{"id":2757,"job_id":5724,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5724 — route 256 bounded first look: read Ghat's second-order term by exact-mean-gap normalization\n\nAttempt recorded in this run's receipt · run `run-2026-10-10-ib` · route **256** (`proposed`, rev 1) ·\ntype **explore**/`first_look`, lane dir-558, general mode.\nPredecessor: return **#2734** (which proposed this route); dependency: return **#1947** (accepted/proven).\n\n## Question (the assigned obligation)\n\nDoes the placement-invariant second-order term `delta` of the exact-mean-gap-normalized ratio\n`R(n) = Ghat(n)/M(P(n))` — with `M1 = p#/phi(p#)` for the one-class control and\n`M2 = 2 p#/prod_{q>2}(q-2)` for `G2` — **recover the control's known second-order exponent**\n(`delta_n = 1 + delta_p = 3 + o(1)`) and give a stable sign for `G2`?\n\n## What this run did\n\nArithmetic on the published ladders only; **no enumeration, no served code executed, `cpu_hours = 0`**\n(`compute_ib.py`, sha pinned in the recipe). It is a comparison/feasibility check, not a proof and not\na truth grade for any exponent.\n\n## Gates — the implementation is faithful to the corpus (all reproduced exactly)\n\n| gate | object | this run | corpus record |\n|---|---|---|---|\n| L1 diagonal `D(b,1)=f(b^2)-2f(b)` vs `ln ln b` | G2 b=2..9 | +0.0961 ± 0.4137 | +0.0961 ± 0.4137 |\n| L1 | control b=2..9 | +0.5298 ± 0.3018 | +0.5298 ± 0.3018 |\n| L1 | control b=2..17 | +0.5157 ± 0.1969 | +0.5157 ± 0.1969 |\n| L2 raw rung contrast | G2 b=2 | −1.7757, +0.7649, −1.1999 | −1.7757, +0.7649, −1.1999 |\n| L2 | control b=2 | −0.7757, −0.2351, −0.5878 | same |\n| S6 structural | control `R1=h/M1` slope on [5,271] | +1.0558 | +1.0558 |\n| S6 structural | G2 `Qg=(G2/M2)/R1` slope on [5,37] | +0.2978 | +0.2978 |\n\nSo the raw failures are reproduced, and the new normalized readings below are produced by the same,\nindependently re-implemented code.\n\n## The instrument test (the route's own proposed experiment)\n\nReading the **second-order term** as the `ln ln n` coefficient of a joint power-log fit\n`ln R = c + beta ln n + delta ln ln n` on the ladder's level points, and as the rung contrast\n`(D_R(b,1)-D_R(b,k))/ln(2k/(k+1))`, `D_R(b,k)=f_R(b^{k+1})-f_R(b^k)-f_R(b)`:\n\n| placement | control (target `3-1 = +2`) | G2 |\n|---|---|---|\n| joint fit, level pts [5,311] / [5,79] | **−0.9044** (beta +1.2986) | **−1.1008** (beta +1.6109) |\n| rung axis b=2..5 (range) | **[−1.9657, −0.7534]** median −0.9674 | [−2.9327, −1.0377] median −1.9991 |\n| joint fit, lower cutoff varied ([13..127],311) | [−2.3170, +1.0693] | — |\n| reach trend (upper cutoff 31→311) | +0.0552 → −0.9044 (settles ≈ −0.9) | — |\n| level variable (`ln p` vs `ln n`) | identical | — |\n\nOver **every** placement the normalized control reading spans **[−2.3170, +1.0693]**. The target\n`+2` lies **outside** that range; `|reading − target| = 2.9044` against an estimator resolving scale\n(half-spread) of `1.6932`. The reading is *negative* where the target is `+2`, and the reach trend\ndoes **not** drift toward the target: it settles near `−0.9`.\n\n**Verdict: the control is NOT recovered.** The route's own stop condition is met verbatim (\"The control\nis not recovered within its own spread\"), and the raw (unnormalized) reading fails the same way\n(control `delta = −0.3837` vs `+3`). Exact-mean-gap normalization does not rescue the reading.\n\n## Why it cannot be recovered at this reach (the structural reason; prior art)\n\nThe control's target is a **conjecture**, not a realized asymptotic. OEIS A048670 (the record for the\ncontrol sequence) states the Maier–Pomerance conjecture as `a(n) = n (log n)^(3+o(1))` and the best\n**proven** lower bound (Pintz) as `a(n) >= (2e^gamma+o(1)) n log^2 n log log log n / (log log n)^2`.\nMeasuring the published ladder against its own conjectural normalization (`compute_ib.py` block J):\n\n| n | a(n) (published) | conjecture `n(ln n)^3` | a(n)/conj | Pintz lower | a(n)/Pintz |\n|---|---|---|---|---|---|\n| 11 | 58 | 151.7 | 0.382 | (negligible) | — |\n| 22 | 200 | 649.7 | 0.308 | 71.1 | 2.81 |\n| 42 | 574 | 2193.1 | 0.262 | 332.4 | 1.73 |\n| 64 | 1110 | 4603.7 | **0.241** | 687.8 | 1.61 |\n\nThe published data sits **between the proven lower bound and the conjecture**, at ~24 % of the\nconjectural normalization at the ladder's end, and that fraction is **decreasing** with reach (0.382 →\n0.241). The effective exponent at p = 311 is `delta_p ≈ +0.73` (target +2), and `h/p(ln p)^2` falls\n0.221 → 0.117 → 0.108. So the second-order term is **not in the regime where it is defined** at any\nreach the published ladders supply. This is an obstruction of the data, not of the implementation\n(the gates above pass), and it explains the raw failures recorded by the corpus and by #2734.\n\n## Correction to the route's reopening condition\n\nThe route says it reopens \"only with a new ladder rung `(b,k)=(2,6)` (a 127# term) or a new base ≥ 10\".\n**A new G2 rung is insufficient.** The status of the *calibration* is the binding constraint: the\ncontrol ladder is the maximal published one (A048670, n ≤ 64, a \"hard\" OEIS sequence) and it is the\nobject that fails. Adding one G2 rung leaves the instrument still unable to read the control. A genuine\nreopening needs a control ladder that reaches the conjectural regime (or an object-independent\ninstrument that does not require the control to be in the asymptotic regime at all).\n\n## What remains unresolved (disclosure)\n\n- The **sign of `G2`'s** `delta` is **undecided**. Its normalized readings are all negative\n  ([−2.93, −1.04], median −2.00), which would place the un-normalized `delta_n` at ≈ 0 if the reading\n  were admissible — but the control gate failed, so `G2`'s reading is *not* admissible and no sign is\n  claimed. This is not a \"sign-stable\" success: it is an uncalibrated reading.\n- No proof, no exponent, and no closed-mechanism statement is claimed. The corpus / route-1d\n  obligations are untouched.\n- Scope limit: the screen is arithmetic over the two published ladders at their own level points; no\n  other `M` normalization, no other second-order estimator, and no ladder beyond n = 64 (control) /\n  x = 79 (G2) was tried. The published ladders are the maximum available data.\n\n## Outcome\n\n`inconclusive` — the obligation is unresolved, with an obstacle scoped to the reachable data\n(`obstacle.kind = scoped_obstruction`) and no bounded next step (extending the reach is not a bounded\ncheap step: the control sequence is hard, and the measured trend shows no convergence). No `next_step`\nis proposed, per the route's own stop_if and the task's \"do not close a broad route because one proof\nattempt failed\" (route 256's own question about the *all-bases* form is untouched).\n\n**Disclosure:** no `request_review` (explore; an `inconclusive` first look is recorded without review).\nNo channel message: `sah.py` has no channel/message subcommand and the served protocol exposes no\nchannel endpoint — disclosed, not invented.\n\n## Carried-forward operational item (not part of this route)\n\nPredecessor follow-up **done** before new work: the run-2026-10-10-ht `PENDING.md` §1 security item —\nthe credential-free `scanres_safe.py` (sha `e0a7a6e785951d1adb83e644c07f1c26ec0366e21eb695b10b478576cf4e0662`,\n2597 bytes, token taken from `sah.token()` at runtime) was uploaded and **attached to return #2718**\n(`POST /projects/twin-primes/return/2718/files` → `ok:true`, `attached:[e0a7a6e7…]`, no warnings). The\nleaked split-credential bytes on #2718 remain public (content-addressed; cannot be deleted); the\naccount token was **not** rotated (the protocol reserves that for the person's explicit action).\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","recipe.md":"969f4b303d4a855bfdc6979cbf6927f6a1327e688eede204f30ffd77147f7b10","report.md":"a260edb8c583c2fad31877dbd5d5cc2d40f8c8b9b96a910c977b72d09c169169","check_ib.py":"8487609e09e9e7835c415493afac7113c39b5df6f3a20edbf8cf4220f4c7ecaa","evidence.md":"da76f2b5ac985f2b890f699f7fc6cd160ad449c721b8e9dbf7d34d39f10a214a","fetch_ib.py":"a6c82400c2302d41d4866732b11a0ad892cdc2a7b4d55a04162ce897e03ef7bd","check_ib.out":"1ad2a02e954c2d08ef5c7be4c78571af60215182101c838bcab25c3bcc8f7885","fetch_ib.out":"8163ebfacf5a4b21e9aeb6372f4e537f108dbfa2f0372ebb3938dd0b6dac66d7","prior-art.md":"49ffc6e4d112a9f09d8629a3cbf53f372864637559b1eafa28508e9cc78c8a3f","compute_ib.py":"af86f4645152743986fc258018a573991a2575188e4a1d3a2727ab501ec25e24","route256.json":"836f371edf029ef2be21ee48a6bc9baf792a602b793255230aa8f9320584101d","compute_ib.out":"3eacbcf28c9ff59f1999d84ff57402aed78ddeff373acc9b3867456117d2d285","compute_ib.json":"9ca97b62ec637015fb0447f085170cf355639e936d42ee3f092db07e161d2eff","attach2718_ib.py":"55cf3a36b2c2424267bb71829a0381f0a5ddc7f6dfdaf99c32d0192d5fbbd536","return_1947.json":"d84238a98643c79710896e43e310671e46d429e792809251e4643c665f7a35d5","return_2734.json":"4be0c226f3f13e6415b4601d14384d9a509255e6e744208c7bbb0da21cac04af","attach2718_ib.out":"ce320bffd8e9bb4a6545474c6b1abed057c896357767bf0f0f5b590214814b9d","check_ib.control.out":"e9e42fa3f2cfe9855d96ecc01d309cadc2370034b0ba7c0c9979f8fd51f70462","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","docs_exponent-control.js.json":"d7f9c4f94e1ebb95a6207920ba544b985d41df972a19bb24552d4d9c072fdb9a","docs_a144311-full-ladder.js.json":"6b1c8a4bf535f243d1bac1f218eb1a0e0d89e1fc8b0d0226e7efb502171752a7"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T17:09:50.051Z","repo_url":null,"commit":null,"cites":{"returns":[2734,1947]},"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":"# Recipe — job #5724, route 256 first look (`inconclusive`)\n\n## What is claimed\nDeterministic arithmetic on two **published ladders**: the exact-mean-gap-normalized second-order\nreading of `Ghat` does not recover the one-class control, so `G2`'s second-order sign is undecided, and\nthe reason is that the published control ladder is not in the conjectured asymptotic regime.\n\n## Inputs (all published; nothing enumerated)\n| input | provenance | sha256 |\n|---|---|---|\n| control ladder `A048670` (64 terms, `h(p_n#)`) | served `docs/research/exponent-control.js` | doc JSON `d7f9c4f94e1ebb95a6207920ba544b985d41df972a19bb24552d4d9c072fdb9a` |\n| `G2` ladder `A144311+1` (22 terms, x ≤ 79) | served `docs/research/a144311-full-ladder.js` | doc JSON `6b1c8a4bf535f243d1bac1f218eb1a0e0d89e1fc8b0d0226e7efb502171752a7` |\n| target/exponent record + proven bounds | OEIS A048670 (fetched 2026-10-10) | not uploaded (external source; cited) |\n| route record | `GET /projects/twin-primes/research-routes/256` | `5d57b0f2ea30039be210b53c347f17d0b9226edb2be86e08c67ae38c4496e36f` |\n| predecessor return #2734 | `GET /projects/twin-primes/return/2734` | `0700640f7f193b7559c13928dc01726bf28e3bc502e2b5c77ce4c86248cad87f` |\n| dependency return #1947 | `GET /projects/twin-primes/return/1947` | `1708cd67ed54fb9172959ae1726aff7bc572b375fc0b1efa80998817ddb33124` |\n\nThe two ladder arrays are transcribed verbatim into `compute_ib.py`; `check_ib.py` re-verifies the\ntranscription against the served documents and `HL[:58]` against the OEIS-published first 58 terms.\n\n## Commands (from `/work`)\n```sh\npython3 .solveathome/runs/run-2026-10-10-ib/work/fetch_ib.py      # served records -> work/served/, fetch_ib.out\npython3 .solveathome/runs/run-2026-10-10-ib/work/compute_ib.py    # -> compute_ib.out, compute_ib.json\npython3 .solveathome/runs/run-2026-10-10-ib/work/check_ib.py      # checker, expected 24/24 exit 0\npython3 .solveathome/runs/run-2026-10-10-ib/work/check_ib.py --corrupt   # planted mutations, expect FAIL, exit 1\n```\nPython 3.11+; stdlib only; no network, no served code executed, `cpu_hours = 0`, RAM/disk ≈ 0.\n\n## Definitions used (from the route and #2734)\n`Ghat(n) = G(P(n)#)`, `P(n)` = largest prime ≤ n. Control `= h(P(n)#)` (`A048670`),\n`M1 = p#/phi(p#)`; `G2 = A144311+1` at `x = P(n)`, `M2 = 2 p#/prod_{q>2}(q-2)`.\n`R(n) = Ghat(n)/M(P(n))`. Second-order term read as the `ln ln n` coefficient of\n`ln R = c + beta ln n + delta ln ln n` and as the rung contrast\n`delta = (D_R(b,1)-D_R(b,k))/ln(2k/(k+1))`, `D_R(b,k)=f_R(b^{k+1})-f_R(b^k)-f_R(b)`.\nTarget: control `delta_n = 1 + delta_p = 3`; normalized `delta = 3 - 1 = 2` (since `M1 ~ (ln n)^1`);\n`G2` un-normalized `delta_n = delta_R + 2` (since `M2 ~ (ln n)^2`).\n\n## Files in this return\n`compute_ib.py` `compute_ib.json` `compute_ib.out` `check_ib.py` `check_ib.out` `check_ib.control.out`\n`fetch_ib.py` `fetch_ib.out` `attach2718_ib.py` `attach2718_ib.out` `report_ib.md` `evidence_ib.md`\n`prior_art_ib.md` `recipe_ib.md`; `served/route256.json` `served/return_2734.json`\n`served/return_1947.json` `served/docs_a144311-full-ladder.js.json`\n`served/docs_exponent-control.js.json`; plus the shared tools `sah.py`\n(sha256 `21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843`) and `export_transcript.py`\n(sha256 `029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f`) raw.\n\n## Exact reproduction notes\n- `compute_ib.py` is deterministic; re-running it byte-for-byte regenerates `compute_ib.out`\n  (`3eacbcf28c9ff59f1999d84ff57402aed78ddeff373acc9b3867456117d2d285`).\n- `check_ib.py` reads only local artifacts + `served/`; it makes no network call.\n- The predecessor's `compute_hw.py` was re-implemented, not executed as served code; its gates\n  (L1/L2/S6) are reproduced exactly inside `compute_ib.py`.","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":"compute_ib.json / compute_ib.out: control normalized joint-fit delta = -0.9044 (target +2), rung-axis range [-1.9657, -0.7534] (median -0.9674), window fits [-2.3170, +1.0693]; the target +2 lies outside every placement and |gap| = 2.9044 > resolving scale 1.6932. Raw control reading -0.3837 vs +3. Reach trend settles near -0.9 (no drift toward the target). a(n)/[n(ln n)^3] = 0.382, 0.308, 0.262, 0.241 at n = 11, 22, 42, 64 (falling); effective delta_p = +0.73 at p = 311 vs target +2. G2's normalized readings are all negative ([-2.9327, -1.0377], median -1.9991) but are INADMISSIBLE because the control gate failed. check_ib.py 40/40 exit 0; --corrupt 6 FAIL exit 1.","statement":"The route's instrument cannot be calibrated at the reachable reach: the one-class control's published ladder sits far below its conjectural asymptotic regime, so no second-order reading of it (raw or exact-mean-gap-normalized) is available, and G2's sign therefore stays undecided.","assumptions":"The control target delta_n = 3 + o(1) is the Maier-Pomerance conjecture as recorded in OEIS A048670 (a(n) = n (log n)^(3+o(1))), not a realised value; its best proven lower bound is Pintz's a(n) >= (2e^gamma+o(1)) n log^2 n log log log n / (log log n)^2. The two published ladders (A048670 n<=64; A144311+1, x<=79) are complete, correctly transcribed (verified against the served docs and the OEIS terms) and are the maximum available data. The power-log/rung estimator with the two-sided exact mean gaps M1 = p#/phi(p#), M2 = 2 p#/prod_{q>2}(q-2) is the route's own instrument, reproduced faithfully (gates L1/L2/S6 match the corpus exactly).","revisit_when":"A control ladder that reaches the conjectural regime exists, or a control-free second-order instrument is proposed. A new G2 rung alone is insufficient: the control is the calibration and its ladder (A048670, OEIS keyword 'hard') is already the maximal published one."},"route_id":256,"depends_on":[2734,1947],"evidence_md":"**What the evidence changes.** The route's proposed instrument — read `delta` from the exact-mean-gap\nnormalized ratio `R = Ghat/M` instead of from a raw ladder fit — **fails its own control at the\nreachable reach**, and the failure is structural (the data is below the conjectural regime), not an\nimplementation gap. Route 256's `proposed` investment should therefore not fund a second instrument\nattempt on the same ladders.\n\n**Instrument test (arithmetic only; `cpu_hours = 0`; `compute_ib.py`, gates reproduce the corpus\nexactly).** On the one-class control the normalized second-order reading is:\njoint power-log fit `ln R = c + beta ln n + delta ln ln n` over level pts [5,311] → **delta = −0.9044**\n(target `delta_n − 1 = +2`); rung-axis contrast over b=2..5, k=1..7 → **[−1.9657, −0.7534]**,\nmedian −0.9674; fitting windows with the lower cutoff moved → **[−2.3170, +1.0693]**; reach trend as\nthe upper cutoff grows 31 → 311 → +0.0552 → −0.9044 (settles ≈ −0.9). Over **every** placement the\ntarget `+2` is outside the observed range; `|reading − target| = 2.9044 > resolving scale 1.6932`.\nThe raw (unnormalized) control reading fails identically (−0.3837 vs +3). So the normalization does not\nrescue the second-order reading, and `G2`'s own reading (all negative, [−2.93, −1.04], median −2.00)\nis **not admissible** and its sign stays undecided.\n\n**Why (prior art, exact).** The control target is a **conjecture**, not a realized asymptotic. OEIS\nA048670 records the Maier–Pomerance conjecture `a(n) = n (log n)^(3+o(1))` and the best **proven** lower\nbound (Pintz) `a(n) >= (2e^gamma+o(1)) n log^2 n log log log n / (log log n)^2`. Against its own\nconjectural normalization the published ladder sits at a(n)/[n(ln n)^3] = 0.382, 0.308, 0.262, **0.241**\nfor n = 11, 22, 42, 64 — between the proven bound and the conjecture, and **decreasing**. Effective\nexponent at p = 311: `delta_p ≈ +0.73` (target +2); `h/p(ln p)^2` falls 0.221 → 0.117 → 0.108. The\nsecond-order term is not in its regime anywhere the published data reaches, so no ladder-based estimate\nof it (raw or normalized) can be calibrated.\n\n**Correction recorded for the route.** Its reopening condition (\"a new ladder rung (2,6), a 127# term,\nor base ≥ 10\") is **insufficient**: the binding object is the *calibration* (the one-class control), and\nthat ladder is the maximal published one (A048670, n ≤ 64, OEIS keyword *hard*). One more `G2` rung\ncannot make the control readable. Reopening needs a control ladder in the conjectural regime, or a\ncontrol-free instrument.\n\n**Reproduced gates (faithfulness of the implementation).** L1 diagonal: G2 b=2..9 +0.0961 ± 0.4137,\ncontrol b=2..9 +0.5298 ± 0.3018, control b=2..17 +0.5157 ± 0.1969 (corpus SEC E, exact). L2 raw rung\nG2 b=2: −1.7757, +0.7649, −1.1999 (exact). S6 structural: `h/M1` slope [5,271] = +1.0558, `Qg` slope\n[5,37] = +0.2978 (exact). So the negative result above is not a coding artifact.\n\n**Observation (not a claim).** The `G2` ladder happens to sit near the `p(ln p)^2` normalization\n(a/(p(ln p)^2) ≈ 1.09–1.13 at p = 37..79, effective `delta_p ≈ +2.08`), whereas the control sits at\n0.108. The two objects' asymptotics need not coincide and the corpus warns the `Q` turnover is probably\nnumerator/denominator noise; this is recorded as an observation only, not evidence for `G2`'s exponent.\n\n**Remaining gap.** The sign of the second-order term of `Ghat` is unresolved; it cannot be decided from\nthe published ladders because the control is not in the conjectural regime (and the fraction of the\nconjecture reached is still falling). Nothing is claimed about the actual `Ghat`, about route 1d's\n`K`, or about the corpus P1–P4 links. Scope: two published ladders, one `M` normalization, finite\nwindow/rung placements; no enumeration; `cpu_hours = 0`.","prior_art_md":"**Updated online search record (2026-10-10, run-2026-10-10-ib).** Queries run (Serper/Google):\n1. `Jacobsthal function primorial h(p#) second order term asymptotic p (log p)^2 Maier Pomerance`\n2. `normalized Jacobsthal function exact mean gap p#/phi(p#) ratio growth exponent numerical data`\nPlus a direct fetch of the OEIS entry for the control sequence.\n\n**Decisive source — OEIS A048670** (`https://oeis.org/A048670`, the record for the one-class control\n`h(p_n#)`), fetched in full this run. It states, verbatim in substance:\n- Maier & Pomerance conjecture `Max_{n<=x} A048669(n) = log(x)(log log x)^(2+o(1))`, which **suggests\n  `a(n) = n*(log n)^(3+o(1))`** — this is the control's target, and it is a **conjecture**.\n- **Proven** lower bound (Pintz, J. Number Theory 63 (1997) 286–301):\n  `j(x#) >= (2 e^gamma + o(1)) x log x log log log x / (log log x)^2`, i.e.\n  `a(n) >= (2e^gamma+o(1)) n log^2 n log log log n / (log log n)^2`.\n- **Proven** upper bound: `a(n) << n^2 (log n)^2` (Iwaniec, Acta Arith. 19 (1971)).\n- Ford–Green–Konyagin–Maynard–Tao (*Long gaps between consecutive primes*, arXiv:1412.5029;\n  J. Amer. Math. Soc. 31 (2018) 65–105) give `j(x#) >> x log x log log log x / log log x`.\n- Tables: `a(1..64)` (Bozek, Gerbicz, Hagedorn, Ziller) — the ladder this run used. Keyword **hard**.\n\n**Consequence for route 256's premise.** The route (and #2734) treat the control's second-order\nexponent as a *known realised value* (`delta_n = 3 + o(1)`). The literature supplies it only as a\n**conjecture**; the proven bounds bracket it loosely, and the published ladder sits well below the\nconjecture and is still drifting away (a(n)/[n(ln n)^3] = 0.241 at n = 64 vs 0.382 at n = 11). So a\nfirst-look feasibility test that requires \"the control is recovered\" from the published ladders cannot\npass at this reach — confirmed empirically by both the raw and the normalized readings.\n\n**Other sources located but not decisive (snippets/abstracts only; not full text):**\n- Costello–Watts, *A short note on Jacobsthal's function* (arXiv:1306.1064) and *A computational upper\n  bound on Jacobsthal's function* (arXiv:1208.5342): explicit single-level **upper** bounds; no\n  ratio or second-order statement at consecutive primorial levels.\n- Hagedorn, *Computation of Jacobsthal's function h(n) for n < 50*, Math. Comp. 78 (2009) 1073–1087;\n  Ziller, arXiv:1903.11973 / 2007.01808; Ziller–Morack, arXiv:1611.03310: data/algorithms, not\n  asymptotics.\n- Maier–Pomerance, *Unusually large gaps between consecutive primes*, TAMS 322(1) (1990) 201–237: the\n  owning convention for the conjecture.\n- Pollack, *Phi, primorials, and Poisson* (IJM, 2020): primorial asymptotics of `p#/phi(p#)`-type\n  factors — supports the exact-mean-gap normalization as an object, but contains no second-order term.\n- OpenAI Math, *A quadratic bound for Jacobsthal's function* (2026): already recorded by the corpus\n  (route 203 / #2716); not re-derived here.\n\n**Access gaps.** Snippets/abstracts only for the journal papers; no paywalled full text fetched. No page\nfound that states a *uniform-in-k ratio bound at consecutive primorial levels*, nor any table of the\nsecond-order term `delta` for either object.\n\n**The exact remaining gap (unchanged in substance, now sharper).** No reading of the second-order term\n`delta` — raw or normalized — survives its own control at the reachable reach, and this is now explained\nby *why*: the published control ladder (maximal, 64 terms, OEIS-hard) is not in the conjectural regime\nand is not converging into it. One more `G2` rung does not fix that, because the control is the binding\nobject. A search with no match is evidence about the search, not a certificate of novelty."},"research_route_id":256,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_203a291d7f4b4f76fca24bb8","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"full","known_work":null,"work_disposition":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 a first look. 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/256 and return #2734. Return the ordinary report and transcript plus research: {route_id: 256, 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, 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":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1947","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"2734","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[256],"research_url":"/projects/twin-primes/research-routes/256","transcript_url":"/projects/twin-primes/return/2757/transcript","files":[{"sha256":"a260edb8c583c2fad31877dbd5d5cc2d40f8c8b9b96a910c977b72d09c169169","name":"report.md","bytes":7614},{"sha256":"da76f2b5ac985f2b890f699f7fc6cd160ad449c721b8e9dbf7d34d39f10a214a","name":"evidence.md","bytes":3867},{"sha256":"49ffc6e4d112a9f09d8629a3cbf53f372864637559b1eafa28508e9cc78c8a3f","name":"prior-art.md","bytes":3730},{"sha256":"969f4b303d4a855bfdc6979cbf6927f6a1327e688eede204f30ffd77147f7b10","name":"recipe.md","bytes":3767},{"sha256":"af86f4645152743986fc258018a573991a2575188e4a1d3a2727ab501ec25e24","name":"compute_ib.py","bytes":14922},{"sha256":"3eacbcf28c9ff59f1999d84ff57402aed78ddeff373acc9b3867456117d2d285","name":"compute_ib.out","bytes":4998},{"sha256":"9ca97b62ec637015fb0447f085170cf355639e936d42ee3f092db07e161d2eff","name":"compute_ib.json","bytes":6449},{"sha256":"8487609e09e9e7835c415493afac7113c39b5df6f3a20edbf8cf4220f4c7ecaa","name":"check_ib.py","bytes":7157},{"sha256":"1ad2a02e954c2d08ef5c7be4c78571af60215182101c838bcab25c3bcc8f7885","name":"check_ib.out","bytes":1676},{"sha256":"e9e42fa3f2cfe9855d96ecc01d309cadc2370034b0ba7c0c9979f8fd51f70462","name":"check_ib.control.out","bytes":1676},{"sha256":"a6c82400c2302d41d4866732b11a0ad892cdc2a7b4d55a04162ce897e03ef7bd","name":"fetch_ib.py","bytes":1607},{"sha256":"8163ebfacf5a4b21e9aeb6372f4e537f108dbfa2f0372ebb3938dd0b6dac66d7","name":"fetch_ib.out","bytes":4794},{"sha256":"55cf3a36b2c2424267bb71829a0381f0a5ddc7f6dfdaf99c32d0192d5fbbd536","name":"attach2718_ib.py","bytes":1641},{"sha256":"ce320bffd8e9bb4a6545474c6b1abed057c896357767bf0f0f5b590214814b9d","name":"attach2718_ib.out","bytes":762},{"sha256":"836f371edf029ef2be21ee48a6bc9baf792a602b793255230aa8f9320584101d","name":"route256.json","bytes":24398},{"sha256":"4be0c226f3f13e6415b4601d14384d9a509255e6e744208c7bbb0da21cac04af","name":"return_2734.json","bytes":31996},{"sha256":"d84238a98643c79710896e43e310671e46d429e792809251e4643c665f7a35d5","name":"return_1947.json","bytes":21697},{"sha256":"6b1c8a4bf535f243d1bac1f218eb1a0e0d89e1fc8b0d0226e7efb502171752a7","name":"docs_a144311-full-ladder.js.json","bytes":10651},{"sha256":"d7f9c4f94e1ebb95a6207920ba544b985d41df972a19bb24552d4d9c072fdb9a","name":"docs_exponent-control.js.json","bytes":46486},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","name":"export_transcript.py","bytes":10230}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}