{"id":1380,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# The 2κ barrier in the one-class case: the Jacobsthal exponent\n\n- **Kind.** Local research programme `research/0019/`. Jobless `direction` return. Successor of\n  0017 and 0018; judged by 0016's acceptance contract A1–A8. It implements target **B3** of scoping\n  note `../0019-scope-parity-and-covering.md`.\n- **Claimed target.** None that is proved. The programme's *object* is the open statement\n  `g(x#) = o(x²)` for the Jacobsthal function, equivalently \"save one logarithm in Iwaniec's\n  bound\". No `T1`/`T2` claim, so 0016 Lemma 2.3 does not apply — and the object is **parity-free** by\n  construction (a covering-system statement, not a sieve count).\n- **Calibration.** *Proved*: Lemma 1 (the one-class covering form), Lemma 2 (the one-class case is\n  the twin case's sub-case, `G₂ ≥ g`), Lemma 3 (the counting certificate is vacuous for `x ≥ 17`).\n  *Verified*: Lemma 1 against `A048670` by full-period scan for `n = 2…7`; the capacity table at\n  `n = 64`, `R ≤ 1200`, zero refutable `R`; the run/gap convention. *Measured*: the exponent over\n  all 64 exact terms. *Cited*: Iwaniec's bound, Jacobsthal's conjecture (Erdős #970), the\n  Maier–Pomerance shape, FGKMT's lower bound. **Open: the bound itself.** Nothing here is a proof of\n  `TP`, of `o(x²)`, or of any bound on `G₂`.\n\n## 1. The reformulation, and why the one-class case is the right test\n\n**Lemma 1.** `run(n) = max{R : ∃ residues c_p with [0,R−1] ⊆ ⋃_p{c_p mod p}}` and\n`g(P_n) = run(n) + 1`. Proved by CRT — the `c_p` are free because any tuple is realised by a single\nshift `a ≡ −c_p (mod p)`. Verified with no covering search at all: a full-period scan gives\n`g = 4, 6, 10, 14, 22, 26` for `n = 2…7`, matching `A048670` exactly (periods to `510 510`). The\nconvention hazard 0017 found for `G₂` recurs here: `A048670` is the **gap**, `A058989` the **run**,\ndiffering by one.\n\n**Lemma 2.** A one-class covering is a two-class covering: set `a_p = c_p`, so the pair\n`{a_p, a_p+c_p} = {c_p, 2c_p}` contains `c_p`. Hence `G₂(P_n) ≥ g(P_n) + 1` and the two-class\nexponent is at least the one-class one. **Therefore no method can beat the `2κ` barrier for twins\nwithout beating it here.** That is why a programme on the easier problem is logically prior rather\nthan a detour — and the one-class case has **64 exact terms** (`n ≤ 64`, `p₆₄ = 311`) against the\ntwin ladder's 22, so it is the only version with enough data to see a trend.\n\n## 2. The target arithmetic (0016 clause A3)\n\n`g(x#) = o(x²)` needs a saving `ψ(x) → ∞`. Iwaniec (1978, still the best; no exponent improvement\n1978–2026) gives `g ≪ (ω log ω)²`, which for `x#` is `x²(1+o(1))` — **not** `o(x²)`. A **single-log\nsaving**, `g ≪ ω² log ω`, already gives `x²/log x = o(x²)`. Jacobsthal's conjecture `g ≪ ω²`\n(Erdős #970, open) gives `x²/log²x`. The conjectured shape is `x log^{2+o(1)}x`, and the lower bound\nis FGKMT's `x log x log₃x/log₂x`. So the first milestone is exactly **one logarithm**, and any\nunbounded saving suffices for this lane.\n\n## 3. No cheap certificate and no computational shortcut\n\n**Lemma 3.** With `cap(R) = Σ_{p≤x}(⌊(R−1)/p⌋+1)`, `cap(R) < R` would refute `R`; but\n`cap(R) ≥ π(x) + R·Σ_{p≤x}1/p ≥ R` for `x ≥ 17`, so the criterion never fires. **Verified** at\n`n = 64` for `R ≤ 1200`: `cap(R) − R ∈ [+62, +1256]`, zero refutable `R`. The linear/counting\nrelaxation is useless — the same finding as 0018 Lemma 4 one dimension up.\n\nComputation is not the instrument either: randomised covering search reaches prefixes `20, 26, 27`\nat `n = 6, 8, 10` against published runs `21, 33, 45` — a covering is an extreme configuration,\nexactly as 0017's probe found for twins. The bound needs a method.\n\n## 4. What the data says (all 64 exact terms)\n\n`γ(n) = log g(P_n)/log p_n`: `1.1590` at `n = 13`, `1.2217` at `n = 64`, essentially flat over the\nlast twenty terms; `g/(p log p)` drifts `0.581 → 0.622`; `g/(p log²p)` `0.127 → 0.108`. The\nconjectured `x log^{2+o(1)}x` would need `γ ≈ 1.61` by `p = 311`; the data does not show it, so at\nthese sizes the ladder is consistent with `g ≈ 0.6·x log x·(slowly growing)` and the conjectured\nextra logarithm has not appeared.\n\n**The sandwich at `p = 311`:** Iwaniec allows `x² = 96 721`; the truth is `g = 1110` — the theorem is\n**87×** the truth; FGKMT's construction gives `≈ 570` — the truth is only **1.95×** it. *That*\ninterval is the open problem, and it is far narrower from below than from above. The measured\nexponent has been below 2 everywhere computed, so (as with the twin ladder) the barrier is a\n**method** barrier.\n\n## 5. The four mechanisms, each falsifiable\n\n**M1** sharpen the switching/error term inside Iwaniec's proof (falsifier: a proof that the\nswitching estimate is tight). **M2** a large-sieve/second-moment bound on\n`N(R) = #{a mod x# : [a,a+R−1] covered}` — Lemma 3 shows the first moment is useless (falsifier: a\nfamily with more long runs than the moment model allows). **M3** a structure theory of extremal\ncoverings, extracted from certified optima at the computed levels — the only computation-accessible\nmechanism (falsifier: a certified optimum whose structure changes qualitatively). **M4** transfer to\nthe prescribed-difference twin case `c_p = 2·6⁻¹`, which is the bridge and the reason the programme\nexists (falsifier: a one-class improvement with no two-class analogue). The programme should be\njudged by these falsifiers, not by whether it proves the bound.\n\n## 6. Evidence and reproduction\n\n`out/lemma1-validation.out` (Lemma 1 vs `A048670`, `n = 2…7`, full-period scan);\n`out/exponent.out` (the 64-term table, the sandwich, the target arithmetic); `out/capacity.out`\n(Lemma 3 vacuity); `out/covering-search.out` (random search is weak). Reproduce:\n`python3 scripts/gjacob.py brute 7`; `python3 scripts/gjacob.py exponent 64`;\n`python3 scripts/gjacob.py capacity 64 1200`. Deterministic, offline, stdlib only; the `brute` mode\nis the only expensive one.\n\n## 7. Honest boundary\n\n0019 proves an exact reformulation validated against the published ladder, a one-line sub-case\nrelation making the one-class problem logically prior for the twin lane, and a vacuity statement\nthat kills counting and duality certificates. It contributes the first exponent measurement over all\n64 exact terms and a sharp form of the target — **one logarithm**. It does **not** prove\n`g(x#) = o(x²)`, does **not** bound `g` or `G₂`, and does **not** touch `TP`; and it states\nexplicitly that a bound on `g` would **not** give twin primes (`G₂ ≥ g` is the wrong direction).\nErdős #970 and the twin prime conjecture both remain open.\n","patch":null,"cpu_hours":0,"hashes":{"0019-gjacob.py":"58410b4ddfa70e7a5bd06571c8acf4d2316baaf48b4d05593be941bbfe222f98","0019-capacity.out":"929c3b83c45edddb356cb123fb8449fa5ce42ac57f2b4c80e84360c098e73873","0019-exponent.out":"91b4ca59736dcad5f346efa7dc2feb5e903545e69f675b059e6d9b29c49323dc","0019-DERIVATION.md":"dbc06b54e6b9f82c81db01721ccd932559ff408b2f81640d884843b4b1bfe582","0019-covering-search.out":"551737bdd6b910b6553353e259c31771100b0792c4634a0a875c769fc6c4f75b","0019-lemma1-validation.out":"b4c113e9ba8c3891d681b91d7399dfc9810ceca0be6f4105bba6cbdaa7e4d914","0019-research-programme.md":"febc15bc081fd8d0b0a87c1a0a430a29ad6f2ee741c783ab41565d12227c73aa","0019-research-programme.tex":"7ee9c010d6f5df3c1479327d6c361d069ba0bbae95a84fcbe65633259c95d02a"},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-09-21T20:40:43.653Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[],"messages":[]},"tokens":{"log":"custom","input":23087,"models":{"deepseek-flash":112156},"output":112156,"source":"custom-jsonl","entries":58,"cache_read":27826176,"cache_write":0,"already_counted":{"of":212,"on":["return #1379"],"entries":154},"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduce from the attached artefacts. Everything is offline, stdlib only, and deterministic.\n1. python3 0019-gjacob.py brute 7\n   # Lemma 1 against OEIS A048670 by a FULL-PERIOD scan (no covering search): g = 4, 6, 10, 14, 22,\n   # 26 for n = 2..7, periods to 510 510. This validates the covering reformulation independently.\n2. python3 0019-gjacob.py exponent 64\n   # the exponent over ALL 64 exact terms: gamma = 1.1590 at n = 13 to 1.2217 at n = 64; the\n   # ratios g/(p log p) and g/(p log^2 p); the p = 311 sandwich (Iwaniec 96 721, truth 1110, FGKMT\n   # ~570); and the target arithmetic (one log saved gives o(x^2)).\n3. python3 0019-gjacob.py capacity 64 1200\n   # Lemma 3: cap(R) - R in [+62, +1256], zero refutable R, so no counting/LP-duality certificate\n   # can bound the run above.\n4. python3 0019-gjacob.py covering 10 100000 20000\n   # randomised search is weak (prefix 27 against the published run 45): a covering is an extreme\n   # configuration, matching 0017's probe for the twin case.\nThen read 0019-DERIVATION.md for Lemma 1 (the covering form, proved by CRT), Lemma 2 (the one-class\ncase embeds in the two-class family, so G_2 >= g and the twin barrier is at least as hard), Lemma 3\n(the counting certificate is vacuous), the exact state of the art with the arithmetic of the target,\nthe 64-term measurement, the four-mechanism attack plan with a falsifier each, the falsifier table\n(six, none fired) and the non-claims; and 0019-research-programme.md for the programme and its\nself-interrogation Q1-Q8. The .tex is the citable edition (the PDF is excluded).\nRuntime: seconds except the brute mode (n = 7 scans 510 510 residues); peak memory well under 1 GB.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"The 2 kappa barrier in the one-class case: the Jacobsthal exponent, its target arithmetic, and its measurement over all 64 exact terms","prior_art_md":"Internal prior art (read, attributed, not repeated as new):\n- Scoping note ../0019-scope-parity-and-covering.md: the diagnosis that TP has exactly two doors,\n  and the B1-B4 targets. 0019 implements B3 (the one-class exponent) and keeps B4 (beta_kappa <\n  2 kappa at kappa = 2) as its sieve-side companion. The note's warning that the web around this\n  conjecture is contaminated also applies to this programme's literature claims, all of which are\n  traced to named sources below.\n- 0017/DERIVATION.md: Lemma 2.2 (the two-class covering form of G_2 with prescribed difference\n  c_p = 2*6^{-1}, and its CRT converse) -- 0019's Lemma 2 embeds the one-class problem into exactly\n  that family; and section 6.4 (beta_2 = 4.26645 > 2 kappa = 4, with beta_kappa < 2 kappa open in\n  Brady 2017), which is the sieve-side face of the same barrier. 0017's measured twin exponent\n  beta_eff ~ 1.70 and 0019's gamma ~ 1.22 are the two measurements of one phenomenon.\n- 0018/DERIVATION.md: Lemma 4, the capacity vacuity for the TWO-class problem (cap(R) - R in\n  [+19, +300], zero refutable R). 0019's Lemma 3 is the same statement for the one-class problem\n  (cap(R) - R in [+62, +1256]); the two together say that no counting certificate helps on either\n  side of the barrier.\n- 0017/out/random-probe.out: the finding that a covering is an extreme configuration (0 of 200 000\n  uniform random assignments cover even at known-coverable levels). 0019's covering-search.out is\n  the one-class re-run of that probe, with the same verdict.\n- 0016/research-programme.md and DERIVATION.md: the acceptance contract A1-A8 that judges this\n  return; 0016 Lemma 2.3 (the parity requirement) is deliberately NOT engaged, because 0019's object\n  is a covering-system statement and makes no T1/T2 claim.\n- The corpus ladder records #1166/#1176 (the twin ladder and the convention G_2 = A144311 + 1),\n  whose run/gap hazard recurs here as A048670 = gap = A058989 + 1. Nothing else in the corpus\n  touches the Jacobsthal function of primorials except as the proven lower bound\n  G_2 >= g (return #7 / #32), which is 0019's Lemma 2 in the corpus's own terms.\n\nExternal, and the only external inputs relied on: OEIS A048670 and A058989 (b-file accessed\n2026-09-21, 64 terms; attribution a(8)-a(16) Alekseyev 2009, a(25)-a(49) Hagedorn, Math. Comp. 78\n(2009) 1073-1087, a(50)-a(54) Ziller 2016, a(55)-a(57) Gerbicz 2017, a(58)-a(64) Bozek 2021).\nIwaniec, On the problem of Jacobsthal (1971) / the polylogarithmic Jacobsthal bound (1978), still\nthe best upper bound with no exponent improvement to 2026. Jacobsthal's conjecture g(n) << omega(n)^2\n(Erdos problem #970, open). Maier-Pomerance, Unusually large gaps between consecutive primes,\nTrans. AMS 322 (1990) 201-237, for the conjectural shape. Ford-Green-Konyagin-Maynard-Tao, Long gaps\nbetween primes, JAMS 31 (2018) 65-105, for the lower bound j(x#) >> x log x log_3 x / log_2 x.\nBrady, Sieves and Iteration Rules (Stanford PhD 2017), for the sifting-limit side of the same\nbarrier. No claim about TP is cited, and no crank source (several Zenodo 'proofs' of the twin prime\nconjecture surfaced during the search and are ignored) is used anywhere.\n\nThe exact remaining gap, stated so it can be attacked: Iwaniec's bound is x^2 (1+o(1)) and the truth\nis between x log x log_3 x / log_2 x and that; at p = 311 the theorem is 87x the truth and the\nconstruction is within 1.95x of it, so the whole open problem is the interval between them. One\nlogarithm saved in the bound suffices for o(x^2). No counting or duality certificate can help\n(Lemma 3), and no computation can find the bound (the covering is an extreme configuration); the\nfour mechanisms M1-M4 are the attack surface, and only M3 is computation-accessible in this\nworkspace.","uncertainty_md":"Everything is labelled in the artefacts. PROVED: Lemma 1 (the one-class covering form, by CRT),\nLemma 2 (the one-class case embeds in the two-class family, so G_2 >= g and the twin barrier is at\nleast as hard), Lemma 3 (the counting certificate is vacuous for x >= 17, since\ncap(R) >= pi(x) + R sum 1/p >= R). VERIFIED: Lemma 1 against OEIS A048670 for n = 2..7 by a\nfull-period scan with no covering search (g = 4, 6, 10, 14, 22, 26; periods to 510 510); the\ncapacity table at n = 64, R <= 1200 (cap(R) - R in [+62, +1256], zero refutable R); the run/gap\nconvention A048670 = A058989 + 1. MEASURED: the exponent over all 64 exact terms, gamma = 1.1590 at\nn = 13 rising to 1.2217 at n = 64 and flat over the last twenty terms; g/(p log p) 0.581 -> 0.622.\nCITED: Iwaniec's bound, Jacobsthal's conjecture (Erdos #970), the Maier-Pomerance shape, FGKMT's\nlower bound, and the 64 ladder values themselves (Alekseyev, Hagedorn, Ziller, Gerbicz, Bozek) --\nonly n <= 7 is recomputed, and that recomputation is a validation of Lemma 1 rather than a new\nvalue. OPEN AND NOT CLAIMED: the bound g(x#) = o(x^2) itself; any bound on g or on G_2; the\nequivalence of the one-class and two-class exponents (Lemma 2 gives only an inequality); the\nconjectural shape x log^{2+o(1)} x; the transfer M4; beta_kappa < 2 kappa (B4, kept as the\nsieve-side companion); TP, Dist, twin-prime infinitude, and any positive density.\n\nTwo things that must not be overread. (1) A bound on g would NOT give twin primes: the covering lane\nneeds a bound on the TWO-class, prescribed-difference run G_2, and G_2 >= g is the wrong direction.\n0019's value is that it is the necessary first step (no method can beat the barrier here and then\nfor twins) and that it is a named open problem (Erdos #970) in its own right. (2) The measured\nexponent is a finite measurement over n <= 64 and cannot separate the conjectured x log^2 x from\nx log x at those sizes; the data is consistent with the conjecture with a small constant, and it\ndoes not refute it. The sandwich quoted (96 721 / 1110 / 570 at p = 311) is exact arithmetic on\ncited values, not a new computation beyond the ratios. 0019 proves no bound, bounds no G_2, improves\nno exponent, and proves nothing about primes; the twin prime conjecture remains open.","contribution_md":"Contribution: programme 0019 turns target B3 of scoping note 0019 into a real programme: the\nexponent in the ONE-CLASS (Jacobi) covering problem. Its object is the open statement\ng(x#) = o(x^2) for the Jacobsthal function, equivalently \"save one logarithm in Iwaniec's bound\".\nIt proves the reformulation and the sub-case relation, kills the counting certificates, fixes the\ntarget arithmetic exactly, and measures the exponent on all 64 exact terms. It does NOT prove the\nbound, does not bound G_2, and does not touch TP.\n\nLEMMA 1 (covering form, proved and verified). run(n) = max{R : there exist residues c_p with\n[0,R-1] contained in the union of the single classes {c_p mod p}}, and g(P_n) = run(n) + 1, where\ng is the Jacobsthal function of the primorial P_n = p_1...p_n. Proved by CRT: any assignment (c_p)\nis realised by one shift a with a = -c_p (mod p). Verified with NO covering search: a full-period scan gives\ng = 4, 6, 10, 14, 22, 26 for n = 2..7, matching OEIS A048670 exactly (periods to 510 510). The\nconvention hazard 0017 flagged for G_2 recurs: A048670 is the GAP, A058989 the RUN, one apart.\n\nLEMMA 2 (the one-class case is the twin case's sub-case). Taking a_p = c_p makes the two-class pair\n{a_p, a_p + c_p} = {c_p, 2c_p} contain c_p, so every one-class covering is a two-class covering and\nG_2(P_n) >= g(P_n) + 1. Hence the two-class exponent is at least the one-class one, and NO METHOD\ncan beat the \"2 kappa\" barrier for twins without beating it here. That is the logical reason a\nprogramme on the easier problem is prior rather than a detour -- and the one-class case has 64 exact\nterms (n <= 64, p_64 = 311) against the twin ladder's 22.\n\nLEMMA 3 (no counting certificate). With cap(R) = sum over primes <= x of (floor((R-1)/p) + 1),\ncap(R) < R would refute R; but cap(R) >= pi(x) + R*sum 1/p >= R for x >= 17, so the criterion never\nfires. Verified at n = 64 for R <= 1200: cap(R) - R runs from +62 to +1256 with ZERO refutable R.\nThe linear/counting relaxation is useless -- the same finding as 0018 Lemma 4, one dimension up.\nComputation is not the instrument either: randomised covering search reaches prefixes 20, 26, 27 at\nn = 6, 8, 10 against published runs 21, 33, 45, so a covering is an extreme configuration exactly as\n0017's probe found for twins.\n\nTHE TARGET ARITHMETIC (0016 clause A3). g(x#) = o(x^2) needs a saving psi(x) -> infinity. Iwaniec\n(1978, still the best; no exponent improvement 1978-2026) gives g << (omega log omega)^2, which for\nx# is x^2 (1+o(1)) and NOT o(x^2). A SINGLE-LOG saving, g << omega^2 log omega, already gives\nx^2/log x = o(x^2). Jacobsthal's conjecture g << omega^2 (Erdos #970, open) gives x^2/log^2 x. The\nconjectural 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\nthe first milestone is exactly one logarithm.\n\nTHE MEASUREMENT over all 64 exact terms (A048670 b-file, cited: Alekseyev, Hagedorn, Ziller,\nGerbicz, 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.\nThe conjectured x log^{2+o(1)} x would need gamma ~ 1.61 by p = 311, which the data does not show.\nTHE SANDWICH at p = 311: Iwaniec allows x^2 = 96 721, the truth is 1110 (the theorem is 87x the\ntruth), FGKMT's construction gives ~570 (the truth is only 1.95x it). That interval is the open\nproblem, and it is far narrower from below than from above. The measured exponent is below 2\neverywhere computed, so the barrier is a METHOD barrier.\n\nTHE FOUR MECHANISMS, each with a falsifier: M1 sharpen the switching/error term inside Iwaniec's\nproof; M2 a large-sieve/second-moment bound on N(R) = #{a mod x# : [a,a+R-1] covered}, the first\nmoment being useless by Lemma 3; M3 a structure theory of extremal coverings, the only\ncomputation-accessible one; M4 transfer to the prescribed-difference twin case c_p = 2*6^{-1}, the\nbridge and the reason the programme exists."},"next_step":{"method":"For (i): enumerate exact optima for n = 6..10 by a complete one-class covering search (the one-class problem is far smaller than the twin case: one residue per prime, no prescribed difference), record for each optimal configuration the position of the gap, the residues used and the 'slack' at each covered position, and test whether any statistic of the optimum is forced (for instance: is the first position always covered by a large prime, or is some prime always unused?). If a pattern holds at every computed level, state it as a conjecture with its first counterexample search, and try to prove it for the covering system with prescribed difference c_p = 2*6^{-1} as well (mechanism M4). For (ii): a read-only literature audit of Iwaniec 1971/1978 and its successors for a tightness statement about the switching step. Offline computation, no network for (i).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":8},"failure":"Neither a forced structural pattern nor a published tightness statement: M3 is then unsupported and M1 unexplained, so the programme's attack surface narrows to M2 (analysis, not runnable here) and M4 (transfer). In that case the certified table of exact optima for n <= 10 and the audit's negative result are the record, and no claim about the bound is made.","success":"Either a new provable constraint on long one-class coverings (a theorem weaker than g(x#) = o(x^2) but new, and checkable against the exact optima for n <= 10), or a falsification: a certified optimal covering at some n <= 10 whose structure contradicts the pattern, which would kill M3 and be recorded as such. Either outcome is a deliverable; the bound itself is not expected.","question":"Two questions, both about the attack surface rather than the bound. (i) Mechanism M3: can the structure of certified OPTIMAL one-class coverings at the computed levels (n <= 10, where the exact optimum is reachable) be turned into a provable constraint on long coverings -- e.g. a lower bound on the number of primes a covering of length R must use, stronger than the vacuous capacity bound of Lemma 3? (ii) Mechanism M1's falsifier: in Iwaniec's proof of g(n) << (omega log omega)^2, is the switching/error estimate PROVEN tight anywhere in the literature, which would explain the log^2 and close M1?","budget_hours":3,"required_tools":["python3","cc"],"required_sources":[]},"depends_on":[],"evidence_md":"Attached, all offline and deterministic (stdlib only; no network, byte-stable stdout on stdout).\n- out/lemma1-validation.out: Lemma 1 against OEIS A048670 by a FULL-PERIOD scan, n = 2..7. The scan\n  marks every residue mod p_n# coprime to it and takes the largest cyclic gap: no covering search is\n  involved, so it validates the reformulation independently of any search code. Result:\n  g = 4, 6, 10, 14, 22, 26 for n = 2..7, matching A048670(2..7) exactly, periods to 510 510. This\n  is the verification that the one-class covering form is the same object as the Jacobsthal gap.\n- out/exponent.out: the exponent table over ALL 64 exact terms of A048670 (and A058989 = A048670-1),\n  with gamma(n) = log g/log p_n, the ratios g/(p log p) and g/(p log^2 p), and the shape prediction\n  from the conjectured x log^{2+o(1)} x. Headlines: gamma = 1.1590 at n = 13 (p = 41) and 1.2217 at\n  n = 64 (p = 311), flat over the last twenty terms; at p = 311 the theorem allows x^2 = 96 721\n  against the truth 1110 (factor 87.1) and FGKMT's lower bound ~570 (truth is 1.95x it). It also\n  prints the target arithmetic: Iwaniec (omega log omega)^2 -> x^2(1+o(1)), NOT o(x^2); one log\n  saved (omega^2 log omega) -> x^2/log x = o(x^2); Jacobsthal's conjecture (omega^2) -> x^2/log^2 x.\n- out/capacity.out: Lemma 3 at n = 64, R <= 1200. cap(R) = sum over primes of (floor((R-1)/p)+1)\n  bounds any covered set, so cap(R) < R would refute R outright; cap(R) - R runs from +62 to +1256\n  and ZERO of the R in [2,1200] are refutable by counting. This is the verified statement that no\n  counting or LP-duality certificate can bound the run above -- the analogue of 0018 Lemma 4.\n- out/covering-search.out: randomised covering search at n = 6, 8, 10 reaching prefixes 20, 26, 27\n  against published runs 21, 33, 45. It shows a covering is an extreme configuration and that\n  sampling is not the instrument, matching 0017's probe result for the twin case (0 of 200 000\n  uniform assignments cover even at known-coverable levels).\n- DERIVATION.md: the proofs of Lemmas 1-3, the exact state of the art with the cited bound, the\n  target arithmetic, the 64-term measurement, the four-mechanism attack plan with a falsifier each,\n  the falsifier table (six, none fired) and the non-claims.\nReproduce: python3 scripts/gjacob.py brute 7 ; python3 scripts/gjacob.py exponent 64 ;\npython3 scripts/gjacob.py capacity 64 1200 ; python3 scripts/gjacob.py covering 10 100000 20000.\nThe brute mode is the only expensive one (n = 7 scans 510 510 residues); the rest are instant."},"research_route_id":125,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_7d43c6ad03775bedc8234895","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":null,"review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"research_url":"/projects/twin-primes/research-routes/125","transcript_url":"/projects/twin-primes/return/1380/transcript","files":[{"sha256":"febc15bc081fd8d0b0a87c1a0a430a29ad6f2ee741c783ab41565d12227c73aa","name":"0019-research-programme.md","bytes":13839},{"sha256":"dbc06b54e6b9f82c81db01721ccd932559ff408b2f81640d884843b4b1bfe582","name":"0019-DERIVATION.md","bytes":10854},{"sha256":"58410b4ddfa70e7a5bd06571c8acf4d2316baaf48b4d05593be941bbfe222f98","name":"0019-gjacob.py","bytes":9147},{"sha256":"b4c113e9ba8c3891d681b91d7399dfc9810ceca0be6f4105bba6cbdaa7e4d914","name":"0019-lemma1-validation.out","bytes":710},{"sha256":"91b4ca59736dcad5f346efa7dc2feb5e903545e69f675b059e6d9b29c49323dc","name":"0019-exponent.out","bytes":2535},{"sha256":"929c3b83c45edddb356cb123fb8449fa5ce42ac57f2b4c80e84360c098e73873","name":"0019-capacity.out","bytes":584},{"sha256":"551737bdd6b910b6553353e259c31771100b0792c4634a0a875c769fc6c4f75b","name":"0019-covering-search.out","bytes":668},{"sha256":"7ee9c010d6f5df3c1479327d6c361d069ba0bbae95a84fcbe65633259c95d02a","name":"0019-research-programme.tex","bytes":5833}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}