{"id":"f7f2e6e9-88cb-4ae4-b9b6-8af9b8d8b5b4","arxiv_id":"1908.09483","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For the semigroup generated by primes at least p_n, the largest atom u_n satisfies u_n ~ 3 p_n; Wilf's inequality holds for these semigroups, and if the Frobenius number f_n is eventually odd then f_n ~ 3 p_n, which would imply Goldbach for large evens.","lead":"The paper studies numerical semigroups generated by all primes from some point onward, proving the largest necessary prime coin grows like three times the starting prime. It also shows that if a computationally suggested property of the largest missing numbers holds, then Goldbach's conjecture would follow for all large even numbers.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the analytic core checks out, with the finite n<429 table as the only external dependency worth verifying.","rationale":"The reader's weakest-assumption analysis pointed to Matomaki-Maynard-Shao [8, Thm 1.1] and the external finite table [16]. My read confirms that the deep theorem is used correctly and that the analytic consequences drawn from it are valid, so the main asymptotic theorem and the n>=429 part of the Wilf proof survive. The finite table is a genuine external input, but it is a small, independently checkable computation rather than a load-bearing mathematical risk. Proposition 6 is terse but not needed for the main claims. Overall the paper's central arguments hold, and the ACCEPT verdict remains appropriate; confidence could be raised by reproducing the finite table, but no verdict change is needed.","tokens_in":7743,"tokens_out":37739,"duration_ms":363718,"concrete_test":"Write a short script to compute f_n, g_n, e_n for 1<=n<=428 (sieving primes up to the needed range and computing the Frobenius number and genus of the semigroup generated by primes at least p_n) and verify Wilf's inequality g_n/(1+f_n) <= (e_n-1)/e_n for every n; compare the results with table [16]. If all n<429 pass, the external dependency in Proposition 5 is discharged; if any case fails, the statement of Proposition 5 would be false as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked the unconditional Theorem, the Corollary behind it, Proposition 5, and Lemma 4. The proof that S_n=S_n^{3+epsilon} for large n is internally sound: the recursive inclusion S_{n+1}^{3+epsilon} subset S_n^{3+epsilon} is uniform in n, and the inequality in case II is valid for epsilon<3. The bound e_n(1+f_n-g_n) >= (pi(3p_n)-n+1)^2 is a legitimate consequence of e_n>=k and 1+f_n-g_n>=k for k=pi(3p_n)-n+1; the Rosser-Schoenfeld estimates give 2n<pi(3p_n)<3n for n>=429, and Selmer's bound together with lambda_2(n)<n+2 gives f_n<n^2, completing the proof of Wilf's inequality. The only non-reproduced step is the finite verification for n<429 delegated to the external table [16]; this is a small finite check rather than a mathematical dependency of the main asymptotic theorem. Proposition 6 is compressed and would benefit from expansion, but it is peripheral and does not affect the central claim. I found no internal inconsistency or unsupported analytic step that would threaten the verdict.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the numerical semigroup S_n generated by all primes at least p_n. The main unconditional result is that the largest irredundant generator u_n satisfies π(u_n) ∼ 3n, the embedding dimension e_n satisfies e_n ∼ 2n, and u_n ∼ 3p_n. The authors also prove that Wilf's inequality g_n/(1+f_n) ≤ (e_n-1)/e_n holds for every S_n and that g_n/p_n converges to 5/2. Conditional statements connect the conjectured asymptotics f_n ∼ 3p_n, f_n odd for large n, and 4p_n > f_{n+1} to the binary Goldbach conjecture, with the Goldbach conjecture following if 4p_n > f_{n+1} for all n.","tokens_in":8002,"tokens_out":29871,"duration_ms":257939,"significance":"The unconditional theorem is a clean, parameter-free asymptotic result for a natural infinite family of numerical semigroups, obtained by combining the Matomäki–Maynard–Shao theorem with elementary semigroup arguments. The proof that Wilf's conjecture holds for all S_n is a substantial advance for a nontrivial family. The conditional bridges to Goldbach are elegant and explicitly non-circular. The analytic core is coherent and the estimates are checkable; the only external dependency is a small finite table used in the proof of Proposition 5.","major_comments":[],"minor_comments":[{"comment":"The proof for n < 429 is delegated entirely to the external table [16]; please include the table or a verification script in the submission so that the finite check is reproducible and not dependent on an external URL.","section":"Section 2, Proposition 5"},{"comment":"The derivation of e_n ∼ 2n is not spelled out; after establishing S_n = S_n^{3+ε}, one should explicitly use that all atoms are primes, that primes in [p_n, 3p_n[ are atoms, and that primes ≥ (3+ε)p_n are not atoms (by Lemma 3), so that e_n is between π(3p_n)−n+1 and π((3+ε)p_n)−n+1, and letting ε → 0 gives e_n ∼ 2n.","section":"Section 1, Corollary and Theorem"},{"comment":"The proof is only a sketch. In particular, the passage from the Coppola–Laporta theorem and [8, Theorem 1.1] to the limits for α_k(n) needs details: for α_2(n), the even numbers in [2p_n, 3p_n) with m close to p_n require a separate treatment, and the exceptional set from [2, Theorem 1, Corollary] has size O(p_n/(log p_n)^A), which is o(p_n). Also, for α_1(n), one should state explicitly that only primes in [p_n, 2p_n) belong to S_n, so the gap density tends to 1.","section":"Section 2, Proposition 6"},{"comment":"There are a few typos: 'similiar' should be 'similar' in the Observations, and in the abstract 'f n+1' should be 'f_{n+1}'.","section":"Throughout"},{"comment":"The line 'pn II ≤ p/(3+ε)' contains a typographical artifact; it should read 'p_n ≤ p/(3+ε)'.","section":"Section 1, proof of the Corollary"},{"comment":"The online tables [16], [17], and [18] are cited with access dates; consider including the relevant data as ancillary files to ensure permanence and reader access.","section":"References and data availability"}],"recommendation":"minor_revision","confidential_remarks":"The paper is mathematically sound and the central claims are well supported. The only point I cannot verify from the manuscript itself is the finite check for n < 429 in Proposition 5, which relies on an external table; including that table or a computation script in the final version would remove this dependency. The proof of Proposition 6 is compressed but the claim is plausible and appears to be fillable; I have asked for an expansion in the minor comments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is better than the average arXiv submission in this area. The unconditional part is real: for S_n generated by all primes at least p_n, they prove pi(u_n) ~ 3n, e_n ~ 2n, and u_n ~ 3p_n, plus Wilf's inequality for every S_n. The proof route through the Matomäki-Maynard-Shao almost-equal-summands theorem is sound, and the consequences (S_n = S_n^{3+epsilon} for large n) check out. The Wilf argument using Eliahou's criterion plus Rosser-Schoenfeld estimates and Selmer's bound is coherent; I followed the chain and found no gap. The stress-test note confirms this: the analytic core holds, and the only non-reproduced step is the finite check for n < 429 delegated to an external table. That is a small dependency, not a structural one.\n\nThe conditional bridges to Goldbach are honestly labeled. Proposition 4 and Proposition 2 are stated as implications, not as theorems about Goldbach, and they are proven correctly. The conjectures C1-C3 are backed by computation but are not premises for the unconditional results. That is how I would want such a paper to behave.\n\nSoft spots, in proportion: Proposition 6 is compressed. The claim g_n/p_n -> 5/2 is asserted with a sketch of the alpha_k(n) asymptotics, and the step from Coppola-Laporta to alpha_2(n) -> 1/2 is not filled in. It is a peripheral result and does not affect the main theorem, but a referee should ask for a fuller proof or a reference. The finite n < 429 table is not reproduced; that is fine for a small check, but it would be better to include the table or a verification script. The paper also has a few minor typographical issues and the prose is a bit rough in places.\n\nThe citation pattern is honest: they attribute the limsup <= 4 argument and the Goldbach connection to MathOverflow and Kløve, and they use published theorems correctly. No invented entities, no fitted parameters.\n\nWho is this for? Anyone working on numerical semigroups and the Frobenius problem, and anyone curious about how close the prime-generated case brings us to Goldbach-type statements. The paper deserves a serious referee. My recommendation: send it to peer review. With an expanded Proposition 6 and a small note on the finite check, it would be a solid publication.","headline":"Genuinely new unconditional results on prime-generated semigroups using deep but legitimate tools, with conditional Goldbach bridges that are clearly labeled; the compressed genus proof and an external finite check are the only real soft spots.","tokens_in":8573,"tokens_out":1098,"would_cite":true,"duration_ms":12045,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11D07","11P32","20M14"],"pacs":[],"model":"deepseek-v4-flash","headline":"The semigroup generated by all primes at least $p_n$ has a rigid asymptotic shape: its largest irredundant generator $u_n$ satisfies $u_n \\sim 3p_n$, and every such semigroup satisfies Wilf's inequality.","keywords":["numerical semigroup","prime-generated semigroup","Frobenius number","embedding dimension","Wilf's conjecture","Goldbach conjecture","almost equal primes","asymptotic semigroup shape"],"falsifier":"Compute the largest atom $u_n$ and Frobenius number $f_n$ directly from the generating interval for a large $n$, say $n=10^5$; if $\\pi(u_n)/(3n)$ is not close to 1 or $u_n/p_n$ is not close to 3, the main theorem fails. A more targeted check of the engine: for $n=10^5$ and $\\varepsilon=0.01$, verify that every odd integer in $[(3+\\varepsilon)p_n,\\,(3+\\varepsilon)p_{n+1}]$ is a sum of three primes each $>p_n$; the first violation would disprove Lemma 3 in the paper.","tokens_in":7556,"feed_emoji":"🔢","tokens_out":10633,"duration_ms":97181,"temperature":0.7,"pith_summary":"The paper studies $S_n$, the numerical semigroup generated by all primes at least the $n$-th prime $p_n$, and asks how the semigroup's shape scales with $n$. Its main unconditional result is that the largest irredundant generator $u_n$ satisfies $u_n \\sim 3p_n$, equivalently $\\pi(u_n)\\sim 3n$, so the minimal generating set has about $2n$ atoms. The same machinery gives $g_n/p_n \\to 5/2$ for the genus, and proves Wilf's inequality $g_n/(1+f_n)\\le (e_n-1)/e_n$ for every $S_n$. The paper also establishes two conditional bridges to the classical even-two-prime problem: if the Frobenius number $f_n$ is odd for all large $n$, then $f_n\\sim 3p_n$, and if $f_n\\sim 3p_n$ then every large even integer is a sum of two primes. A sympathetic reader should care because a naturally defined infinite family of semigroups is shown to have rigid asymptotics, and Wilf's generally open question is settled on this family, with its limiting ratio $5/6$ explicit.","feed_headline":"Prime semigroups obey a 3p_n asymptotic law","feed_subtitle":"For the semigroup generated by primes at least p_n, the largest atom grows like 3 p_n and Wilf's inequality holds.","key_machinery":"The load-bearing object is the almost-equal-summands form of the ternary-prime theorem, which makes the generators dense enough that the region just below $(3+\\varepsilon)p_n$ is filled by sums of three small primes. Around it the argument builds a self-contained inclusion $S_{n+1}^{3+\\varepsilon}\\subseteq S_n^{3+\\varepsilon}$; iterating gives $S_n=S_n^{3+\\varepsilon}$, meaning every element of $S_n$ is generated inside $[p_n,(3+\\varepsilon)p_n)$. The lower bound $f_n\\ge 3p_n-6$ is the elementary counterpart: the odd composite $3p_n-6$ cannot belong to $S_n$. For Wilf's inequality, the decisive estimate is $f_n<n^2$ for $n\\ge429$, derived from a classical Frobenius bound for semigroups generated by a long interval of integers, paired with standard explicit prime-counting estimates to control $\\pi(3p_n)$.","core_discovery":"The central discovery is an asymptotic rigidity theorem for prime-generated semigroups. Writing $e_n$ for the embedding dimension and $\\pi$ for the prime-counting function, the paper proves $\\pi(u_n)\\sim 3n$, $e_n\\sim 2n$, and $u_n\\sim 3p_n$. The engine is a strong three-prime theorem with almost equal summands: every sufficiently large odd $N$ is $q_1+q_2+q_3$ with $\\lvert q_i - N/3\\rvert\\le N^\\theta$ for any $\\theta>11/20$. This proves Lemma 3, that every odd $N\\ge(3+\\varepsilon)p_n$ is in $S_{n+1}$ for large $n$, which combines with the elementary lower bound $f_n\\ge 3p_n-6$ to give the Frobenius asymptotics under the parity condition, and yields the corollary $S_n=S_n^{3+\\varepsilon}$ that drives the atom count. The paper then proves unconditionally that Wilf's inequality holds for every $S_n$, using the $f_n<3p_n$ criterion together with an explicit bound $f_n<n^2$ for $n\\ge429$ obtained from a general Frobenius bound and standard prime estimates.","pith_inferences":["I infer that the constant $3$ is not special to primes but is set by the ternary representation threshold: the same almost-equal-three-summand machinery, applied to primes in a fixed arithmetic progression $a\\bmod d$, would predict $f_n/p_n\\to d+1$ for even $d$ and $2d+1$ for odd $d$, matching the paper's numerical observations.","A testable extension would be an effective version of the parity bridge: the proof only needs oddness of $f_n$ along a positive-density subsequence, not all large $n$; if one could show even Frobenius numbers occur with density zero using the two-prime almost-equal estimate, the $f_n\\sim3p_n$ conjecture would follow without a full parity proof.","I infer that the lower bound $f_n\\ge3p_n-6$ is asymptotically sharp in a strong sense: any proof that $f_n<3p_n$ happens infinitely often would immediately produce infinitely many twin prime pairs inside $[p_n,3p_n+4]$, so the paper's framework explains why $\\liminf f_n/p_n=3$ cannot be improved without resolving the twin-prime question."],"forward_implications":["For all sufficiently large $n$, the minimal generating set of $S_n$ has exactly about $2n$ atoms, so the semigroup is generated by a thin but long interval of primes, two atoms per prime on average.","Wilf's inequality holds for every $S_n$; the limiting ratio $g_n/(1+f_n)=5/6$ sits strictly below the limiting upper bound $(e_n-1)/e_n\\to1$.","If the conjectured parity condition $f_n$ odd for all $n\\ge5$ holds, then $f_n\\sim 3p_n$ follows unconditionally from the theorem's engine.","If $f_n\\sim3p_n$ holds, then every sufficiently large even integer $x$ is a sum of two primes with one summand in $(x/4,x/2]$.","If the sharper conjecture $4p_n>f_{n+1}$ holds for all $n\\ge1$, the even-two-prime conjecture follows with the same strong one-summand-in-$(x/4,x/2]$ property."],"supporting_citations":[{"why":"Supplies the almost-equal-summands three-prime theorem that drives Lemmas 2 and 3, the corollary $S_n=S_n^{3+\\varepsilon}$, and the atom asymptotics.","marker":"[8]"},{"why":"Gives the criterion that Wilf's inequality holds whenever $f<3p$, used as the starting point of Proposition 5.","marker":"[4]"},{"why":"Provides the explicit prime-counting estimates used to bound $\\pi(3p_n)$ between about $2n$ and $3n$ for $n\\ge429$.","marker":"[10]"},{"why":"Gives the general Frobenius-number bound for a long interval of generators, from which $f_n<n^2$ follows for $n\\ge429$.","marker":"[11]"},{"why":"Supplies the almost-equal two-prime representation for even numbers used to compute the gap proportions and $g_n/p_n\\to5/2$.","marker":"[2]"},{"why":"External computed table used for the $n<429$ check in Proposition 5's proof of Wilf's inequality.","marker":"[16]"}],"fun_headline_variants":["Prime semigroups show 3p_n atom growth","Goldbach tie: prime semigroup Frobenius ~3p_n","Wilf's inequality proven for prime semigroups","Prime atom count: u_n ~3p_n, Wilf holds","Frobenius for primes: f_n ~3p_n and Goldbach"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole asymptotic engine rests on the strong three-prime theorem with almost equal summands at exponent $\\theta>11/20$; if that theorem failed in the required range, the Frobenius upper bound, the corollary $S_n=S_n^{3+\\varepsilon}$, and the atom asymptotics would all lose their support, while the small-$n$ check of Wilf's inequality additionally depends on an external table the paper does not reproduce.","fun_headline_variants_meta":{"raw":{"variants":["Prime semigroups show 3p_n atom growth","Goldbach tie: prime semigroup Frobenius ~3p_n","Wilf's inequality proven for prime semigroups","Prime atom count: u_n ~3p_n, Wilf holds","Frobenius for primes: f_n ~3p_n and Goldbach"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000204,"raw_usage":{"total_tokens":1443,"prompt_tokens":1049,"completion_tokens":394,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":665,"completion_tokens_details":{"reasoning_tokens":303}},"tokens_in":665,"tokens_out":394,"duration_ms":3866,"temperature":1.0,"reasoning_tokens":303,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:11:56.919999+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the largest atom $u_n$ and Frobenius number $f_n$ directly from the generating interval for a large $n$, say $n=10^5$; if $\\pi(u_n)/(3n)$ is not close to 1 or $u_n/p_n$ is not close to 3, the main theorem fails. A more targeted check of the engine: for $n=10^5$ and $\\varepsilon=0.01$, verify that every odd integer in $[(3+\\varepsilon)p_n,\\,(3+\\varepsilon)p_{n+1}]$ is a sum of three primes each $>p_n$; the first violation would disprove Lemma 3 in the paper.","supporting_citations":[{"cited_title":"Matomäki, J","cited_arxiv_id":null,"evidence_quote":"Supplies the almost-equal-summands three-prime theorem that drives Lemmas 2 and 3, the corollary $S_n=S_n^{3+\\varepsilon}$, and the atom asymptotics."},{"cited_title":"Eliahou,Wilf’s conjecture and Macaulay’s theorem, J","cited_arxiv_id":null,"evidence_quote":"Gives the criterion that Wilf's inequality holds whenever $f<3p$, used as the starting point of Proposition 5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the explicit prime-counting estimates used to bound $\\pi(3p_n)$ between about $2n$ and $3n$ for $n\\ge429$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the general Frobenius-number bound for a long interval of generators, from which $f_n<n^2$ follows for $n\\ge429$."},{"cited_title":"Coppola, M","cited_arxiv_id":null,"evidence_quote":"Supplies the almost-equal two-prime representation for even numbers used to compute the gap proportions and $g_n/p_n\\to5/2$."},{"cited_title":"Date Accessed: October 21, 2019","cited_arxiv_id":null,"evidence_quote":"External computed table used for the $n<429$ check in Proposition 5's proof of Wilf's inequality."}],"review_version":1}