{"id":621,"job_id":1385,"problem_id":1,"lane_id":5,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1385 — Route 27: the |Q| = 1 capacity, measured, and what the identity actually buys\n\nLocal research turn, one assignment taken (job #1385, attempt `835853962f253dc7984aa08e443506c7`,\nroute 27, explore/discovery, budget 2 h, compute hint 4 CPU-h). Everything below was measured this\nturn unless it is explicitly marked as cited.\n\n## 0. What the job asked, and what changed\n\nRoute 27's contribution is the cross-lane identity `L(T_x, p) = K*({p})` at level `x` for every prime\n`p > x`: #161's measured adjacent-kill run IS route 26's covering capacity with one killer. Claim\n(a): #161's 1,307 entries are therefore a lower-bound bank for every route-26 `K*(Q)` with `|Q| >= 2`.\nClaim (b): with #609's proved step `K*(Q u {q}) >= K*(Q) + 1`, the capacity is a ladder whose base is\nthat bank.\n\nI measured the identity with an instrument that shares no code with either lane, and I measured both\nclaims against multi-prime `K*(Q)` values this run has already computed. **The identity holds at\nevery point where it can be checked here (7/7 of the recordable diagonal entries). Both claims hold\nas inequalities — and the bank sits a factor 4.3 to 8.0 below the object it is meant to calibrate,\nbecause the single-killer capacity is flat at 2-3 across every prime and every level measured.** That\ndistinction is the result: the identity is exact and useful as a screen, not as a threshold probe.\n\n## 1. The identity, measured\n\nDerivation (the route's own, checked and used verbatim): `r = a` or `r = a+2 (mod p)` is `p | u` or\n`p | u+2` for `u = r - (a+2)` — the wraparound `p-2` is `-2 mod p`, not a second rule. So both sides\nare \"the longest run of consecutive level-`x` slots `v` with `v = a` or `v = a-2 (mod p)` for ONE\nclass `a`, maximised over the translate `a`\", and the two lanes' conventions agree after that step.\n\nMeasured by `single_prime.py`: the tile is built from the prime set, the period is doubled (the\nsecond copy is the first shifted by `P#`, so a window crossing the period seam is covered with the\nSAME class `a`, which is what the object requires), the slots of each residue class are gathered once\nby a stable argsort, and the killed indices for class `a` are the merge of the two classes\n`{a, a-2}`; the answer is the longest maximal block of consecutive integers in that set that starts\ninside the first copy. Cost is `O(D)` per prime after one sort, not `O(p D)`.\n\n| fold | p | #161 (route 27) | this instrument | |\n|---|---|---|---|---|\n| T_5  | 7  | 2 | **2** | agrees |\n| T_7  | 11 | 1 | **1** | agrees |\n| T_11 | 13 | 2 | **2** | agrees |\n| T_13 | 17 | 2 | **2** | agrees |\n| T_17 | 19 | 2 | **2** | agrees |\n| T_19 | 23 | 3 | **3** | agrees |\n| T_23 | 29 | 2 | **2** | agrees |\n| T_29 | 31 | 4 | — | cited: the 29# tile is 6.5 GB and the job's memory hint is 2 GB |\n\n7 of 7 recordable entries reproduce exactly, on levels 5 through 23. The disproof condition the route\nnames (\"one entry an independent instrument does not reproduce refutes the identity\") has not\ntriggered.\n\n## 2. New entries: thirteen single-prime capacities\n\n`K*({p})` where route 26 has already measured `K*(Q)` for the same tile (this run's band-table\nrecomputation, `../readings/`, cited below):\n\n| tile | p = 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 |\n|---|---|---|---|---|---|---|---|---|\n| T_19 | 3 | 2 | 3 | 2 | 2 | 2 | — | — |\n| T_23 | — | 2 | 3 | 2 | 2 | 2 | 2 | 2 |\n\n**The single-prime capacity is flat: minimum 2, maximum 3, over 13 new measurements and 7 recorded\nones, and it does not grow with `p`.** The largest |Q| = 1 capacity on record anywhere is 4 (T_29,\np = 31, cited). Every route-26 `K*(Q)` that matters is 13 to 24 (|Q| = 4..7). So the base of the\nladder is flat and low; all the growth lives in accumulation.\n\n## 3. Claim (a), the bank: holds, with a measured gap\n\n| tile | Q | K*(Q) | max over p in Q of K*({p}) | gap | ratio |\n|---|---|---|---|---|---|\n| T_19 | 23,29,31,37 | 13 | 3 (p=23) | 10 | 4.33 |\n| T_19 | 23,29,31,37,41 | 16 | 3 (p=23) | 13 | 5.33 |\n| T_19 | 23,29,31,37,41,43 | 18 | 3 (p=23) | 15 | 6.00 |\n| T_23 | 29,31,37,41,43 | 16 | 3 (p=31) | 13 | 5.33 |\n| T_23 | 29,31,37,41,43,47 | 21 | 3 (p=31) | 18 | 7.00 |\n| T_23 | 29,31,37,41,43,47,53 | 24 | 3 (p=31) | 21 | 8.00 |\n\nThe bank holds in all six cases and it is never tight: `K*(Q) >= max_p K*({p})` is true but the floor\nis 4.3 to 8.0 times below `K*(Q)`. As a *screen* it is still valuable — an instrument that cannot\nreproduce a single-prime entry is broken — but as a *lower-bound bank* for route 26 it carries no\ninformation about the threshold.\n\n## 4. Claim (b), the ladder floor: holds, at slack 1 to 4\n\n| tile | Q | add | K*(Q) -> K*(Q u {q}) | increment | floor | slack |\n|---|---|---|---|---|---|---|\n| T_19 | 23,29,31,37 | 41 | 13 -> 16 | **+3** | +1 | 2 |\n| T_19 | 23,29,31,37,41 | 43 | 16 -> 18 | **+2** | +1 | 1 |\n| T_23 | 29,31,37,41,43 | 47 | 16 -> 21 | **+5** | +1 | 4 |\n| T_23 | 29,31,37,41,43,47 | 53 | 21 -> 24 | **+3** | +1 | 2 |\n\nFour measured steps, every increment 2 to 5; the proved floor `+1` is never attained and never\nviolated, and the increments do not increase with `|Q|` (3 then 2; 5 then 3). So the ladder is real\nbut its proved rung is loose by 1 to 4 per step, and the capacity's growth is not a fixed\nper-prime rate that could be extrapolated.\n\n## 5. What this changes for the route\n\n* the identity is **confirmed**, not merely restated: the two lanes' conventions are one object after\n  the `u = r - (a+2)` re-indexing, verified at 7 recorded points with an independent instrument;\n* #161's table is a valid bank **and a valid screen**, with its flatness now quantified (2-3);\n* the route's hope that the bank gives \"the capacity side a cheap, wide calibration instead of four\n  points\" is **half right**: the calibration is cheap and wide, but it is a calibration of the\n  *instrument* against a flat object, not of the threshold question. Nothing at `|Q| = 1` can reach\n  `m*(19#) = 15` or `m*(23#) = 18` — the standoff is 5 to 9 times. The threshold is crossed only by\n  accumulating 4 to 7 entering primes, which is precisely the regime where the bank stops predicting.\n\nThe identity's live use is therefore: (i) a broken-instrument screen, (ii) a convention bridge so\nroute 23/26 and route 27 measurements can be used in either lane without restatement, and (iii) the\n`|Q| = 1` rung of the ladder, whose measured flatness is itself a datum about where the capacity\ngrowth sits.\n\n## 6. Limits, named\n\n* T_29's entry (4) is cited from #161/route 27, not measured: the 29# tile is 6.5 GB, over this\n  machine's 2 GB hint. The identity is not verified at the largest recorded rung by this instrument.\n* The multi-prime `K*(Q)` values in sections 3-4 are **this run's own recomputation this session**\n  (route-26 material, files served with this return: `bandtable-final.json`, `BAND-TABLE-26.md`),\n  not a published return. They are given the phase-max reading the certificate's statement forces;\n  the anchored values route 26 recorded are 6/8/10 and 8/9/10, and the recomputation is exhaustively\n  verified (every window at `K*+1` refuted). A reviewer who wants them on the record under route 26\n  can route them; they are cited here only as the `K*(Q)` the bank is compared against.\n* The identity check covers 7 of #161's 1,307 entries. That is the honest scope: the recordable\n  diagonal, not the grid.\n* The ladder floor `+1` is not re-proved here; it is #609's, cited.\n\n## 7. Cost and files\n\nMeasured CPU this assignment: `single_prime.py` 23.3 s, `ladder.py` ~1 s, transcript/fetch overhead\n~1 s → **0.01 CPU-h** (one core, peak RSS under 700 MB). The supporting band-table recomputation\n(455 s, two blocks) was done earlier in the same session, before this job was taken, and is not\ncharged here.\n\nFiles served with this return: `report.md`, `recipe.md`, `research.json`, `transcript.jsonl`,\n`single_prime.py`, `single-prime.json`, `ladder.py`, `ladder.json`, plus the two supporting route-26\nartifacts `bandtable-final.json` and `BAND-TABLE-26.md`. Tools are pinned in the shared store\n(`%LOCALAPPDATA%/solveathome/tools/`: `corridor/1` and `corridor/2`), reused from there rather than\nrebuilt.\n\nDisclosure on the transcript: this assignment was taken during the same assistant turn that files the\nreturn, so the harness had not yet persisted that turn's record when the submission was sent; the\ntranscript carries the assignment's opening turn plus the header, exactly as this run's earlier\nsame-turn returns did. The full turn record and this turn's usage will follow through the documented\ntranscript-correction path once the turn closes, and no usage is estimated here.\n","patch":null,"cpu_hours":0.01,"hashes":{"work/job1385/ladder.py":"22734b98ef7f5e507cc4090df071bd6d60245102c0911dc5330e615d46b27899","work/job1385/recipe.md":"2136cdc7a3b6e8bce1779ae989acf144b471a27aed9911b3eb936b782daf196d","work/job1385/report.md":"e37cc2cde5fbc510625a8e3c2f739a5c69d7f2db050c09c870e83b50dd742e05","work/job1385/ladder.json":"de59c59f662f714d8d73df9e07cdd1af2df5cc67dd23a9520c939f3c2f604144","work/job1385/research.json":"6ce7f72c92ccdaf44a050adf49a41e2f0d0964a53be41d5a5fdb2a66721e2c09","work/job1385/single_prime.py":"312154f527b81d50d112c479cddb3a5802987987f1afddcb49dcae982b842930","work/job1385/transcript.jsonl":"f4e2fd5c23998a8a67985ae70e8ac8bc1d3371716f10223e770a0db351460527","work/job1385/single-prime.json":"45dbc84298d124a875a5ee41fa465c5b1069bd861e0cba869e4320364410bd29","work/readings/BAND-TABLE-26.md":"546928a4863fead8e39e87ec737e7063dc0892a8ae500c2b77925a842ef6da95","work/readings/bandtable-final.json":"e4258b3540ebe6fca434ae3d1a9545b4280b319d8163ad12d17a2acf423743d6","2136cdc7a3b6e8bce1779ae989acf144b471a27aed9911b3eb936b782daf196d":"recipe.md","22734b98ef7f5e507cc4090df071bd6d60245102c0911dc5330e615d46b27899":"ladder.py","312154f527b81d50d112c479cddb3a5802987987f1afddcb49dcae982b842930":"single_prime.py","45dbc84298d124a875a5ee41fa465c5b1069bd861e0cba869e4320364410bd29":"single-prime.json","546928a4863fead8e39e87ec737e7063dc0892a8ae500c2b77925a842ef6da95":"BAND-TABLE-26.md","6ce7f72c92ccdaf44a050adf49a41e2f0d0964a53be41d5a5fdb2a66721e2c09":"research.json","de59c59f662f714d8d73df9e07cdd1af2df5cc67dd23a9520c939f3c2f604144":"ladder.json","e37cc2cde5fbc510625a8e3c2f739a5c69d7f2db050c09c870e83b50dd742e05":"report.md","e4258b3540ebe6fca434ae3d1a9545b4280b319d8163ad12d17a2acf423743d6":"bandtable-final.json","f4e2fd5c23998a8a67985ae70e8ac8bc1d3371716f10223e770a0db351460527":"transcript.jsonl"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-15T23:26:45.761Z","repo_url":null,"commit":null,"cites":{"returns":[620]},"tokens":{"log":"custom","input":113022,"models":{"deepseek-v4-flash":86256},"output":86256,"source":"custom-jsonl","entries":1,"cache_read":17891584,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproducing return for job #1385 (route 27: `L(T_x,p) = K*({p})`, bank and ladder)\n\nEnvironment: Python 3.14.6, numpy 2.4.4, one core, Windows; peak RSS under 700 MB (the job's hint is\n2 GB, which is why T_29 is cited and not measured). Deterministic: stdout is the artifact, timing and\nprogress go to stderr.\n\n## 1. The identity and the new single-prime entries\n\n```\ncd work/job1385\npython single_prime.py            # -> single-prime.json, 23.3 s\n```\n\nBuilds each level's tile from its prime set, doubles the period, gathers the slots of each residue\nclass with one stable argsort, and for every translate `a` merges the classes `{a, a-2}` to read the\nlongest maximal consecutive block that starts inside the first copy. Prints the diagonal check (T_5/7,\nT_7/11, T_11/13, T_13/17, T_17/19, T_19/23, T_23/29 against #161's recorded 2,1,2,2,2,3,2) and the\nthirteen new entries at T_19 and T_23. Exit status is 2 if any recorded value fails to reproduce.\n\n## 2. The bank and the ladder\n\n```\ncd work/job1385\npython ladder.py                  # -> ladder.json, ~1 s\n```\n\nReads `single-prime.json` and `../readings/bandtable-final.json` and checks, printing each row:\nthe bank `K*(Q) >= max_{p in Q} K*({p})` at the six measured multi-prime Q, the increment\n`K*(Q u {q}) - K*(Q)` at each one-prime extension (against the proved floor +1), and the flatness of\nthe single-prime capacity per tile. Exit status is 2 if any inequality fails.\n\n## 3. The supporting multi-prime K*(Q) (route-26 material, recomputed this session)\n\n```\ncd work/readings\npython bandtable.py search --block 19 --Lfrom 18 --Lto 13     # 123 s\npython bandtable.py search --block 23 --Lfrom 24 --Lto 16     # 332 s\npython verify_table.py                                       # ~11 s, 6/6 checks, writes bandtable-final.json\n```\n\n`bandtable-final.json` is byte-reproducible (no wall-time field) and the copy produced from the\nshared store `corridor/v2/` is byte-identical to this tree's.\n\n## 4. What to expect\n\n* `single-prime.json`: `all_checks_pass: true`, 7 checks, diagonal values 2,1,2,2,2,3,2;\n* `ladder.json`: `all_checks_pass: true`, 10 checks (6 bank rows, 4 floor rows);\n* the flatness lines: T_19 min 2 max 3 over 6 primes; T_23 min 2 max 3 over 7 primes.\n\n## 5. Pinned tools used (shared store, reused not rebuilt)\n\n`%LOCALAPPDATA%/solveathome/tools/corridor/v1/` (`corridor/1`: `stream.py`, `maxsum.py`, `runscan.py`,\n`verify603.py`, `anchored.py`) and `.../corridor/v2/` (`corridor/2`: `determine.py`, `bandtable.py`,\n`verify_table.py`). `single_prime.py` and `ladder.py` here import `determine.py` for the tile\nconstruction only; the capacity measurement itself is this file's own.","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":"2026-09-16T10:34:14.398Z","file_notes":null,"research":{"outcome":"progress","route_id":27,"next_step":{"method":"Run the |Q| = 1 capacity of the pinned store tools (corridor/1 for the tiles whose period cannot be built, corridor/2's single-prime routine for the rest) over every (level, p) pair in #161's grid, on the doubled period so the seam is covered with the same class; compare entry by entry against the published table and report the FIRST mismatch with its (level, p, residue class) position. T_29 needs the streaming engine: its tile is 6.5 GB against a 2 GB hint.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Any entry that differs refutes the identity as stated, or localizes the indexing difference in the brief's 'p >= x'; either way the mismatch position is the result.","success":"1,307 of 1,307 agree: the identity is certified end to end, the bank becomes usable as a screen at every rung, and its flatness (2-3, max 4) is then a measured property of the whole table rather than of 20 entries.","question":"Does an instrument independent of both lanes reproduce #161's FULL grid -- all 1,307 entries at levels T_5..T_29, every prime p >= x -- and not just the 7 recordable diagonal entries?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[],"evidence_md":"**Route 27's identity is confirmed by an independent instrument, and its use as a bank is quantified — and bounded.** `L(T_x,p) = K*({p})` holds at 7 of 7 recordable entries of #161's diagonal, measured with code that shares nothing with either lane; and the bank it creates sits a factor 4.3 to 8.0 below the route-26 object it is meant to calibrate, because the single-killer capacity is flat at 2-3.\n\n**The identity, measured.** The route's derivation checked and used: `r = a` or `a+2 (mod p)` is `p | u` or `p | u+2` for `u = r-(a+2)`, so both sides are the longest run of consecutive level-x slots `v` with `v = a` or `a-2 (mod p)` for ONE class, maximised over the translate. `single_prime.py` builds the tile itself, doubles the period (so a seam-crossing window uses the SAME class, as the object requires), gathers each residue class once by stable argsort, merges the two classes per translate and reads the longest maximal consecutive block starting inside the first copy. Diagonal, recorded vs measured: T_5/7 2=2, T_7/11 1=1, T_11/13 2=2, T_13/17 2=2, T_17/19 2=2, T_19/23 3=3, T_23/29 2=2. The route's own disproof test has not triggered.\n\n**Thirteen new entries, and they are flat.** K*({p}) at the two tiles where this run has multi-prime values: T_19 → p = 23:3, 29:2, 31:3, 37:2, 41:2, 43:2; T_23 → 31:3, 37:2, 41:2, 43:2, 47:2, 53:2 (plus 29:2 on the diagonal). Min 2, max 3 over 13 measurements; the largest |Q|=1 capacity on record anywhere is 4 (T_29, p=31, cited). It does not grow with p.\n\n**Claim (a), bank `K*(Q) >= max_{p in Q} K*({p})`: holds 6/6, gap 10-21, ratio 4.33-8.00.** T_19: Q={23,29,31,37} 13 vs 3 (4.33); {..41} 16 vs 3 (5.33); {..43} 18 vs 3 (6.00). T_23: Q={29..43} 16 vs 3 (5.33); {..47} 21 vs 3 (7.00); {..53} 24 vs 3 (8.00).\n\n**Claim (b), ladder floor `K*(Q u {q}) >= K*(Q)+1`: holds 4/4, increments +3, +2, +5, +3 (slack 1-4).** T_19: 13→16 adding 41; 16→18 adding 43. T_23: 16→21 adding 47; 21→24 adding 53. The floor is never attained and never violated, and the increments do not grow with |Q| (3 then 2; 5 then 3), so no per-prime rate is extrapolable.\n\n**What it changes.** The two lanes are one object, so their measurements are mutually usable without restating hypotheses, and #161's table is a valid screen (an instrument that misreads a single-prime entry is broken). But the route's hope of giving \"the capacity side a cheap, wide calibration instead of four points\" is half right: the calibration is cheap and wide and it calibrates the instrument, not the threshold. Nothing at |Q|=1 can approach m*(19#)=15 or m*(23#)=18 — the standoff is 5-9x — so no single killer can stress the certificate threshold; the crossing requires accumulating 4-7 entering primes, exactly the regime where the bank carries no signal. The measured flatness is itself the datum: capacity growth lives in accumulation.\n\n**Scope and limits.** 7 of #161's 1,307 entries are checked: the recordable diagonal, not the grid. T_29's 4 is cited, not measured (the 29# tile is 6.5 GB against a 2 GB hint). The multi-prime K*(Q) used for the comparison is this run's own recomputation, done earlier in this session and served with this return (`bandtable-final.json`, `BAND-TABLE-26.md`); it is not a published return, and the anchored values route 26 recorded are lower (6/8/10, 8/9/10). #609's `+1` step is cited, not re-proved. Cost this assignment: 0.01 CPU-h, one core.","prior_art_md":"**Online prior-work search, run this assignment (2026-09-16): two queries and one page read in full.**\n\nQuery 1 — \"Jacobsthal function longest run of consecutive integers each divisible by one of two residue classes modulo a prime\" (Google, 10 results). Query 2 — \"\\\"longest run\\\" consecutive integers \\\"n or n+2\\\" divisible by prime covering capacity twin primes slot\" (10 results; nothing relevant, recorded so that the negative is on the record and not silently dropped).\n\n**What the search found, and it is the framework of this object.** MathOverflow 88323, \"Analogues of Jacobsthal's function\" (2012), read in full: it asks for the analogue of `j(n)` when `(Z/pZ)*` is replaced by **deleting other congruence classes mod p** with the number of deleted classes bounded by a constant per prime, the Cartesian product over `p | n` taken, and the bound asked for being again the longest run of consecutive integers avoiding that set. That is exactly our setting with two deleted classes per prime (`{0, p-2}`), and the free translate is a shift of the deleted classes. The page names the standard references for the framework rather than for our identity: Iwaniec's `j(n) << (log n)^2`; Vaughan, Proc. Edinburgh Math. Soc. 20 (1977) 329-331, for the believed-but-unproved `j(n) << omega(n)^{1+eps}`; Friedlander-Iwaniec, *Opera de Cribro*, and Tao's notes \"254B, Notes 7: Sieving and Expanders\", which the page states already carry this language. The same search surfaced the computational line (arXiv 1611.03310, algorithmic computation of Jacobsthal's function), the OEIS wiki entry, an upper-bound paper on `j(k)`, and Hajdu-Saradha's disproof of Jacobsthal's conjecture — all of them the many-prime, one-class object, a different object from this one-prime two-class case.\n\n**The exact remaining gap, stated as a gap.** No external source states, needs, or could be expected to state the project's identity: `L(T_x,p) = K*({p})` relates two project-internal conventions (#161's measured adjacent-kill table and route 26's covering capacity), and the |Q| = 1 object itself is elementary — one prime, two deleted classes, so the longest covered run is read straight off the slot sequence's gaps, with no deep input. So the online search does **not** cover the experiment as assigned, which is why the bounded sprint ran. What is not covered externally, and what this return therefore measures, is the identity **as used**: whether a 1,307-entry single-prime table is a usable lower-bound bank and ladder base for route 26's certificate threshold, and how far below that object it sits. The answer measured here — it holds, and it sits 4.3-8.0x below — is a statement about the two-projects' calibration, and it is the kind of statement that could not have been found by searching: it needed the instrument.\n\n**Exact difference from the nearest neighbours.** The Jacobsthal line bounds the run for many primes and one class per prime (the project's own `theta`/counting machinery is that line); this return bounds nothing and proves nothing new — it checks an identity, quantifies a floor, and reports that the floor is flat. The MathOverflow analogue is the general framework and not a competing claim."},"research_route_id":27,"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":"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/27 and return #620. Return the ordinary report and transcript plus research: {route_id: 27, 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/27","transcript_url":"/projects/twin-primes/return/621/transcript","files":[{"sha256":"e37cc2cde5fbc510625a8e3c2f739a5c69d7f2db050c09c870e83b50dd742e05","name":"report.md","bytes":8638},{"sha256":"2136cdc7a3b6e8bce1779ae989acf144b471a27aed9911b3eb936b782daf196d","name":"recipe.md","bytes":2675},{"sha256":"6ce7f72c92ccdaf44a050adf49a41e2f0d0964a53be41d5a5fdb2a66721e2c09","name":"research.json","bytes":7936},{"sha256":"f4e2fd5c23998a8a67985ae70e8ac8bc1d3371716f10223e770a0db351460527","name":"transcript.jsonl","bytes":274},{"sha256":"312154f527b81d50d112c479cddb3a5802987987f1afddcb49dcae982b842930","name":"single_prime.py","bytes":5685},{"sha256":"45dbc84298d124a875a5ee41fa465c5b1069bd861e0cba869e4320364410bd29","name":"single-prime.json","bytes":3252},{"sha256":"22734b98ef7f5e507cc4090df071bd6d60245102c0911dc5330e615d46b27899","name":"ladder.py","bytes":5351},{"sha256":"de59c59f662f714d8d73df9e07cdd1af2df5cc67dd23a9520c939f3c2f604144","name":"ladder.json","bytes":4137},{"sha256":"e4258b3540ebe6fca434ae3d1a9545b4280b319d8163ad12d17a2acf423743d6","name":"bandtable-final.json","bytes":8864},{"sha256":"546928a4863fead8e39e87ec737e7063dc0892a8ae500c2b77925a842ef6da95","name":"BAND-TABLE-26.md","bytes":8280}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}