bibkey: fordmaynardtao2015primechains authors: Kevin Ford, James Maynard, Terence Tao year: 2015 title: Chains of large gaps between primes doi: null url: https://arxiv.org/abs/1511.04468v1 claim: The primary proof supplies a positive proportion of widely spaced prime rows; its quantitative abundance supports deletion of sparse higher-layer CA events, but supplies no root-conditioned prefix balance. strata_touched: [] license: citation-only triage: anchor

Many prime rows and the actual CA isolation interface

The inspected primary is arXiv:1511.04468v1, 13 November 2015, by Ford, Maynard and Tao. Its 16-page PDF has 200,200 bytes and SHA-256 4c5709180f4b427eac3525411618534a6d5ab03c1a79d3faeee3318d2ad7ecce. Locators refer to this version and its printed pages. This note cites the primary rather than redistributing it; it does not assert that this version is the latest literature on prime gaps.

Reuse the abundance retained in the proof

Theorem 1, p. 2, gives chains of any fixed number of consecutive large prime gaps. For removing a second sparse family, the relevant input is stronger than the existence of one such chain: use Theorem 2, pp. 5–6, Lemma 3.1, pp. 6–7, and the proof of Theorem 1 on p. 7, equation (3.10) and its following close-pair estimate.

Fix the source’s sieve parameter sufficiently large that its good rows contain at least three primes. Its equation (3.10) has probability bounded below by a positive constant independent of the small separation parameter. The subsequent probability of a close pair is $O(A^2\epsilon )$; choose $\epsilon$ so this error is smaller than that lower bound. Thus the source proof directly supplies fixed $c_0,\epsilon ,c>0$ and arbitrarily large $x$ with at least $c_{0}Z$ good rows

$$I_z=zP+m+(x,y],1\le z\le Z,y=c\frac{x\log x{\log }_{3}x}{{\log }_{2}x},$$

each containing at least three primes whose pairwise distances exceed $\epsilon y$. Here $P$ is the product of primes up to $x$ after removal of the source’s possible exceptional prime, and $m\in [1,P]$ is its CRT shift. Choose a fixed integer $D>1$ at least as large as the constant in Lemma 3.1 and put $Z=P^D$; that lemma holds for every $Z$ beyond its lower threshold. Constants may depend on this fixed $D$.

Keeping the positive proportion is an application of the published proof, not a new large-gap or sieve theorem. The source also gives $\log P\asymp x$, $y=o(P)$, and $y/x\to \infty$.

The additional project correspondence

Use the actual unrestricted CA price $g(t)=1/(t\log t)$ and events $g({\eta }_{p,j})=F(p,j)$, with

$$F(p,j)=\frac{\log (1+1/(p+\cdots +p^j))}{\log p}.$$

The project’s §420 combines the supplied many-row result with two elementary consequences of this exact activation equation:

$$p<{\eta }_{p,1}<p+1,\#\{(q,j):j\ge 2,{\eta }_{q,j}\le T\}=O\left(T^{1/2}(\log T{)}^{3/2}\right).$$

The enlarged rows $J_z=[zP+m+x-1,zP+m+y+2]$ are disjoint and lie below $T\asymp P^{D+1}$. Deleting every row containing a higher-layer event costs $o(Z)$ rows because $D>1$. An interior prime in any remaining good row then supplies an actual untied first-layer event with no other layer within $K\log p$, for any prescribed fixed $K>0$, once $x$ is sufficiently large. All layer counts retain multiplicities, including ties.

This CA correspondence is a source-derived paper application. The source does not discuss CA activations, regular returns, their signed moments or Robin’s inequality; no Lean verification or literature novelty claim is made for this additional interface.

What isolation leaves unpaid

Isolation does not locate a self-matching return. For the same untied event, write ${\eta }_p=p+{\delta }_p$ and retain the exact earlier higher-layer prefix

$$B_{\ge 2}(p)=\sum _{\begin{array}{c}q\text{prime},j\ge 2\\ F(q,j)>F(p,1)\end{array}}\log q.$$

Its post-event surplus is $s_p=\vartheta (p)-p+B_{\ge 2}(p)-{\delta }_p$. The sufficient singleton family in §420 additionally requires, at the same isolated primes,

$$\frac{3}{4}\log p\le s_p\le \frac{7}{8}\log p.$$

The many-row estimate supplies no such logarithmic-width prefix-placement count. Even a positive and then negative surplus in one clean row can skip this entire strip: between consecutive row primes $p<q$, $s_q-s_p=\log q-({\eta }_q-{\eta }_p)$, with the second term much larger than $\log q$. Separate oscillation and large-gap results therefore do not establish this joint assertion.

The existing packet source remains the authority for the singleton criterion and packet-margin direction. Even an unbounded winning singleton family would leave the selection obligation at potentially nonpositive minima and the full Robin/RH sign unresolved.