{"id":"4d930d45-b4f0-4a1e-bfbc-ca5604f021dc","arxiv_id":"2508.18786","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every n ≥ 106 and every x ≥ 1, there is always a prime in the interval [x, x + x^(1−1/n)], making several classical prime-gap results fully explicit.","lead":"The paper proves explicit, fully worked out guarantees that a prime exists in a short interval starting at any number, once the interval is chosen wide enough. It combines known results to give concrete thresholds instead of 'sufficiently large', and shows that for a wide interval exponent (n at least 106) the guarantee holds for every starting point.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The all-x≥1 claim for n≥106 rests on the unproved strengthening (2.5); the cited 90th-power theorem, as stated for integer i, does not imply the real-interval form, so the lower-bound coverage is not secure.","rationale":"The reader's weakest assumption is exactly the unverified strengthening (2.5)/(2.10); my independent reading confirms this is the most load-bearing gap. The derivation of the final n≥106 claim depends on (2.5) to start the 90th-power lower bound at 1.65×10^1165, which lies below the linear-interval endpoint 3.46×10^1208. Without (2.5), the ordinary integer-power theorem gives a much larger threshold, and the two coverages need no longer overlap. I also note the paper contains obvious internal slips, e.g., (2.18) drops the x≥exp(exp(33)) condition, but these do not affect the final integer-n theorem as severely. Since the reader already marked the verdict CONDITIONAL pending substantiation of these strengthenings, my stress-test does not change the verdict; it sharpens the reason. A positive outcome would require the proposed re-derivation to confirm (2.5).","tokens_in":5661,"tokens_out":11192,"duration_ms":112041,"concrete_test":"Re-derive (2.5) from the proof of Cully-Hugill–Johnston [6, Thm 1.4]: write the ψ-difference ψ(x+90x^{89/90})−ψ(x) and verify each explicit inequality in their proof with the real variable x replacing the integer i (or x^{1/90}). If the proof only supplies the integer-i statement, compute the least T such that the integer-power theorem plus k=ceil(x^{1/90}) forces a prime in [x,x+x^{1−1/106}]; if T>3.46×10^1208, the claimed coverage in §5 fails and (5.2) is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The final claim (5.2) is obtained by covering x∈[1,~3.46×10^1208] via maximal gaps plus linear intervals, and x≥[90^90]^{106/16}≈1.65×10^1165 via the 90th-power result. The latter coverage needs (2.5): ∀x≥1, [x,x+90x^{1−1/90}] contains a prime. But [6, Thm 1.4] is stated only as (2.4): for every integer i≥1, [i^90,(i+1)^90] contains a prime. This does not formally imply (2.5): for x∈[i^90,(i+1)^90], the prime guaranteed by (2.4) may lie below x, and the next power interval's prime may lie beyond x+90x^{89/90}; the elementary inclusion argument in §2.1 only compares endpoints and does not control the location of the prime inside the power interval. The paper's assertion that 'inspection of the proof' yields (2.5) is unsupported: no details, equation numbers, or error-term estimates from [6] are given. The same issue affects (2.10) from Dudek's cube theorem, though the final n≥106 claim depends primarily on (2.5). If (2.5) is not actually established, the threshold [90^90]^{n/(n−90)} may be far too small; using only the integer-power theorem (with k=ceil(x^{1/90})) gives a threshold around x∼10^1345 for n=106, which exceeds the linear-interval endpoint (Δmax−1)^106≈3.46×10^1208, so the interval coverage would fail and (5.2) would be unproved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper combines several known effective results—primes between 90th powers (Cully-Hugill–Johnston), primes between cubes (Dudek/Cully-Hugill), the table of maximal prime gaps, and explicit linear intervals due to Cully-Hugill–Lee—to produce explicit ranges in which the interval [x, x + x^{1-1/n}] is guaranteed to contain a prime. The main claims are: for all n ≥ 4 and x ≥ exp(exp(33)); for all n ≥ 91 and x ≥ [90^90]^{n/(n-90)}; and, for all n ≥ 106 and every x ≥ 1, the interval contains a prime. The final all-x statement, if established, gives a fully explicit prime-gap bound of size x^{1-1/106}.","tokens_in":6191,"tokens_out":10164,"duration_ms":108870,"significance":"If the main theorem is correct, it is a clean and useful explicit contribution: it makes classical sub-linear prime-in-short-interval results fully effective for all x ≥ 1, at the cost of a large exponent (n ≥ 106). The method is elementary and transparently combines several strong external results. The paper also gives explicit numerical thresholds in Tables 1–3, which are easily checked. However, the central derivation rests on an unproved strengthening of a cited theorem, and one stated auxiliary claim is false as written. The significance is therefore conditional on closing those gaps.","major_comments":[{"comment":"Equation (2.5) is load-bearing, but the justification 'Inspection of the proof of [6, Theorem 1.4] shows' is not supplied. The cited theorem, as stated in (2.4), guarantees a prime in (i^90, (i+1)^90) for each integer i. This does not formally imply the interval statement [x, x+90x^{89/90}] for every real x ≥ 1, because the prime guaranteed by the power-interval theorem may lie below x. The derivation of (2.8), and eventually (5.2), depends on (2.5). The author must either reproduce the argument from [6] that yields the stronger statement, or quote an explicitly stated stronger theorem from [6] with a precise location. The same issue affects (2.10) from Dudek's cube theorem, although the final n ≥ 106 claim depends primarily on (2.5).","section":"§2.1, Eq. (2.5)"},{"comment":"Equation (3.1) is false as stated: it claims for all n ≥ 2 and all x < 2×10^19 that [x, x+x^{1-1/n}] contains a prime. For n = 2 and x = 8, the interval is [8, 8+√8] ≈ [8, 10.828], which contains no prime. The later use for n ≥ 106 can be repaired because x^{1-1/106} is much larger than √x for the x-values in question, and the maximal-gap table appears to suffice there; but the universal assertion must be corrected, e.g. by restricting the claimed n-range. In addition, 'easily verified by inspecting a table' is not a proof; the author should state the exact verified property (which gaps are bounded by p_i^{1-1/n}) and cite the exhaustive computation or include enough data to make the verification reproducible.","section":"§3, Eq. (3.1)"},{"comment":"The final all-x coverage depends on three external inputs: the maximal-gap verification for x < 2×10^19, the linear-interval data from [11] as reproduced in Table 2, and the unproved strengthening (2.5). Even if (2.5) is accepted, the paper should make explicit how the maximal-gap bound and the (x_Δ, Δ) pairs are combined for n = 106, and in particular why the union of intervals [x_Δ, (Δ−1)^n] indeed covers [4×10^18, (Δ_max−1)^n]. The overlap is plausible from Tables 1 and 3, but the verification is asserted rather than demonstrated. Given the central claim's dependence on this numerical covering, a short verification script or a more detailed table would materially strengthen the paper.","section":"§5, Eq. (5.1)–(5.2)"}],"minor_comments":[{"comment":"The abstract states x ≥ exp(3 exp(33)), while the body consistently uses exp(exp(33)). This discrepancy should be resolved in favor of the body's value unless the abstract's larger threshold is intended.","section":"Abstract"},{"comment":"Equation (2.9) says i ≥ exp(exp(33)), but (2.10) says x ≥ exp(exp(33)); for consistency, state whether the endpoints are inclusive and whether the inequalities are strict or non-strict in all displayed results.","section":"§2.2"},{"comment":"References [8]–[10] rely on Wikipedia, Prime Pages, and software code. For a formal explicit-number-theory paper, it would be preferable to cite a peer-reviewed computation or provide the full table and a description of the verification method.","section":"References"},{"comment":"There are several minor typographical/formatting issues: 'Cully-Hugill+Johnston' should be 'Cully-Hugill and Johnston'; some displayed equations have inconsistent spacing and missing parentheses in the extracted text; the threshold in Table 1 for n=91 is written with a period in '1.762594084 · 1016005' and should be formatted consistently.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The central idea is reasonable and the final theorem, if properly supported, would be a worthwhile explicit result. The most important issue is whether (2.5) is actually proved in [6]; if not, the main theorem may be unsupported. This should be checked carefully during revision, possibly by asking the authors of [6] to confirm the stronger interval statement. The false claim in (3.1) and the informal maximal-gap verification are also fixable but should not be left as they are in a journal submission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper takes known effective prime-between-powers results, a table of maximal prime gaps, and Cully-Hugill–Lee linear intervals, and assembles them into explicit statements about primes in [x, x+x^(1−1/n)]. The specific package is new: the n≥106, x≥1 claim and the n≥4, x≥exp(exp(33)) claim are not in the cited literature. As a compilation, it is clean and honest, and the paper explicitly notes where the argument stops short (n=105 has an unresolved gap). No new analytic method is introduced, but that is not a flaw if the assembly is correct.\n\nThe soft spots are real. First, (3.1) is false as stated: for n=2 and x=8, the interval [8, 8+√8] ≈ [8,10.83] contains no prime. The statement may hold for n≥106, but the universal quantifier over n≥2 is overbroad and should be corrected. Second, and more seriously, the derivation of (2.5) from Cully-Hugill and Johnston's Theorem 1.4 is asserted by 'inspection' without any detail. The cited theorem only guarantees a prime between consecutive 90th powers, i.e., in [i^90,(i+1)^90] for integer i. The stronger claim that for every real x≥1 the interval [x, x+90x^(89/90)] contains a prime is not a formal consequence of that statement by simple algebra; it depends on where inside the power interval the prime actually sits. The same issue affects (2.10) from Dudek's cube theorem. If this strengthening is not actually established, the coverage for large x in (5.2) collapses: using only the integer-power theorem with k=ceil(x^(1/90)) gives a threshold around x≈10^1345 for n=106, which exceeds the endpoint of the linear-interval coverage, so the final all-x≥1 claim would be unproved. The author needs to provide the missing details or quote a theorem that directly gives the real-x statement.\n\nI should note that the reader's concern about (2.18) does not survive close reading: for n in (3,n1], the [27]^{n/(n−3)} term dominates exp(exp(33)), so the condition is not actually omitted. That is a minor point in the author's favor.\n\nWho is this for? Readers who need explicit prime-gap bounds for computations or for removing 'sufficiently large' conditions will find it useful, once the gaps are fixed. It deserves a serious referee, but the referee should insist on a full justification of (2.5) (or a revised statement that does not depend on it) and a correction of (3.1). My own inclination is skeptical until the 'inspection' claim is made reproducible.","headline":"A useful but uneven assembly of explicit short-interval prime bounds; the headline claim for n≥106 depends on an unproved 'inspection' strengthening that needs to be substantiated before the result can be trusted.","tokens_in":6604,"tokens_out":4558,"would_cite":false,"duration_ms":45953,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every x >= 1, a prime lies in [x, x + x^(1 - 1/106)].","keywords":["primes in short intervals","effective bounds","prime gaps","primes between powers","sub-linear intervals","explicit estimates","universal prime gap bound"],"falsifier":"Find one real x >= 1 with no prime in [x, x + x^(1 - 1/106)]; since the theorem is universal, that single counterexample would settle it. Short of that, show that the proofs behind references [6] and [7] only establish prime-between-powers for integer bases, not all real x, which would break the 'by inspection' steps (2.5) and (2.10).","tokens_in":5582,"feed_emoji":"🔢","tokens_out":12085,"duration_ms":127464,"temperature":0.7,"pith_summary":"The paper's aim is to turn a century-old ineffective existence theorem for primes in short intervals into a fully explicit, universal statement. It combines three independent effective inputs: results on primes between 90th powers and between cubes, verified maximal prime gaps, and explicit high-precision linear interval data. The strongest claim is that for every n >= 106 and every real x >= 1, the interval [x, x + x^(1 - 1/n)] contains at least one prime. If correct, this means the next prime after any x is no farther than x^(0.9906), and the classical 1930/1933 sub-linear interval theorems hold for all x rather than only for 'sufficiently large' x. The paper also gives explicit astronomical thresholds for smaller exponents, such as n >= 4 with x >= exp(exp(33)).","feed_headline":"Every x has a prime within x^0.9906","feed_subtitle":"An explicit n>=106 argument turns century-old short-interval theorems into a bound that holds for all real x >= 1.","key_machinery":"The load-bearing object is the family of sub-linear intervals [x, x + x^(1 - 1/n)], with n as a dial controlling interval length. The argument's mechanism is a crossover identity: an effective prime-between-powers result [i^m, (i+1)^m] is equivalent, by the binomial expansion of (i+1)^m, to an interval of length m x^(1 - 1/m) ending at x; comparing that length with x^(1 - 1/n) gives an explicit threshold x = [m^m]^(n/(n-m)). For global coverage at n = 106, the paper stitches three regimes: maximal prime gaps below 2 x 10^19, tabulated effective linear intervals [(1 - 1/Delta)x, x] up to (Delta_max - 1)^106, and the 90th-power interval above that threshold.","core_discovery":"The paper's central claim is that old existence theorems for primes in short intervals can be made fully explicit by splicing together three independent effective inputs. Using a cited 90th-power theorem, interpreted by inspection as [x, x + 90 x^(1 - 1/90)] for all real x >= 1, plus the cited cube theorem with its improved threshold, plus the known table of maximal prime gaps and explicit linear interval data, the paper derives three effective statements: for n >= 4 and x >= exp(exp(33)) the interval [x, x + x^(1 - 1/n)] contains a prime; for n >= 91 and x >= [90^90]^(n/(n-90)) it contains a prime; and for n >= 106 the statement holds for every x >= 1. The last statement yields the universa","pith_inferences":["A direct check of the two 'by inspection' strengthenings in the cited proofs is a natural next step; if either fails, the universal claim would still likely hold for all x above some finite threshold, but not exactly as stated.","Any future reduction of the 90th-power exponent would automatically lower the all-x n through the crossover formula x = [m^m]^(n/(n-m)).","The n = 105 middle gap is a concrete computational target: a few more (x_Delta, Delta) pairs would either close it or reveal a genuine boundary.","The same splice-and-cover recipe could be applied to other explicit interval data, such as primes in arithmetic progressions, whenever analogous tables exist."],"forward_implications":["Every starting point x >= 1 is guaranteed to contain a prime within distance x^(1 - 1/106), giving an explicit and checkable stopping rule for prime searches.","The classical 1930/1933 interval exponents hold unconditionally for all real x, not just asymptotically.","For any n >= 106 the same universal guarantee holds, so one can trade a slightly longer allowable gap for exactly the same coverage.","For n between 4 and 106, explicit but astronomical thresholds are supplied; for example n = 4 requires x >= exp(exp(33)).","The n = 105 case leaves a bounded middle range undecided, and only finitely many additional explicit linear-interval pairs would be needed to close it."],"supporting_citations":[{"why":"Provides the 90th-power prime-between-powers theorem that the paper interprets as the stronger interval statement (2.5).","marker":"[6]"},{"why":"Provides the cube prime-between-powers theorem with the threshold exp(exp(33.217)), the basis for the stronger statement (2.10).","marker":"[7]"},{"why":"Updates the cube threshold to exp(exp(32.892)), which the paper rounds to exp(exp(33)).","marker":"[4]"},{"why":"Provides the tabulated (x_Delta, Delta) pairs used to cover the middle range up to (Delta_max - 1)^106.","marker":"[11]"},{"why":"One source for the maximal prime gap data that covers x < 2 x 10^19.","marker":"[8]"},{"why":"Table of known maximal prime gaps used for the low-x coverage.","marker":"[9]"},{"why":"Software used to verify or extend the maximal prime gap table.","marker":"[10]"}],"fun_headline_variants":["Every x≥1 has a prime in x^0.9906","For all real x, a prime lies within x^0.9906","Prime exists in every x≥1 interval of length x^0.9906","Century-old prime gap theorem now explicit: prime within x^0.9906"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is the paper's 'by inspection' claim that the cited 90th-power theorem and cube theorem actually prove the stronger all-real-x interval statements (2.5) and (2.10); if that reading is wrong, the n >= 106 all-x result would not be established by this paper.","fun_headline_variants_meta":{"raw":{"variants":["Every x≥1 has a prime in x^0.9906","For all real x, a prime lies within x^0.9906","Prime exists in every x≥1 interval of length x^0.9906","Century-old prime gap theorem now explicit: prime within x^0.9906"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001868,"raw_usage":{"total_tokens":7258,"prompt_tokens":925,"completion_tokens":6333,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":669,"completion_tokens_details":{"reasoning_tokens":6248}},"tokens_in":669,"tokens_out":6333,"duration_ms":50757,"temperature":1.0,"reasoning_tokens":6248,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T16:13:03.122893+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find one real x >= 1 with no prime in [x, x + x^(1 - 1/106)]; since the theorem is universal, that single counterexample would settle it. Short of that, show that the proofs behind references [6] and [7] only establish prime-between-powers for integer bases, not all real x, which would break the 'by inspection' steps (2.5) and (2.10).","supporting_citations":[{"cited_title":"An Explicit Result for Primes Between Cubes","cited_arxiv_id":"1401.4233","evidence_quote":"Provides the cube prime-between-powers theorem with the threshold exp(exp(33.217)), the basis for the stronger statement (2.10)."},{"cited_title":"Primes between consecutive powers","cited_arxiv_id":"2107.14468","evidence_quote":"Updates the cube threshold to exp(exp(32.892)), which the paper rounds to exp(exp(33))."},{"cited_title":"Explicit Interval Estimates for Prime Numbers","cited_arxiv_id":"2103.05986","evidence_quote":"Provides the tabulated (x_Delta, Delta) pairs used to cover the middle range up to (Delta_max - 1)^106."},{"cited_title":"Prime gap","cited_arxiv_id":null,"evidence_quote":"One source for the maximal prime gap data that covers x < 2 x 10^19."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Table of known maximal prime gaps used for the low-x coverage."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Software used to verify or extend the maximal prime gap table."}],"review_version":1}