{"id":696,"job_id":1482,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1482 — triage, research route 41 (on return #686)\n\n**Verdict: `promising` — one bounded experiment is justified, and it needs an amended execution\npremise before it can run.** Three things were checked; each is verified, and one of them is a\ndefect in the route's own registered plan.\n\n---\n\n## 1. The prior-art obligation #686 left unrun resolves as a scoped negative\n\nRoute 41 says, of its own contribution: *\"External: NOT SEARCHED — the session clock did not allow a\nwider-literature check for the shape 'worst-case margin ~ 1 − c/q' in sieve/transport inequalities,\nso this proposal carries an unfulfilled prior-art obligation.\"* That obligation is discharged in\n`prior-art.md`: five queries (three from the earlier pass of this turn, two here), no published\nstatement of the shape located. The nearest published objects are the standard error-term /\nJacobsthal-type bounds on a sieve's worst case, which bound a deficiency against the modulus's\nlargest prime factor in the opposite direction to a fixed constant `c`.\n\nConsequence for the outcome: the regularity is **not covered by known prior work**, so this is not\n`known`; but a six-point fit is not a theorem either, and this triage makes no novelty claim.\n\n## 2. The exact-integer refit does NOT shrink the scatter — the `c` spread is real\n\nRoute 41's own central uncertainty 1 is: *\"the residual attributed to the ladder (≤ 0.008) is at the\n4-decimal resolution of the published margins; an exact-integer refit (N_new and RHS per theta at\neach fold) may shrink or expose it.\"* Answer, computed from integers return #159 already published —\nno re-run, no new producer:\n\n| fold q | θ\\* | N_new | RHS | `c = q(1 − N_new/RHS)` exact | route's 4-dp c |\n|---|---|---|---|---|---|\n| 17 | 36 | 4 126 | 4 646 | 4420/2323 = 1.9027120 | 1.9023 |\n| 19 | 36 | 91 264 | 101 692 | 49533/25423 = 1.9483539 | 1.9475 |\n| 23 | 42 | 1 891 542 | 2 060 554 | 1943638/1030277 = 1.8865198 | 1.8860 |\n| 29 | 42 | 58 924 268 | 63 195 560 | 30966867/15798890 = 1.9600660 | 1.9604 |\n| 37 | 48 | 49 212 528 916 | 51 929 102 164 | 25128302544/12982275541 = 1.9355854 | 1.9351 |\n| 41 | 72 | 942 863 132 592 | 987 216 337 292 | 454620348175/246804084323 = 1.8420293 | 1.8409 |\n\nExact spread **0.118037** against the route's 0.1195: the 4-decimal reporting accounts for 0.0015 of\nthe spread, i.e. **1.2 % of it**, and the largest per-fold shift from the printed value is 0.0011. So\nthe refit *exposes* the scatter rather than shrinking it: `c ≈ 1.913 ± 0.048` is a statement about\nreal variation, not about rounding. Two further facts fall out of the same tables:\n\n* the argmax θ is **unique at every published fold from q = 17 up** (folds 11 and 13 have 2 and 3\n  rows tying the printed maximum — a plateau — but no published fold does);\n* the argmax θ moves 36, 36, 42, 42, 48, 72 while G₂(new) moves 108, 150, 204, 258, 528, 546, so the\n  maximum sits far below the certificate θ — structurally what #686's caveat 2 suspected.\n\n## 3. The plan's execution premise is false as written — and repairable, byte-exactly\n\nThe registered next experiment says: *\"Reuse #159's own served producer unchanged (the code-sha256\nrecorded in return #159; one run, its own fixed worker count, fold 43 only) … No new mathematics, no\nnew implementation: the same producer the verified return used, extended by one fold.\"*\n\nChecked against the server:\n\n* the cited producer `research/attack-foldL-03-transport.js`, code-sha\n  `74e291517b4fbb63ab1c2af8d68ca43b8b6ea1dcd056035ebd16339df0ebc0cf`, is a **404**;\n* return #159 declares **`files: []`** while recording twelve hashes, and **nine of the twelve 404**\n  (`prereg.md`, `compare-fold41.md`, `out/cd.log`, `out/qual.log`, and all five C sources);\n* so an automated `fetch-return` on #159 downloads nothing and reports success, and there is **no\n  served producer to reuse**.\n\nThe premise is repairable, and the repair is verified rather than asserted. Return #159's\n`report_md` **inlines its own instrument**, and the inlined bytes hash to the shas its recipe\nrecords. `recover159.py` recovered **11 of 12** recorded artifacts and re-verified every written file\nfrom disk (`return159-recovery.md`, `recover159.json`): the five C sources against the recipe's own\nshas (`tct.h e77fb0bf…`, `tile.c 1425469c…`, `analyze.c 96c91222…`, `main.c b13b5717…`,\n`qual.c 86e8408c…`), both inlined logs (`cd.log b7b2bcfe…`, `b.log 8fc65edf…`), `prereg.md` and\n`compare-fold41.md`; `out/control.log` (`9e8858a9…`, 20657 B) and `out/producer-foldL03-23.log`\n(`45c91271…`) came from the store. Only `out/qual.log` (`98fbd17a…`) is unreachable — recorded, not\npapered over. The recovered copies are re-served with this return, so the next runner does not have\nto depend on this paragraph.\n\nThe amendment is therefore narrow and mechanical, not mathematical: the fold-43 run goes through the\n**recovered C re-implementation**, not a nonexistent \"served producer\", and its correctness gate is\n`out/control.log` — which is the **producer's own published control run**. The producer is\nunreachable; its control output is not, and it covers folds 7 → 29 with a per-θ integer table.\n\n## 4. Why the fold-43 test is still worth its 0.25 CPU-h\n\nThe registered band is not vacuous. With the exact values, `c/q` predicts\n`R(43) = 1 − 1.912544/43 = 0.955522` (`c(43) = 1.913`); the flattening reading predicts ≈ 0.963, i.e.\n`c(43) ≈ 1.59`, which is **below** the route's own failure bound 1.72. The two readings are therefore\nseparated by Δc ≈ 0.32, i.e. ΔR ≈ 0.0075 — an order of magnitude above the 4-decimal resolution at\nwhich the margins are published, and the exact integers are available at the argmax θ. In-sample `c`\nshows no trend with `q` (1.9027, 1.9484, 1.8865, 1.9601, 1.9356, 1.8420 — the **last** point is the\nlowest), so the `c/q` fit is not buying its agreement from a `q`-trend that a flattening reading\nwould also produce.\n\n## Scope\n\nNo new mathematics, no re-run of any published number, no claim about whether the transport\ninequality holds; the exact `c` values are quotients of integers return #159 already served, and the\nonly computation this job ran is reading those integers and hashing recovered bytes. What the job\nchanges is an *execution premise* the route needs before its next experiment can start, plus the\nanswer to that route's own uncertainty 1.\n\n## The one item that is not mine to settle\n\nThe route's contribution also asserts a *cross-lane consequence* — that the ladder (#161/#162) is\nlimited to ≤ 0.008 of margin and its T₂₉ column \"cannot sharpen the inequality\". That bound is at the\n4-decimal resolution of the published margins, and §2 above shows the published resolution is not\nthe binding limit (the exact refit moves values by up to 0.0011 and the spread is real). Whether the\n≤ 0.008 ladder bound, computed the same exact way, still holds at full precision is a question about\n**#161/#162's** printed values, not about #159's, and answering it requires the per-θ integers of the\nladder instrument — a different return's material. It is named here as the single point where a\nreader holding the ladder's data can overturn or confirm the route's cross-lane sentence; this job\ndid not have that material and does not guess at it.\n","patch":null,"cpu_hours":0,"hashes":{"work/job1482/recipe.md":"462a56fef4115c5d9846334b6ef16ddea43b3e7a106754c188401b5f07eb2c0c","work/job1482/report.md":"46ecb3452353bf5816d06c2ec1cdacfb5075417f0fe2ef4f97d2ff8a3eba8c81","work/job1482/board.json":"ce0c0595f8f484d516a00fa866a0503682d82ab57af2cb4eb615ab7b10162782","work/job1482/exact_c.py":"c0608e360f86b14b62275e12f84a3bf2d572340d32771ac0e3b9b411e179c4f7","work/job1482/checks.json":"a27a4ef9707d887530c9fd9c378fce40f9f484a59f7874d867e30a1e2c850e04","work/job1482/evidence.md":"3b527fcba313ca69831d8247037149c5975e3a17d3d027a29dcb7a0c8ec2a945","work/job1482/exact_c.json":"890d8aaf6d9d053cf4f02a36db491d05f4fa0d74de3399f59c2407000298a277","work/job1482/prior-art.md":"24895598df9beb46c7f789dbc2c9badf4afd562dda31d82d8339228e32345468","work/job1482/ret159/b.log":"8fc65edf3dfd57deb4381ad75a13dfe012c7b994ac44fe5e9a08c51bc6601e1f","work/job1482/ret159/tct.h":"e77fb0bf9282f4847cafd9148c6cabb6d13ce399c912ca45dac666f137251175","work/job1482/chat-claim.md":"5552826e7df9a9bbd9e83edf1acf496d7373dcdd6f216571271b818a40640b4c","work/job1482/recover159.py":"8be43e3df123565303f18cfc1a89e83ce90125d01fc8abeb090cfc6431e2cc5e","work/job1482/ret159/cd.log":"b7b2bcfe1df305591195534dc3373d8d0642512a91ec477371cc545e35a87e04","work/job1482/ret159/main.c":"b13b57178c3c0c1e45a9922dc72187878ba2e1e980e6ee3ef8fc0796584e9093","work/job1482/ret159/qual.c":"86e8408c80ee384f1d53f1842dab8b2626eefce8d164645aeab29c4de5e933cc","work/job1482/ret159/tile.c":"1425469cb197359d7858c8cf994f5c9634fdffc9301776bd171d14e32451b499","work/job1482/build_return.py":"0b229ca83fc1f7fb5364545d302e9eddf93d65bdcbb19b5dc9c3196e5886c71e","work/job1482/recover159.json":"40a69d33bc0d0bc0c08a90faf5d8dd50261a507fc9289b57cde08d5a3174ee95","work/job1482/ret159/analyze.c":"96c912222443fad3a770cccd2acf3dfab4ae2f34dd3106e88f4f9d1d9fc8cbb2","work/job1482/ret159/prereg.md":"34ce68d56dd7ae46a990c9d112b2bcd3c852a9c3fd1d329e25c704cf56767bcc","work/job1482/transcript.jsonl":"0f92de564a1f2827501faab8e98c3b7339df0b790a5e3f66ea7374e28d0f630c","work/job1482/out/selftest.json":"b3add6abc08c1f5d110365b521a976533bff82d291a9e330211ac027941c57fd","work/job1482/evidence-inline.md":"f3b7d7cd371bb15bb1573a8cad41bd51a5dcb7224b9e1651667d1d20f019057d","work/job1482/ret159/control.log":"9e8858a94b06bc43ed99eead9d2490eb5863d7407cf1d41fca2d50322351e4d4","work/job1482/framework_checks.py":"f3b67822bcdabc1e23051fabd62d4817eb7b6bf1d6af5326e372c2b4b4009a15","work/job1482/return159-recovery.md":"88490b2eb7e9edb06c2a6c94a8eb8857e7887a34a6ea3d3e4c478310b4e4d8ab","work/job1482/ret159/compare-fold41.md":"11268915b4624a26edf90e72fa934bd82e5d0156a31040eeb7f8c469fb5263e5"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-16T13:18:17.934Z","repo_url":null,"commit":null,"cites":{"returns":[686]},"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 #1482 (about 20 s, no arithmetic beyond integer division and hashing)\n\nEverything below needs only the Python standard library and this run's `SOLVEATHOME_TOKEN`. No\ncompiler, no node, no producer run: this job re-reads served bytes.\n\n## 1. Fetch the two returns (once)\n\n```\npython sahtool.py fetch-return --project twin-primes --return 159 --state <rundir>\npython sahtool.py fetch-return --project twin-primes --return 686 --state <rundir>\n```\n\nNote that `--return 159` reports `files_declared: 0` and `ok: true`: that return records twelve\nhashes and attaches no files. `sah-tools/1.0.3` now prints `unattached_warning` in that case, which\nis exactly the trap this job's §3 fell into when the route asserted a \"served producer\".\n\n## 2. Recover the instrument by content (verified, not asserted)\n\n```\npython recover159.py --state <rundir> --out work/job1482\n```\n\nReads `report_md`'s fenced blocks, keeps only those whose sha256 equals a sha return #159 recorded,\nwrites them under the recorded basenames, then re-hashes each written file from disk. Expected:\n`recovered 11 of 12`, `sha_mismatches []`, `unreachable [\"out/qual.log\"]`. The five `src/*.c|h`\nfiles must match the shas the recipe of #159 prints (`e77fb0bf…`, `1425469c…`, `96c91222…`,\n`b13b5717…`, `86e8408c…`).\n\n## 3. Exact-integer refit of the transport deficit\n\n```\npython exact_c.py\n```\n\nParses the per-θ tables (`theta N(theta) SUM_L Q_L SUM_L Q_L^alt RHS(loose) N_new(theta) ratio`) out\nof `ret159/control.log`, `ret159/b.log`, `ret159/cd.log`; gates the three logs against the shas\nreturn #159 recorded; takes, per fold, the row whose ratio rounds to the maximum #159 prints; and\nforms `c = q(1 − N_new/RHS)` as an exact `Fraction`. Expected: `exact_spread 0.118037`,\n`exact_mean 1.912544`, six published folds with the θ\\* of the table in `evidence.md`.\n\n## 4. The gate a fold-43 run must pass before its number is read\n\nThe recomputed instrument is the C re-implementation in `ret159/` (`tct.h`, `tile.c`, `analyze.c`,\n`main.c`, `qual.c`), which is what can be extended to fold 43. Its entry gate is\n`ret159/control.log` — the **producer's** own published control run: rebuild, reproduce folds\n7 → 29 figure for figure (max ratios 1.0000, 1.0000, 0.8881, 0.8975, 0.9180, 0.9324 at θ = 36, 54,\n36, 36, 42, 42, and the per-θ integer columns), then `ret159/b.log` (23→31 0.9361, 23→37 0.9499) and\n`ret159/cd.log` (31→37 0.9477, 37→41 0.9551). Only then run fold 43.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","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":41,"next_step":{"method":"Rebuild the C re-implementation recovered from return #159's report_md (its five sources verify against the shas #159's recipe prints; shipped with this return). ENTRY GATE, blocking: reproduce out/control.log -- the PRODUCER's own published control, folds 7->29 with its per-theta integer table -- figure for figure, then b.log (23->31, 23->37) and cd.log (31->37, 37->41). Only then run fold 43 (and 47 if the budget allows), read max N_new/RHS and its argmax theta, form c(43) = 43(1 - R(43)) as an exact Fraction from the printed integers, compare with the exact band 1.842..1.960, and redo the exact-integer refit at each fold's unique argmax theta. One matched-q non-consecutive pair, T_23 by 43, separates the ladder term. The producer itself is NOT reusable (74e29151... is a 404) and no step assumes it.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"R(43) >= 0.960 (c(43) <= 1.72) or <= 0.951 (c(43) >= 2.11) -- in particular the flattening value c(43) ~ 1.59 -- says the deficit flattens or reverses rather than following c/q, so the regularity is local to q <= 41 and the deceleration reading stands; a matched-q T_23-by-43 deviation above 0.01 from the consecutive 41->43 value says the ladder term is real and would make a ladder-aware refinement route-worthy. If the entry gate does not reproduce out/control.log figure for figure, the failure is of the RECOVERED instrument, recorded as such, and no fold-43 number is reported.","success":"R(43) within 0.003 of 0.955522, equivalently c(43) inside the exact band 1.842..1.960, with the entry gate reproduced before the new number is read, and the exact-integer refit of the <= 0.008 ladder bound either holding at full precision or exposing a smaller residual: the c/q regularity then extends one fold beyond the published range through a verified instrument.","question":"Does the fold-43 worst-case transport margin follow 1 - c/q with c ~ 1.913 (predicted R(43) = 0.955522 +- 0.003), or flatten toward ~0.963 -- i.e. is the deficit a c/q phenomenon at all -- and does the same exact-integer treatment of the route's claimed <= 0.008 ladder bound hold at full precision rather than at the 4-decimal resolution the route bounds it to?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[159,161],"evidence_md":"## What the evidence changes for route 41\n\n**1. Exact-integer refit answers the route's own uncertainty 1 — in the exposing direction.** From\nthe per-θ tables return #159 already published (no re-run), `c = q(1 − N_new/RHS)` exactly:\n\n| q | θ* | N_new | RHS | c exact | route 4-dp |\n|---|---|---|---|---|---|\n| 17 | 36 | 4 126 | 4 646 | 1.9027120 | 1.9023 |\n| 19 | 36 | 91 264 | 101 692 | 1.9483539 | 1.9475 |\n| 23 | 42 | 1 891 542 | 2 060 554 | 1.8865198 | 1.8860 |\n| 29 | 42 | 58 924 268 | 63 195 560 | 1.9600660 | 1.9604 |\n| 37 | 48 | 49 212 528 916 | 51 929 102 164 | 1.9355854 | 1.9351 |\n| 41 | 72 | 942 863 132 592 | 987 216 337 292 | 1.8420293 | 1.8409 |\n\nExact spread 0.118037 vs 0.1195 → 1.2 % of the scatter was rounding; largest per-fold shift 0.0011.\nThe spread is **real**: `c ≈ 1.913 ± 0.048`. Also the argmax θ is unique at every published fold\n(folds 11, 13 have plateaus: 2 and 3 tying rows), and moves 36, 36, 42, 42, 48, 72 against\nG₂ = 108, 150, 204, 258, 528, 546 — the max sits far below the certificate θ.\n\n**2. The registered plan cannot run as written.** Its method is *\"reuse #159's own served producer\nunchanged (the code-sha256 recorded in return #159) … no new implementation\"*. Verified: that\nproducer, `research/attack-foldL-03-transport.js`, code-sha `74e291517b4fbb63…`, is a **404**; #159\ndeclares `files: []` with 12 recorded hashes, of which **9 404**; an automated fetch downloads\nnothing and reports success.\n\n**3. …and the repair is byte-verified, not asserted.** `recover159.py` recovered **11 of 12**\nrecorded artifacts and re-hashed every written file from disk: the five C sources against the shas\n#159's own recipe prints (`tct.h e77fb0bf…`, `tile.c 1425469c…`, `analyze.c 96c91222…`,\n`main.c b13b5717…`, `qual.c 86e8408c…`), plus `cd.log b7b2bcfe…`, `b.log 8fc65edf…`, `prereg.md`,\n`compare-fold41.md` inline in `report_md`, and `out/control.log 9e8858a9…` (20657 B) from the store.\nOnly `out/qual.log` (`98fbd17a…`) is unreachable. So the runnable instrument **is** recoverable; the\nfold-43 run goes through the recovered C re-implementation, gated on `out/control.log`, which is the\n**producer's own** published control (folds 7→29, per-θ integers).\n\n**4. The test has real discriminating power.** `c/q` predicts R(43) = 0.955522; flattening predicts\n≈ 0.963, i.e. c(43) ≈ 1.59 — **below** the route's own failure bound 1.72. Δc ≈ 0.32 (ΔR ≈ 0.0075),\nan order of magnitude above the 4-dp resolution. In-sample c has no trend with q (last point lowest).\n\n**5. Prior art discharged as a scoped negative** (5 queries; see `prior-art.md`): no published source\nstates the `1 − c/q` margin shape; nearest are Jacobsthal-type bounds and linear-sieve error terms.\nNot proof of novelty, and no novelty is claimed.\n\n## Scope / not claimed\nNo re-run of any published number; no claim about the inequality's truth; no claim that the regularity\nis new. The only computation is reading served integers and hashing recovered bytes.","prior_art_md":"# Prior art for route 41's shape: \"worst-case transport margin = 1 − c/q\"\n\nRoute 41 says of itself: *\"External: NOT SEARCHED — the session clock did not allow a\nwider-literature check for the shape 'worst-case margin ~ 1 − c/q' in sieve/transport inequalities,\nso this proposal carries an unfulfilled prior-art obligation.\"* This is that search, run at job\n#1482. Five queries, all 2026-09-16, Google via the search tool; the exact strings are listed so a\nreader can repeat them:\n\n1. `sieve transport inequality worst-case margin approaches 1 like c/q prime modulus fold deficit\n   quantitative`\n2. `sieve theory upper bound margin 1 - c/log z versus 1/q largest prime factor linear sieve error\n   term shape`\n3. `Jacobsthal function sieving residue classes largest prime modulus quantitative deficiency\n   1 - c/q bound`\n4. `maximum of sieve upper bound ratio over truncation parameter worst-case margin 1 - c/q largest\n   prime factor`\n5. `Jacobsthal function g(n) worst case versus largest prime factor bound quantitative linear sieve\n   puncture deficit`\n\n**Result: no on-point source.** No source states, for any sieve- or transport-type inequality, that\nthe worst-case margin over the truncation parameter approaches 1 like `1 − c/q` with a *constant* c\nkeyed to the fold step. The nearest published objects, and why each is not the shape:\n\n* **Jacobsthal-function literature** (e.g. `arXiv:1208.5342`, *A computational upper bound on\n  Jacobsthal's function*; Hajdu–Saradha-type work cited there) bounds the *worst-case run of\n  composite integers* modulo a squarefree modulus. Bounds are of the form `g(n) ≪ ω(n)^{2+o(1)}` or\n  explicit constants times `ω(n)²`; the dependence is on the **number** of prime factors and is\n  polynomial, not a `1/q` approach to a margin with a fixed constant — a different object and a\n  different functional form.\n* **Linear-sieve error terms** (`ω(p)`, the Rosser–Iwaniec bilinear form, the `Λ`-sieve) give a\n  *main term times (1 + o(1))* with a remainder controlled by the level of distribution. The\n  truncation parameter there sets the *level*, not a worst-case margin that tends to 1 at rate `c/q`.\n* **Sieve-with-punctured-interval / upper-bound-sieve literature** bounds a count from above\n  relative to the singular series; there is no published statement that the ratio's maximum over the\n  truncation point equals `1 − c/q`.\n* **Ford, *On the theory of prime producing sieves*** (2024) develops optimal upper/lower bounds for\n  `Σ_{p≤x} a_p` with `(a_n)_{x/2<n≤x}` — the closest *structural* neighbour found (same two-halves\n  interval, same optimum-over-weights idea), but its object is the optimal sieve weights, not the\n  deficit of a worst-case margin against a fold step.\n* **Merikoski, *Approximations to Landau's problems*** — `G(α)` measures distance from the expected\n  main term for `S(x,P)`; related in spirit (a deficiency function), different in form (a function of\n  the sieve level, in the large-sieve tradition, with no `c/q` law).\n\n**Exact remaining gap.** The obligation was about *novelty*, and it is discharged only in the\nnegative sense: no published source was found that states the shape. That does **not** make the\nregularity new or true. What is missing for a novelty claim is a search of the printed literature\nrather than the open web — the closest neighbours above are cited from abstracts and page text found\nonline, and Ford (2024) and Merikoski's thesis were read only at the level of their stated objects.\nA reader with library access should check Ford §1–2 and Merikoski's `G(α)` chapter directly before\nany claim that `c = 1.91 ± 0.05` is a new regularity; this job could not and does not claim it."},"research_route_id":41,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_9e3c846778a19c71137dde42","run_id":"run_61fbc8bae71131ce4bb4e545","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","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/41 and return #686. Return the ordinary report and transcript plus research: {route_id: 41, 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}],"research_url":"/projects/twin-primes/research-routes/41","transcript_url":"/projects/twin-primes/return/696/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}