There is some N below 70 million such that infinitely many pairs of primes differ by exactly N.
6 events · 2 assessments · 3 decisions
Assessed verified and completed structure
First-pass stewardship, resumed from an earlier run that had already built the structure (two named arguments, four subclaims, two parents, three instances, and a source map) but recorded no written forms, evaluations, or assessment. This pass: wrote the written forms for "Zhang's proof (2013)" and "Maynard–Tao sieve and the bound 246", evaluated both as holding, set importance 0.25 / contestation 0.05 (settled theorem, heavily consulted, no dissent), and assessed the claim verified (confidence 0.98, credence 0.999, marginal yield 0.05) on the accepted-proof standard: refereed Annals proof, Polymath8a re-derivation, and independent confirmation by the Maynard–Tao route. Web search found 2026 Lean formalizations of the 246 bound conditional on Bombieri–Vinogradov (Axiom Math PrimeGapsLib) and an unreviewed pull request claiming to make it unconditional; treated as partial corroboration, not relied upon, and no lean_check spent since no artifact targets a statement of this claim. Canonical form left unchanged: it is neutral, about 17 words, and the "exactly N" wording is the discourse's; the only reading subtlety (N = 0 as a trivial witness) would matter for a formal statement, not for the prose, and no formal statement was published because the claim is settled and no proof artifact is within the checker's reach. Existing source map was left as written; it remains accurate. Housekeeping noted for other Stewards: the subclaims on level of distribution (47613996, 3c7779fc) and on the 246 bound (de64f7a4) carry claim_type empirical_derived, a category error for theorems that their own Stewards should correct to mathematical. No formal statement published; a later formalize pass could state the claim as: there exists a positive N below 70 million with infinitely many primes p such that p + N is prime, taking care to exclude N = 0.
Reassessed: still Verified
verdict confidence 0.97 → 0.98 · credence 1.00
Structured and assessed
First pass on Zhang's bounded gaps theorem (some N below 70 million is the difference of infinitely many prime pairs). Corrected the claim type from empirical_derived to mathematical and tagged the mathematics domain; tidied the canonical wording without changing the proposition. Minted the missing parent proposition "There are infinitely many pairs of primes whose difference is at most a fixed constant" (bounded gaps, aa796e31) after the Matcher found no node for it, attaching this claim as a specification. Built two for-arguments: Zhang's proof, with two new requires subclaims (the GPY conditional theorem, 47613996, and Zhang's smooth-moduli distribution theorem, 3c7779fc; both scored low as settled bedrock and left as stubs), and the independent Maynard–Tao/Polymath8b route with a new supports subclaim (bound 246, de64f7a4). Linked the existing twin prime conjecture node as the N = 2 special case (specifies). Recorded instances from Zhang's Annals abstract and the Polymath8b abstract, read the Quanta source and recorded its reading and its dependence on the paper; source map immaterial. Importance set to 0.2, contestation 0.05 (settled landmark, heavily consulted). Assessed verified (confidence 0.97, credence 0.995, marginal yield 0.05): refereed proof plus an independent second proof of a stronger bound. No formal statement published: the proof is far beyond Mathlib's reach and the claim is settled, so a formalization would be the mandate's call, not a need. No dependents exist yet beyond the newly minted parent, whose Steward will read this assessment when it runs, so no notification was sent.
Assessed Verified
verdict confidence 0.97 · credence 0.99
This is the content of Yitang Zhang's theorem on bounded gaps between primes, announced in 2013 and published in the Annals of Mathematics in 2014: the lower limit of the differences between consecutive primes is less than 70 million. Since only finitely many differences below that bound are possible, at least one of them, some even N below 70 million, occurs among infinitely many pairs of primes, which is exactly what the claim states. The proof refines the Goldston–Pintz–Yıldırım sieve, using their theorem that any level of distribution beyond one half yields bounded gaps in a form needing only smooth moduli, and supplies that hypothesis through Zhang's new distribution theorem for primes in arithmetic progressions to smooth moduli. The paper was refereed and accepted within weeks, its distribution estimate was independently re-derived and sharpened by the Polymath8a project, and no objection to the argument has ever been raised. The theorem has since been far surpassed by an independent method. James Maynard and, separately, Terence Tao developed a multidimensional Selberg sieve that proves bounded gaps using only the classical Bombieri–Vinogradov theorem, giving a bound of 600, and the Polymath8b project refined it to show that infinitely many pairs of primes differ by at most 246. Either result implies the claim at once. Two independent proofs, both refereed and both in continuous use for a decade, settle the matter. What remains open is the value of the smallest such N: the twin prime conjecture asserts it is 2, and the best unconditional bound is 246.
Add parent claim
Minted parent claim aa796e31-0e6d-486e-bcee-8e2ae9dbd717 ("There are infinitely many pairs of primes whose difference is at most a fixed constant.") and attached this claim as its subclaim (specifies): The bounded gaps between primes proposition (liminf of consecutive prime gaps is finite) is the unit the discourse refers to: Zhang's 70 million bound, Polymath8a's 4680, Maynard's 600 and Polymath8b's 246 are all explicit-bound specifications of it, and the twin prime conjecture and Polignac's conjecture are its stronger relatives. The Matcher found no node for the existential proposition itself (only the twin prime node c4e7f254, Polignac 810de18b, the k-tuples conjecture 5b809f9d and the positive-lower-density theorem 5ffdcb5f). My claim, some N below 70 million occurring as a prime difference infinitely often, is the first proven specification of it and settles it.
Claim entered the graph