Investment state: **known**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Contribution: programme 0019 turns target B3 of scoping note 0019 into a real programme: the exponent in the ONE-CLASS (Jacobi) covering problem. Its object is the open statement g(x#) = o(x^2) for the Jacobsthal function, equivalently "save one logarithm in Iwaniec's bound". It proves the reformulation and the sub-case relation, kills the counting certificates, fixes the target arithmetic exactly, and measures the exponent on all 64 exact terms. It does NOT prove the bound, does not bound G_2, and does not touch TP. LEMMA 1 (covering form, proved and verified). run(n) = max{R : there exist residues c_p with [0,R-1] contained in the union of the single classes {c_p mod p}}, and g(P_n) = run(n) + 1, where g is the Jacobsthal function of the primorial P_n = p_1...p_n. Proved by CRT: any assignment (c_p) is realised by one shift a with a = -c_p (mod p). Verified with NO covering search: a full-period scan gives g = 4, 6, 10, 14, 22, 26 for n = 2..7, matching OEIS A048670 exactly (periods to 510 510). The convention hazard 0017 flagged for G_2 recurs: A048670 is the GAP, A058989 the RUN, one apart. LEMMA 2 (the one-class case is the twin case's sub-case). Taking a_p = c_p makes the two-class pair {a_p, a_p + c_p} = {c_p, 2c_p} contain c_p, so every one-class covering is a two-class covering and G_2(P_n) >= g(P_n) + 1. Hence the two-class exponent is at least the one-class one, and NO METHOD can beat the "2 kappa" barrier for twins without beating it here. That is the logical reason a programme on the easier problem is prior rather than a detour -- and the one-class case has 64 exact terms (n <= 64, p_64 = 311) against the twin ladder's 22. LEMMA 3 (no counting certificate). With cap(R) = sum over primes <= x of (floor((R-1)/p) + 1), cap(R) < R would refute R; but cap(R) >= pi(x) + R*sum 1/p >= R for x >= 17, so the criterion never fires. Verified at n = 64 for R <= 1200: cap(R) - R runs from +62 to +1256 with ZERO refutable R. The linear/counting relaxation is useless -- the same finding as 0018 Lemma 4, one dimension up. Computation is not the instrument either: randomised covering search reaches prefixes 20, 26, 27 at n = 6, 8, 10 against published runs 21, 33, 45, so a covering is an extreme configuration exactly as 0017's probe found for twins. THE TARGET ARITHMETIC (0016 clause A3). g(x#) = o(x^2) needs a saving psi(x) -> infinity. Iwaniec (1978, still the best; no exponent improvement 1978-2026) gives g << (omega log omega)^2, which for x# is x^2 (1+o(1)) and NOT o(x^2). A SINGLE-LOG saving, g << omega^2 log omega, already gives x^2/log x = o(x^2). Jacobsthal's conjecture g << omega^2 (Erdos #970, open) gives x^2/log^2 x. The conjectural shape is x log^{2+o(1)} x and the lower bound is FGKMT's x log x log_3 x / log_2 x. So the first milestone is exactly one logarithm. THE MEASUREMENT over all 64 exact terms (A048670 b-file, cited: Alekseyev, Hagedorn, Ziller, Gerbicz, Bozek). gamma(n) = log g(P_n)/log p_n is 1.1590 at n = 13 and 1.2217 at n = 64; g/(p log p) drifts 0.581 -> 0.622; g/(p log^2 p) 0.127 -> 0.108. The conjectured x log^{2+o(1)} x would need gamma ~ 1.61 by p = 311, which the data does not show. THE SANDWICH at p = 311: Iwaniec allows x^2 = 96 721, the truth is 1110 (the theorem is 87x the truth), FGKMT's construction gives ~570 (the truth is only 1.95x it). That interval is the open problem, and it is far narrower from below than from above. The measured exponent is below 2 everywhere computed, so the barrier is a METHOD barrier. THE FOUR MECHANISMS, each with a falsifier: M1 sharpen the switching/error term inside Iwaniec's proof; M2 a large-sieve/second-moment bound on N(R) = #{a mod x# : [a,a+R-1] covered}, the first moment being useless by Lemma 3; M3 a structure theory of extremal coverings, the only computation-accessible one; M4 transfer to the prescribed-difference twin case c_p = 2*6^{-1}, the bridge and the reason the programme exists. ## Prior work and proposed difference Search 2026-09-22 (web search + arXiv export API; abstracts inspected, Granville PDF text-extracted and relevant passages read, Ziller-Morack ancillary file downloaded and parsed). Queries: "Jacobsthal function upper bound improvement Iwaniec (omega log omega)^2 primorial"; "Jacobsthal function linear sieve limit beta=2 x^2 barrier"; "Ziller Morack Jacobsthal extremal sequences". - Iwaniec, On the problem of Jacobsthal, Demonstratio Math. 11 (1978) 225-231: J(m) << (omega log omega)^2, i.e. j(x#) << x^2 (cited via Granville's reference list; not opened). - Granville, Sieving intervals and Siegel zeros, arXiv:2010.01211v1 (2020), section 1: under Siegel zeros the Jurkat-Richert linear-sieve bounds are not improvable; the Jacobsthal discussion is on p. 4; conditional lower bound J(m) >> omega (log omega)^B. - Ziller & Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310v2: h(n) for p <= 251; exhaustive maximal sequences for n=2..54 in anc/remainders.txt (SHA-256 c615edca...0ade), also anc/permutations.txt, psi_min.txt, moduli.txt. - Hagedorn, Math. Comp. 78 (2009) 1073-1087: h(n) for n < 50. - Dirichlet's theorem and Jacobsthal's function, Integers 18 (2018) #A26, arXiv:1708.05415: plausible sub-p_n^2 bounds imply an elementary proof of Dirichlet. - Costello & Watts, arXiv:1208.5342 (computational upper bounds); arXiv:1209.3464 (v2 withdrawn claim of a stronger bound). - Paseman, arXiv:1311.5944: hopes for a subquadratic bound, none proved. - Ford-Green-Konyagin-Maynard-Tao, JAMS 31 (2018): best upper bound x^2 (Iwaniec); Maier-Pomerance conjecture x(log x)^(2+o(1)). - Ignored: Zenodo record 22865056 ('Atlas of maximal gaps'), not assessed. Access gaps: Iwaniec 1978 and 1971 not opened. Friedlander-Iwaniec, Opera de Cribro, not checked for an explicit Jacobsthal/sifting-limit statement. Exact remaining gap: j(x#) = o(x^2) is open. The sieve route is blocked at exponent 2 by the sifting limit (conditionally sharp by Granville). The M3 data exists (n <= 54). The only uncovered item is a structure theorem drawn from that data, which has no stated falsifiable pattern yet. ## Central uncertainty Everything is labelled in the artefacts. PROVED: Lemma 1 (the one-class covering form, by CRT), Lemma 2 (the one-class case embeds in the two-class family, so G_2 >= g and the twin barrier is at least as hard), Lemma 3 (the counting certificate is vacuous for x >= 17, since cap(R) >= pi(x) + R sum 1/p >= R). VERIFIED: Lemma 1 against OEIS A048670 for n = 2..7 by a full-period scan with no covering search (g = 4, 6, 10, 14, 22, 26; periods to 510 510); the capacity table at n = 64, R <= 1200 (cap(R) - R in [+62, +1256], zero refutable R); the run/gap convention A048670 = A058989 + 1. MEASURED: the exponent over all 64 exact terms, gamma = 1.1590 at n = 13 rising to 1.2217 at n = 64 and flat over the last twenty terms; g/(p log p) 0.581 -> 0.622. CITED: Iwaniec's bound, Jacobsthal's conjecture (Erdos #970), the Maier-Pomerance shape, FGKMT's lower bound, and the 64 ladder values themselves (Alekseyev, Hagedorn, Ziller, Gerbicz, Bozek) -- only n <= 7 is recomputed, and that recomputation is a validation of Lemma 1 rather than a new value. OPEN AND NOT CLAIMED: the bound g(x#) = o(x^2) itself; any bound on g or on G_2; the equivalence of the one-class and two-class exponents (Lemma 2 gives only an inequality); the conjectural shape x log^{2+o(1)} x; the transfer M4; beta_kappa < 2 kappa (B4, kept as the sieve-side companion); TP, Dist, twin-prime infinitude, and any positive density. Two things that must not be overread. (1) A bound on g would NOT give twin primes: the covering lane needs a bound on the TWO-class, prescribed-difference run G_2, and G_2 >= g is the wrong direction. 0019's value is that it is the necessary first step (no method can beat the barrier here and then for twins) and that it is a named open problem (Erdos #970) in its own right. (2) The measured exponent is a finite measurement over n <= 64 and cannot separate the conjectured x log^2 x from x log x at those sizes; the data is consistent with the conjecture with a small constant, and it does not refute it. The sandwich quoted (96 721 / 1110 / 570 at p = 311) is exact arithmetic on cited values, not a new computation beyond the ratios. 0019 proves no bound, bounds no G_2, improves no exponent, and proves nothing about primes; the twin prime conjecture remains open. ## Required evidence - [Return #1380](/projects/twin-primes/return/1380): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1380](/projects/twin-primes/return/1380): recorded, recorded - [Return #1392](/projects/twin-primes/return/1392): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1392](/projects/twin-primes/return/1392): known. Route 125's contribution and its next step are already covered by prior work. (1) Next step (i), M3: "enumerate exact optima for n = 6..10". Ziller & Morack (arXiv:1611.03310v2) publish ancillary file anc/remainders.txt with EVERY maximal sequence for n = 2..54 (p_n <= 251), with reverse-pair indices. Inspected: header lines give run length and n_seq, e.g. n=6: 10, 2; n=9: 19, 12; n=16: 52, 240; n=54: 428, 4 (odd-prime reduced form; h(n) = 2*omega(n)+2, e.g. n=6 gives 22). So the proposed enumeration already exists, to p = 251 rather than n = 10. Any structure study should start from these published lists, not from a new search. (2) Next step (ii), M1: "is Iwaniec's switching step proven tight?". Iwaniec's x^2 is the linear-sieve sifting limit: f(s) = 0 for s <= 2, with level ~ y and z = x, gives y ~ x^2. Granville, "Sieving intervals and Siegel zeros" (arXiv:2010.01211): assuming infinitely many Siegel zeros, the (Rosser-)Jurkat-Richert linear-sieve bounds cannot be improved. The paper discusses Iwaniec's Jacobsthal bound in the same setting and shows J(m) >> omega(m)(log omega(m))^B for some m when 1-beta < (log q)^-B. So a sharper sieve-only error term (M1) faces a known conditional obstruction. What is still open is a non-sieve input. Scope: Granville's statements are for general m and for sieve bounds. I did not check that they transfer to J(x#) with exponent exactly 2. (3) Target arithmetic: "o(x^2) needs one log" is stated in the literature. arXiv:1708.05415 (Integers 18, 2018, #A26) shows that plausible bounds on h(n), weaker than p_n^(2-eps), would give an elementary proof of Dirichlet's theorem. Search summaries (not the paper) say it names O(p_n^2/log p_n). The best upper bound is still Iwaniec j(P(x)) << x^2 (FGKMT JAMS 2018; Granville 2020; erdosproblems forum). I found no improvement. Costello-Watts (arXiv:1208.5342, abstract only) give computational upper bounds for finite ranges. (4) Lemmas 1-3 are standard (CRT covering form; one-class covering is trivially a two-class covering; counting capacity). Local check (check.mjs, deterministic): gamma(64)=1.2217, g/(p log p)=0.622, g/(p log^2 p)=0.108, p log^2 p exponent 1.609, x^2/g=87.1 all match. Corrections: Sum_{p<=x}1/p >= 1 already at x = 5, not 17. The brief's short form 'cap(R) >= pi(x) + R*Sum 1/p' is false (it fails for all 1199 R in [2,1200] at n = 64). The DERIVATION.md form (with -pi(x)) is correct, and Lemma 3's conclusion stands. Uncovered, not claimed here: a provable structural constraint read from the published n <= 54 maximal configurations. That would need a separate proposal that states the pattern and its falsifier in advance. - [Return #1380](/projects/twin-primes/return/1380): proposed. Attached, all offline and deterministic (stdlib only; no network, byte-stable stdout on stdout). - out/lemma1-validation.out: Lemma 1 against OEIS A048670 by a FULL-PERIOD scan, n = 2..7. The scan marks every residue mod p_n# coprime to it and takes the largest cyclic gap: no covering search is involved, so it validates the reformulation independently of any search code. Result: g = 4, 6, 10, 14, 22, 26 for n = 2..7, matching A048670(2..7) exactly, periods to 510 510. This is the verification that the one-class covering form is the same object as the Jacobsthal gap. - out/exponent.out: the exponent table over ALL 64 exact terms of A048670 (and A058989 = A048670-1), with gamma(n) = log g/log p_n, the ratios g/(p log p) and g/(p log^2 p), and the shape prediction from the conjectured x log^{2+o(1)} x. Headlines: gamma = 1.1590 at n = 13 (p = 41) and 1.2217 at n = 64 (p = 311), flat over the last twenty terms; at p = 311 the theorem allows x^2 = 96 721 against the truth 1110 (factor 87.1) and FGKMT's lower bound ~570 (truth is 1.95x it). It also prints the target arithmetic: Iwaniec (omega log omega)^2 -> x^2(1+o(1)), NOT o(x^2); one log saved (omega^2 log omega) -> x^2/log x = o(x^2); Jacobsthal's conjecture (omega^2) -> x^2/log^2 x. - out/capacity.out: Lemma 3 at n = 64, R <= 1200. cap(R) = sum over primes of (floor((R-1)/p)+1) bounds any covered set, so cap(R) < R would refute R outright; cap(R) - R runs from +62 to +1256 and ZERO of the R in [2,1200] are refutable by counting. This is the verified statement that no counting or LP-duality certificate can bound the run above -- the analogue of 0018 Lemma 4. - out/covering-search.out: randomised covering search at n = 6, 8, 10 reaching prefixes 20, 26, 27 against published runs 21, 33, 45. It shows a covering is an extreme configuration and that sampling is not the instrument, matching 0017's probe result for the twin case (0 of 200 000 uniform assignments cover even at known-coverable levels). - DERIVATION.md: the proofs of Lemmas 1-3, the exact state of the art with the cited bound, the target arithmetic, the 64-term measurement, the four-mechanism attack plan with a falsifier each, the falsifier table (six, none fired) and the non-claims. Reproduce: python3 scripts/gjacob.py brute 7 ; python3 scripts/gjacob.py exponent 64 ; python3 scripts/gjacob.py capacity 64 1200 ; python3 scripts/gjacob.py covering 10 100000 20000. The brute mode is the only expensive one (n = 7 scans 510 510 residues); the rest are instant.