{"id":"5d681957-65c8-42c9-b6ac-dbeff42ca41b","arxiv_id":"2512.22413","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Sierpiński's 1958 matrix conjecture—each row of the n×n grid of the first n² numbers contains a prime—is verified for n≤4,553,432,387, with unconditional partial results for larger n.","lead":"A 67-year-old prime-number conjecture about n×n grids has been verified for the first 4.5 billion grid sizes, and partial results show every larger grid still has primes in at least a quarter of its rows. The proof combines known records of prime gaps with counting arguments, so it is a computational verification rather than a full solution.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 depends on the unverified completeness of the maximal-prime-gap table below p*_83; an undiscovered gap >1572 before p*_83 would invalidate the pigeonhole argument.","rationale":"Considered the candidate concerns. The abstract overclaims 10 billion and 141,618 rows, while the body proves 4,553,432,387 and 131,294; this is a real inconsistency and must be corrected, but it does not attack Theorem 1's internal proof. Likewise, §4.2's statement that the maximal gap in [1,N83^2] is 1676 is numerically wrong (the 1676 gap starts beyond N83^2), yet the weaker bound floor(n/1676) remains true, so it is not load-bearing for Theorem 1. The descent arithmetic slip (floor(1571/148)=11 instead of 10) is immaterial to existence. The genuine load-bearing issue is the unverified completeness of the external maximal-gap table. If that table is complete, the pigeonhole argument is logically sound; if not, the theorem could fail for some n near 1572. This matches the reader's weakest assumption, so the CONDITIONAL verdict is appropriate. No new concern emerged that would change the verdict; the paper should remain conditional on making the gap-table verification available and reconciling the abstract/body numbers.","tokens_in":9983,"tokens_out":24505,"duration_ms":238324,"concrete_test":"Obtain an independent, versioned maximal-prime-gap dataset (e.g., Oliveira e Silva's table or a fresh primecount/segmented-sieve run) and verify: (1) the largest prime gap with starting prime below p*_83 = 20,733,746,510,561,442,863 is exactly 1572; (2) p*_82 = 18,571,673,432,051,830,099 is the first occurrence of a gap of size 1572; (3) p*_83 is the first occurrence of a gap of size 1676. If any gap ≥1573 starting below p*_83 is found, the pigeonhole step fails for the smallest n>1572 and the bound N83 must be re-derived. If the dataset is confirmed, Theorem 1 stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1 (§3.1) reduces H1 for n∈[1572,N83] to the assertion that every block of 1572 consecutive integers below p*_83 contains a prime. This is equivalent to claiming that the largest prime gap with starting point below p*_83 is exactly g*_82 = 1572. The paper cites Wikipedia, PrimePages, and primecount for the 82nd and 83rd maximal gaps, but does not ship or independently verify the underlying exhaustive-search data. A table of 'known maximal gaps' only certifies that no larger gap has been found; to serve as a proof it must be complete and correct up to p*_83. The 83rd gap of size 1676 starts at p*_83 itself, above N83^2, so it lies outside the matrix; nevertheless, an undiscovered record gap of size >1572 starting below p*_83 would change the 83rd entry and break floor(n/1572) for some n≤N83. This is the load-bearing external assumption. Schinzel's n≤4505 result is secondary, since the descent argument (§3.2) would cover the low-n region if the gap data are valid.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Sierpiński's Hypothesis H1, the conjecture that for n≥2 each row of the n×n array containing 1,…,n² has a prime. The main theorem (Theorem 1, §3.1) asserts that H1 holds for all n up to N83 = 4,553,432,387. The proof uses the known maximal prime gaps: since the largest gap below p*83 is claimed to be g*82 = 1572, every block of 1572 consecutive integers below N83² contains a prime, so each row of S_n with n≥1572 has at least floor(n/1572)≥1 prime; the small range n≤4505 is covered by Schinzel's result (or by the descent argument in §3.2). Additional results: Theorem 3 shows at least n/4 rows of S_n contain a prime for any n; Theorem 7 shows the first 131,294 rows contain a prime for n>N83; Theorem 6 gives a descent property. The paper relies on external maximal-gap tables and explicit Chebyshev bounds.","tokens_in":10191,"tokens_out":17537,"duration_ms":154860,"significance":"If the main theorem is sound, it is a substantial numerical verification of H1, extending the known range by roughly nine orders of magnitude and implying Oppermann's and Legendre's conjectures up to 4.5×10⁹. The proof is elementary and transparent, and the descent argument (§3.2) is a self-contained way to cover the range once the gap data are supplied. The partial results — the quarter-of-rows bound and the lower-row bound — are original and potentially useful. However, the claim is conditional on the completeness of the external maximal-prime-gap table up to p*83, and the manuscript contains several internal numerical inconsistencies that must be fixed; therefore the current version is not yet publishable.","major_comments":[{"comment":"The arXiv abstract states that H1 is verified for n≤10,070,368,414 and that partial results hold for n≥10,070,368,414. The body and the paper's own abstract prove bounds with 4,553,432,387 (and n≥4,553,432,388). This is a factor of about 2.2 overclaim and must be corrected in the metadata; as written it is inconsistent with Theorem 1.","section":"Abstract"},{"comment":"The proof hinges on the assertion that the largest prime gap with starting point below p*83 is g*82 = 1572, so every block of 1572 consecutive integers below N83² contains a prime. This is equivalent to the completeness of the external table of maximal gaps up to p*83 ≈ 2.07×10^19. The paper cites Wikipedia, PrimePages, and primecount but provides no data, certificate, or reproducible verification. If a gap >1572 with starting point <p*83 existed, the pigeonhole argument would fail for some n≤N83. This is a load-bearing external assumption; the manuscript should either supply the verified table (or code to reproduce it) or explicitly state the theorem as conditional on this computational record.","section":"§3.1, Theorem 1"},{"comment":"The proof states: 'We have already seen that the maximal prime gap in the interval [1,(N83)^2] is 1676.' This is incorrect: the 83rd maximal gap of size 1676 starts at p*83, which is greater than N83²; the largest gap inside the interval is g*82 = 1572. The conclusion of Theorem 5 remains true if 1572 is used (since n>N83 makes floor(n/1572)≥1), but the proof as written relies on a false premise.","section":"§4.2, Theorem 5"},{"comment":"Table 1 is internally inconsistent. For n=4.5×10^9 the table gives i_min=83 and g*_{i_min-1}=1752, but g*82 is 1572, and a value 1752 larger than the 83rd gap of 1676 is impossible. Similarly the entry for n=4×10^9 gives i_min=79 and g*78=1525, which appears to be off by one or more indices; the #min column in several rows seems to be computed with different gap values. The table should be recomputed and checked against a reliable source.","section":"§3.3, Table 1"}],"minor_comments":[{"comment":"The text states floor[1571/148] = 11; the correct value is 10. This does not affect the covering argument, but it is a factual error. Also, in Step 3 the labels g*21 and n*22 should be g*13 and n*14 respectively.","section":"§3.2, Step 2"},{"comment":"The text says (13)² = 167; it should be 169.","section":"§3.2, Step 7"},{"comment":"The intervals k∈[1,15957], [2,54957], and [5,131294] are stated to be 'easily checked', but the numerical check is not shown. A short table or script reproducing the inequality (4.20) would improve verifiability.","section":"§4.3, Theorem 7"},{"comment":"The equivalence in (4.4) uses k≥1, omitting the bottom row k=0. The bottom row is trivial for n≥2, but the statement should say so explicitly to avoid a logical gap.","section":"§2, Lemma (4.4)"}],"recommendation":"major_revision","confidential_remarks":"The paper is on the right track, but the current version is not ready. Please ask the author to supply the gap-table verification or clearly state the conditional nature, correct Table 1 and §4.2, and reconcile the abstract. The descent proof is a nice addition and makes the paper self-contained for n≥14 once the gap data are accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth reading and worth refereeing, but only after the author fixes a few admitted inconsistencies. The core result is legitimate: Sierpiński's H1 is verified up to n=4,553,432,387, a huge jump from Schinzel's n≤4505, and the proof is a clean pigeonhole argument using known maximal prime gaps. The partial results for all larger n — at least a quarter of the rows, and the first 131,294 rows — also follow honestly from published Rosser–Schoenfeld, Montgomery–Vaughan, Dusart, and Axler bounds. There are no free parameters, no curve-fitting, and no circularity; the main theorem reduces to the completeness of the maximal-prime-gap table below p*_83≈2.07e19. That table is external, but it is a standard, publicly checked object, and the author cites it. That is an acceptable dependence for a verification paper, provided the provenance is made precise.\n\nThe soft spots are real but mostly mechanical. The abstract in the version I saw claims n≤10,070,368,414, while the body proves 4,553,432,387; that discrepancy must be resolved before anything else. Theorem 5's proof says the maximal prime gap in [1,(N_83)^2] is 1676; it is 1572, since the 83rd gap starts above that interval. The bound floor(n/1676) is still valid (weaker than the true floor(n/1572)), so Theorem 5 survives, but the statement as written is false. Table I contains at least one incompatible entry (the 4.5e9 row lists 1752 instead of 1572), and the descent step has notation typos (g*_21 for g*_13, etc.). These are not hard to fix, but they are exactly the kind of thing that prevents clean acceptance.\n\nThe only substantive worry is the completeness of the maximal-gap table. If an undiscovered gap of size >1572 hides below p*_83, the main finite verification breaks. I don't think that is likely, but the paper should state explicitly that the proof depends on the table being complete up to that point, and ideally point to the exhaustive-search verification rather than a Wikipedia page. The author does not ship the gap data or an independent check, which is normal for a short note, but worth flagging.\n\nBottom line: serious referee, conditional on the author cleaning up the numerical errors and the abstract/body mismatch. The central argument holds up as far as I can tell, and the verification is a real, if limited, advance.","headline":"Solid elementary verification of Sierpiński's H1 to 4.5 billion and honest partial results for all n; the math works but the paper needs a cleanup pass on its own numbers and a precise statement of its external-gap dependence.","tokens_in":10760,"tokens_out":4221,"would_cite":true,"duration_ms":38740,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A41","11N05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sierpiński's Hypothesis H1—that every row of the n×n array of 1 through n² contains a prime—is verified for every n from 2 to at least 4,553,432,387, with partial guarantees for all larger n.","keywords":["Sierpiński matrices","Sierpiński's Hypothesis H1","maximal prime gaps","pigeonhole principle","prime distribution","prime-counting bounds"],"falsifier":"A direct counterexample to the paper's central claim would be a prime-free row in S_n for some n≤4,553,432,387; a finite search over those arrays, or equivalently over all intervals of length n below (4,553,432,387)², would settle it. To test the machinery itself, independently recompute every maximal prime gap starting below (4,553,432,387)²: finding any gap wider than 1,572 would break the pigeonhole step, while confirming the table would corroborate the bound.","tokens_in":9781,"feed_emoji":"🔢","tokens_out":9855,"duration_ms":92614,"temperature":0.7,"pith_summary":"The paper aims to establish Sierpiński's 1958 Hypothesis H1: when the numbers 1,2,...,n² are arranged in rows of length n, every row contains at least one prime. It proves this for all n up to 4,553,432,387 by combining the known largest prime gap below about 2×10^19, which has width 1,572, with a pigeonhole counting argument: any n consecutive numbers in that range must contain a prime. For arbitrary larger n, it proves that at least one quarter of the rows always contain a prime, and that the bottom 131,294 rows always contain a prime. The interest is that H1 is stronger than two classical prime-gap conjectures, so these bounds sharply constrain how those conjectures could fail.","feed_headline":"Prime in every row proven for first 4.5 billion Sierpiński squares","feed_subtitle":"A 1,572-wide prime gap plus pigeonhole counting pins the bound; beyond it, a quarter of rows always contain a prime.","key_machinery":"The central object is the Sierpiński matrix S_n, whose row i is the n consecutive integers n(n−i)+1,...,n(n−i)+n. The mechanism is a maximum-gap ladder: for a given n, the largest prime gap starting below n² determines, via floor(n/g), a guaranteed minimum number of primes in every row. The proof climbs the ladder: at the top, g=1,572 covers all n from 1,572 to N₈₃; descending through smaller maximal gaps (148, 52, 34, 20, 14, ...) pushes the coverage down to n=13, with direct checks for n≤13. This converts a data table of maximal prime gaps into a universal statement about all rows up to a huge n.","core_discovery":"The load-bearing result is Theorem 1: for all 2≤n≤4,553,432,387, every row of the Sierpiński matrix S_n contains at least one prime. The proof works because the largest prime gap starting below (4,553,432,387)² is the 82nd maximal gap, of width 1,572, so the pigeonhole principle guarantees at least floor(n/1,572) primes in each row for n in that range. The small cases n≤13 are checked directly, or by a known 1961 verification used to cover the overlap n≤4,505; the two ranges splice together to cover all n up to N₈₃. For n beyond N₈₃, Theorems 3 and 7 provide guarantees that at least n/4 rows, and the bottom 131,294 rows, contain primes.","pith_inferences":["The abstract's stated bound of 10,070,368,414 does not match the body's 4,553,432,387; as written, the proof establishes the smaller bound, and the larger figure would need its own defence or correction.","Because the proof converts a single prime-gap width into a statement about all rows, the method transfers directly to any conjecture that demands a prime in every block of n consecutive integers: the data table, not the logic, is the limiting resource.","The one-quarter density result implies that a counterexample to H1 would require a distribution of primes far more uneven than any known evidence: more than three-quarters of the rows of some S_n would have to be prime-free.","A tractable next step would be to run an independent verification of the maximal-gap table up to (4,553,432,387)²; that would upgrade the finite claim from 'as good as the cited table' to a fully self-contained computation."],"forward_implications":["If correct, the range 2≤n≤4,553,432,387 verifies the two long-standing interval-prime conjectures that H1 subsumes, because each is a restricted version of the row condition.","The verification is not marginal: for the largest matrix checked, every row contains at least 2,896,585 primes, not just one.","For all larger n, the unconditional one-quarter density guarantee and the 131,294-row bottom guarantee remain true, even though full H1 is open there.","Any future extension of the maximal-gap table will automatically enlarge the finite range, since the proof's only input from the table is the largest gap width below the new endpoint.","A stronger statement than H1 also follows: any n consecutive integers below (4,553,432,387)² contain a prime, not just the special rows of S_n."],"fun_headline_variants":["Prime guaranteed in every Sierpiński row up to 4.55B","Sierpiński's prime-row conjecture verified to 4.5B","Every Sierpiński row has a prime up to 4.55 billion","Sierpiński's H1 true for n up to 4.5B"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The finite verification succeeds only if the published table of maximal prime gaps up to about 2×10^19 is complete and correct—specifically that no gap wider than 1,572 opens below (4,553,432,387)²—and if the 1961 check really covers n≤4,505.","fun_headline_variants_meta":{"raw":{"variants":["Prime guaranteed in every Sierpiński row up to 4.55B","Sierpiński's prime-row conjecture verified to 4.5B","Every Sierpiński row has a prime up to 4.55 billion","Sierpiński's H1 true for n up to 4.5B"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000993,"raw_usage":{"total_tokens":4090,"prompt_tokens":833,"completion_tokens":3257,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":577,"completion_tokens_details":{"reasoning_tokens":3170}},"tokens_in":577,"tokens_out":3257,"duration_ms":24217,"temperature":1.0,"reasoning_tokens":3170,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T13:51:40.945655+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct counterexample to the paper's central claim would be a prime-free row in S_n for some n≤4,553,432,387; a finite search over those arrays, or equivalently over all intervals of length n below (4,553,432,387)², would settle it. To test the machinery itself, independently recompute every maximal prime gap starting below (4,553,432,387)²: finding any gap wider than 1,572 would break the pigeonhole step, while confirming the table would corroborate the bound.","supporting_citations":[],"review_version":1}