# The two-class Jacobsthal function: bounds, data, and distance from the twin prime conjecture *Draft, 2026-09-05; audited 2026-09-10 and 2026-09-19. Internal to the research repository; the publication moratorium is in force and this document is not submission copy. Prose follows `paper/writing-style-math.md`. Nothing is re-derived here: every theorem points at the file that proves it, every number at the script that produced it.* **Author.** Chris Benjaminsen. **Methods and AI disclosure** (the statement adopted for the suite, `paper/PAPERS.md`): the framework, vocabulary and driving questions are the author's, developed over six years of independent work. Formal derivations, literature audits, computations and manuscript drafting were carried out using AI assistants under the author's direction. Computations have reproducible code and recorded outputs; asymptotic arguments require their stated mathematical inputs and are not proved by finite checks. Refuted intermediate claims are retained in the record. --- ## Abstract Let $P(x) = \prod_{p \le x} p$ and call $r$ a twin slot mod $P(x)$ if $\gcd(r(r+2), P(x)) = 1$. Let $G_2(x\#)$ be the largest cyclic gap between consecutive twin slots: the two-class analogue of Jacobsthal's function $g$, where each odd prime removes the two classes $0$ and $-2$ instead of one. We record what is known about $G_2$. Above: $G_2(x\#) \ll_\varepsilon x^{\beta_2 + \varepsilon}$ with $\beta_2 = 4.26645\ldots$ the sifting limit of the Diamond–Halberstam–Richert two-dimensional sieve, a corollary of that sieve that we did not find recorded. Below: $G_2 \ge g$ pointwise, and $G_2(P(y)) \gg y (\ln y)^3 (\ln\ln\ln y)^2/(\ln\ln y)^4$ by carrying the two-class kill set through Kalmynin and Konyagin's construction. Between: twenty-two exact terms to $x = 79$, fourteen computed here and eight from OEIS A144311. The eighteen normalized terms with $x \ge 11$ run at about $0.446$ to $0.594$ times $m \ln D$, with $m$ the reciprocal slot density and $D$ the census. A fixed upper exponent below $2$ would imply twin primes and $g(x\#)=o(x^2)$, whereas a constant bound at exponent $2$ alone does not give that little-o conclusion. The present DHR argument does not reach either sufficient target. We make no claim that its threshold is optimal among all sieves or that the exact tile structure excludes a different argument. ## 1. The object Fix a prime $x$ and write $W = x\#$. The reduced residues mod $W$ are the holes of the primorial wheel; a **twin slot** is a residue $r$ with $r$ and $r+2$ both holes. For $p = 2$ one class is forbidden ($r$ even) and for each odd $p \le x$ two classes are forbidden, $r \equiv 0$ and $r \equiv -2$. The number of twin slots per period is $D_x = \prod_{3 \le q \le x}(q-2)$, OEIS A059861 (the Schemmel totient $\varphi_2$ evaluated at $W/2$), and $$G_2(x\#) = \max\{\text{cyclic gap between consecutive twin slots of } W\}.$$ Three facts fix the object's place. - **It is a maximum-gap function of a sieve of dimension two.** The forbidden classes are set by the arithmetic, not chosen: $\omega(2) = 1$, $\omega(p) = 2$ for odd $p$, and $\prod_{p \le z}(1 - \omega(p)/p) \asymp 1/\ln^2 z$. The linear sieve, with its sifting limit exactly $2$, is unavailable (`paper/beta2-note.md` §2). - **It brackets the one-class function.** Twin slots are a subset of holes, so $G_2(x\#) \ge g(x\#)$ for Jacobsthal's $g$ (OEIS A048670 at primorials); and $G_2 \le h_2$, Ziller and Morack's function where each prime chooses its two classes freely (Ziller and Morack 2017, OEIS A288815). Both hold at every shared term, $g \le G_2$ at all 22 and $G_2 \le h_2$ at all 21 to $x = 73$ (`research/G2-STATE.md` §2). - **It is OEIS A144311 shifted by one.** A144311 counts the longest run of consecutive integers each $\equiv \pm 1$ mod some prime $\le x$; that is $G_2 - 1$ in the mirror convention. The entry dates from 2008 and its wording contains none of this programme's words, which is why our own search missed it for a week (`research/SEARCH-CONVENTIONS.md` §1). The elementary connection to primes: $W - 1$ and $W + 1$ are both holes, so the tile carries a twin slot at its edge, and every $x$-rough integer below $x'^2$ is prime ($x'$ the next prime). Hence $$G_2(x\#) < x'^2 - 2 \quad\Longrightarrow\quad \text{a twin prime pair lies in } (x, x'^2),$$ for every $x$ where it holds (`research/ZONE-POSTULATE.md` §3). Since $\ln W \sim x$, the window is $(\ln W)^2$: an exponent below $2$ for $G_2$ would settle the conjecture. Section 5 says why that observation is a calibration and not a plan. ## 2. The upper bound > **Theorem 1** (`paper/beta2-note.md`; the one input is the Diamond–Halberstam > book's Theorem 9.1, cited and read at the page). > Let $\beta_2 = 4.26645028\ldots$ be the sifting limit of the DHR sieve of > dimension $2$. For every $\varepsilon > 0$ there is $C(\varepsilon)$ with > $$G_2(x\#) \le C(\varepsilon)\, x^{\beta_2 + \varepsilon} \quad \text{for all primes } x.$$ > Equivalently, in $q = x\#$: consecutive $r$ with $\gcd(r(r+2), q) = 1$ are > spaced $\ll_\varepsilon (\ln q)^{4.267 + \varepsilon}$ apart. *Shape of the proof.* Take $A = \{r(r+2) : u < r \le u + H\}$ on an interval of length $H$ and sift by all $p \le x$. This is the Diamond–Halberstam book's own Example 1.2 at $g = 2$, $L(n) = n(n+2)$: $|A_d| = (\omega(d)/d) H + r_d$ with $|r_d| \le \omega(d) \le 2^{\nu(d)}$ and the dimension condition $\Omega(\kappa)$ holding at $\kappa = 2$ with an absolute constant, by Mertens. Their Theorem 9.1 gives $S(A, z) \ge H V(z)\{f_2(u) - o(1)\} - 2\sum_{m 0$ exactly for $u > \beta_2$. With $H = z^{\beta_2 + \varepsilon}$ and level $y = z^{\beta_2 + \varepsilon/2}$ the remainder is $\ll y \ln^7 y$ and the main term wins by $z^{\varepsilon/2}$, so every interval of length $z^{\beta_2 + \varepsilon}$ holds a twin slot, uniformly in position. Taking $u$ over a period gives the theorem. The sieve input was read line by line from the primary source with page images archived (`research/dhr-verification.md`); the constant is inexplicit, as in Iwaniec's one-class theorem. *What the theorem is and is not.* It is the dimension-two sieve applied to an interval, with no idea added; the note that proves it says so. The $\varepsilon$ absorbs the $8^{\nu(m)}$ remainder that makes the one-class exponent-$2$ result hard (Iwaniec 1978 had no $\varepsilon$ to spend). Our search in the owning convention, bounded number of residue classes per prime, found no earlier upper bound at any exponent for the two-class problem (`research/PRIOR-ART.md` row "Iwaniec-type upper bound for 2 omitted classes"; `research/SEARCH-CONVENTIONS.md` §3). The field of $\kappa = 2$ sifting limits in print is Rosser–Iwaniec $4.834$, Ankeny–Onishi $4.42$, Selberg's $\Lambda^2\Lambda^-$ $4.516$ (Franze 2011), Blight $4.45$, and DHR $4.26645$; the theorem uses the smallest. ## 3. The lower bounds > **Theorem 2a** (pointwise, elementary; `research/two-class-lower-bounds.md` §1, §3). > $G_2(x\#) \ge g(x\#)$ for every $x$. In particular every published lower bound > on Jacobsthal's function at primorials transfers: Ford, Green, Konyagin, > Maynard and Tao give $G_2(x\#) \gg x \ln x \ln\ln\ln x/\ln\ln x$. > **Theorem 2b** (`paper/kk-lower-bound.md` §9, Theorem A; published statements only, not refereed). > $G_2(x\#) \gg x \ln x$. > **Theorem 2c** (`paper/kk-lower-bound.md` §3, Theorem B; derived here, internally checked three times, on 2026-08-19, 2026-09-07 and 2026-09-08, and not refereed). > There is an absolute $y_0$ such that for $y \ge y_0$ > $$G_2(P(y)) \gg \frac{y (\ln y)^3 (\ln\ln\ln y)^2}{(\ln\ln y)^4}.$$ *Shape of 2c.* By the Chinese remainder theorem, $G_2(P(y)) - 1$ is the longest interval $[1, m]$ that can be covered by choosing, for each prime $p \le y$, a pair of classes $\{a_p, a_p - 2\}$ (band 1 fixes $a_p = 0$, where the free translate and the sieve's own pair coincide; the covering freedom is spent in band 2). Kalmynin and Konyagin (*A polynomial analogue of Jacobsthal function*, Izv. Math. 88:2, 2024) publish a multi-class Erdős–Rankin construction with a trichotomy of primes; substituting the two-class kill set into it, their case 2 hypothesis quantifies over an empty set because the two-linear-factor system has $h_f = 0$, and the rest goes through at sieve dimension $\kappa = 4$ with the fundamental lemma, Mertens with the Rosser–Schoenfeld error, the smooth-number estimate in Hildebrand's range, and CRT. The seven asymptotic hypotheses (H1 to H7 of that note's §8) first hold together at $y_0 = 10^{134.1}$ with every implied constant set to one, a floor on the true $y_0$; band 2 is empty at every reachable $y$, so no computation exhibits the construction. The substitution was re-derived by a second reader from $200$ dpi page images of the source and its finite content brute-forced with zero counterexamples (`research/history/staging/verify-kk-substitution.md`). *Where the logs come from.* The Erdős–Rankin base gives one power of $x$; the two-class survivor density after the small primes, $1/\ln^2 z$ in place of $1/\ln z$, gives one log; the band-2 device of the source gives one more. The Maier–Pomerance multi-kill upgrade, conjectural in both dimensions, would add a fourth: the conjectural ceiling is $G_2(x\#) = x (\ln x)^{4 + o(1)}$, still $x^{1 + o(1)}$ (`research/two-class-lower-bounds.md` §4c). *A caution on the one-class floor.* The Erdős Problems page for #687, edited 2026-08-31, reports a one-class bound $Y(x) \gg x \ln x/\ln\ln\ln x$ that improves FGKMT by a factor $\ln\ln x/(\ln\ln\ln x)^2$. Read at source on 2026-09-07 and again on 2026-09-10, unchanged: the attribution is not a paper. The page credits "GPT 5.6 Pro" and links to Erdős Problem #4, whose proof-claims thread (claim 224) leads to the artifact; the artifact is an anonymous AI-authored PDF posted to GitHub on 2026-08-26, not on arXiv and not refereed, with its covering theorem formalised in Lean by a third party (plby/lean-proofs); the page itself carries only the attribution and the link to #4, and the artifact and the formalisation were reached through that thread (`research/history/reviews-0907/03-erdos-687-source-read.md`). The statement is the one-class primorial form $Y(X) \ge c_0 X\ln X/\ln\ln\ln X$, so it transfers to $G_2$ through $G_2 \ge g$ at exactly that calibration. Theorem 2b sits above it in either case and Theorem 2c far above. ## 4. The data Fourteen exact terms computed here: $x \le 23$ by `research/05-twin-jacobsthal.js`, $29\#$ and $31\#$ by the segmented `research/05b-twin-jacobsthal-segmented.js`, $37\#$ by a mod-30 lattice walk, and $41\#$ and $43\#$ each twice on disjoint natal masks on 2026-08-18 (provenance table in `research/oeis-G2-submission.md`; `research/G2-STATE.md` §2). The values and their lower certificates are re-checked at the artifact by `research/exact-g2-ladder.js`, and a wheel-210 segmented striking sieve sharing no code with either producer re-derived $x = 11$ to $37$ exhaustively on 2026-09-04 (`research/history/staging/measure-0904-argmax.md`). All fourteen agree with A144311, whose remaining eight terms (Alekseyev 2009 for $47\#$ and $53\#$, Wang 2024 for $59\#$ to $79\#$, a public C++ program read here as a branch-and-bound depth-first search, sharing no code with ours) are adopted as trusted (`research/a144311-full-ladder.js`). $G_2(37\#) = 528$ additionally carries an exhaustive maximality certificate from a third engine over all $217{,}929{,}355{,}875$ gaps (`research/history/staging/scanstat-t37.md`). | $x$ | $G_2(x\#)$ | $c_2' = G_2/(m \ln D)$ | $G_2/x^2$ | $x$ | $G_2(x\#)$ | $c_2'$ | $G_2/x^2$ | |---|---|---|---|---|---|---|---| | 2 | 2 | | 0.500 | 41 | 546 | 0.512 | 0.325 | | 3 | 6 | | 0.667 | 43 | 618 | 0.492 | 0.334 | | 5 | 12 | | 0.480 | 47 | 708 | 0.484 | 0.321 | | 7 | 30 | | 0.612 | 53 | 870 | 0.518 | 0.310 | | 11 | 42 | 0.500 | 0.347 | 59 | 966 | 0.506 | 0.278 | | 13 | 66 | 0.447 | 0.391 | 61 | 1080 | 0.502 | 0.290 | | 17 | 108 | 0.471 | 0.374 | 67 | 1284 | 0.534 | 0.286 | | 19 | 150 | 0.456 | 0.416 | 71 | 1398 | 0.523 | 0.277 | | 23 | 204 | 0.458 | 0.386 | 73 | 1530 | 0.519 | 0.287 | | 29 | 258 | 0.446 | 0.307 | 79 | 1710 | 0.528 | 0.274 | | 31 | 348 | 0.479 | 0.362 | | | | | | 37 | 528 | 0.594 | 0.386 | | | | | Here $m = W/D$ is the reciprocal twin-slot density and $\ln D$ is the log of the census, so $m \ln D$ is the Cramér-type scale, the mean gap times the log of the number of gaps. Three readings, all MEASURED (`research/G2-STATE.md` §3c, `research/maxgap-law.md`): - $c_2'$ sits in $[0.446, 0.534]$ at every term except $x = 37$, whose $0.594$ is the one outlier of the ladder and overshoots a blind seven-term extreme-value forecast at $z = +6.58$, unexplained. The band drifts upward along the ladder, $0.446$ to $0.500$ over $x \le 31$ against $0.484$ to $0.534$ over $x \ge 41$. - The scale $m \ln D$ is $x \ln^2 x$ up to a constant, so the ladder lives at $x^{1 + o(1)}$, as the one-class function does. A power-law fit through the 20 terms with $x \ge 5$ of the 22-term ladder ($p = 2, 3$ excluded, as in every fit of that note) returns $1.50 \pm 0.05$, statistical error only with the systematic unquantified, after correcting the estimator's bias on a one-class control (`research/exponent-control.md` §5). That figure is a local slope over one decade of $x$ and carries no asymptotic content. - The unconditional floor $x \ln^3 x$ and the conjectural ceiling $x \ln^4 x$ both sit above the $x \ln^2 x$ the data run at. This is the same situation as the one-class function, where the data run at $x \ln x$, the FGKMT floor sits just below, and the Maier–Pomerance conjecture $x (\ln x)^{2 + o(1)}$ sits a log above: the Rankin-type gains are asymptotic and the accessible range does not see them. The ladder therefore tests nothing about either bound. $G_2/x^2$ falls from $0.347$ at $x = 11$ to $0.274$ at $x = 79$, ten descents in seventeen steps. The zone statement of §1 needs $G_2 < x'^2 - 2$; the measured ratio $x'^2/G_2$ runs $3.2$ to $4.5$ over the ladder and the zone $(x, x'^2)$ holds a twin prime pair at all $78{,}497$ zones with $x'^2 \le 10^{12}$ ($x = 2$ through $999{,}979$; the prime $999{,}983$ is below $10^6$ but its zone ends at $1{,}000{,}003^2 > 10^{12}$ and is outside the sweep), the sweep to $X = 10^{12}$ (`research/history/staging/zonegap-03-score.md`, rows S4 and S5, from `research/zonegap-04-sweep-1e12.js`). None of this is evidence for an asymptotic statement, since $\ln^2 x/x \to 0$ makes the truth of the inequality overwhelmingly likely and its proof no easier. ## 5. Distance from the twin prime conjecture The implication of §1 is correct and it is why the object was studied. We distinguish three claims. 1. **Quantifiers.** The safe sufficient inequality is $G_2(x\#)0$, would give little-o as well as twin primes. These are different targets. 3. **Sieve scope.** DHR achieves the sifting threshold $4.266450284\ldots$. Its current application gives no positive lower bound at the required quadratic scale. Neither this threshold nor Selberg's conjectured value $4$ is a proved optimal universal floor. See `research/OUTCOMES.md` for the withdrawn floor-at-4 claim. Classical parity obstructions limit particular methods using specified arithmetic statistics and errors. Exact residue arrangements determine the survivors and, below the square frontier, primality; they are not thereby excluded as a language for a future proof. Any broader impossibility claim would need its own hypotheses and argument. Type II estimates are useful additional inputs, not a proved necessary form for every possible proof. Likewise a raw shifted-prime Liouville average needs factor-count restrictions before its sign can certify a prime. This manuscript supplies no such new estimate and no proof of the conjecture. ## 6. What would move each statement | statement | rung | what would move it | has the check run | |---|---|---|---| | Theorem 1 | PROVEN, citing DHR Theorem 9.1 | a defect in the primary-source reading | yes: read line by line, page images archived | | Theorem 2a | PROVEN | nothing; elementary | yes: verified at all 22 shared terms | | Theorem 2b | DERIVED, not refereed | a misread of Kalmynin–Konyagin Corollary 1 | adversary-confirmed at the page image, once | | Theorem 2c | DERIVED, internally checked three times (2026-08-19 second reader with brute force; 2026-09-07 re-read at the arXiv text; 2026-09-08 source review with the chain re-derived and finite CRT checks, `research/history/reviews-0907/11`), not refereed | a hypothesis of the source that the substitution does not satisfy; Halberstam and Richert Theorem 2.2 printed in a form that does not give Kalmynin and Konyagin's Lemma 1 (`paper/kk-lower-bound.md` §11.2) | second reader plus brute force of the finite content (2026-08-19); both §11.1 readings repeated at the arXiv text (2026-09-07) and at the TeX source, every other input read at source (2026-09-08, `research/history/reviews-0907/11-two-class-theorem2c-source-review.md`); the 1974 page unread after a second bounded access pass (2026-09-08, thirteen channels, `research/history/reviews-0907/12-halberstam-richert-second-access.md`); a referee has not | | the $c_2'$ band | MEASURED, 18 normalized terms of 22 | a term at $x \ge 83$ outside $[0.44, 0.60]$ | no; the exact ladder and A144311 both end at $x = 79$ | | "no earlier two-class upper bound" | CALIBRATED ABSENCE | one citation in the owning convention | searched 2026-08-18 in the owning convention (`research/SEARCH-CONVENTIONS.md` §3, row "Published upper bound on $G_2$", under the section header "all dated 2026-08-18 unless noted"; `research/PRIOR-ART.md` row "Iwaniec-type upper bound for 2 omitted classes", dated 2026-08-18) | | §5 | elementary implications and scoped method limits | an independently proved estimate at the stated target scale | none supplied here; no universal impossibility claim | ## 7. Relation to the rest of the repository This note supersedes nothing. `paper/beta2-note.md` carries Theorem 1 with its verification record; `paper/kk-lower-bound.md` carries Theorems 2b and 2c with a per-number provenance appendix; `research/G2-STATE.md` carries the full state of the object with a calibration marker on every line; `research/OUTCOMES.md` holds the routes tried toward a lower exponent and closed, one line each; `research/G2-STATE.md` §0's recount of 2026-08-30 puts them at 72, though its own breakdown there (67, then 2, then 4 added) sums to 73, so the count is quoted as the register's and not re-derived here. The current campaign benchmarks classical Chen input on long intervals before proposing a signed extension; its scope is in `README.md` §Status. ## References - H. G. Diamond, H. Halberstam, with W. F. Galway, *A Higher-Dimensional Sieve Method*, Cambridge Tracts in Mathematics 177, CUP 2008. Theorem 9.1, Definition 1.3, Example 1.2, §6.5. - A. Booker, T. D. Browning, *Square-free values of reducible polynomials*, Discrete Analysis 2016:8, DOI 10.19086/da.732; ancillary table giving $\beta_2$ to twenty decimals. - C. S. Franze, *Sifting limits for the $\Lambda^2\Lambda^-$ sieve*, J. Number Theory 131 (2011), DOI 10.1016/j.jnt.2011.04.008, arXiv:1012.3809. - H. Iwaniec, *On the problem of Jacobsthal*, Demonstratio Math. 11 (1978). - K. Ford, B. Green, S. Konyagin, J. Maynard, T. Tao, *Long gaps between primes*, J. Amer. Math. Soc. 31 (2018), arXiv:1412.5029. - A. Kalmynin, S. Konyagin, *A polynomial analogue of Jacobsthal function*, Izv. Math. 88:2 (2024) 225–235, DOI 10.4213/im9467e, arXiv:2302.00459. - M. Ziller, J. F. Morack, *Algorithmic concepts for the computation of Jacobsthal's function*, arXiv:1611.03310. - M. Ziller, J. F. Morack, *Divisibility in paired progressions, Goldbach's conjecture, and the infinitude of prime pairs*, arXiv:1706.00317 (2017); the paired Jacobsthal function $h_2$, OEIS A288815. - OEIS A144311 (A. Carter 2008; M. Alekseyev 2009; J. Wang 2024), A059861, A048670. - Erdős Problems #687, https://www.erdosproblems.com/687, as edited 2026-08-31; the new lower bound's source is the proof-claims thread of #4 (https://www.erdosproblems.com/forum/thread/4/proof-claims, claim 224) and https://github.com/DottedCalculator/ai-math (Erdos_4_GPT_5.6_Sol.pdf), Lean formalisation https://github.com/plby/lean-proofs (Erdos4Tilted.lean).