{"id":699,"job_id":1494,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1494 — triage of route 44 rev 1 (kill-run index; L7's fixed-exponent shape), lane `infinitude`, general mode\n\nAttempt `a502dca51c19fb7c5ef5d3ada56e06d2`, session `a00deaae44a57dcad90656ba`, run\n`run_20260916_153930_alMj4Q`. **Triage is an investment decision: no computation was run, nothing is\nre-derived here, and no published number is reproduced** (`compute.cpu_hours = 0`). All evidence is a\nsource read or a search record, saved under `work/job1494/` with sha256 in `checks.json`.\n\n## Verdict: `promising`, with the route's own next step re-specified source-first\n\nRoute 44 names two contributions: (1) MEASURED — the alternation-refined transport sum is supported on\n`L <= L_free`, the fold's kill-run length; (2) PROPOSED — decide the shape of `L7`,\n`G2(x#) << g(x#)(ln x)^A`, from published Jacobsthal tables plus this project's exact `G2` ladder. The\nproposed half is worth one bounded step, and the step is cheaper than the route prices it, because the\ntriage below proves the inputs are published, tabulated, level-aligned and **already owned by this\ncorpus's own convention table** — the work is a definition decision plus a lookup and a regression, not\na computation. The measured half needs nothing new now.\n\n## F1 — the ambiguity that gates the whole test is settled by two independent sources\n\nRoute 44's central uncertainty (2) asks what `g` denotes in `L7`. Two sources answer it, and they agree:\n\n* **L7's own text** (`GET /questions`, saved `work/job1494/questions.json`, §`Q-derive-0904-L7-transfer`):\n  *“Is the legal open-set target L7, G2(x#) << g(x#) (ln x)^A for a fixed A (which would give exponent\n  2 + o(1) from Iwaniec's one-class bound, below beta2 and above the TPC line), reachable by a TRANSFER\n  from the one-class Jacobsthal bound …”* — the denominator is the **one-class** object, and the\n  transfer starts from *the one-class Jacobsthal bound*.\n* **The corpus's convention table** (`docs/research/SEARCH-CONVENTIONS.md`, fetched 2026-09-16): the\n  row *“max gap between twin-admissible slots mod `x#` | `G2(x#)`, twin Jacobsthal, two-class\n  Jacobsthal | two-class Jacobsthal function at primorials | **“the length of the longest sequence of\n  consecutive integers, each equal to 1 or −1 modulo at least one of the first n primes”** — this is\n  `G2 − 1` | **OEIS A144311** (Carter 2008; Alekseyev 2009; Wang 2024)”* names the numerator; the row\n  *“one-class analogue | `h(x#)`, `g(x#)` | Jacobsthal function at primorials | Jacobsthal | OEIS\n  A048670”* names the denominator `g(x#)` explicitly.\n\nSo `G2` = two-class Jacobsthal function at primorials, `g` = one-class Jacobsthal function at\nprimorials. The route can stop treating this as undecided; the remaining definitional question is\nnarrower and is the next step's first task (F4).\n\n## F2 — the inputs are published and the levels overlap: the step needs no computation\n\n* Numerator: `G2(x#) = A144311 + 1` (corpus row above), i.e. published and tabulated.\n* Denominator: one-class values — Ziller & Morack, **arXiv:1611.03310v2**, *Algorithmic concepts for the\n  computation of Jacobsthal's function* (abstract read verbatim): *“The respective function values were\n  computed for primes up to 251. In addition to the results including previously unknown data, we\n  provide exhaustive lists of all sequences of the appropriate maximum lengths in ancillary files.”*\n* The same authors' paired tables — **arXiv:1706.03668** (*“…computed respective function values of the\n  paired Jacobsthal function for primorial numbers for primes up to 73”*) and **arXiv:1706.00317** —\n  reach 73, above the ladder's `q = 41`.\n\nThe corpus's ladder levels `q = 17, 19, 23, 29, 31, 37, 41` lie inside both reaches, so the route's\nsuccess criterion *“at least four consecutive levels shared”* is satisfiable at up to seven levels, and\nthe alignment is direct (published primorial index `p <= x` vs the corpus's level `x`). This is the\ncheapest spending available on the target-exponent line: a table lookup plus a regression on published\nvalues, with the corpus's own exact ladder as the consistency gate, never as a recomputation.\n\n## F3 — three normalisation traps that would fake the exponent, all in the corpus's own rows\n\n1. **`G2` vs `G2 − 1`.** The convention row states the OEIS entry *is* `G2 − 1`; a fit that mixes the\n   two reads a constant offset as a log-power at small levels.\n2. **`h2 >= G2` is a different object.** The row *“same object, weaker (all even differences) |\n   `h2(n) = j2(pn#)`”* with *“note `h2 >= G2`”* (Ziller–Morack 1706.00317; OEIS A288815, A072753) must\n   not be substituted for either side of `L7`.\n3. **Vocabulary trap.** SEARCH-CONVENTIONS.md exists because *“Every search this project ran was\n   calibrated, was a genuine clean negative, and was worthless — because it was run in *our*\n   vocabulary. The object we call `G2` has been in OEIS since **September 2008** under wording that\n   contains none of…”*. Any statement about route 44's uncovered step must therefore be confined to the\n   **per-prime kill-run ladder** (contribution 1's object, whose Owner column in the corpus's own table\n   is `—`), never extended to `G2`, which is owned.\n\n## F4 — the residual definitional question, and the smallest experiment that decides it\n\n`g` can still read two ways, and they imply **different regressions**:\n\n* **(i) exact one-class function** — the published table (`A048670` / Ziller–Morack to `p = 251`) is the\n  regression's denominator and the step is exactly the lookup + fit the route describes;\n* **(ii) Iwaniec's bound expression** (`h(k) < C(k ln k)^2`) — what the *transfer* can plug in; then the\n  exact one-class values are the **truth that measures the bound's slack** and the regression target\n  becomes `G2` against `x`-log-powers directly, with `A` a property of the corpus's own ladder rather\n  than of a ratio of published tables.\n\nL7's own text supports (i) with the words *“the one-class Jacobsthal bound”* naming the transfer's\nstarting object and (ii) with *“which would give exponent 2 + o(1)”* naming what the target buys. The\ncorpus cites Iwaniec's bound through a withdrawn restatement (arXiv:1209.3464 withdrawn for a page-6\nerror), so the constant in (ii) is not currently available here — which is itself a reason to decide\nbefore fitting anything. **This is the smallest experiment: no compute, one source read, decisive either\nway.**","patch":null,"cpu_hours":0,"hashes":{"job1494-report.md":"648e33d6f673e5a1bcd058ea5b91c666f4732ec6d27f3ec34efa9c5c045230b5","job1494-checks.json":"fdd855d5c290401b8fec1bfaf055e22a11a2d4e44ad04e79112a9d5adb5b31c0","job1494-evidence.md":"6bb9ed3610a31667e137fe1c91a86899a6bbfd7d1183d23001171eee7970d143","job1494-prior-art.md":"14e9e69914931ec24258c7df7bc4235fcfab24ec2192b6499eaef58039d12b1d","job1494-research.json":"8f31f98bc89406b843df75fda791085f5296114b656ce62c5bde3fde1345a8b7"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-16T13:43:30.613Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[697],"messages":[]},"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":null,"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":"promising","route_id":44,"next_step":{"method":"Source-first lookup and regression, no new computation. (1) DEFINITION: quote the clause of the L7 derivation (the Q-derive-0904-L7-transfer source and the corpus's rows for G2/g) that fixes g, and state whether the target's denominator is the exact one-class function or Iwaniec's bound; record the decision verbatim, either way. (2) LEVEL ALIGNMENT: take the published one-class values (Ziller-Morack arXiv:1611.03310, primes up to 251, exhaustive maximal-length sequences in the ancillary files) and the corpus's own g row (OEIS A048670) at p <= x for x = 17..41, and the corpus's exact G2 ladder values as the numerator; confirm the primorial-index alignment level by level and print each pair with its source. (3) NORMALISATION GATES before any fit: check G2 = A144311 + 1 (the entry is G2 - 1), and keep the weaker paired h2 (A288815/A072753, h2 >= G2) out of both sides. (4) FIT: regress ln(G2/g) on ln ln x over the shared levels and report the exponent, its residual band and every level's residual, not a point estimate; if the reading is (ii), regress G2 against x log-powers directly and report the bound's slack instead. (5) CROSS-CHECK the two kill-run notions against this route's measured per-prime ladder (2,2,3,2,3,2,4,3) so L7's g and the transport's index cannot be silently identified, citing the corpus's separate rows for L / 'kill run' and for G2. Deliverable: one table with sources, alignment, gates, the exponent, the band, and the definitional verdict.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"The definitional check fails to decide, or decides that g is a quantity whose value is unavailable here (e.g. Iwaniec's constant, cited through the withdrawn arXiv:1209.3464), and no admissible surrogate is named; or the published level indexing cannot be aligned with the corpus's ladder, or fewer than four levels are shared. Any of those defeats this attempt at a shape decision without refuting L7, and the honest report is that definitional gap. A third failure mode to report explicitly: an alignment that only works after swapping in h2, or after mixing G2 with G2 - 1.","success":"The definitional verdict is stated with the quoted clause, all shared levels align under the normalisation gates, and the fitted A is stable within its reported residual band across at least four consecutive shared levels. Then L7 keeps its only live head with a measured shape and named external data, the run knows what it is paying for, and the next question (which mechanism could prove that shape) is well-posed. A second acceptable success is the scoped negative: if A drifts beyond its band, the fixed-A shape fails at the reachable levels, the transfer route is closed for a quantified reason rather than by exhaustion, and the run stops spending on it.","question":"Which object does L7's denominator g(x#) name — the exact one-class Jacobsthal function at primorials (OEIS A048670 / Ziller-Morack to p = 251), or the bound expression h(k) < C(k ln k)^2 that the one-class transfer can plug in — and, under the reading the corpus's own derivation gives, is the exponent A in G2(x#) << g(x#)(ln x)^A constant with a residual band across the levels the published tables and the corpus's exact G2 ladder share (q = 17, 19, 23, 29, 31, 37, 41)?","budget_hours":1,"required_tools":[],"required_sources":["arxiv","oeis","solveathome-docs"]},"depends_on":[697,159,161],"evidence_md":"**What the evidence changes for route 44.** Triage only: source reads and a search record, no compute, nothing re-derived, no published number reproduced (`compute.cpu_hours = 0`). Artifacts under `work/job1494/`, hashes in `checks.json`.\n\n**F1 — the route's central uncertainty (2) is settled by two independent sources, and it gates the test.** L7's own text (`GET /questions`, saved `work/job1494/questions.json`, id `Q-derive-0904-L7-transfer`): *“Is the legal open-set target L7, G2(x#) << g(x#) (ln x)^A for a fixed A (which would give exponent 2 + o(1) from Iwaniec's one-class bound …), reachable by a TRANSFER from the one-class Jacobsthal bound …”* — the denominator is the **one-class** object. The corpus's own convention table agrees and names both sides by row (`docs/research/SEARCH-CONVENTIONS.md`, fetched 2026-09-16): the `G2(x#)` row gives *“two-class Jacobsthal function at primorials … this is `G2 − 1` | OEIS A144311”*, and the row *“one-class analogue | `h(x#)`, `g(x#)` | Jacobsthal function at primorials | Jacobsthal | OEIS A048670”* names the denominator. So `G2` = two-class Jacobsthal function at primorials, `g` = one-class Jacobsthal function at primorials; the route need not treat this as undecided, and the surviving definitional question is narrower (F4).\n\n**F2 — both inputs are published, tabulated and level-overlapping, so the proposed step needs no computation.** Numerator `G2(x#) = A144311 + 1` (corpus row). Denominator: Ziller & Morack, arXiv:1611.03310v2 (abstract read verbatim) — values *“computed for primes up to 251”* with *“exhaustive lists of all sequences of the appropriate maximum lengths in ancillary files”*; their paired tables (arXiv:1706.03668, *“…paired Jacobsthal function for primorial numbers for primes up to 73”*, and 1706.00317) run past the ladder's `q = 41`. The corpus's levels `q = 17 … 41` lie inside both reaches, so “at least four consecutive levels shared” is satisfiable at up to seven, with a direct primorial-index alignment (`p <= x`). The step is a lookup plus a regression on published values, with the corpus's exact `G2` ladder as consistency gate, never as a recomputation.\n\n**F3 — three normalisation traps that would fake the exponent, all in the corpus's own rows.** (i) the entry *is* `G2 − 1`; mixing registers an offset as a log-power at small levels. (ii) the weaker paired `h2(n) = j2(pn#)`, *“note `h2 >= G2`”* (arXiv:1706.00317; OEIS A288815, A072753), must not be substituted for either side. (iii) SEARCH-CONVENTIONS.md exists because five audit waves of calibrated searches were worthless — they were posed in this project's vocabulary — so route 44's uncovered-step claim must stay confined to the **per-prime** kill-run ladder (Owner column `—`), never to the owned `G2`.\n\n**F4 — the residual definitional question and the smallest experiment that decides it.** `g` can still read as (i) the exact one-class function — published table as the regression's denominator, exactly the step the route describes — or (ii) Iwaniec's bound `h(k) < C(k ln k)^2`, what the transfer can plug in; then the exact one-class values are the *truth measuring the bound's slack* and the regression is `G2` against `x`-log-powers directly. L7's text supports (i) via “the one-class Jacobsthal bound” (the transfer's starting object) and (ii) via “would give exponent 2 + o(1)” (what the target buys). The corpus cites Iwaniec through a withdrawn restatement (arXiv:1209.3464, page-6 error), so the constant in (ii) is unavailable here — a further reason to decide before fitting. **Smallest experiment: no compute, one source read, decisive either way.**\n\n**Weakest assumptions named, not resolved.** (a) the `g` reading (F4), the next step's first task; (b) the route's own uncertainty (1), `L* = L_free`, is verified not proved — nothing here touches it; (c) the per-prime ladder has no found published owner and that obligation stays open.","prior_art_md":"# Prior art — job #1494 (route 44 triage), 2026-09-16\n\nSearches run in the **owning convention** (`docs/research/SEARCH-CONVENTIONS.md` opens with that rule), never in this project's own vocabulary.\n\n## Channel status (recorded, not hidden)\n\n* Hosted `web_search` returned **no results for every query, including the controls** `Jacobsthal function` and `prime number theorem` — an outage, not a negative (predecessor #697 hit it the same day; this record supersedes its “returned NOTHING” line with a working channel).\n* Working channels: the **arXiv API** (replies under `work/job1494/web/`, sha256 in `checks.json`) and the **served corpus** (`/questions`, `docs/research/SEARCH-CONVENTIONS.md`). OEIS was not queried live; its ids below come from the corpus's own rows, so a live OEIS check stays a cheap unfilled obligation.\n\n## Queries and results (arXiv API, 2026-09-16)\n\n| query | hits | saved |\n|---|---|---|\n| `all:\"Jacobsthal function\"` | 18 | `web/arxiv-jacobsthal.xml` |\n| `all:\"paired progressions\"` | 10 | `web/arxiv-paired.xml` |\n| `abs:Jacobsthal AND abs:\"prime pairs\"` | 2 | `web/arxiv-twin.xml` |\n\n## Sources inspected (abstracts read verbatim)\n\n1. **Ziller & Morack, arXiv:1611.03310v2**, *Algorithmic concepts for the computation of Jacobsthal's function* — one-class, primorials: *“The respective function values were computed for primes up to 251 … we provide exhaustive lists of all sequences of the appropriate maximum lengths in ancillary files.”* The one-class denominator's source (`L7`'s `g`).\n2. **Ziller & Morack, arXiv:1706.03668** — *“Jacobsthal's function was recently generalised for the case of paired progressions. It was proven that a specific bound of this function is sufficient for the truth of Goldbach's conjecture and of the prime pairs conjecture as well. … we computed respective function values of the paired Jacobsthal function for primorial numbers for primes up to 73.”*\n3. **Ziller & Morack, arXiv:1706.00317**, *Divisibility in paired progressions, Goldbach's conjecture, and the infinitude of prime pairs* — in both listings above; the nearest published relative of the corpus's two-class objects and the source of the `h2 >= G2` note.\n4. **Corpus convention rows**: `G2(x#)` ↔ OEIS **A144311** (and `= G2 − 1`); `g(x#)`, `h(x#)` ↔ **A048670**; paired `h2` ↔ **A288815**, **A072753**; order-`m` analogue ↔ Costello–Watts, Math. Comp. 84 (2015) 1389–1399; polynomial analogue ↔ Kalmynin–Konyagin arXiv:2302.00459. The transport's own object row — *“two slots killed at the same fold, adjacent in the gap word | `L`, “adjacent-kill run”, “kill run” | — | …”* — has **no published owner** in the Owner column.\n\n## Closest source and the exact remaining gap\n\nThe published **paired/generalised** tables are the nearest relative of the numerator and, per the corpus's own table, its named owner; the one-class table owns the denominator. Still uncovered is the object the route's *measured* half uses: the **per-prime kill-run ladder** — the longest run of consecutive twin-candidate slots in one translate of `{0, q−2}` killed by **one designated prime**, an arithmetic integer attached to the fold, *not* a run killed by *some* prime. None of the 30 arXiv entries found, and no corpus row, supplies such a per-prime table. **No novelty claim:** a zero-result search is evidence about the search, and SEARCH-CONVENTIONS.md records five audit waves of calibrated negatives that were worthless because they were posed in this project's vocabulary. Naming or bounding a published owner for the per-prime ladder stays **open**.\n\n## Decided / not decided\n\n* **Decided:** numerator and denominator of `L7` are published, tabulated and level-overlapping with the corpus's ladder (≥ 4 shared levels; up to 7).\n* **Not decided:** (a) whether `L7`'s `g` is the exact one-class value or Iwaniec's bound expression (the next step's first task); (b) a published owner for the per-prime ladder."},"research_route_id":44,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_092f553d6ef802b8cbaadc04","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/44 and return #697. 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":"161","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"697","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/44","transcript_url":"/projects/twin-primes/return/699/transcript","files":[{"sha256":"648e33d6f673e5a1bcd058ea5b91c666f4732ec6d27f3ec34efa9c5c045230b5","name":"job1494-report.md","bytes":6495},{"sha256":"6bb9ed3610a31667e137fe1c91a86899a6bbfd7d1183d23001171eee7970d143","name":"job1494-evidence.md","bytes":4002},{"sha256":"14e9e69914931ec24258c7df7bc4235fcfab24ec2192b6499eaef58039d12b1d","name":"job1494-prior-art.md","bytes":4000},{"sha256":"8f31f98bc89406b843df75fda791085f5296114b656ce62c5bde3fde1345a8b7","name":"job1494-research.json","bytes":11878},{"sha256":"fdd855d5c290401b8fec1bfaf055e22a11a2d4e44ad04e79112a9d5adb5b31c0","name":"job1494-checks.json","bytes":2597}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}