged asymptotics forcing gaps above $g((c\\xi-o(1))(\\log x)^2)$, and they show that the existence of exceptional zeros would force $W_y$ infinitely often to be as small as $o(y/\\log y)$, making the normalized largest gap unbounded.","pith_inferences":["The paper suggests a concrete numerical target: if the folklore $g(u)\\sim u$ holds, $G_P(x)/(\\log x)^2$ should tend to $2e^{-\\gamma}\\approx1.1229$; the $10^{18}$ record ratio is about $0.9206$, so the approach to the asymptotic may be slow and worth tracking computationally.","The same construction applies to admissible tuples: one can define an extremal survivor statistic $W_y^{(H)}$ for shifted tuples and expect a $g_H$ law for maximal gaps between twin primes or other constellations, a direction the paper lists as an open problem.","The connection to exceptional zeros means the sharp form of the prime-gap prediction cannot be separated from the exceptional-zeros problem: proving $G_P(x)=O((\\log x)^2)$ under the model transfer would implicitly rule out the exceptional-zero scenario, and conversely a slow decay of exceptional zeros would push the largest gaps above any fixed multiple of $(\\log x)^2$.","The deterministic converses imply that future work establishing the singular-series asymptotics in the required uniform range would immediately yield unconditional gap lower bounds of the same shape for the primes, with no probabilistic model needed."],"forward_implications":["Almost surely, $G_R(x)$ lies between $g((\\xi-o(1))(\\log x)^2)$ and $g((\\xi+o(1))(\\log x)^2)$; with the current bounds on $g$, this ranges between $\\xi(\\log x)^2$ and $\\xi(\\log x)^2\\log_2 x/(2\\log_3 x)$.","Conjecture 1.2 transfers the same bracket to the primes, so $G_P(x)\\sim g(\\xi(\\log x)^2)$; under the folklore $g(u)\\sim u$, this becomes $G_P(x)\\sim 2e^{-\\gamma}(\\log x)^2\\approx1.1229(\\log x)^2$.","Theorem 1.3 shows that the random model obeys the strong singular-series asymptotics with error $O(x^{1/2+o(1)})$ for admissible tuples of size up to $(\\log x)^{1/2}$ in the stated range, so the model is consistent with both the Riemann Hypothesis analog and prime $k$-tuple statistics.","Any set satisfying the uniform singular-series asymptotic (1.17) must have gaps $\\gg \\kappa(\\log x)^2/\\log_2 x$, and the averaged version forces gaps above $g((c\\xi-o(1))(\\log x)^2)$.","If exceptional zeros exist, $W_y$ is infinitely often $o(y/\\log y)$, and then $G_R(x)/(\\log x)^2\\to\\infty$ almost surely; under Conjecture 1.2 the same unboundedness would hold for prime gaps."],"supporting_citations":[{"why":"Supplies the classic independent-probability prime model whose largest gap scales like $(\\log x)^2$, the baseline the new model is designed to correct.","marker":"[6]"},{"why":"Introduces the small-prime-bias modification from which the constant $\\xi=2e^{-\\gamma}$ enters the largest-gap scale.","marker":"[16]"},{"why":"Provides the linear-sieve theory used for the lower bound on $W_y$ in the fundamental estimate (1.12).","marker":"[12]"},{"why":"Proves the linear-sieve estimate underlying the same lower bound for $W_y$.","marker":"[19]"},{"why":"Gives the upper-bound sieve and standard sieve estimates used throughout the sieving lemmas.","marker":"[28]"},{"why":"States the singular-series asymptotics that motivate the random-sieve interpretation of the model and drive Theorem 1.3.","marker":"[17]"},{"why":"Gives the earlier result that uniform singular-series asymptotics imply an exponential gap distribution, which Theorems 1.5 and 1.6 extend.","marker":"[14]"},{"why":"Supplies the prime number theorem in arithmetic progressions used to build small $W_y$ under exceptional zeros.","marker":"[13]"}],"fun_headline_variants":["Random sieve sets largest prime gap scale at (log x)^2","Probabilistic primes: gaps fixed by random residue deletion","New model rigorous on prime gaps: (log x)^2 law","Sieve model predicts prime gaps, rigorous bounds","Random residue sieve yields sharp prime gap bounds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the primes' local structure is faithfully modeled by independent random residue classes up to the cutoff $z(n)\\sim n^{1/e^\\gamma}$; the rigorous results for the random model do not need this, but Conjecture 1.2 transfers them to the actual primes only if the premise holds.","fun_headline_variants_meta":{"raw":{"variants":["Random sieve sets largest prime gap scale at (log x)^2","Probabilistic primes: gaps fixed by random residue deletion","New model rigorous on prime gaps: (log x)^2 law","Sieve model predicts prime gaps, rigorous bounds","Random residue sieve yields sharp prime gap bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000241,"raw_usage":{"total_tokens":1494,"prompt_tokens":887,"completion_tokens":607,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":503,"completion_tokens_details":{"reasoning_tokens":527}},"tokens_in":503,"tokens_out":607,"duration_ms":6182,"temperature":1.0,"reasoning_tokens":527,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:34:29.949855+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For some explicit large $x$, find a prime-free interval of length greater than $g((\\xi+\\varepsilon)(\\log x)^2)$, with $g$ computed from the extremal problem (1.11): Theorem 1.1 says the random model almost surely has no such gap, so such a prime gap would disprove Conjecture 1.2 in the upper direction; conversely, showing that for infinitely many $x$ no prime-free interval of length $g((\\xi-\\varepsilon)(\\log x)^2)$ exists would disprove it in the lower direction.","supporting_citations":[{"cited_title":"Cram´ er,Some theorems concerning prime numbers, Acta Arith","cited_arxiv_id":null,"evidence_quote":"Supplies the classic independent-probability prime model whose largest gap scales like $(\\log x)^2$, the baseline the new model is designed to correct."},{"cited_title":"Granville, Harald Cram´ er and the distribution of prime numbers, Harald Cram´ er Sym- posium (Stockholm, 1993)","cited_arxiv_id":null,"evidence_quote":"Introduces the small-prime-bias modification from which the constant $\\xi=2e^{-\\gamma}$ enters the largest-gap scale."},{"cited_title":"Friedlander and H","cited_arxiv_id":null,"evidence_quote":"Provides the linear-sieve theory used for the lower bound on $W_y$ in the fundamental estimate (1.12)."},{"cited_title":"Iwaniec, On the error term in the linear sieve , Acta Arith","cited_arxiv_id":null,"evidence_quote":"Proves the linear-sieve estimate underlying the same lower bound for $W_y$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the upper-bound sieve and standard sieve estimates used throughout the sieving lemmas."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the singular-series asymptotics that motivate the random-sieve interpretation of the model and drive Theorem 1.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the earlier result that uniform singular-series asymptotics imply an exponential gap distribution, which Theorems 1.5 and 1.6 extend."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the prime number theorem in arithmetic progressions used to build small $W_y$ under exceptional zeros."}],"review_version":1}