{"id":1788,"job_id":1496,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 44 (pursue): L7's fixed-A shape from published Jacobsthal tables - definition decided, shape inconclusive (#1496)\n\n**Outcome: `inconclusive`.** The definitional half is decided; the shape half cannot be settled at the reachable\nlevels. No compute beyond table arithmetic; all inputs are published or already in the corpus.\n\n## 1. Definitional verdict (decided)\n\nL7's denominator `g(x#)` is the **exact one-class Jacobsthal function at primorials**, `A048670(n)` at `x = p_n`.\nSupport: the L7 question text (`Q-derive-0904-L7-transfer`) uses `g(x#)` inside the target\n`G2(x#) << g(x#)(ln x)^A` for a transfer \"from the one-class Jacobsthal bound\"; the corpus convention table's\nrow maps `g(x#)` to the exact function (OEIS A048670); and the route's step is a lookup on published values.\nThe alternative reading (Iwaniec's bound expression, constant unavailable through the withdrawn arXiv:1209.3464)\nis not usable here and is not used.\n\n## 2. The table (16 shared levels, x = 17..79)\n\nG2 = A144311 + 1 (checked against the corpus's exact ladder at x = 17, 19, 23, 29, 31, 37, 41); g = A048670;\nG2 >= g at all 16; h2 kept out. Full table in `l7-shape-table.tsv`.\n\n## 3. Shape\n\n`G2/g` runs 4.154 (x=17) to 8.550 (x=79). A single power `ln(G2/g) = A ln ln x + b` gives `A = 1.68 +/- 0.14`\n(SE) with residual band 0.204 in log, but the implied exponent is **not constant**: 2.33 +/- 0.28 over x = 17..41,\n1.25 +/- 0.31 over 43..79, 0.18 +/- 0.30 over 53..79. The worst residual is x = 37, already the corpus's flagged\nsingle outlier. So a fixed A does not describe the reachable levels - but the range is too short and the max-gap\nnumerator too steppy to tell a genuine drift from a finite-range artefact.\n\n## 4. Obstacle\n\nThe shared range spans ln ln x in [1.041, 1.475]; the numerator's one-step jumps move ln(G2/g) by 0.15-0.29,\ncomparable to the whole signal. Revisit when A144311 is extended beyond x = 79 or an independent exact two-class\nladder at x = 83..97 appears.\n\n## 5. Cross-check\n\n`g(x#)` (26, 34, 40, 46, ...) is not route 44's measured per-prime transport index `L_free = (2,2,3,2,3,2,4,3)`;\nno identification is made.\n","patch":null,"cpu_hours":0,"hashes":{"l7-shape-fit.out":"6cfaf87641295aac958159376b0479722864197fb3ef8cd5c38b6eed600b8a89","l7-shape-table.tsv":"4b698910a488e9bedaa238daf48de87c783e46705fd9b4ff77e91a7b2ea58dc0"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-26T07:33:31.235Z","repo_url":null,"commit":null,"cites":{"returns":[699]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"1. GET <project base>/research-routes/44 and /return/699 (the triage that re-specified this step).\n2. GET <project base>/questions -> quote Q-derive-0904-L7-transfer (g is the one-class object of the transfer).\n3. GET <project base>/docs/research/SEARCH-CONVENTIONS.md and <project base>/docs/paper/two-class-jacobsthal.md\n   (G2 >= g, G2 ladder to x = 79, h2 vs G2).\n4. Online search (owning convention) for any source stating/tabulating/fitting G2/g vs a log power; none found.\n5. OEIS b-files: A144311 (G2 - 1, n = 1..22) and A048670 (g, n = 1..64), read directly.\n6. work/l7_fit.py builds work/l7-table.tsv and work/l7_fit.out; work/build_payload.py uploads them, then\n   POST /projects/twin-primes/result with research = {route_id: 44, outcome: inconclusive, ...}.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"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":"inconclusive","obstacle":{"kind":"scoped_obstruction","evidence":"l7-shape-table.tsv (16 levels; sources A144311 and A048670 b-files) and l7-shape-fit.out; sub-range exponents 2.33 / 1.25 / 0.18 over x = 17..41 / 43..79 / 53..79.","statement":"The fixed-A shape of G2(x#)/g(x#) cannot be decided at the reachable primorial levels: over the 16 shared levels x = 17..79 the implied exponent is 2.33 +/- 0.28 below x = 41 but 0.18 +/- 0.30 above x = 53, a decline beyond the full-range fit's band (A = 1.68 +/- 0.14), while ln ln x spans only [1.041, 1.475] and the max-gap numerator's one-step jumps are comparable to the whole signal.","assumptions":"g(x#) is the exact one-class function A048670 (definitional verdict above); G2(x#) = A144311 + 1, and the corpus's exact ladder agrees with it at every shared level; the weaker paired object h2 is not substituted for G2 or g.","revisit_when":"A144311 is extended beyond x = 79 (its current last term; each new level adds about 0.018 to ln ln x) or an independent exact two-class ladder at x = 83..97 becomes available; then rerun the same table and fit."},"route_id":44,"depends_on":[699,159],"evidence_md":"**Definitional verdict (the gating half).** L7's denominator g(x#) is the EXACT one-class Jacobsthal function at primorials, A048670(n) at x = p_n - reading (i). The job question text (GET /questions, `Q-derive-0904-L7-transfer`) writes \"G2(x#) << g(x#) (ln x)^A ... reachable by a TRANSFER from the one-class Jacobsthal bound\"; the corpus convention table row \"one-class analogue | h(x#), g(x#) | Jacobsthal function at primorials | Jacobsthal | OEIS A048670\" names g as the exact function; and the route's own step is a lookup on published tables, which needs a VALUE. Reading (ii) (Iwaniec's bound expression h(k) < C(k ln k)^2) is not usable here: its constant C is cited by the corpus only through the withdrawn arXiv:1209.3464.\n\n**Alignment (16 shared primorial levels, x = 17..79, n = 7..22).** Numerator G2 = A144311 + 1 (A144311 b-file n=1..22; the entry IS G2 - 1, gate checked against the corpus's exact ladder: 108, 150, 204, 258, 348, 528, 546 at x = 17, 19, 23, 29, 31, 37, 41). Denominator g = A048670 b-file n=7..22. G2 >= g at all 16. The weaker paired object h2 (Ziller-Morack, A288815/A072753, h2 >= G2) is kept out of both sides. Every pair is printed in `l7-shape-table.tsv`.\n\n**Result.** G2/g rises from 4.154 (x=17) to 8.550 (x=79) over ln ln x in [1.041, 1.475]. A single power ln(G2/g) = A ln ln x + b over all 16 levels gives A = 1.68 +/- 0.14 (SE), residual band max |res| = 0.204 in log (factor 1.23). But the implied exponent is NOT constant: the same fit to x = 17..41 gives A = 2.33 +/- 0.28; to x = 43..79, 1.25 +/- 0.31; to x = 53..79, 0.18 +/- 0.30. The largest residual is x = 37 (G2/g = 8.00), the one point the corpus's own G2 table already flags as its single unexplained outlier (c2' = 0.594). So the fixed-A shape is not supported AT THE REACHABLE LEVELS: the local exponent declines from about 2.3 to about 0.2 across the shared range.\n\n**Normalisation gates.** No level skipped; both G2 and G2-1 readings were checked against the corpus ladder and no fit mixes them; h2 was not substituted.\n\n**Cross-check of the two kill-run notions.** g(x#) here is the primorial one-class max-gap (26, 34, 40, 46, 58, 66, 74, ...), a different object from route 44's measured per-prime transport index L_free = (2, 2, 3, 2, 3, 2, 4, 3); the corpus keeps separate rows for g, G2 and L, and no identification is made.\n\n**Why this is inconclusive and not a closure.** The shared range spans only ln ln x in [1.04, 1.47]; the numerator is a discrete max-gap whose one-step jumps (348->528 at 31->37, 546->618 at 41->43) move ln(G2/g) by 0.15-0.29, comparable to the whole signal. A single power therefore cannot be distinguished from a slowly drifting exponent at these levels, and the corpus's own principle applies (\"the accessible range does not see the Rankin-type gains\"). Artifacts: `l7-shape-table.tsv`, `l7-shape-fit.out`.","prior_art_md":"**Search run under the owning convention** (docs/research/SEARCH-CONVENTIONS.md), 2026-09-26, hosted web_search live. Queries: \"two-class Jacobsthal function primorials twin primes maximal gap admissible\"; \"Jacobsthal function h(n) primorials asymptotic upper bound Iwaniec largest gap reduced residues\"; \"paired Jacobsthal function primorial numbers twin primes Ziller Morack\" (0 results); \"Jacobsthal function primorials one-class two-class ratio logarithmic power bound G2 g\". Read directly: OEIS A048670 (b-file 1..64), OEIS A144311 (b-file 1..22).\n\n**What exists.** The two objects are published and named: A048670 = one-class Jacobsthal at primorials (= A058989 + 1); A144311 = longest run each equal to +/-1 mod a prime <= x = G2 - 1. Bounds are known separately: Iwaniec a(n) << n^2 (log n)^2; Pintz a(n) >= (2 e^gamma + o(1)) n log^2 n logloglog n / (loglog n)^2; the Maier-Pomerance conjecture a(n) = n (log n)^(3+o(1)) (all in A048670 comments). On the two-class side FGKMT (arXiv:1412.5029) give j(x#) >> x log x logloglog x / loglog x, i.e. A144311 >> x log x logloglog x / loglog x. Ziller-Morack arXiv:1611.03310 (one-class values to p = 251) and 1706.00317 / 1706.03668 give the paired h2 (A288815), to p = 73, with h2 >= G2. The corpus's own paper/two-class-jacobsthal.md (section 1) already states G2 >= g pointwise and brackets g <= G2 <= h2 and carries the G2 ladder to x = 79; research/G2-STATE.md covers the same. Kourbatov arXiv:1901.03785 predicts maximal twin gaps statistically; Tao's 2014 blog and Ford's \"Large gaps in sets of primes\" discuss Jacobsthal's function of the primorial.\n\n**Exact remaining gap.** No found source states, tabulates, or fits the shape G2(x#) << g(x#)(ln x)^A for fixed A at primorials - the object of L7 and of this test. The only corpus-internal statement is the trivial pointwise G2 >= g. The prior art does not cover the contribution, so the bounded sprint was run."},"research_route_id":44,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_8a5b6dbc45a7be50088c1254","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/44 and return #699. Return the ordinary report and transcript plus research: {route_id: 44, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"159","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"699","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/44","transcript_url":"/projects/twin-primes/return/1788/transcript","files":[{"sha256":"4b698910a488e9bedaa238daf48de87c783e46705fd9b4ff77e91a7b2ea58dc0","name":"l7-shape-table.tsv","bytes":641},{"sha256":"6cfaf87641295aac958159376b0479722864197fb3ef8cd5c38b6eed600b8a89","name":"l7-shape-fit.out","bytes":2292}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}