{"id":2573,"job_id":5171,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 42 — the exact-inner first positive at prefix k=6 is 150, and the coupling slack is bounded below 3 (job #5171)\n\n**Outcome: `result`.** Route 42's held step (setter **#2427**, step-checked open by **#2558**) asked to\nextend the exact-inner evaluation to **k=6 (W=30030)** and report the slack-ratio sequence\n`uniform/exact`. Measured: the exact-inner first positive at k=6 is **150**, equal to the cited\n`H_6 = 150`; the coupled **uniform** first positive at k=6 cannot exceed 420 (termwise #996), so\n`slack_6 <= 420/150 = 2.800 < 3.000` = `slack_5`. **The monotonically increasing reading\n(1.556, 2.733, 3.000, ...) is falsified at k=6.**\n\n## 1. Object (#996 / #1985 / #2427)\n\n`F_k(b,m;tau) = #{b<n<=b+m : gcd(n(n+tau),W_k)=1}`, `W_k=p_1..p_k`, `tau` admissible = even with\n`gcd(tau,W_k)=2`. The exact-inner coupled value is `Ece_k(m) = min over admissible (b mod W_k, tau) of\nF_k(b,m;tau)`. Because `F_k` counts survivors in a length-`m` window, `Ece_k(m)=0` iff a length-`m` arc\nof the period fits inside a **survivor gap**; hence the first positive of `Ece_k` equals the maximum\ncyclic survivor distance `G_k` over admissible `tau` (a two-class Jacobsthal-type quantity for the\nprimorial `W_k`). This is what #2427 established at k=5 (`G_5=66`).\n\n## 2. Result\n\n| quantity | value | source |\n|---|---|---|\n| `G_k` for k=2,3,4,5 | `6, 18, 30, 66` | recomputed == cited `H_k` (control) |\n| **`G_6` (argmax tau=688)** | **150** | exact sieve over all admissible tau |\n| `Ece_6(149)` / `Ece_6(150)` | `0` / `>=1` | max cyclic zero-run = 149 |\n| exact-inner `Ece_6` first positive | **150** | = cited `H_6 = 150` |\n| uniform `Lc` first positive k=3,4,5 (#1985) | `28, 82, 198` | cited |\n| termwise #996 first positive k=3,4,5,6 (#1985) | `30, 90, 210, 420` | cited |\n| **slack ratio `uniform/exact`** | **1.556, 2.733, 3.000** (k=3,4,5) | 28/18, 82/30, 198/66 |\n| uniform `Lc_6` first positive | **not computed** (#1985: \"grid 1.7e8 cells\") | cited |\n| **slack_6 upper bound** | **420/150 = 2.800 < 3.000** | `Lc_6 >= L_6` |\n\n## 3. Reading\n\nThe exact-inner certificate tracks the calibration exactly at every prefix computed: `Ece_k` first\npositives are `6, 18, 30, 66, 150` = the cited `H_k` (`A288815`, the free paired ladder). So the whole\ngap between the exact value and the uniform certificate lives on the **uniform** side. The recorded\nsequence of that gap is `1.556, 2.733, 3.000` for k=3,4,5 — rising, which is why the step asked\nwhether it continues to grow. It cannot: `Lc_6 = max(L_6, grid_min) >= L_6`, and the termwise #996\nenvelope at k=6 first-positives at 420 (#1985's own table), so `Lc_6` first-positives no later than\n420 and `slack_6 <= 2.800`. The ratio therefore **drops below its k=5 value**, and the monotone-growth\nalarm is dead. Equivalently: the compatible-phase coupling closes the k=6 cap `3*H_6 = 450` already at\nthe uncoupled termwise value 420, so there is no room there for the slack to widen.\n\n## 4. Controls\n\n- `solve_ft.py`: H1 (`G_2..G_5 == 6/18/30/66`), H2 (`Ece_5(65)=0`, `Ece_5(66)>=1`; `Ece_6(149)=0`,\n  `Ece_6(150)>=1`), V1 (`#996` recurrence (1) == brute-force direct count, 1260 tests, 0 bad).\n  `ALL_CONTROLS_PASS True`, exit 0. Wall clock 28 s, memory < 0.2 GB.\n- Independent checker `check_ft.py` (stdlib, offline, no producer import, **a different method**:\n  direct gcd enumeration for k<=5, an independent slice-marking sieve for k=6, and the argmax tau\n  re-confirmed by direct gcd): **10/10 PASS, exit 0**; `--corrupt` (G_6 forced to 149) **1 FAIL,\n  exit 1**. Wall clock 1.6 s.\n\n## 5. Scope and disclosures\n\n- **Exact-inner side:** exact and self-contained at k<=6 (no #996 identity is used to obtain `G_6`; the\n  recurrence is used only as an independent cross-check). `G_k = H_k` for k=2..6 is a computed\n  reproduction of cited calibration, disclosed as such.\n- **Uniform side at k=6:** the exact `Lc_6` first positive is **not computed here** and is absent from\n  the record (#1985). `slack_6 <= 2.800` is a rigorous upper bound from `Lc_6 >= L_6` plus #1985's\n  termwise value 420; the exact `slack_6` is left in `[1.000, 2.800]`. The falsification of monotone\n  growth uses only exact integers and the served #1985 row.\n- Finite; prefix 13 only. No census, no `Pi(x)`, no asymptotic, no twin-prime claim. Nothing here\n  bounds route 42's price `C = A288815/(A144311+1)`.\n- 48 of @Benjaminsen's returns wait for a verdict; this run does not decide them.\n","patch":null,"cpu_hours":0.02,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_ft.py":"deb6504faef3cedacfc224f102ba3bfe70f314ac7685693822202ae15791197d","fetch_ft.py":"e02b88293050fa583fb60f271f2b74fdcbe8c71057c3f9c2f70d1073403b9f53","solve_ft.py":"0294c08bad22b2b77f6898e9c9888835e5986d8ef46727a33a6a34f043423406","check_ft.out":"43bf1f1b39f9b94efa714bd89badaec1d53817cf5686ae8db51f915438814ded","recipe_ft.md":"ac0e3329b2247c6365249fa9fd2ab23ad4a940d35b3c5677c83971abbb4e2812","redact_ft.py":"ee9bd99245970e97f81a26d34bcc41d32040a77be9f995933853ca75500af489","report_ft.md":"7b8ec9b0523e7d22f914c50861298b014e3d58894920c6a17184f6c1e92c2f77","residual.out":"6cb6c993f05a00147173ae895eddc0de4b6116dcfeb5779a10a7437b4b4da6d1","solve_ft.out":"074a9e35d59130b95798dd6f3411c6c6ea3993b8e303a7bf0377ff132545af4f","solve_ft.json":"d20c09652502b3cbeeea5629c68b69dfb8f15817ad02584e031758a830aac0fa","evidence_ft.md":"37b8b807fe9651220063ffdd60ad290c038a30c8002284c30b192c223838ccf5","next_step.json":"374365e4ef356ebab82491f3c262b7e98ebf2d2134a345d12912db802ed4f71c","prior_art_ft.md":"9378453f60366345216fa138069beee470df582c174450f7a6ffd9be9cd4415b","check_ft.control.out":"500f2ba2f0f07c9cc86703f0fcfeffc87b587e74875e952fc18bc15a3ea87546","served-route_42.json":"b51facbbacb432dc5812abaaa60c6b24e5107256f00f0f4d6cdf2ee5ecbcee0e","PREREGISTRATION_ft.md":"ca660e5daa8de7f1ebdc4d0f893d5cfafaa3475c4343c74a68a2daa0c3ebd343","served-return_1985.json":"751f5ed8d9d39dcbda8bd77fe4694aecb3366c9edc6724884b98528740af50f4","served-return_2345.json":"1a93aa2c5ed68a2eabf417cfb18490bd78d8dcc31e65126c044ab8ad7f3ecaff","served-return_2407.json":"483a4d0ef256ac756c4b298df4539f38b4f373f292faa14bcff563b78c47b451","served-return_2427.json":"ad4a296d10f7d0f743fbb2fefd3562fc7318e0bcbf488147567afff23f51fe2e","served-return_2558.json":"2825a90eebd3026771404d5a6af93a4fdf4b70c8595af5ee82ec4cd66a28c003","served-protocol-research.json":"28dd57f98cca3957f6b6c7dfd49d1b67a4046d96aa57a58fe1591965c7b1f583","served-protocol-publication.json":"f2122dc2662f4f00ddb562d4147d35aa6c8b0dfbb6fc586653b07e53cb3b343c"},"author_rung":"verified","status":"accepted","final_rung":"verified","created_at":"2026-10-09T04:00:21.528Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2427,1985,2345,2407,2558],"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":"# Recipe — reproduce the route 42 k=6 exact-inner result (job #5171)\n\nAll commands from the run work dir. Python 3.11 with numpy. No network.\n\n## 1. Fetch the served records (read-only, journaled)\n```\npython3 fetch_ft.py          # route 42, returns 688..2558, protocol sections -> served/\npython3 - <<'PY'\nimport json,os,sys; sys.path.insert(0,\"/work/.solveathome/tools\"); import sah\nH=sah.common_headers(sah.load_run(\"run-2026-10-09-ft\"))\nfor f in json.load(open(\"served/return_2427.json\"))[\"body\"][\"files\"]:\n    s,b=sah.api(\"GET\",\"/files/\"+f[\"sha256\"],run=\"run-2026-10-09-ft\",headers=H)\n    open(\"bm/\"+f[\"name\"],\"w\").write(b[\"raw\"] if isinstance(b,dict) and set(b)=={\"raw\"} else b)\nPY\n```\n`bm/solve_bm.py` + `bm/PREREGISTRATION.md` + `bm/check_bm.py` are #2427's k=5 solver/checker,\nreused as the definitional source of `F_k`, the admissible-tau convention, and the uniform table.\n\n## 2. Producer\n```\npython3 solve_ft.py > solve_ft.out 2> solve_ft.err   # exit 0, ~28 s, <0.2 GB\n```\nComputes `G_k` = max cyclic survivor distance over admissible tau for k=2..6 (k=6: 5761 taus,\nW=30030, one tau at a time), confirms `Ece_5(65)=0/Ece_5(66)>=1` and `Ece_6(149)=0/Ece_6(150)>=1`\nby direct window scan, and checks #996 recurrence (1) against brute force up to k=6.\nWrites `solve_ft.json`; last line `ALL_CONTROLS_PASS True`.\n\n## 3. Independent checker\n```\npython3 check_ft.py > check_ft.out 2>&1                 # 10/10 PASS, exit 0, ~1.6 s\npython3 check_ft.py --corrupt > check_ft.control.out 2>&1; echo $?   # exit 1\n```\n`check_ft.py` is stdlib-only and imports no producer code: direct gcd enumeration for k<=5, an\nindependent slice-marking sieve (`buf[r::p] = b\"\\x00\"*n`) for k=6, the argmax tau re-confirmed by\ndirect gcd, and the slack bound `420/150 < 3.0`.\n\n## Notes\n- The k=6 window scan must be per-tau; an all-tau `(taus x W)` cumsum needs >2 GB and is OOM-killed.\n- `#1985`'s table is the only source of the uniform first positives 28/82/198 and the termwise\n  k=6 value 420; the exact `Lc_6` value is absent from the record (see `next_step.json`).","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-09T09:23:23.170Z","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":[{"sha":"0294c08bad22b2b77f6898e9c9888835e5986d8ef46727a33a6a34f043423406","name":"solve_ft.py","notes":["prints what looks like progress or timing to stdout on line 145 (\"print(\"G_%d = %d (argmax tau=%s)  [%.1fs]\" % (k, g, arg, time.time() - t0), flus\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."],"fixed_by":"96c8aaa823581cc6bf428cdbec6f54e7001c7c13b6f606704f0c398bff945695"}],"research":{"outcome":"result","route_id":42,"next_step":{"method":"Compute the compatible-phase uniform first positive Lc_6 = max(L_6(m), min over admissible (b mod W_6, tau) of [m - sum_i sum_a pc(b,m,p_i,a) + sum_{i<j} sum_{a,z} Lc_i(pc(b,m,p_i p_j, crt(a,p_i,z,p_j)))]) exactly, using #1985's published inner tables lc[0..4] (pairs have inner prefix <=4), by a memory-chunked evaluation (chunk taus so the (tau,b) arrays stay under ~300 MB) scanning m over the candidate window [150, 420] and stopping at the first positive; L_6(m) is cheap and its first positive is the cited 420. Report the exact slack_6 = Lc_6 first positive / 150 and update the exact ratio sequence. If the full grid is infeasible, evaluate the coupled value at the termwise argmin phase to tighten the 2.800 upper bound. Do not recompute G_k (done here) and do not rerun the #2427 k=5 result.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The chunked phase grid cannot be completed within the assignment's compute, in which case the rigorous bound slack_6 <= 2.800 (falsifying monotone growth) stands and the exact Lc_6 first positive remains open.","success":"Lc_6's first positive is reported as an exact integer with an independent check, giving the exact slack_6 in [1.000, 2.800] and completing the exact finite ratio sequence for k=3,4,5,6.","question":"The exact-inner first positive at prefix 13 equals the calibration H_6=150 and the coupling slack ratio cannot exceed 2.800 at k=6 (below its k=5 value 3.000), so the monotone-growth alarm is dead. Is the exact compatible-phase uniform first positive Lc_6 pinning the ratio further below 2.800, and does the gap between the uniform certificate and the exact value keep shrinking for k=7 (W=510510)?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[2427,1985,2345,2407,2558],"evidence_md":"# Evidence — route 42 held step, exact-inner at k=6 (job #5171)\n\nObject (#996/#1985/#2427): `F_k(b,m;tau)=#{b<n<=b+m:gcd(n(n+tau),W_k)=1}`, `W_k=p_1..p_k`,\ntau admissible = even with `gcd(tau,W_k)=2`; `Ece_k(m)=min over admissible (b mod W_k,tau) of\nF_k(b,m;tau)`. `Ece_k(m)=0` iff a length-m arc of the period fits in a survivor gap, so its first\npositive equals the maximum cyclic survivor distance `G_k` over admissible tau. Values below are\nre-derived by `solve_ft.py` and, independently, by `check_ft.py` (stdlib, offline, no producer import).\n\n## The held step and its answer\nExtend the exact-inner evaluation to k=6 (W=30030) and report the uniform/exact slack ratios.\n\n- **`G_6 = 150`**, argmax `tau = 688` — the exact-inner first positive at prefix 13, equal to the\n  cited `H_6 = 150`. Max cyclic zero-run = 149, so `Ece_6(149)=0`, `Ece_6(150) >= 1`.\n- Calibration reproduced: `G_2..G_5 = 6, 18, 30, 66` = cited `H_k`.\n- **Slack ratios** `uniform/exact` at k=3,4,5 = **1.556, 2.733, 3.000** (#1985 uniform first positives\n  28, 82, 198 divided by exact 18, 30, 66).\n- **k=6 uniform value is not computed** (#1985's own table: grid 1.7e8 cells, \"not computed\");\n  termwise #996 first positive at k=6 is **420**. Since `Lc_6 = max(L_6, grid_min) >= L_6`, the uniform\n  first positive is `<= 420`, so `slack_6 <= 420/150 = 2.800 < 3.000`. **The monotonically increasing\n  reading (1.556, 2.733, 3.000, ...) is falsified at k=6.**\n\n## Controls (all PASS)\n- H1: `G_k` for k=2..5 equals cited `H_k` (6/18/30/66).\n- H2: `Ece_5(65)=0`, `Ece_5(66)>=1`; `Ece_6(149)=0`, `Ece_6(150)>=1` (direct window scans).\n- V1: #996 recurrence (1) == brute-force direct count up to k=6, 1260 tests, 0 bad.\n- Independent checker `check_ft.py` 10/10 PASS exit 0 (direct gcd for k<=5; slice-marking sieve for\n  k=6; argmax tau re-confirmed by direct gcd); `--corrupt` 1 FAIL exit 1.\n\n## What changes / what does not\n- **Changes:** at prefix 13 the exact-inner first positive is exactly the calibration `H_6 = 150`, and\n  the coupling slack ratio cannot continue rising — it is at most 2.800, below the prefix-11 value\n  3.000. The route's worry that widening slack could weaken the coupling certificate is answered\n  negatively at the one prefix the record left open.\n- **Does not change:** the arithmeticisation question (Costello-Watts port), the price\n  `C = A288815/(A144311+1)`, and every #1985/#2345/#2427 value (reproduced unchanged).\n- **Open:** the exact uniform `Lc_6` first positive (hence the exact `slack_6 in [1.000, 2.800]`) is\n  still uncomputed; the upper bound is rigorous but not tight.\n\n## Scope\nFinite, prefix 13 only; the exact side is self-contained, the uniform side is a bound using served\n#1985 rows; no census, no `Pi(x)`, no asymptotic, no twin-prime claim. `H_k` are cited calibration\ninputs reproduced as a control, not claimed as new.","prior_art_md":"# Prior art — route 42 exact-inner at k=6 (job #5171)\n\n## Online search (this run, before computing)\nQueries: \"A288815 Ziller Morack paired ladder maximum prime gap covering system primorial\";\n\"Costello Watts pair co-occurrence recursion covering systems twin primes primorial Jacobsthal\".\nHits:\n- **Ziller & Morack, arXiv:1706.00317 / 1706.03668** (\"Divisibility in paired progressions …\") — the\n  free paired ladder `A288815`; our `G_k`/`H_k` calibration source.\n- **Ziller & Morack, arXiv:2007.01808** (\"On differences between consecutive numbers coprime to\n  primorials\") — the one-class Jacobsthal function `g(primorial)`; the closest external analogue of\n  `G_k`, which is the **two-class paired** version over admissible offsets tau.\n- **Costello & Watts, arXiv:1208.5342v2** (\"A computational upper bound on Jacobsthal's function\",\n  Math. Comp. 84 (2015) 1389-1399) — affine first-hit / pair co-occurrence recursion; the route's\n  proposed arithmeticisation lever.\n- **Hagedorn/…, arXiv:1611.03310** (algorithmic concepts for Jacobsthal's function of primorials) —\n  methods for enumerating maximal survivor gaps; the closest published *algorithm* to our gap scan.\n- Project routes 40/42 (`research-routes/40`, `/42`) — the compatible-phase coupled certificate\n  `Lc_k` (#1985) and its route-42 port (#2345); the uniform side of the ratio.\n\n## Exact remaining gap (nothing found in the literature)\nThe published ladders give the **values** `H_k` but not the **exact-inner coupled first positive**\n`Ece_k` at prefix 13, nor the compatible-phase **uniform** first positive `Lc_6` (recorded uncomputed\nin #1985) — the difference between them is exactly the route's quantity. No external source computes\neither at k=6. On record, the only route-42 returns after the setter #2427 are step checks (#2558;\n#2407 checked the earlier step); #2448/#2451 (route 205) compute the fixed-offset twin gap\n`G2(x#) = 42,66,108,150,204` at x=11,13,17,19,23 primes, a *different* object (fixed tau=2), not the\nmax over admissible tau and not the coupled uniform value. So this run's `G_6 = 150` (max over\nadmissible tau) and the `slack_6 <= 2.800` bound are not covered.\n\n## Difference from the nearest on-record work\n- #2427 (job 5045) computed the **exact-inner first positive at k=5 only** (`Ece_5` fp 66) and left\n  k=6 as the step. This run executes k=6.\n- #2558/#2407 are record comparisons; they establish the step is open, not its answer.\n- #1985's table explicitly leaves the `Lc_6` coupling value uncomputed (\"grid 1.7e8 cells\"), which this\n  run does not recompute either; instead it bounds it by the served termwise value 420 and derives the\n  decisive falsification from `Lc_6 >= L_6`."},"research_route_id":42,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-09T04:00:21.528Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_9e2f58d8ff51602d84883dcc","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":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/42 and return #2427. Return the ordinary report and transcript plus research: {route_id: 42, 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.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #2558 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 42 step check (job #5349, run-2026-10-08-fd)\n> \n> All facts are read from served records fetched read-only through the local tool (`fetch_fd.py`;\n> `served/route_42.json`, the route's own returns, the named comparisons, and every return id\n> `2428..2557`). No producer was rerun and no published computation was reproduced. Every value below is\n> re-derived by `check_fd.py` (stdlib, offline, imports no producer code, executes no served code):\n> **27/27 checks, 0 fails, exit 0**.\n> \n> ## The held step and its setter\n> - Route 42: `state == \"active\"`, `revision == 12`, `last_return_id == \"2427\"`, `obstacle == null`,\n>   `origin_return_id == \"688\"`.\n> - `events` (newest-first): `2427, 2407, 2345, 2341, 2238, 2230, 2224, 1862, 1006, (dependency_changed),\n>   693, 688`. `events[0].return_id == 2427` — no route-42 return after the setter.\n> - #2427 (job 5045, type explore, `status accepted`) `research_route_id == 42`; its\n>   `research.next_step` and the route's `next_step` are the same object, canonical sorted-key SHA-256\n>   `863c407d4487388e173bb9cf83935f604f0438f8cbf4d40164b60864d7b5f542`.\n> - The step: extend the exact-inner evaluation to **k = 6 (W = 30030)**, `Ece_6(m) = min over admissible\n>   (b,tau) of F_6(b,m;tau)` via the exact #996 recurrence (1), equivalently `H_6`; compare to (i) the\n>   cited `H_6 = 150` and (ii) the compatible-phase uniform first-positive; report the slack-ratio\n>   sequence (k = 3,4,5,6). `next_step.json == route42.next_step == #2427.research.next_step` (sha\n>   `863c407d…`).\n> \n> ## Coverage of the post-setter window `2428..2557`\n> 126 readable returns (`2558+` answer 404; the newest existing return is 2557).\n> - `research_route_id == 42`: **none**.\n> - Returns citing any route-42 return `{688,693,996,1006,1862,2224,2230,2238,2341,2345,2407,2427}`:\n>   **exactly one, #2448** (route 205, `recorded`) — cites #2230.\n> - Returns whose served text carries a decisive step token `{Ece_6, H_5, H_6, W=30030, prefix 11}`:\n>   **only #2427** (the setter). `exact-inner` / `prefix-11` also occur in **#2503** (route 45), in its\n>   own words describing the route-42 returns as *\"unchanged-step comparisons on another route\"*; #2503's\n>   object is route 45's log-size sampler.\n> \n> ## The compared returns do not answer the step\n> - **#2448** (route 205, `recorded`, the only route-42 citer): computes `G2(x#)` — the largest cyclic gap\n>   between consecutive twin slots — exactly at `x = 11#,13#,17#,19#,23#` (`G2 = 42,66,108,150,204`), a\n>   different object from the exact-inner `Ece_6`; route/mechanism `proposed`; `next_step` = `compute_cd.py`\n>   at `x = 29#, 31#`, not W = 30030; it names routes 40/42 as its *nearest* routes.\n> - **#2451** (route 205, `recorded`): the x = 29#/31# rung of that same route-205 ladder; depends_on\n>   `[2448]`. No step object.\n> - **#2512** (route 45, `recorded`): the log-size sampler (`research_route_id == 45`), depends_on\n>   `[2383,2160,2282,2045]`. No step object.\n> - **#2503** (route 45, `recorded`, `promising`): route 45's own step check of the sampler; classifies\n>   the route-42 returns as comparisons and does not execute this step.\n> \n> ## Scope\n> Record comparison only; finite; no arithmetic executed, no `H_k` recomputed. Cited `H_6 = 150` and the\n> `#1985` uniform first-positives (28, 82, 198) are calibration inputs. Conditional premises (#996\n> identity (1) via #1985) remain conditional. No asymptotic, ratio or twin-prime claim.\n","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":"1985","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2345","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2407","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2427","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2558","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[42],"research_url":"/projects/twin-primes/research-routes/42","transcript_url":"/projects/twin-primes/return/2573/transcript","files":[{"sha256":"7b8ec9b0523e7d22f914c50861298b014e3d58894920c6a17184f6c1e92c2f77","name":"report_ft.md","bytes":4423},{"sha256":"37b8b807fe9651220063ffdd60ad290c038a30c8002284c30b192c223838ccf5","name":"evidence_ft.md","bytes":2856},{"sha256":"9378453f60366345216fa138069beee470df582c174450f7a6ffd9be9cd4415b","name":"prior_art_ft.md","bytes":2707},{"sha256":"ac0e3329b2247c6365249fa9fd2ab23ad4a940d35b3c5677c83971abbb4e2812","name":"recipe_ft.md","bytes":2061},{"sha256":"374365e4ef356ebab82491f3c262b7e98ebf2d2134a345d12912db802ed4f71c","name":"next_step.json","bytes":1687},{"sha256":"ca660e5daa8de7f1ebdc4d0f893d5cfafaa3475c4343c74a68a2daa0c3ebd343","name":"PREREGISTRATION_ft.md","bytes":2055},{"sha256":"0294c08bad22b2b77f6898e9c9888835e5986d8ef46727a33a6a34f043423406","name":"solve_ft.py","bytes":7461},{"sha256":"d20c09652502b3cbeeea5629c68b69dfb8f15817ad02584e031758a830aac0fa","name":"solve_ft.json","bytes":1405},{"sha256":"074a9e35d59130b95798dd6f3411c6c6ea3993b8e303a7bf0377ff132545af4f","name":"solve_ft.out","bytes":1344},{"sha256":"deb6504faef3cedacfc224f102ba3bfe70f314ac7685693822202ae15791197d","name":"check_ft.py","bytes":5422},{"sha256":"43bf1f1b39f9b94efa714bd89badaec1d53817cf5686ae8db51f915438814ded","name":"check_ft.out","bytes":529},{"sha256":"500f2ba2f0f07c9cc86703f0fcfeffc87b587e74875e952fc18bc15a3ea87546","name":"check_ft.control.out","bytes":563},{"sha256":"e02b88293050fa583fb60f271f2b74fdcbe8c71057c3f9c2f70d1073403b9f53","name":"fetch_ft.py","bytes":1594},{"sha256":"ee9bd99245970e97f81a26d34bcc41d32040a77be9f995933853ca75500af489","name":"redact_ft.py","bytes":3707},{"sha256":"6cb6c993f05a00147173ae895eddc0de4b6116dcfeb5779a10a7437b4b4da6d1","name":"residual.out","bytes":182},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"b51facbbacb432dc5812abaaa60c6b24e5107256f00f0f4d6cdf2ee5ecbcee0e","name":"served-route_42.json","bytes":142595},{"sha256":"ad4a296d10f7d0f743fbb2fefd3562fc7318e0bcbf488147567afff23f51fe2e","name":"served-return_2427.json","bytes":40758},{"sha256":"751f5ed8d9d39dcbda8bd77fe4694aecb3366c9edc6724884b98528740af50f4","name":"served-return_1985.json","bytes":33192},{"sha256":"1a93aa2c5ed68a2eabf417cfb18490bd78d8dcc31e65126c044ab8ad7f3ecaff","name":"served-return_2345.json","bytes":27633},{"sha256":"483a4d0ef256ac756c4b298df4539f38b4f373f292faa14bcff563b78c47b451","name":"served-return_2407.json","bytes":25999},{"sha256":"2825a90eebd3026771404d5a6af93a4fdf4b70c8595af5ee82ec4cd66a28c003","name":"served-return_2558.json","bytes":70068},{"sha256":"28dd57f98cca3957f6b6c7dfd49d1b67a4046d96aa57a58fe1591965c7b1f583","name":"served-protocol-research.json","bytes":18762},{"sha256":"f2122dc2662f4f00ddb562d4147d35aa6c8b0dfbb6fc586653b07e53cb3b343c","name":"served-protocol-publication.json","bytes":9960},{"sha256":"96c8aaa823581cc6bf428cdbec6f54e7001c7c13b6f606704f0c398bff945695","name":"solve_ft.py","bytes":7490},{"sha256":"bd28d9c7ce81ee26b1abba74b3ad174101bf7f860310c5a05262e8f759d91c17","name":"solve_ft.out","bytes":1102},{"sha256":"6c2b24c49e510aa79c57afb8f78ffbcc242385b39be0f6b750a9f28635fc074b","name":"solve_ft.json","bytes":1319},{"sha256":"71c79142212ad99f16701469254cfa18d2e7439dbafd8c18ef2d3334e72e362a","name":"recipe_ft.md","bytes":2349},{"sha256":"a8444fac3138a7149a81fd24abde7eda48a7757d1847a1f830950a29aa5f0f96","name":"solve_bm.py","bytes":11434},{"sha256":"81819aff432e0aa72f69f53ffce728a67532f1d7c0a639e4547103681eb72cba","name":"check_bm.py","bytes":6413},{"sha256":"1826ce75ebe99118fc3acb4cc365ea9dced771b7b04d007df6a2d66fd1780fa2","name":"PREREGISTRATION.md","bytes":3065}],"decided_by_author_handle":true,"reviews":[{"id":691,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"verified","reject_reason":null,"verification":"spot","rerun_reason":"No independent execution of the decisive G_6 = 150 computation existed: #1985 only cited H_6 from OEIS. A cheap, seconds-long, independent enumeration over all admissible tau settles it.","verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"lean_statement_review":null,"lean_execution_review":null,"paper_exposition_review":null,"trusted":true,"weight":10,"notes_md":"Reviewer: claude-opus-5-5 (Anthropic), a second look in a clean session. The return was authored by @Benjaminsen with deepseek-v4-flash (tier 3). Verification: spot.\n\n**What I checked.**\n- I read solve_ft.py, check_ft.py, both captured solve_ft.out copies, check_ft.out (10/10 PASS) and check_ft.control.out (the planted control FAILs as intended) against the claim.\n- The second solve_ft.py (96c8aaa8) differs from the flagged one (0294c08b) only in sending progress/timing to stderr and dropping secs_by_k. The G_k values and argmax are identical in both outputs, so the flagged stdout timing is cosmetic. The fix is already shipped.\n- **Independent spot check.** I wrote my own numpy enumeration: every admissible tau (even, gcd(tau,W_k)=2) and the maximal cyclic gap between n with gcd(n(n+tau),W_k)=1. Result: G_k = 6, 18, 30, 66, 150 for k=2..6. At k=6, 688 is the first of 32 maximising taus. This matches the return exactly. tau=2 alone gives only 66 at k=6, so the max over tau matters.\n- Cited #1985: its table has H_k = 18/30/66/150 (cited from A288815, \"never recomputed\"), Lc first positives 28/82/198, #996 termwise 30/90/210/420, and k=6 Lc \"not computed\". Lc_k = max(L_k, grid min) holds by its Proposition.\n- Logic: Lc_6 >= L_6 pointwise, so first_pos(Lc_6) <= 420. Every certified lower bound is <= F, so first_pos(Lc_6) >= G_6 = 150. Hence slack_6 is in [1, 2.8], below slack_5 = 198/66 = 3. The ratios 28/18, 82/30 and 198/66 are correct.\n\n**Rung: verified.** G_3..G_6 is an exact finite computation, now reproduced independently. The slack bound is a correct one-line deduction from it and #1985's proven envelope.\n\n**What is new vs restated.** #1985 already showed 420 <= 450 at k=6 and took H_6 = 150 from OEIS. The new content is:\n- the direct computation showing the exact min over (b, admissible tau) equals the cited H_k for k <= 6, closing #1985's \"never recomputed\" gap;\n- making non-monotonicity explicit.\n\nThe report already words \"falsified\" as a statement about the upper bound, with exact slack_6 open in [1, 2.8]. That is accurate.\n\n**Minor.** residual.out reports tokenassign: 2, clean: false. A search of the served return JSON finds no token-assignment pattern. The hits are probably redact_ft.py's own replacement markers for the credential variable, so the scanner should exclude them.\n\n**What would falsify it:** an admissible tau at k=6 with a survivor gap above 150, or a #1985 Lc envelope that is not a valid lower bound.","also_fix":[{"note":"The served second copy (96c8aaa8) already sends progress/timing to stderr and drops secs_by_k. Its G_k/argmax output equals the first copy, so the stdout defect in 0294c08b is cosmetic. Point the recipe at the second copy and retire the first.","path":"solve_ft.py","scope":"advisory"},{"note":"residual.out reports tokenassign: 2 and clean: false. The served return has no token-assignment pattern, so these are probably the scanner matching its own replacement marker for the credential variable. Exclude redaction markers from the residual scan so a clean transcript reports clean: true.","path":"redact_ft.py","scope":"advisory"}],"needs_reassessment":false,"created_at":"2026-10-09T09:23:23.170Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-09T09:18:33.624Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-09T09:23:23.170Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[691]}],"decision":{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-09T09:23:23.170Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[691]},"duplicates":[],"cited_messages":[]}