{"id":"bcff83b8-87ae-4cf4-bf2d-5caba2da1b51","arxiv_id":"2508.12139","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Infinitely many three-term prime progressions exist in the thin Bohr set {p prime : ||alpha p|| <= 1/p^tau} for tau in (0,1/8).","lead":"This paper claims that for any irrational alpha, infinitely many three-term arithmetic progressions exist among primes p with the fractional part of alpha p within 1/p^tau, for tau below 1/8. A generalist might care because it tests how much additive structure survives in extremely sparse, alpha-dependent subsets of the primes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central claim not supported by supplied text: the full manuscript is a different combinatorial paper, so the required exponential-sum estimate for primes in Bohr neighborhoods is neither stated nor proved.","rationale":"The paper advertises a number-theoretic theorem, but the supplied full text is a different paper. Under the reviewing rule, I treated the mismatch as in-scope: it means the manuscript contains no derivation of the central claim. The claim itself is plausible and within the range of current analytic number theory—thin Bohr sets around primes have been studied with exponential-sum methods, and tau < 1/8 is a conservative exponent consistent with type-I/II estimates—but plausibility is not proof. The load-bearing component is the hidden analytic estimate. If it is proved unconditionally, the main theorem likely follows; if it is assumed or derived from an unproved conjecture, the theorem is conditional. The abstract does not even name the input. Therefore the only honest verdict is unverified, matching the reader. I propose a concrete verification step rather than a substantive mathematical objection, because the supplied body is missing. No independence or correctness issue can be assessed until the actual full text is inspected.","tokens_in":15634,"tokens_out":4197,"duration_ms":51597,"concrete_test":"Obtain the actual arXiv:2508.12139 full text. Locate the key estimate (likely labelled Proposition/Lemma 3.1 or 4.1) bounding S(theta) = sum_{p <= X, ||alpha p|| <= p^{-tau}} e(p theta) for the level needed to count 3-APs. Check whether the proof is fully present and unconditional; in particular, verify that the exponent tau < 1/8 emerges from a Type-I/Type-II decomposition via Vaughan's identity and that no step invokes an unproved distribution hypothesis. Re-run the proof's threshold numerology for 1/8. If the key lemma is assumed or conditional, downgrade the theorem to conditional.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract's central claim—infinitely many nontrivial 3-term APs in primes with ||alpha p|| <= p^{-tau} for fixed irrational alpha and tau in (0,1/8)—would require an unconditional additive-combinatorial input: a nontrivial bound on exponential sums over primes weighted by the Bohr condition ||alpha p|| <= p^{-tau}, strong enough to control the ternary (or binary) count over p <= x. The supplied full text is not this paper; it is arXiv:2508.12135, a lozenge-tiling reflection principle paper. Consequently, the manuscript as provided contains no statement (let alone proof) of the necessary analytic estimate, no verification that tau < 1/8 is the correct threshold, and no indication whether the result is conditional on GRH, Elliott–Halberstam, or an unproved level-of-distribution hypothesis. This is the single load-bearing link: the claimed theorem is exactly as strong as that hidden estimate. Without seeing and checking that estimate, the abstract alone cannot support the conclusion. The proper disposition is not acceptance or rejection but unverified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The abstract of arXiv:2508.12139 claims that for a fixed irrational alpha and for tau in (0, 1/8), there exist infinitely many nontrivial three-term arithmetic progressions consisting of primes p satisfying ||alpha p|| <= 1/p^tau. A binary Goldbach-type variant is also announced. The supplied full text, however, is not the corresponding number theory paper: it is arXiv:2508.12135, a combinatorics paper on a reflection principle for nonintersecting paths and lozenge tilings with free boundaries. It contains no statement about primes, Bohr sets, exponential sums, or the parameter tau. Thus, as submitted, the manuscript's central claim is unsupported by any proof or even by a theorem statement beyond the abstract.","tokens_in":15916,"tokens_out":2874,"duration_ms":33787,"significance":"If the abstract's claim were proved unconditionally, it would be a noteworthy additive number theory result: it would show additive structure inside a thin Bohr-type subset of the primes for arbitrary fixed irrational alpha, with a nontrivial exponent range tau < 1/8, together with a binary Goldbach-type analogue. Such a result would likely require a delicate exponential-sum or level-of-distribution estimate for primes restricted by ||alpha p|| <= p^{-tau}, and the threshold 1/8 would be a substantive quantitative feature. However, the manuscript as supplied gives the referee no way to assess the derivation: the full text is an unrelated combinatorics paper, and the abstract contains no estimates, no hypotheses beyond alpha irrational and tau in (0,1/8), and no indication whether the claim is unconditional or conditional on an unproved hypothesis. The potential significance cannot currently be credited.","major_comments":[{"comment":"The body of the submission is not the paper described by the abstract. It is a lozenge-tiling and nonintersecting-paths paper (arXiv:2508.12135) with no mention of primes, Bohr sets, the condition ||alpha p|| <= 1/p^tau, or additive problems. The central claim of the abstract therefore has no proof in the submitted manuscript. This is a load-bearing omission, not a presentational issue: the claimed theorem is exactly as strong as the missing analytic estimate, and no such estimate is stated or proved anywhere in the supplied text.","section":"Full text"},{"comment":"Even treating the abstract as the only mathematical content, it omits the analytic input necessary to make the claim checkable. A result of this type requires a nontrivial bound on exponential sums over primes weighted by a Bohr-neighborhood condition of vanishing width p^{-tau}, or an equivalent level-of-distribution statement. The abstract does not state such an estimate, does not say whether it is proved or assumed, and does not indicate whether the theorem is unconditional or conditional on, for example, GRH or a conjecture. Consequently the claimed exponent range tau < 1/8 cannot be verified from the manuscript as submitted.","section":"Abstract"},{"comment":"The announced binary Goldbach-type result is not formulated. It is unclear whether the statement is an analogue for sums of two primes satisfying the Bohr condition, whether it is an asymptotic or an infinitude statement, and what error terms are claimed. Without a precise statement and proof, this part of the abstract is not assessable.","section":"Abstract"}],"minor_comments":[{"comment":"The phrase 'non-trivial three-term arithmetic progressions' is not defined; presumably it means progressions with distinct primes, but this should be stated.","section":"Abstract"},{"comment":"The manuscript gives no references to prior work on primes in Bohr neighborhoods, Diophantine approximation with primes, or related exponential-sum estimates. Such context would be needed in a complete submission.","section":"General"},{"comment":"The full text appears to be a different article with its own title, abstract, and numbering. This mismatch should be resolved editorially; the current submission is not suitable for normal peer review as it stands.","section":"Full text"}],"recommendation":"uncertain","confidential_remarks":"The decisive issue is procedural: the full text supplied with arXiv:2508.12139 is actually arXiv:2508.12135, a combinatorics paper. I cannot evaluate the claimed number-theoretic theorem because its proof is absent. I recommend that the editor return the submission to the authors to supply the correct manuscript, and that the scientific assessment begin afresh once the actual paper, including the required exponential-sum or level-of-distribution estimate, is provided. I found no evidence of misconduct; the mismatch appears to be a submission error. My 'uncertain' verdict reflects the fact that the claim is neither confirmed nor refuted by the supplied material."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, this one is easy in one way and frustrating in another. The abstract of 2508.12139 states a plausible and genuinely new-sounding theorem: infinitely many 3-term APs among primes p with ||alpha p|| <= p^{-tau} for fixed irrational alpha and tau<1/8, plus a binary Goldbach analogue. If it's true, it's a meaningful extension of Green-Tao to a thin alpha-dependent prime set. But I cannot assess it. The supplied full text is not this paper; it is Byun's lozenge-tiling reflection principle paper, arXiv:2508.12135. So every load-bearing detail of the number theory claim—the exponential sum estimate over primes in short Bohr neighborhoods, the level of distribution, whether the result is unconditional or depends on GRH or Elliot-Halberstam—is simply absent from what I was given. The reader's stress test is right: the theorem is exactly as strong as that hidden estimate, and the hidden estimate is not there.\n\nWhat can be said in the paper's favor: the abstract is clearly written, the target result is not obviously in the literature, and the threshold tau<1/8 is specific enough to be checkable. No circularity is visible in the statement itself. But that's about all.\n\nThe soft spot is not a subtle argument gap; it's a missing manuscript. A desk editor who receives number-theory abstract plus a combinatorics PDF should return it as defective. If the real full text exists, it needs to be evaluated on its own merits, and I'd want to see the proof of the analytic input before believing the theorem. As it stands, this is a desk reject, not because the idea is bad but because there is nothing to referee.\n\nRecommendation: return to authors for the correct full text; if the actual paper matches the abstract, then it deserves serious refereeing. Right now, no.","headline":"The abstract is a plausible new result about primes in thin Bohr sets, but the supplied full text is an unrelated lozenge-tiling paper, so there is nothing to referee.","tokens_in":16318,"tokens_out":1909,"would_cite":false,"duration_ms":21089,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N05","11B25","11P32","11K60"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that for any irrational $\\alpha$ and $\\tau<1/8$, the set of primes $p$ with $\\lVert\\alpha p\\rVert\\le p^{-\\tau}$ contains infinitely many nontrivial three-term arithmetic progressions.","keywords":["three-term arithmetic progressions","primes","Bohr sets","irrational rotations","Diophantine approximation","binary Goldbach problem","thin prime subsets"],"falsifier":"For a fixed irrational $\\alpha$ with known continued fraction expansion, enumerate primes $p\\le X$ and count triples $(p_1,p_2,p_3)$ in arithmetic progression with $p_1<p_2<p_3\\le X$ and $\\lVert\\alpha p_i\\rVert\\le p_i^{-\\tau}$ for a fixed $\\tau\\in(0,1/8)$. If this count does not grow without bound as $X$ increases—or vanishes for some $\\tau<1/8$—the infinitude claim is false; comparing the count's order with the claimed lower bound would test the proof mechanism directly.","tokens_in":15565,"feed_emoji":"🔢","tokens_out":11189,"duration_ms":114963,"temperature":0.7,"pith_summary":"The paper is trying to establish that for any irrational $\\alpha$ and any fixed $\\tau\\in(0,1/8)$, the primes $p$ satisfying $\\lVert\\alpha p\\rVert\\le p^{-\\tau}$ still contain infinitely many three-term arithmetic progressions with nonzero common difference. The condition defines a thin Bohr set: each selected prime must land extremely close to a multiple of $\\alpha$, with the allowed distance shrinking as $p$ grows. A sympathetic reader cares because this tests whether additive patterns in the primes survive a density-zero selection governed by an irrational rotation, and gives a quantitative range $\\tau<1/8$ for which the pattern persists. The paper also announces a binary Goldbach-type problem for the same kind of thin prime set.","feed_headline":"Primes near multiples of an irrational still form infinite 3-term APs","feed_subtitle":"For every irrational α and τ below 1/8, the selected primes remain rich enough to contain infinitely many progressions.","key_machinery":"The operative object is the thin Bohr set $\\mathcal{P}_\\tau(\\alpha)$, defined by the distance condition $\\lVert\\alpha p\\rVert\\le p^{-\\tau}$; the threshold $\\tau<1/8$ is the range in which the paper claims the required analytic control holds. The argument turns on distribution estimates for primes in these short Bohr neighborhoods—how uniformly the primes spread over intervals of multiples of $\\alpha$—since that distribution is what lets one count three-term progressions inside the selected set. The binary Goldbach variant is carried by the same distribution mechanism.","core_discovery":"The central claim is that for an irrational $\\alpha$ and fixed $\\tau\\in(0,1/8)$, the set $\\mathcal{P}_\\tau(\\alpha)=\\{p\\text{ prime}:\\lVert\\alpha p\\rVert\\le p^{-\\tau}\\}$ contains infinitely many nontrivial three-term arithmetic progressions. The novelty is that membership depends on a single irrational multiplier and the selected primes become sparser as $p$ grows, yet additive configurations of length three still occur infinitely often. The paper further states a binary Goldbach-type result for sums of two primes from the same thin Bohr set.","pith_inferences":["One testable extension is to compute, for a concrete irrational $\\alpha$ such as $\\sqrt{2}$, the number of three-term progressions up to $X$ in $\\mathcal{P}_\\tau(\\alpha)$ and compare its growth with the order implied by the claimed proof; a collapse for some $\\tau<1/8$ would indicate a hidden dependence on the Diophantine quality of $\\alpha$.","The threshold $1/8$ may come from the available exponential-sum bounds rather than from a structural barrier; a plausible stronger statement would be that the same infinitude holds for all $\\tau$ below the irrationality-measure exponent of $\\alpha$, though the paper does not claim this.","The body text supplied with this submission concerns nonintersecting paths, Pfaffians, and lozenge tilings, and contains none of the prime-counting argument; the summary above follows the abstract. If the body is the intended manuscript, the abstract's claims are not supported by the visible text."],"forward_implications":["For every irrational $\\alpha$, prime three-term progressions exist even after restricting to primes within $p^{-\\tau}$ of a multiple of $\\alpha$, for every $\\tau<1/8$.","The same thin selection is compatible with additive representations: the announced binary Goldbach-type result would give infinitely many binary representations using two primes from the same thin Bohr set.","The exponent range $\\tau\\in(0,1/8)$ provides a quantitative benchmark for how sparse a Diophantine selection can be before additive structure disappears.","If the underlying distribution estimate is proved for the stated range, it supplies a new counting input for additive problems over density-zero prime subsets defined by rotations."],"supporting_citations":[],"fun_headline_variants":["Primes near irrational multiples have infinite APs for τ<1/8","Sparse primes near α-multiples still form infinite APs","Infinite 3-term APs in primes selected by a thin Bohr set","For τ<1/8, irrational-thinned primes contain infinite APs"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The theorem rests on a nontrivial analytic estimate controlling how many primes fall into the thin Bohr neighborhoods of multiples of $\\alpha$; if that estimate is assumed rather than proved, the infinitude result is conditional.","fun_headline_variants_meta":{"raw":{"variants":["Primes near irrational multiples have infinite APs for τ<1/8","Sparse primes near α-multiples still form infinite APs","Infinite 3-term APs in primes selected by a thin Bohr set","For τ<1/8, irrational-thinned primes contain infinite APs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000409,"raw_usage":{"total_tokens":1881,"prompt_tokens":592,"completion_tokens":1289,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":336,"completion_tokens_details":{"reasoning_tokens":1208}},"tokens_in":336,"tokens_out":1289,"duration_ms":13044,"temperature":1.0,"reasoning_tokens":1208,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T19:34:11.317927+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed irrational $\\alpha$ with known continued fraction expansion, enumerate primes $p\\le X$ and count triples $(p_1,p_2,p_3)$ in arithmetic progression with $p_1<p_2<p_3\\le X$ and $\\lVert\\alpha p_i\\rVert\\le p_i^{-\\tau}$ for a fixed $\\tau\\in(0,1/8)$. If this count does not grow without bound as $X$ increases—or vanishes for some $\\tau<1/8$—the infinitude claim is false; comparing the count's order with the claimed lower bound would test the proof mechanism directly.","supporting_citations":[],"review_version":1}