{"id":"50364e06-cedb-4b49-8de2-9f23c3eefd6c","arxiv_id":"2605.14871","paper_version":5,"verdict":"REJECT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims every prime gap g_n satisfies g_n < (51/16) log^2 p_n, but the derivation breaks down in Lemma 2.2.","lead":"A short preprint claims a record-strong upper bound on prime gaps: g_n < (51/16) log^2 p_n, which would imply several famous open conjectures. The proof appears to contain a load-bearing algebraic error, and the abstract even states a different constant than the main text.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.2's key lower bound (2.9) uses an invalid division: from A_k > log(k-1)+2/k it concludes A_k/(k-1) > log(k-1)/(k-1)+2/k, which is algebraically false.","rationale":"The reader's weakest_assumption identifies exactly the same algebraic error in (2.9). I re-read the proof of Lemma 2.2 in good faith: the derivation of A_k from Rosser's bound, the definition of B_n and T_n, and the structure of the argument are clear, but the step from A_k > log(k-1)+2/k to the summed inequality is plainly incorrect unless one divides incorrectly. The corrected term 2/[k(k-1)] telescopes rather than growing like log n, so the strong lower bound (2.10) is not justified. This is not merely a cosmetic typo: the proof's contradiction relies on the magnitude of the 2/k sum. The manuscript contains no formal verification or independent computational evidence for the main theorem, and the abstract/full-text constant mismatch (13/3 vs 51/16) further underscores the lack of careful checking. I agree the paper should be rejected as written, although I note this does not formally disprove the theorem; it means the proof is invalid at a load-bearing point.","tokens_in":6900,"tokens_out":7195,"duration_ms":53669,"concrete_test":"Evaluate both sides of equation (2.9) for a few n ≥ 22 using actual primes, e.g. n=22, 30, 100. Compute LHS = Σ_{k=2}^{n-1} A_k/(k-1) with A_k=(p_{k+1}-2)/k and RHS = Σ_{k=2}^{n-1} log(k-1)/(k-1) + Σ_{k=7}^{n-1} 2/k. If LHS < RHS in any case, inequality (2.9) is false and Lemma 2.2's proof is invalid. Equivalently, re-derive (2.9) with the algebraically correct term 2/[k(k-1)] and check whether the resulting version of (2.10) still contradicts g_n/(n-1) ≤ p_n/(n-1) for all n≥22; if not, the finite base case of the lemma must be rechecked.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim depends on Lemma 2.2, which asserts B_n < 2B_{n-1} for all n≥22. In the proof, inequality (2.9) states\n    Σ_{k=2}^{n-1} A_k/(k-1) > Σ_{k=2}^{n-1} log(k-1)/(k-1) + Σ_{k=7}^{n-1} 2/k.\nTo obtain this from the preceding bound A_k > log(k-1)+2/k, one must divide by k-1, which gives A_k/(k-1) > log(k-1)/(k-1) + 2/[k(k-1)], not +2/k. Equivalently, the claimed inequality would require A_k > log(k-1)+2(k-1)/k, which is not established by (2.8) and is generally false for moderate k. The extra 2/k terms are exactly what produce the log(n-1) term in (2.10); without them the claimed contradiction for all n≥22 does not follow. Since Lemma 2.2 is the engine for Lemma 2.3 and Theorem 2.5, this invalid step is load-bearing. The proof as written therefore fails to establish B_n < 2B_{n-1}, so the main theorem is unsupported. I find this the single most decisive flaw; it is an internal algebraic error, not a disagreement with consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims an unconditional bound for every prime gap: g_n = p_{n+1} - p_n < (51/16) log^2 p_n for all n (the arXiv metadata abstract states 13/3 instead of 51/16). The proof introduces averages A_k, B_n, and T_n, proves elementary identities in Theorem 2.1, then uses a chain of lemmas: Lemma 2.2 asserts B_n < 2B_{n-1} for n >= 22, Lemma 2.3 converts this into g_n < (51/16) B_{n+1}, Lemma 2.4 bounds A_n by 2 log(n-1), and Theorem 2.5 assembles the final gap bound. Consequences claimed include Oppermann's conjecture (Theorem 3.1) and a strengthened Andrica bound (Theorem 3.2).","tokens_in":7301,"tokens_out":17680,"duration_ms":133226,"significance":"If the proof were correct, the result would be a landmark: it would improve all known unconditional upper bounds on prime gaps (currently of size about x^{0.525}) to O(log^2 x), prove Oppermann's conjecture, and establish a strong Andrica-type inequality. Unfortunately, the central derivation is not valid. The main algebraic step in Lemma 2.2 is simply incorrect, and Lemma 2.3 contains a further unjustified inequality. The paper does contain some correct elementary identities and makes standard use of Rosser-type bounds, but these do not support the claimed theorem. The finite 'direct calculation' ranges are not documented with code or tables, which is an additional reproducibility concern for a result of this magnitude.","major_comments":[{"comment":"The step from (2.8) to (2.9) is algebraically false. From A_k > log(k-1) + 2/k one may divide by k-1 to obtain A_k/(k-1) > log(k-1)/(k-1) + 2/[k(k-1)], not log(k-1)/(k-1) + 2/k. The displayed sum in (2.9) with the extra sum_{k=7}^{n-1} 2/k is therefore not justified. This erroneous term is exactly what produces the log(n-1) term in (2.10) and drives the contradiction for all n >= 22. With the correct convergent sum sum 2/[k(k-1)], the lower bound loses the logarithmic growth and the claimed contradiction does not follow. Since Lemma 2.2 is the engine for Lemma 2.3 and Theorem 2.5, the main theorem is unsupported.","section":"Lemma 2.2, Eq. (2.9)"},{"comment":"Even accepting Lemma 2.2, the proof of Lemma 2.3 contains an invalid inequality. From the order relations T_n < B_n < B_{n+1} < 2B_n the paper asserts 16T_n - 4√(B_{n+1}T_n) ≤ [B_{n+1}+T_n - √Δ]/(2·1/16) ≤ B_n. This is not a consequence of those order relations; for example, take T_n = 1/2 and B_{n+1} = 1, which are compatible with the order relations. Then 16T_n - 4√(B_{n+1}T_n) ≈ 5.17, while [B_{n+1}+T_n - √Δ]/[2·1/16] ≈ 0.34, so the first inequality fails. No additional argument is supplied. This inequality is load-bearing for the contradiction that yields g_n < (51/16)B_{n+1}.","section":"Lemma 2.3, Eq. (2.14)"},{"comment":"The proof depends on several claimed finite verifications: 1 ≤ n ≤ 21 in Theorem 2.5, 2 ≤ a ≤ 488 in Theorem 3.1, and 31 ≤ n ≤ 21141 in Theorem 3.2. These are asserted as 'direct calculation' with no code, tables, or reproducible method. Since the theorem is meant to be an unconditional result about all prime gaps, these ranges are part of the proof. This is secondary to the algebraic errors above, but it should be addressed in any revision.","section":"Theorems 2.5, 3.1, 3.2"}],"minor_comments":[{"comment":"The arXiv metadata abstract states the bound as (13/3) log^2 p_n, while the body and the abstract in the full text state (51/16) log^2 p_n. These must be reconciled.","section":"Abstract"},{"comment":"The sentence 'But (2.11) implies that 21 ≤ n' appears to have the inequality direction backwards: the derivation requires that (2.11) be impossible for n ≥ 22, i.e. it should imply n ≤ 21 (or fail for all n ≥ 22), not n ≥ 21.","section":"After Eq. (2.11)"},{"comment":"The text says that (2.8) implies A_k > log(k-1)+2/k 'for all k ≤ 7' and A_k > log(k-1) 'for all k ≤ 6'. The inequalities and the use of the sums later indicate these should be k ≥ 7 and k ≥ 6 (or similar); as written the directions are inconsistent.","section":"Eq. (2.8) surrounding text"},{"comment":"There are several typographical issues, including 'Betrand's postulate' in Section 1 and the hyphenation of Erdős's name. These do not affect the mathematics but should be corrected.","section":"General"}],"recommendation":"reject","confidential_remarks":"This is an independent attempt at a longstanding open problem, and the claimed theorem would be a breakthrough. The two flaws I identify are internal algebraic errors in the main chain of lemmas, not disagreements with consensus. The finite computations are not documented, and the proof as written cannot be repaired by a local correction. I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the claimed bound g_n < 51/16 log^2 p_n would resolve Oppermann's conjecture and strengthen Andrica's inequality, but the proof as written fails at a single algebraic step in Lemma 2.2. This is not a salvageable draft; the main theorem is unsupported.\n\nWhat's genuinely there: the paper has a clear structure built around the prime-gap means A_k and the sums B_n. Theorem 2.1's summation identities are correct and could be useful in another context. The references are relevant and honestly cited — Bertrand, Rosser's explicit bounds, Dusart, Baker–Harman–Pintz, Granville's critique of Cramér. The idea of attacking the log^2 barrier with elementary averages is not absurd; it just doesn't survive contact with the details.\n\nThe soft spot is the whole lower spine of the proof. In Lemma 2.2, from A_k > log(k-1)+2/k, the proof concludes that A_k/(k-1) > log(k-1)/(k-1)+2/k. That's false; dividing by k-1 gives 2/[k(k-1)], not 2/k. The extra 2/k terms are exactly what produces the log(n-1) contribution in (2.10), which drives the contradiction for n≥22. Without those terms, the lower bound on Σ A_k/(k-1) is roughly log^2(n-2)/2 + O(1), far too weak to contradict p_n < n(log n + log log n). The stress-test note is right on target.\n\nThere are other signs of haste: the abstract states 13/3 log^2 p_n while the full text proves 51/16 log^2 p_n; Lemma 2.4's 'bounded above by calculation' is asserted rather than shown; and Lemma 2.3's use of Bernoulli's inequality is under-explained. But the algebraic error is decisive.\n\nThe author is serious and the exposition is readable; it's not crank work. But the central claim is unsupported, and the error is elementary enough that no referee is going to find a hidden proof. If you want a 'find the error' exercise for a class, it's useful. As a research contribution, I would not cite it. A serious editor could desk-reject immediately or send it to a referee for a formal confirmation; either way, rejection is the outcome.","headline":"An elementary division error in Lemma 2.2 invalidates the main bound, so the paper does not establish what it claims.","tokens_in":7745,"tokens_out":5529,"would_cite":false,"duration_ms":43606,"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":"This paper claims a new explicit upper bound on prime gaps: for every n, p_{n+1} − p_n < (51/16) log² p_n, a bound that would immediately imply Oppermann's conjecture and a strengthened Andrica conjecture.","keywords":["prime gaps","Oppermann's conjecture","Andrica's conjecture","Cramér-type bounds","explicit estimates for primes","weighted prime-gap sums","log-squared bounds","elementary number theory"],"falsifier":"Check the termwise inequality in Lemma 2.2 that leads to equation (2.9). For k = 7, A_7 = 17/7, so A_7/(k−1) = 17/42 ≈ 0.4048, while log(6)/6 + 2/7 ≈ 0.5843: the claimed step 'A_k/(k−1) > log(k−1)/(k−1) + 2/k' does not hold term by term. Whether the sum bound in (2.9) still holds by another route is the question that decides Lemma 2.2 and hence Theorem 2.5.","tokens_in":6794,"feed_emoji":"🔢","tokens_out":9728,"duration_ms":82637,"temperature":0.7,"pith_summary":"The paper sets out to prove that consecutive primes are never too far apart: the gap g_n = p_{n+1} − p_n is always less than (51/16) times the square of the natural logarithm of p_n. This is an explicit constant-factor bound of the kind conjectured by Cramér, and it would settle Oppermann's century-old conjecture that a prime lies between a(a−1) and a² and between a² and a(a+1). It would also prove a strengthened form of Andrica's inequality: sqrt(p_{n+1}) − sqrt(p_n) < 1/2 for all n ≥ 31. The proof is elementary: it rewrites each gap in terms of weighted sums of averaged gaps, shows those sums double only slowly, then converts that doubling control into a log-squared estimate.","feed_headline":"New bound: every prime gap is under 51/16 log² p_n","feed_subtitle":"If correct, it settles Oppermann's conjecture and sharpens Andrica's inequality on consecutive roots.","key_machinery":"The central objects are the running mean A_k = (p_{k+1} − 2)/k of the first k prime gaps and the weighted sums B_n = sum_{k=2}^n g_k/(k−1) and T_n = sum_{k=2}^n A_k/(k−1) + (p_n−2)/n − A_1. Theorem 2.1 gives exact identities such as g_n = n(B_n − T_n) and n − 1 = (B_n − T_n)/(T_n − B_{n−1}). The crucial mechanism is Lemma 2.2, which uses lower bounds for A_k and integral estimates to prove B_n < 2B_{n−1} for n ≥ 22; this doubling bound limits how fast the weighted gap sums can grow and is what eventually produces the constant 51/16.","core_discovery":"On the paper's own terms, the central discovery is Theorem 2.5: g_n < (51/16) log² p_n for every n ≥ 1. The engine is a set of summation identities (Theorem 2.1) expressing each gap g_n through B_n = sum_{k=2}^n g_k/(k−1) and T_n = sum_{k=2}^n A_k/(k−1) + (p_n−2)/n − A_1, where A_k is the average of the first k gaps. Lemma 2.2 asserts that B_n < 2B_{n−1} for every n ≥ 22; Lemma 2.3 converts that into g_n < (51/16) B_{n+1}; and explicit classical bounds on the nth prime, n log n < p_n < n(log n + log log n), let the author bound B_{n+1} by a logarithm-squared expression. The cases n ≤ 21 are checked directly.","pith_inferences":["The method leaves the constant 51/16 dependent on the explicit bounds for the nth prime; tightening those bounds would lower the constant directly, so the paper implicitly offers a parameterized route toward Cramér-type constants without pursuing it.","The same summation-identity machinery could be applied to other sparse sequences wherever a Bertrand-type bound is available; the paper does not explore this extension.","Because 51/16 is far larger than the heuristic lower limit near 1.12 for the limsup of g_n/log² p_n, the proved bound does not discriminate between competing conjectures about the true limiting ratio."],"forward_implications":["If correct, Oppermann's conjecture follows: for every integer a ≥ 2 there is at least one prime in (a(a−1), a²) and one in (a², a(a+1)).","If correct, a strengthened Andrica inequality follows: sqrt(p_{n+1}) − sqrt(p_n) < 1/2 for all n ≥ 31, whereas the classical conjecture only needs < 1.","The maximal prime gap among the first n primes is at most (51/16) log² p_n, an explicit constant-factor Cramér-type bound.","Any prime gap of size g must occur at a prime below e^(√(16g/51)), giving an explicit numerical relation between gap size and location."],"fun_headline_variants":["Prime gaps now bounded: g_n < (51/16) log² p_n for all n","Sharp new prime gap bound: g_n < (51/16) log² p_n","Every prime gap under 51/16 log² p_n, new proof","Prime gap bound tightened to (51/16) log² p_n","All prime gaps < 51/16 log² p_n"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire chain rests on Lemma 2.2's assertion that the weighted sums B_n grow by less than a factor of two for every n ≥ 22; if the numerical inequality behind that doubling statement is not valid, the proof cannot exclude B_n ≥ 2B_{n−1}, and the log-squared bound does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Prime gaps now bounded: g_n < (51/16) log² p_n for all n","Sharp new prime gap bound: g_n < (51/16) log² p_n","Every prime gap under 51/16 log² p_n, new proof","Prime gap bound tightened to (51/16) log² p_n","All prime gaps < 51/16 log² p_n"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001138,"raw_usage":{"total_tokens":4518,"prompt_tokens":656,"completion_tokens":3862,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":400,"completion_tokens_details":{"reasoning_tokens":3757}},"tokens_in":400,"tokens_out":3862,"duration_ms":26328,"temperature":1.0,"reasoning_tokens":3757,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T13:59:17.718421+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the termwise inequality in Lemma 2.2 that leads to equation (2.9). For k = 7, A_7 = 17/7, so A_7/(k−1) = 17/42 ≈ 0.4048, while log(6)/6 + 2/7 ≈ 0.5843: the claimed step 'A_k/(k−1) > log(k−1)/(k−1) + 2/k' does not hold term by term. Whether the sum bound in (2.9) still holds by another route is the question that decides Lemma 2.2 and hence Theorem 2.5.","supporting_citations":[],"review_version":3}