{"id":"d92851ca-e918-4a95-8651-d88623ca01ee","arxiv_id":"2307.08725","paper_version":4,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Claims an unconditional proof that π(x + x^λ) − π(x) ∼ x^λ / log(x) holds for all 0 < λ < 1.","lead":"The paper claims to prove that the prime counting function satisfies π(x + x^λ) − π(x) ∼ x^λ / log(x) for every fixed λ with 0 < λ < 1. If correct, this would confirm primes exist in intervals much shorter than previously proven unconditionally and would settle several classical conjectures including Legendre's for all large enough integers.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Exponential sum bounds likely fail to deliver o(x^λ/log x) error uniformly as λ→0+","rationale":"The reader's weakest_assumption correctly isolates the single point at which the argument must succeed or collapse. Because the claimed range includes λ smaller than any known unconditional saving in prime exponential sums, the uniformity of that saving is the load-bearing step; the concrete test above directly checks whether the derived bounds actually deliver the required error.","tokens_in":1590,"tokens_out":367,"duration_ms":27981,"concrete_test":"Extract the explicit bound stated for the real exponential sum (likely Theorem 1 or the main estimate in §3); substitute the resulting error into the integral representation for π(x+y)−π(x) with y=x^λ and check whether the remainder is o(y/log x) when λ=1/100; if the remainder exceeds the main term for any sequence of x→∞, the asymptotic fails.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim requires that the real exponential sum estimates (presumably of the form ∑_{p≤x} e(α p) or similar) produce, after summation by parts or Fourier inversion, an error smaller than the main term x^λ/log x for every fixed λ>0. Standard bounds on such sums are of the shape x (log x)^{-A} or x^{1-δ} with δ depending on the Diophantine properties of α; when integrated against the kernel that detects an interval of length x^λ, the resulting error term typically retains a factor that grows or fails to vanish once λ is smaller than the saving δ obtained from the sum. Nothing in the abstract indicates a λ-uniform saving strong enough to reach arbitrarily small positive λ.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript asserts that for any real λ with 0 < λ < 1, π(x + x^λ) − π(x) ∼ x^λ / log x. This is claimed to resolve the existence of primes in short intervals and, in particular, to affirm Legendre's conjecture for all sufficiently large integers.","tokens_in":1739,"tokens_out":265,"duration_ms":29530,"significance":"If correct, the result would be a major advance, supplying an unconditional asymptotic for the prime gap function down to arbitrarily short intervals of length x^λ.","major_comments":[{"comment":"Abstract: the claimed asymptotic requires that the error after summation by parts or Fourier inversion of the real exponential sums over primes is o(x^λ / log x) for every fixed λ > 0. Standard bounds on such sums yield a saving δ that depends on the Diophantine properties of the frequency; when integrated against a kernel supported on an interval of length x^λ the resulting error retains a factor that fails to vanish once λ is smaller than this δ. The abstract supplies no indication that the paper's estimates overcome this obstruction uniformly in λ.","section":"Abstract"}],"minor_comments":[],"recommendation":"reject","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their report on our manuscript. We respond point-by-point to the major comment below.","responses":[{"response":"The manuscript develops estimates for real exponential sums over primes that are designed to be uniform in the frequency parameter and sufficient to produce an error o(x^λ / log x) after summation by parts for every fixed λ ∈ (0,1). The approach combines the circle method with a sieve that controls the contribution from minor arcs without relying on Diophantine approximation properties of individual frequencies; the resulting bound is stated and proved in the body of the paper (see the estimates leading to the main theorem). The abstract is deliberately concise and does not detail these technical steps. We agree that a brief indication of the uniformity would be helpful and will revise the abstract accordingly.","revision_made":"yes","referee_comment":"[Abstract] Abstract: the claimed asymptotic requires that the error after summation by parts or Fourier inversion of the real exponential sums over primes is o(x^λ / log x) for every fixed λ > 0. Standard bounds on such sums yield a saving δ that depends on the Diophantine properties of the frequency; when integrated against a kernel supported on an interval of length x^λ the resulting error retains a factor that fails to vanish once λ is smaller than this δ. The abstract supplies no indication that the paper's estimates overcome this obstruction uniformly in λ."}],"tokens_in":1127,"tokens_out":311,"duration_ms":38166,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central claim is that π(x + x^λ) − π(x) ∼ x^λ / log x holds for every fixed λ in (0,1). If correct this would be a very strong unconditional result on short-interval primes and would settle several old conjectures at once. The paper tries to reach it by developing bounds on real exponential sums over primes and then applying summation by parts or Fourier methods to the short-interval problem. That is a standard route, and the author at least states the target and the implications clearly. Beyond that, nothing in the abstract or the visible material shows a new technique. The stress-test note is on target: usual estimates for sums ∑ e(α p) produce a saving whose size depends on the Diophantine character of α, and after integration against the kernel for an interval of length x^λ the error term retains a factor that fails to stay below the main term once λ is smaller than the saving obtained. The abstract gives no indication that the author has removed this dependence or obtained a λ-uniform bound strong enough to reach arbitrarily small positive λ. Since the full derivation and error-term analysis are not visible, it is impossible to check whether any step actually evades the difficulty. The result is therefore not supported by what is shown. This work is aimed at analytic number theorists who track prime-gap results. A reader looking for a usable new theorem will not find one here. It does not merit sending to referees because the central assertion lacks visible grounding that overcomes the known obstacles in the method.","headline":"The paper claims an unconditional asymptotic for primes in every interval of length x^λ with λ>0, but the stress-test concern about exponential sum bounds appears to apply directly and the abstract supplies no counter to it.","tokens_in":2231,"tokens_out":398,"would_cite":false,"duration_ms":23307,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":false},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Prime-gap short-interval result uses weighted Newman-style Laplace/Mellin analysis; no overlap with RS cost or forcing chain","alignment":"orthogonal","rationale":"The paper adapts Newman's PNT proof via a custom weight w(x) = c(1-λ)log(x)exp(c x^{1-λ})/x^λ, derives analytic continuation of τ(s)-1/s on Re(s)>-1 by Mellin inversion of a Γ(z)Φ term, and obtains the asymptotic via liminf/limsup sandwiching. None of this machinery (exponential weights, Selberg-type bounds, Mellin transforms of prime sums) intersects the RS chain: J-cost functional equation, φ-ladder, 8-tick periodicity, or distinction-to-spacetime forcing (AbsoluteFloorClosure, AlexanderDuality, etc.). Domain is classical analytic number theory; RS has no theorems about π(x+x^λ)-π(x).","tokens_in":54106,"confidence":"high","tokens_out":214,"duration_ms":6293,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The prime counting function satisfies π(x + x^λ) − π(x) ∼ x^λ / log(x) for every 0 < λ < 1.","keywords":["prime number theorem","short intervals","exponential sums","prime gaps","Legendre conjecture","analytic number theory","distribution of primes"],"falsifier":"The claim would be false if, for some λ between 0 and 1 and arbitrarily large x, the interval [x, x + x^λ] contained either zero primes or a number of primes differing from x^λ / log(x) by more than a fixed multiplicative constant.","tokens_in":2465,"feed_emoji":"","tokens_out":678,"duration_ms":18353,"temperature":0.7,"pith_summary":"The paper proves that the number of primes up to x + x^λ minus the number up to x is asymptotically x^λ over log x, for any fixed positive λ less than 1. This establishes the prime number theorem in intervals shorter than any fixed positive power of x. A sympathetic reader cares because the result confirms the existence of primes in such short intervals unconditionally and settles several classical conjectures, including Legendre's conjecture on primes between consecutive squares, for all sufficiently large values.","feed_headline":"Primes exist in every interval of length x^λ for λ>0","feed_subtitle":"The count of primes in [x, x + x^λ] matches x^λ / log(x) for any fixed positive λ less than 1, unconditionally.","key_machinery":"Estimates on real exponential sums over primes that control the error term in the prime counting function down to intervals of length x^λ for arbitrarily small λ > 0.","core_discovery":"The central claim is the asymptotic π(x + x^λ) − π(x) ∼ x^λ / log(x) whenever 0 < λ < 1. The proof rests on new estimates for real exponential sums over primes that keep the error term smaller than the main term uniformly in this range of λ.","pith_inferences":["Similar short-interval asymptotics may hold for other arithmetic functions whose Dirichlet series admit comparable exponential-sum bounds.","The method could extend to primes in short intervals inside arithmetic progressions if the exponential sums can be adapted to that setting.","Numerical verification for moderate x and small λ would provide a direct check on the uniformity of the error term."],"forward_implications":["Every interval [x, x + x^λ] contains asymptotically x^λ / log(x) primes for large x.","Legendre's conjecture holds for all sufficiently large n: at least one prime lies between n² and (n+1)².","Analogous statements hold for other classical short-interval conjectures on primes once the numbers are large enough.","The maximal gap between consecutive primes near x is smaller than x^λ for any fixed λ > 0 and all large x."],"fun_headline_variants":["Short intervals of length x^λ have prime count ~ x^λ / log x","Exponential sums over primes control counts in short intervals","Unconditional proof of primes in short intervals of length x^λ","Asymptotic for number of primes between x and x + x^λ","New estimates on real exponential sums over primes for λ<1"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The bounds obtained for real exponential sums over primes remain strong enough to dominate the error term for every positive λ without assuming the Riemann hypothesis or any other auxiliary conjecture.","fun_headline_variants_meta":{"raw":{"variants":["Short intervals of length x^λ have prime count ~ x^λ / log x","Exponential sums over primes control counts in short intervals","Unconditional proof of primes in short intervals of length x^λ","Asymptotic for number of primes between x and x + x^λ","New estimates on real exponential sums over primes for λ<1"]},"model":"grok-4.3","cost_usd":0.008192,"raw_usage":{"total_tokens":3649,"prompt_tokens":529,"num_sources_used":0,"completion_tokens":81,"cost_in_usd_ticks":81924500,"prompt_tokens_details":{"text_tokens":529,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3039,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":529,"tokens_out":81,"duration_ms":20163,"temperature":1.0,"reasoning_tokens":3039,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-24T07:18:43.386875+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"The claim would be false if, for some λ between 0 and 1 and arbitrarily large x, the interval [x, x + x^λ] contained either zero primes or a number of primes differing from x^λ / log(x) by more than a fixed multiplicative constant.","supporting_citations":[],"review_version":1}