{"id":"583462b5-1838-44a9-af13-532c6fc02bd9","arxiv_id":"2610.15674","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"partial","parameter_count":0,"one_line_summary":"The conditional implication from smooth shifted primes to Erdős Problem #824 is formalized in Lean, and a sign-change check adapts a recent preprint to the needed case.","lead":"This paper shows that if many primes p have p+1 free of large prime factors, then the number of coprime pairs a<b<x with equal sum of divisors is at least x^{2-\\varepsilon}. It formalizes this conditional proof in Lean and argues that a recent OpenAI preprint, if correct, implies the needed prime condition for p+1, yielding h(x)=x^{2-o(1)} conditionally.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"No significant objection identified; the conditional claim is honestly stated and the formalization is sound.","rationale":"The paper is a model of conditional results: it separates the unconditional formalized implication from the unverified preprint dependency. The stress-tester finds no additional flaws. The formalization appears robust, the sign-change argument is plausibly correct, and the paper explicitly and repeatedly caveats the reliance on [10]. Therefore the reader's ACCEPT verdict stands.","tokens_in":7735,"tokens_out":2628,"duration_ms":21256,"concrete_test":"Verify that the Lean formalization's definition of SmoothSuccessorHypWeak correctly encodes the quantifier order (∀δ>0, δ<1/4, ∀η>0, ∃y0, ∀y≥y0, ...) as required in Hypothesis A+. If a quantifier is misplaced, the formal theorem may not align with the paper's claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper transparently states that its main conclusion (Corollary 1.5) depends on the correctness of the unrefereed OpenAI preprint [10], which the authors have not independently verified. This is not a hidden flaw but an acknowledged limitation. The Lean formalization of the unconditional implication (Theorem 1.3) is machine-checked, and the sign-change argument in §2 appears logically sound based on the description. No new load-bearing vulnerability beyond this dependence is identified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"This paper addresses Erdős Problem #824, which asks whether h(x), the number of coprime pairs 1 ≤ a < b < x with equal sum of divisors, satisfies h(x) > x^{2-o(1)}. The authors prove unconditionally that if Hypothesis A+ holds (for every δ>0 there are y^{1-o(1)} primes p ≤ 5y with p+1 free of prime factors exceeding y^δ), then h(x) > x^{2-ε} for every ε>0 and all sufficiently large x. This implication is formalized in Lean 4. The authors then argue that a sign-change adaptation of the recent OpenAI preprint [10] (which claims the analogous statement for p-1) would imply A+, leading to the conditional conclusion h(x)=x^{2-o(1)}. The paper is transparent about the dependence on the unrefereed preprint [10] and about the fact that the Lean formalization was produced by ChatGPT without human review.","tokens_in":7965,"tokens_out":6843,"duration_ms":63814,"significance":"If the conditional conclusion holds, it settles a long-standing problem of Erdős. The unconditional implication (Theorem 1.3) is nontrivial and cleanly proven; its formalization in Lean is a valuable contribution to the growing body of verified analytic number theory. The paper's honest handling of its reliance on unverified, AI-generated work is a model of scientific transparency. The sign-change adaptation, while not independently verified, is a plausible extension that the authors have partially checked. The paper is a well-structured, self-aware piece that makes the best possible use of a result it cannot itself confirm.","major_comments":[{"comment":"The bound (k choose t) ≤ 2k is incorrect; for k large and t ≈ ηk, the binomial coefficient is exponential in k. The subsequent logarithmic calculation uses (k+1)log2, which suggests the intended bound is 2^k, consistent with a corrected argument (and with the Lean formalization). The manuscript should correct the displayed inequality and the following line to use 2^k (or 2^{k+1}) instead of 2k. The error does not affect the formalization but could confuse readers.","section":"§3.3","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The title and abstract contain a typographical artifact: \"P AIRS\" should be \"PAIRS\" (the space after P is a formatting glitch). The same appears in the header of §1.","section":"Title and abstract","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"In the sentence beginning \"By an argument of Pollack and Pomerance\" in the abstract, the phrase \"sufficiently\" is a misspelling of \"sufficiently\".","section":"§1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The determinant parametrization derivation states mh' - m'h = k. The paper should clarify that k is the same integer that appears in the shift-cancellation theorem, and note that the original argument in [10] already treats both signs of k. A brief reference to the relevant equation in [10] would help the reader.","section":"§2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Equation (3) for U: the bound U ≤ (1+L/log2)^{L^{1/2}} is correct but unnecessarily crude; for completeness, the exponent in the smooth count could be made more precise (though the final log U = o(L) is what matters). This is not a flaw.","section":"§3.2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The sentence \"By (1), (3) and M ≥ L^{B(1-η)}\" should reference the bound M ≥ L^{B(1-η)} which follows from A+. The text as written says \"M ≥ L^{B(1-η)}\" but the superscript formatting may be ambiguous; ensure it is typeset correctly.","section":"§3.3","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The Lean lemma SmoothSuccessorHypWeak uses real exponent arithmetic; the choice of ℝ for the exponent is natural, but the paper should note that y^δ is defined as the power of a real number, which is non-Archimedean in ℕ contexts. The formalization likely handles this via coercion; the description could be more explicit.","section":"§4","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"The paper's central claim depends on an unrefereed, AI-generated preprint, but the authors are completely open about this. The unconditional implication and its Lean formalization are the paper's own contribution and are sound. The minor error in the binomial bound is a presentation issue; the Lean code presumably uses the correct exponential bound. I recommend acceptance after the authors correct the typographical and bounding errors noted in the minor comments. The paper is a good fit for the journal and follows proper norms of conditional reasoning."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"This paper does exactly what it says on the tin: it formalizes in Lean the implication that if there are enough primes with smooth successors (Hypothesis A+), then h(x) > x^{2-ε} for Erdős Problem #824. The formalization is the main new contribution, and it is clean—the code compiles, the axioms are standard, and the proof follows the Pollack–Pomerance argument with explicit parameters. The sign-change check in §2 is competent, though it is a straightforward adaptation of the OpenAI preprint's sign. The paper is scrupulously honest about its limitations: the conclusion depends on [10], an unrefereed AI-generated preprint whose core estimates (about 2800 lines) the authors have not independently verified. They also disclose that the Lean code was produced by ChatGPT without human mathematical review. So the formalization verifies the logic of the implication, but it does not verify the premise.\n\nWhere are the soft spots? The central claim (Corollary 1.5) is entirely contingent. If the OpenAI preprint turns out to be wrong, the paper's main advertised result collapses—though the formalized Theorem 1.3 remains a valid conditional theorem regardless. The novelty of the mathematical reduction itself is low: it is a repackaging of Pollack–Pomerance with a slightly weaker hypothesis (y^{1-o(1)} rather than y/(log y)^{O(1)}). The sign-change check is necessary but not deep. The paper also does not attempt to verify the OpenAI preprint's new method, which is a large unresolved piece.\n\nThat said, the paper is not overclaiming. It is a short, honest note that formalizes a known reduction and checks a sign change. The formalization has value: it gives a machine-checked backbone to the conditional claim. A reader who wants to understand how the Pollack–Pomerance argument works with a y^{1-o(1)} hypothesis can find a clean exposition here.\n\nI would send this to a journal. It is a valid contribution as a conditional result with formal verification. The editor should treat it as a short note; the peer review should focus on the correctness of the formalization and the sign-change adaptation, not on the OpenAI preprint. The paper deserves a serious referee, even though its headline result is only as strong as its unverified premise.","headline":"Honest conditional result with a machine-checked implication, but the main conclusion rides entirely on an unrefereed preprint the authors did not verify.","tokens_in":8409,"tokens_out":902,"would_cite":false,"duration_ms":9621,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["11N25","11N05","11A25","11N36","03B35"],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"Erdős Problem #824 conditionally resolved if unrefereed preprint's smooth-prime estimates hold.","keywords":["Erdős problem #824","sum-of-divisors function","coprime pairs","smooth numbers","shifted primes","conditional proof","Lean formalization","combinatorial number theory"],"falsifier":"The paper's central claim would be falsified if the OpenAI preprint's core argument is found to contain a fatal error. One concrete point raised by the authors is the handling of ℓ∈N versus ℓ≥1 in the definition of prime groups; if this leads to an inconsistency in the proof, the preprint's conclusions—and thus the corollary—would be invalid.","tokens_in":9287,"feed_emoji":"","tokens_out":9300,"duration_ms":88157,"temperature":0.7,"pith_summary":"The paper addresses Erdős Problem #824, which asks whether there are nearly $x^{2}$ coprime pairs a<b<x with equal sum of divisors. It formalizes a reduction: if sufficiently many primes have a smooth successor (p+1 free of large prime factors), then the answer is yes. This reduction is written with explicit parameters and verified in Lean 4. The authors then argue that a recent unrefereed OpenAI preprint, which proves the needed smooth-prime estimate for p−1, can be adapted to p+1. Consequently, the problem is conditionally solved with exponent 2−o(1), pending verification of the preprint's core estimates.","feed_headline":"Coprime sum-of-divisors pairs reach exponent 2−o(1)","feed_subtitle":"Erdős #824 is conditionally resolved if an AI preprint's smooth-prime estimates hold.","key_machinery":"The key machinery is the combinatorial pigeonhole argument of Pollack and Pomerance, which the paper rewrites with explicit bounds (parameters B, η, δ, and k). From a set P of primes with smooth successor, one forms A, the set of all products of k distinct primes from P. Every σ(a) is then a smooth integer, so the range of σ on A is bounded by U, the count of smooth integers ≤x. By Cauchy–Schwarz and careful counting, one shows that many distinct a∈A share the same σ-value. Removing a small common prime divisor yields many coprime pairs with equal σ. The paper also checks a technical point in the OpenAI preprint: the determinant parametrization in the Type II estimate works for p+1 after a sign change, because the shift-cancellation theorem applies to both signs of the shift k.","core_discovery":"The paper's central contribution is twofold. First, it makes explicit the argument (due to Pollack and Pomerance) that Hypothesis A+—the existence of $y^{{1-o(1)}}$ primes p≤5y with p+1 free of prime factors exceeding y^δ for every δ>0—implies h(x)>$x^{{2-ε}}$ for every ε>0, with the pairs squarefree. This implication is written out with explicit constants and formalized in Lean 4, providing an unconditional machine-checked proof of the conditional statement. Second, the paper checks that the OpenAI preprint's construction of smooth shifted primes for p−1 extends to p+1 by a sign change in two sections. On that basis, it states that if the preprint's core estimates (about 2800 lines, unverified) are correct, then Hypothesis A+ holds and consequently h(x)=$x^{{2-o(1)}}$. The authors do not vouch for the preprint and emphasize that its main new method remains unverified.","pith_inferences":["The paper's conditional resolution underscores the growing role of AI-generated mathematics in open problems; the burden of verification now rests on the community to either validate or refute the OpenAI preprint's core estimates.","If the preprint's method is verified, it could likely be extended to other shifted primes (e.g., p+a for fixed a) or to Euler's totient function φ, where analogous problems exist.","The Lean formalization, while covering only the combinatorial reduction, may serve as a reusable framework for future conditional results in smooth-prime contexts."],"forward_implications":["If the OpenAI preprint's estimates are correct, Erdős Problem #824 is resolved: h(x)=x^{2-o(1)}.","The Lean formalization proves the implication unconditionally, so any future proof of Hypothesis A+ automatically yields the same bound for h(x).","The pairs may be taken squarefree, a refinement that follows from the argument.","The explicit parametrization highlights how large the set of smooth shifted primes must be to achieve the exponent 2−ε."],"supporting_citations":[{"why":"Supplies the original combinatorial reduction from smooth shifted primes to coprime pairs with equal σ; the paper rewrites and formalizes this argument.","marker":"[7]"},{"why":"Claims the smooth shifted prime estimate for p−1; the paper adapts it to p+1 via a sign change, making the main result conditional on its correctness.","marker":"[10]"}],"fun_headline_variants":["Erdős #824 conditionally solved: sigma pairs hit exponent 2","Coprime sigma-equal pairs reach exponent 2−o(1) conditionally","Lean checks that smooth primes imply sigma pair growth","Erdős #824 hinges on unverified smooth prime estimates","Coprime sigma pairs: exponent 2 assuming smooth primes"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The entire conditional result depends on the correctness of an unrefereed OpenAI preprint whose core estimates (Sections 3–5, about 2800 lines) the authors have not independently verified.","fun_headline_variants_meta":{"raw":{"variants":["Erdős #824 conditionally solved: sigma pairs hit exponent 2","Coprime sigma-equal pairs reach exponent 2−o(1) conditionally","Lean checks that smooth primes imply sigma pair growth","Erdős #824 hinges on unverified smooth prime estimates","Coprime sigma pairs: exponent 2 assuming smooth primes"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":7.9e-05,"raw_usage":{"cost":7.8638e-05,"is_byok":false,"cost_details":{"upstream_inference_cost":7.8638e-05,"upstream_inference_prompt_cost":3.621184e-05,"upstream_inference_completions_cost":4.242616e-05},"total_tokens":1497,"prompt_tokens":944,"completion_tokens":553,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":0,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":462}},"tokens_in":944,"tokens_out":553,"cost_source":"openrouter_usage","duration_ms":5882,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":462,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-14T23:01:14.773267+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"The paper's central claim would be falsified if the OpenAI preprint's core argument is found to contain a fatal error. One concrete point raised by the authors is the handling of ℓ∈N versus ℓ≥1 in the definition of prime groups; if this leads to an inconsistency in the proof, the preprint's conclusions—and thus the corollary—would be invalid.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the original combinatorial reduction from smooth shifted primes to coprime pairs with equal σ; the paper rewrites and formalizes this argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Claims the smooth shifted prime estimate for p−1; the paper adapts it to p+1 via a sign change, making the main result conditional on its correctness."}],"review_version":1}