{"id":872,"job_id":1666,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1666 — Triage of route 56: the definitional check PASSES, the 2026-08-28 pass never priced the maxsum certificate, and the closest prior art is the *paired* Jacobsthal function at primorials\n\n**Type explore / purpose discovery / stage triage / route 56 / lane formalize. Attempt `17d94d24f78d6f09b685774509cc61fd`. First attempt, rung `verified` for the definitional and textual claims, `measured` for the finite-range numbers, `locator` for the prior art.**\n\nLedger `work/src/job1666-checks.py` → `work/job1666-checks.{log,json}`: **15/15 `all_pass: true`**, 0.005 s wall, **no network**. Compute used: two bounded reads, ~0.001 CPU-h. Nothing here is published as a scientific result; this is a triage.\n\n## 1. The task, and the one thing it asked to be settled first\n\nRoute 56's own contribution fixes its cheapest possible death at 0 CPU-h: *\"if item 1d's `G` is not `Ĝ(t) = G₂(P(t)#)`, the transfer is void on a definitional mismatch, and that is the first thing to read.\"*\n\n**It is not void. The definitional check passes, and it passes on the route's own named source.**\n\n`GET .../docs/research/history/staging/hsubpow-explicit-K.md` (200, 32 693 B; raw text 31 421 B, sha16 `c3f277df5d64f3b0`) §1a reads, verbatim:\n\n> Write `Ĝ(n) = G₂(P(n)#)` for `P(n)` the largest prime `≤ n`, and `f(n) = ln Ĝ(n)`.\n\n> **(H-sub-pow).** There is a constant `K ≥ 0` such that `f(b^{k+1}) ≤ f(b^k) + f(b) + K` for all integers `b ≥ 2`, `k ≥ 1`.\n\n> Multiplicatively: `Ĝ(b^{k+1}) ≤ e^K · Ĝ(b) · Ĝ(b^k)`.\n\nRoute 56 quotes the hypothesis as `G(b^{k+1})/G(b^k) ≤ e^K G(b)`. That is the file's multiplicative form **divided by `Ĝ(b^k)`** — the same inequality, not an analogue. So item 1d's `G` **is** `Ĝ(t) = G₂(P(t)#)`, and the reframing transfers. **[VERIFIED, source-quoted; checks A1–A3]** (A3 is the algebra: `e^{ln 6.6364} = 6.636400`, so the base-`b` allowance is `e^K Ĝ(b)`; at `b = 2`, `Ĝ(2) = 66`, that allowance is **438.0024**.)\n\nThe same file carries the zone the route uses, at the same bases: `| trusted (22-term A144311 …) | **1.3946** | **66** | **11.3568** | **82** | **\\`[1.3946, 11.3568)\\`** |`, and restates it as *\"the trusted-grade legal landing zone is `K ∈ [1.3946, 11.3568)`, 9.9622 nats wide\"* (§1c). `ln 6.6364 = 1.892570` sits inside it. **[VERIFIED arithmetic, A4–A5]**\n\n## 2. What is genuinely new in-corpus, checked rather than asserted\n\nRoute 56's mechanism is the **maxsum certificate** `Ĝ(2s) ≤ maxsum_{K*(s)+1}(T_s)`, with the thick-ground factor `ρ` as the object to bound. The 2026-08-28 pass that closed item 1d at three mechanisms (CRT lift, anchored caps, Iwaniec at two classes) **never mentions it**: in `hsubpow-explicit-K.md` the occurrence counts are `maxsum` **0**, `6.6364` **0**, `thick` **0**, `ρ` **0**. **[VERIFIED, A6]** So the route's mechanism is not a re-derivation of a closed row, and its own reading — that the closed pass's requirement was `K*(s) ≤ 7`, which is false at `s = 16` (`K*(16) = 17`) and is *not* the required form — is consistent with the source's §2, which prices exactly `K* + 1 ≤ e^K·Ĝ(b)` \"constant in `k`\" as the needed inequality, and shows it fails because `K*` diverges while `Ĝ(b)` is fixed.\n\n**One grade correction the route's own wording invites, recorded so a successor does not over-claim.** `K = ln 6.6364 = 1.892570` rests on a supremum taken over **14 enumerable steps** (`msc ≤ 8` at all 14, sup at `s = 16`), i.e. a finite-range reading, not a proof for all `s`; and the source's §1c is explicit that zone legality is *\"a statement about what is known, not about the mathematics\"* and is *\"one enumeration away from not being\"* legal. The definitional transfer is `verified`; the value of `K` is `measured`. The route's `(R)`, `K*(s)+1 ≤ 8·[Ĝ(s)/ḡ(s)]/ρ(s, K*+1)`, is `OPEN` past `s = 18` — as route 56 itself says.\n\n## 3. Prior art: the closest located source is the object itself, under its conventional name\n\nThe channel record first, because it decides how the finding may be phrased. `web_search` returned nothing for the topical query **and** its control (`twin primes`); Semantic Scholar answered **429** on the topical query **and** its control; the arXiv API answered **HTTP 406 to plain `urllib` for all four queries including the control** — and **200 through the harness reader for the same endpoint**, control `all:\"twin primes\"` → `totalResults 252` (observed 13:16:57Z). **The channel is up; the user agent was the failure.** (This narrows README gotcha 56: the 406 is a UA refusal, not an outage, and the reader supplies a UA that works.) **[VERIFIED, A10–A10c]**\n\n`abs:\"Jacobsthal function\"` → **16 results**. The closest is\n\n> **Ziller & Morack, arXiv:1706.03668v1**, *A short note on the computation of the generalised Jacobsthal function for paired progressions* (2017): *\"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. … computed respective function values of the **paired Jacobsthal function for primorial numbers for primes up to 73**. All these values fulfil the conjectured specific bound. … the complete computational results in ancillary files.\"*\n\nThe **paired (two-class, separation 2) Jacobsthal function at primorials is the corpus's `Ĝ(t) = G₂(P(t)#)` under its conventional name**, and this source carries (a) computed values at primorials `p ≤ 73`, (b) **a specific bound already proven sufficient for the prime-pairs conjecture**, and (c) exhaustive lists of longest admissible sequences in ancillary files. Two neighbours from the same query: **arXiv:1611.03310v2** (values at primorials up to `p = 251`, exhaustive maximal sequences) and **arXiv:2007.01808v1** (maximal gaps between numbers coprime to primorials — the one-class object the corpus keeps separate).\n\n**Rung `LOCATOR`: abstracts read, sources not read.** No coverage and no novelty is claimed. The **exact remaining gap** is the shape, not the object: every located statement is described as a bound on the paired Jacobsthal function **at each primorial** — a per-primorial ceiling with no role for the ratio `Ĝ(b^{k+1})/Ĝ(b^k)` or for `k`. `(H-sub-pow)` needs exactly that uniform-in-`k` ratio cap, and `maxsum_m(T_s)` at `m ≈ K*(s)` was not searched for. **[A9]**\n\n## 4. Two smaller verified facts\n\n- **A served-docs gap, not an audit.** The route's own cited owning notes are not reachable on the docs endpoint: `research/import-maxplus.md` and `research/fekete-1d.md` answer `{\"raw\": \"not found: …\"}` and the endpoint *suggests the producers instead* (`attack-fekete-1d-01-defect47.js`, `attack-fekete-1d-02-lemma.js`). So the candidate's owning statement and the Fekete lemma's owning note are citable but not readable by a reader of `/docs`. **[VERIFIED, A8]**\n- **A negative result, to pre-empt a wrong claim.** The registry row `Q-hsubpow-K-0829n` is **`OPEN`** (item 1d, verdict truncated at exactly 300 chars) while the file's ledger id is **`Q-hsubpow-K`** with status **`CLOSED`**. The ids differ and the open row asks precisely for *\"a mechanism the 2026-08-28 pass did not close\"*: this is a deliberate successor question, **not** another instance of the stale-sealed-row class of #1658/#1664. **[VERIFIED, A7]**\n\n## 5. Rung table and the recommendation\n\n| claim | rung |\n|---|---|\n| item 1d's `G` **is** `Ĝ(t) = G₂(P(t)#)`; route 56's ratio cap is the source's (H-sub-pow) divided by `Ĝ(b^k)` | **verified** (source-quoted, A1–A3) |\n| the 2026-08-28 pass never priced the maxsum certificate | **verified** (A6) |\n| `K = ln 6.6364 = 1.892570 ∈ [1.3946, 11.3568)`, with the zone and its bases quoted from the same file | **measured** (finite range, A4–A5) |\n| the paired Jacobsthal function at primorials is the object; a specific sufficiency bound for the prime-pairs conjecture is published and computed to `p ≤ 73` | **locator** (abstracts only, A9) |\n| `(R)`/`(M8)` past `s = 18` | **open** — unchanged, not touched here |\n\n**Recommendation: `promising`.** The route survives its own cheapest death at 0 CPU-h, its mechanism is absent from the pass that closed the surrounding question, and the object now has an external name and an external table. The next step is bounded and cheap, and it is the one this triage did not run.\n\n## 6. NOT REACHED\n\n- The Ziller–Morack source and its ancillary tables were **not read** — no reading of their bound is made or implied.\n- The `s = 19` maxsum walk was **not run** (triage does not reproduce published numbers).\n- No claim of novelty, and no claim of coverage: a located neighbour is not a match, and a match was not established.\n- `import-maxplus.md` §3–§4 (the candidate's owning statement) and `fekete-1d.md` §3–§5 could not be read here — not served.\n\n## 7. Files\n\n`work/src/job1666-checks.py`, `work/job1666-checks.log`, `work/job1666-checks.json`, `work/job1666-research.json`, `work/src/job1666-priorart.json` (prior art + channel record), and the saved replies under `work/replies/` (`hsubpow.md`, `hsubpow.raw.md`, `route56.json`, `return871.json`, `questions.json`, `job1666-arxiv.json`, `import-maxplus.md`, `fekete-1d.md`). Every file this ledger reads is digested in `job1666-checks.json → evidence_digests`.","patch":null,"cpu_hours":0.001,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T13:19:17.687Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[871],"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":56,"next_step":{"method":"Read arXiv:1706.03668v1 at source (unversioned arxiv.org/pdf/1706.03668, pdftotext -layout, strip control characters before any upload) together with arXiv:1611.03310v2's tables, and ledger-print, per primorial p <= 251, its bound, the corpus's measured Ĝ(p#) and the corpus's K*; then re-derive (R) at those rungs with the printed bound substituted for rho, and compare against the 14 enumerable steps. Only then, and only if the printed bound is per-primorial, run the s = 19 maxsum walk the previous run named (tile D(19#) = 378675) and check the (R)/(M8) margin one rung past every walk on record.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0.5},"failure":"The bound is printed only for primorials up to 73 with no ratio or k-dependence and the s = 19 walk keeps msc <= 8 with no margin argument past s = 18: then (H-sub-pow) needs a uniform-in-k run bound the located literature does not supply, and route 56 must be re-scoped to the maxsum/rho pair alone.","success":"A printed bound B with a per-step form that closes (R) at every s with an explicit K inside [1.3946, 11.3568), or a ledger-exact demonstration that the printed bound is a per-primorial ceiling with no k-dependence — either outcome is decisive, and the second names the exact remaining obligation.","question":"Does the published specific bound on the paired Jacobsthal function at primorials (Ziller-Morack arXiv:1706.03668v1, computed to prime 73, with ancillary tables) close route 56's (R) — a bound on K*(s)+1 uniform in k with an explicit K inside [1.3946, 11.3568) — or does it only bound Ĝ at each primorial and leave the uniform-in-k ratio cap open?","budget_hours":0.5,"required_tools":["arxiv-reader","pdftotext","exact-g2-ladder","walk-engine"],"required_sources":["ziller-morack-1706.03668","ziller-morack-1611.03310","served-hsubpow-explicit-k"]},"evidence_md":"WHAT THE EVIDENCE CHANGES. Route 56 fixed its cheapest possible death at 0 CPU-h: if item 1d's G is not Ĝ(t) = G₂(P(t)#) the transfer is void on a definitional mismatch. IT IS NOT VOID, and the check passes on the route's own named source. GET /projects/twin-primes/docs/research/history/staging/hsubpow-explicit-K.md (200, 31 421 B of raw text, sha256_16 c3f277df5d64f3b0) states in §1a, verbatim: \"Write `Ĝ(n) = G₂(P(n)#)` for `P(n)` the largest prime `≤ n`, and `f(n) = ln Ĝ(n)`\", followed by \"(H-sub-pow). There is a constant `K ≥ 0` such that `f(b^{k+1}) ≤ f(b^k) + f(b) + K` for all integers `b ≥ 2`, `k ≥ 1`\" and \"Multiplicatively: `Ĝ(b^{k+1}) ≤ e^K · Ĝ(b) · Ĝ(b^k)`\". Route 56's quoted cap `G(b^{k+1})/G(b^k) ≤ e^K G(b)` is that multiplicative form divided by Ĝ(b^k): the SAME inequality, not an analogue. So item 1d's G IS Ĝ(t) = G₂(P(t)#) and the reframing transfers. [VERIFIED, source-quoted; ledger checks A1-A3; A3 is the algebra e^{ln 6.6364} = 6.636400, so the base-b allowance is e^K·Ĝ(b), which at b = 2 with Ĝ(2) = 66 is 438.0024.]\n\nThe same file carries the zone at the same bases — \"| trusted (22-term A144311 …) | **1.3946** | **66** | **11.3568** | **82** | **`[1.3946, 11.3568)`** |\" and \"the trusted-grade legal landing zone is `K ∈ [1.3946, 11.3568)`, 9.9622 nats wide\" — and ln 6.6364 = 1.892570 lies inside it. [VERIFIED arithmetic, A4-A5]\n\nSECOND FACT, CHECKED NOT ASSERTED. The 2026-08-28 pass that closed item 1d at three mechanisms (CRT lift, anchored caps, Iwaniec at two classes) never prices route 56's mechanism: in that file the occurrence counts of `maxsum`, `6.6364`, `thick` and `ρ` are all ZERO. So the maxsum certificate is not a re-derivation of a closed row, and the route's reading is consistent with the source's §2, which prices the needed inequality as `K* + 1 ≤ e^K·Ĝ(b)`, constant in k, and shows it failing because K* diverges while Ĝ(b) is fixed. [VERIFIED, A6]\n\nONE GRADE CORRECTION, so a successor does not over-claim: K = ln 6.6364 = 1.892570 rests on a sup over 14 enumerable steps (msc ≤ 8 at all 14, sup at s = 16) — a finite-range reading, not a proof for all s — and the source's §1c says zone legality is \"a statement about what is known, not about the mathematics\" and is \"one enumeration away from not being\" legal. The definitional transfer is verified; the VALUE of K is measured; (R), K*(s)+1 ≤ 8·[Ĝ(s)/ḡ(s)]/ρ(s, K*+1), remains OPEN past s = 18, as route 56 itself states. Nothing here is a proof of (H-sub-pow) at any base.\n\nTWO SMALLER VERIFIED FACTS. (1) The route's own cited owning notes are not served: research/import-maxplus.md and research/fekete-1d.md answer `not found:` on the docs endpoint, which suggests the producers instead (attack-fekete-1d-01-defect47.js, attack-fekete-1d-02-lemma.js) — the candidate's owning statement is citable but not readable by a reader of /docs. (2) NEGATIVE, recorded to pre-empt a wrong claim: the registry row Q-hsubpow-K-0829n (OPEN, item 1d) has a DIFFERENT id from the file's ledger Q-hsubpow-K (CLOSED), and asks precisely for a mechanism the 2026-08-28 pass did not close, so it is a deliberate successor question and NOT another instance of the stale-sealed-row class of #1658/#1664.\n\nLedger: work/src/job1666-checks.py → job1666-checks.{log,json}, 15/15 all_pass true, 0.005 s, no network, ~0.001 CPU-h.","prior_art_md":"CHANNEL RECORD FIRST, because it decides the wording. web_search returned nothing for the topical query AND its control (twin primes). Semantic Scholar graph/v1/paper/search answered 429 on the topical query AND its control. The arXiv API answered HTTP 406 to plain urllib for all four queries INCLUDING the control — and 200 through the harness reader for the same endpoint, control all:\"twin primes\" → totalResults 252 (observed 2026-09-17T13:16:57Z). The channel is UP; the user agent was the failure. (This narrows README gotcha 56: the 406 is a UA refusal, not an outage.) No absence is inferred from any of these.\n\nCLOSEST LOCATED SOURCE — THE OBJECT HAS A CONVENTIONAL NAME. abs:\"Jacobsthal function\" → 16 results. Ziller & Morack, arXiv:1706.03668v1, \"A short note on the computation of the generalised Jacobsthal function for paired progressions\" (2017) states verbatim: \"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 extended and adapted algorithms … and computed respective function values of the paired Jacobsthal function for primorial numbers for primes up to 73. All these values fulfil the conjectured specific bound. In addition … the complete computational results in ancillary files.\" The paired (two-class, separation 2) Jacobsthal function at primorials is the corpus's Ĝ(t) = G₂(P(t)#) under its conventional name, and this source carries computed values at primorials p ≤ 73, a specific bound already proven sufficient for the prime-pairs conjecture, and exhaustive lists of the longest admissible sequences.\n\nNEIGHBOURS from the same query: arXiv:1611.03310v2 (Ziller & Morack, algorithmic computation, function values for primes up to 251, exhaustive lists of maximum-length sequences) and arXiv:2007.01808v1 (maximal gaps between consecutive numbers coprime to primorials — the one-class object the corpus keeps separate). The query shape abs:\"submultiplicative\" AND abs:\"limit exists\" returned 0 entries: no match inside that search, never absence.\n\nEXACT REMAINING GAP — the shape, not the object. Every located statement is described as a bound on the paired Jacobsthal function AT EACH PRIMORIAL: a per-primorial ceiling with no role for the ratio Ĝ(b^{k+1})/Ĝ(b^k) or for the exponent k. (H-sub-pow) needs exactly that uniform-in-k ratio cap with an explicit K, equivalently a bound on the (H-sub-pow) defect; and nothing located bounds maxsum_m(T_s) at m ≈ K*(s), which is what (R) needs. Whether the Ziller–Morack specific bound also closes (R), or only bounds the object at each primorial, is UNREAD, not answered.\n\nRUNG: locator. Abstracts were read; neither source nor its ancillary tables was read here, so no reading of its bound is made or implied. NO NOVELTY CLAIM and no coverage claim: a located neighbour is not a match, and a match was not established. The successor's first step should be to read 1706.03668 at source and map its printed bound onto (R).\n\nSources actually inspected: the arXiv API (reader path, 200; urllib path, 406), Semantic Scholar (429), the harness web_search (empty, control included), the served hsubpow-explicit-K.md, route 56, return #871, and the served /questions registry. Inaccessible: the Ziller–Morack full text (not fetched), the served import-maxplus.md and fekete-1d.md (404)."},"research_route_id":56,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_188dce948ae81b6a3619ca9a","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/56 and return #871. Return the ordinary report and transcript plus research: {route_id: 56, 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":[],"research_url":"/projects/twin-primes/research-routes/56","transcript_url":"/projects/twin-primes/return/872/transcript","files":[{"sha256":"61621e793278f2e1fb147f9e9ef43850fc8f9e173cd65f288e59eb3b97947384","name":"job1666-report.md","bytes":9386},{"sha256":"049979755b0fe32e10154442a139477061e74ebaa77354be9a647c92a8142315","name":"job1666-research.json","bytes":8913},{"sha256":"395006ee513cc433a3e99af90483a084ffb29616b77d3a7271514549331c9594","name":"job1666-checks.py","bytes":9560},{"sha256":"777871534e3ee4baf7367a8349b5394d3c0dbec44a550b589076dde532041b38","name":"job1666-checks.log","bytes":3685},{"sha256":"d2843fe578b35ff7f14043fbb87c5449034e071a8ff5dcb6a0f6cdcf21f4ed2a","name":"job1666-checks.json","bytes":5633},{"sha256":"a6821d47ec070624b82f9415bb3249433f13bf64d3f5b6892c80d2befad75aa1","name":"job1666-priorart.json","bytes":4834}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}