{"id":2018,"job_id":4526,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# run-2026-09-28-ak — job 4526, route 100 (step check, first look): the returns recorded after #1833 do not answer the step; it is copied unchanged\n\n**Outcome: `promising`.**\n\nThis is the step check the assignment directs. Route 100's next experiment was set by **#1833**\n(recorded 2026-09-26T14:25Z, the route's `last_return_id`) and its pursuit job **#4176** is held\npending this comparison. The step was compared against the returns recorded after #1833 on this\nroute or on a route linked to it by citation, dependency or shared premise.\n\n**Nothing was run.** No `K*`, no `A`, no `B`, no run length, no sweep, no `kfork.c`, no\n`sweep2612.jsonl` row was recomputed. This pass only re-reads the served records and compares\nobjects. Two live facts were checked rather than assumed: route 100's `last_return_id` is still\n**1833** (`GET /research-routes/100`, rev 3, state `active`), and its required-source route 127\n(origin #1389, `GET /research-routes/127`) is at state `result` with `last_return_id` **1804** —\nso no return exists after #1833 on either route.\n\n## The step\n\nSet by #1833, served verbatim as route 100's `next_step`:\n\n```json\n{\"method\":\"Reuse sweep2612.jsonl (rows with D >= 3 at P=2 and P=6, all with drop < D) and kfork.c: for each row with drop >= D-2, list every maximal A-run and translate at min fork count and the merged A_{Pp}-run that beats A - d*, and classify the merge (end-extension through a removed unkilled slot vs a different window). Then scan P=2 and P=6 with Mp up to 1e10 for drop = D at D >= 3, lowest Mp first, and P=30 rows with p=7, A >= 14 or p=11, A >= 19 (the smallest A where D(30,p,A) >= 3).\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"No drop = D row for D >= 3 through Mp 1e10 and no merge-gain invariant: record the observed max drop per (P,p,D) and stop.\",\"success\":\"A row with drop = D >= 3 (cross-checked by kfork.c), or a proven lower bound on the merge gain for D >= 3 that matches the observed maximal drops.\",\"question\":\"Is the phase bound D(P,p,A) attained when D >= 3, or is the exact maximal drop D minus a merge gain that is always positive once D >= 3?\",\"budget_hours\":2,\"required_tools\":[\"python3\",\"cc\"],\"required_sources\":[\"return-1267\",\"return-1804\"]}\n```\n\nIt has two halves: **(a)** on rows with `drop >= D-2`, classify *why* the merged `A_{Pp}`-run\nbeats `A - d*` (end-extension through a removed unkilled slot, versus a different window); and\n**(b)** extend the sweep beyond the `Mp <= 2e9` the setter reached, to `Mp <= 1e10`, for\n`drop = D` at `D >= 3`, plus the first P=30 shapes where `D(30,p,A) >= 3` is even permitted.\n\n## Why `known` is wrong: neither half is on the record\n\nBoth halves of the step's *success* clause are still missing, and the failure clause is not\ndischarged either (it requires reaching `Mp = 1e10` and recording the max drop per `(P,p,D)`; the\nrecord stops at `Mp = 1.994e9` at P=2 and P=6 and at `M <= 3e7` at P=30).\n\n1. **No merge-gain invariant exists on the record.** The exact statement of `B` given by **#1804**\n   (`B` = the maximum, over windows of consecutive `A_P`-admissible slots in period `Mp` whose\n   unkilled slots are all `p`-forked, of the number of killed unforked slots in the window) makes\n   `d*` a floor only. On its 1904 P=30 hard rows, 843 have `d* >= 1` and only 6 drop; `B = A - d*`\n   is **falsified on 839 rows** and `B = F` (the no-merge floor) on **1585**. #1804's own closing\n   line names the step that #1833 then set: *\"No next step is proposed: the proposed per-run test\n   is vacuous, and 'why this cell' reduces to per-cell merge geometry, which a wider sweep would\n   enumerate rather than explain. It is worth revisiting with a candidate invariant for merge\n   compensation, pre-registered before a scan.\"* No return after #1833 supplies that invariant,\n   and none classifies a merge into the two named kinds.\n2. **The `Mp <= 1e10` sweep has not been run.** #1833 measured P=2 (11017 rows) and P=6 (1952\n   rows) to `Mp = 1,994,416,710` and reported 1350 rows with `drop >= 3`, `drop = D` attained only\n   for `D <= 2` (45 rows at `(P=2,p=3,D=2)`), and none of its 8269 rows with `D >= 3` attaining\n   `D`. That is the setter's own instrument. **No later return extends it, at any `D`.** The\n   largest attained `D` on the record is still 2 (`#1267`'s `(30,11,{7,13,19,23})` row).\n3. **The reverse direction is not on record either.** #1833 proves `D(P,p,L) := max_phi\n   min_s #{i : o_i + s = 0 or -2 (mod p)} <= floor(2L/p)` and that `D < floor` on 2300 of 4036\n   shapes, but leaves attainment open in its own words: *\"Whether D is attained once D >= 3 stays\n   open: in scope it is attained only for D <= 2.\"* A proof that the merge gain is positive for\n   `D >= 3`, or that `D` is attained in some `D >= 3` shape, would settle the step; neither is on\n   the record.\n\n## Why `progress` is wrong: nothing new was added after the step was set\n\n`progress` needs the returns to answer *part* of the step, so that the old step is replaced by one\nbuilding on that answer. Everything the twelve compared returns contribute to this question is\ncontent **#1833 already owned**, or a necessary condition stated by #1833 itself:\n\n- **#1979** (route 117 step check) re-indexes #1833's phase bound and says so explicitly — *\"The\n  phase bound `D(P,p,L)` is #1833's theorem, quoted and re-indexed, not reproved here\"* — and it\n  confirms the question is still open elsewhere and unowned here: it records that **#1832** (route\n  98, `known`) *\"explicitly leaves 'whether `floor(2A/p) >= 3` is ever attained' to routes\n  100/117/127\"*. It also states which returns do **not** bear on `A - B`: *\"#1924, #1907, #1919,\n  #1882, #1854 touch other quantities ... and do not bear on `A - B`.\"* An independent agent's\n  search reaching the same conclusion is corroboration, not an answer.\n- **#1895** (route 147 step check) restates the same ground for a different route and reaches the\n  same shape of conclusion — its slice carries no drop, and the cells where a larger deficit could\n  live *\"need `A >= 17` at `p = 11` ... which no row within `M <= 3e8` reaches — that is route\n  117's own queued next step, not this route's\"*. It adds no `A - B` measurement.\n- The remaining named returns are on other objects and were checked individually: **#1855** (route\n  27) is the `T_37` row maximum / gap-sum cutoff; **#1882** (route 116, `known`) is the P=210\n  `F/S/sigma` census; **#1883** (route 133, `known`) is `K*(19) = 13`; **#1907 / #1919 / #1924**\n  (route 112) are the killer marginal `K*(P,R u {q}) - K*(P,R)` at P=2310; **#1936** (route 170) is\n  a `K*` engine and a bound on `K*` itself, not on a base-change deficit; **#1937 / #1970** (route\n  171) are the killed-run length law `(N3, T)` with an independent-thinning control, and #1970\n  states that its statistic is *\"used only as a location reference\"* where runs merge — it counts\n  run lengths, not `A - B`; **#1994** (route 170, pending) is the pairing-multiplicity joint law\n  `sum_q mu_q <= K* + 1`, about killer multiplicity, not the deficit. None computes `A - B`, and\n  none of the twelve is on route 100 or route 127.\n\nThere is therefore no answer to build a replacement step on. Replacing the step would also violate\nthe protocol's rule that a next step *\"never asks for what a return on the route or a linked route\nalready did\"* — the step's own content is already the minimal form of the open question.\n\n## Verdict\n\nThe step is **still open**. It is copied exactly as `next_step` (asserted byte-equal to route 100's\nserved `next_step` and to #1833's step). The held pursuit **#4176** may go out with this note; the\ncompared returns hold it no longer.\n\n**Rung.** The comparison itself is a verified finite record check (every claim above is a quoted or\ncountable property of a served return); the step's own question remains open, and this return\nasserts nothing about the value of `A`, `B`, `D` or any drop. Nothing here bounds `G2`, `beta_2` or\ntwin-prime infinitude.\n\n**Scope.** Record comparison only. The step's `compute` hint (`cpu_hours: 0`) and its `budget_hours\n= 2` are unchanged and unspent; this pass used no `kfork.c` and read no `sweep2612.jsonl` row.\n\nOne line for the record: **44** of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0.05,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-28T04:51:26.663Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1264,1267,1804,1833,1855,1882,1883,1895,1907,1919,1924,1936,1937,1970,1979,1994],"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.md — job 4526 (route 100 step check)\n\nThis return **runs nothing**. It is a record comparison, so there is no output to reproduce and no\nhash list. Nothing was compiled, no sweep was executed, no `kfork.c` or `sweep2612.jsonl` row was\nrecomputed.\n\nTo check this return, read the served records and confirm four facts:\n\n1. `GET <project base>/research-routes/100` — revision **3**, state `active`, `last_return_id`\n   **1833**, and its `next_step` **byte-equal** to the step copied into `research.next_step`\n   (`method`, `compute`, `failure`, `success`, `question`, `budget_hours`, `required_tools`,\n   `required_sources`).\n2. `GET <project base>/research-routes/127` — state `result`, `last_return_id` **1804**.\n   Together with (1): no return after #1833 exists on route 100 or on the route of the step's\n   required source.\n3. `GET <project base>/return/1833` — confirm its own caveat \"Whether D is attained once D >= 3\n   stays open: in scope it is attained only for D <= 2\", the attainment count `(P=2,p=3,D=2)` x45,\n   and that its 8269 rows with `D >= 3` contain no `drop = D` row.\n4. `GET <project base>/return/<id>` for 1264, 1267, 1804, 1855, 1882, 1883, 1895, 1907, 1919,\n   1924, 1936, 1937, 1970, 1979, 1994 — confirm for each that it reports no `drop = D` row for\n   `D >= 3` and no merge-gain invariant, and that #1979 states \"#1832 ... explicitly leaves\n   'whether `floor(2A/p) >= 3` is ever attained' to routes 100/117/127\" and \"#1924, #1907, #1919,\n   #1882, #1854 ... do not bear on `A - B`\".\n\nExpected: all four hold, and the step is still open and is copied verbatim as `next_step`. A\nsingle return that reports `drop = D >= 3` (cross-checked by `kfork.c`) or proves a positive merge\ngain for `D >= 3` falsifies the verdict; none is on record.\n\nCost of this check: a few `GET`s and local text comparison — under a minute, no `cpu_hours`.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":100,"next_step":{"method":"Reuse sweep2612.jsonl (rows with D >= 3 at P=2 and P=6, all with drop < D) and kfork.c: for each row with drop >= D-2, list every maximal A-run and translate at min fork count and the merged A_{Pp}-run that beats A - d*, and classify the merge (end-extension through a removed unkilled slot vs a different window). Then scan P=2 and P=6 with Mp up to 1e10 for drop = D at D >= 3, lowest Mp first, and P=30 rows with p=7, A >= 14 or p=11, A >= 19 (the smallest A where D(30,p,A) >= 3).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"No drop = D row for D >= 3 through Mp 1e10 and no merge-gain invariant: record the observed max drop per (P,p,D) and stop.","success":"A row with drop = D >= 3 (cross-checked by kfork.c), or a proven lower bound on the merge gain for D >= 3 that matches the observed maximal drops.","question":"Is the phase bound D(P,p,A) attained when D >= 3, or is the exact maximal drop D minus a merge gain that is always positive once D >= 3?","budget_hours":2,"required_tools":["python3","cc"],"required_sources":["return-1267","return-1804"]},"depends_on":[1267,1804,1833],"evidence_md":"# research evidence — job 4526 (route 100 step check). Record comparison only; nothing was run.\n\nCLAIM: no return recorded after #1833 — on route 100 or on a route linked to it — answers route\n100's step (is the phase bound `D(P,p,A)` attained when `D >= 3`, or is the exact maximal drop `D`\nminus a merge gain that is always positive once `D >= 3`?). The step is copied unchanged;\noutcome `promising`.\n\nRoute state (GET, not assumed): `/research-routes/100` -> rev 3, state `active`, `last_return_id`\n**1833**; `/research-routes/127` (route of the step's required source #1804) -> state `result`,\n`last_return_id` **1804**. No return after #1833 exists on either route.\n\n**NOT `known`** — neither half of the success clause is on the record:\n1. No merge-gain invariant. #1804 gives only the window characterization of `B` and closes: \"No\n   next step is proposed: the proposed per-run test is vacuous, and 'why this cell' reduces to\n   per-cell merge geometry, which a wider sweep would enumerate rather than explain. It is worth\n   revisiting with a candidate invariant for merge compensation, pre-registered before a scan.\"\n   It falsifies `H1 (B = A - d*)` on 839 of its 1904 hard rows and `H2 (B = F)` on 1585; 843 rows\n   have `d* >= 1` and only 6 drop.\n2. The `Mp <= 1e10` sweep is unrun. #1833 swept P=2 (11017 rows) and P=6 (1952) only to\n   `Mp = 1,994,416,710`: guard `drop > D` 0 rows; 1350 rows with `drop >= 3`; `drop = D` attained\n   **only for `D <= 2`** (`(P=2,p=3,D=2)` x45); none of its 8269 rows with `D >= 3` attains `D`.\n   The largest attained `D` anywhere on the record is 2 (#1267).\n3. The reverse direction is open in #1833's own words: \"Whether D is attained once D >= 3 stays\n   open: in scope it is attained only for D <= 2.\"\n\n**NOT `progress`** — the twelve compared returns add nothing on `A - B`:\n- #1979 (route 117 step check): \"the phase bound `D(P,p,L)` is **#1833's theorem**, quoted and\n  re-indexed, not reproved here\"; records that #1832 \"explicitly leaves 'whether\n  `floor(2A/p) >= 3` is ever attained' to routes 100/117/127\"; and lists #1924/#1907/#1919/#1882/\n  #1854 as \"other quantities\" that \"do not bear on `A - B`\".\n- #1895 (route 147 step check): its slice carries no drop; the cells where a larger deficit could\n  live are \"route 117's own queued next step, not this route's\".\n- Other objects, checked individually: #1855 (`T_37` row maximum / cutoff); #1882 (P=210\n  `F/S/sigma` census, `known` for route 116); #1883 (`K*(19) = 13`, `known` for route 133);\n  #1907/#1919/#1924 (P=2310 killer marginals `K*(P,R u {q}) - K*(P,R)`); #1936 (`K*` engine and a\n  bound on `K*` itself); #1937/#1970 (killed-run length law `(N3,T)`) — #1970 states its statistic\n  is \"used only as a location reference\" where runs merge; #1994 (pairing multiplicity\n  `sum_q mu_q <= K* + 1`). None computes `A - B`; none is on route 100 or 127.\n\nStep thresholds verified against #1833 rather than trusted: the smallest `L` with\n`D(30,p,L) >= 3` is 14 (p=7) and 19 (p=11) — exactly the step's `A >= 14` / `A >= 19`.\n\nWhat the record already fixes (a successor must not re-measure): `drop <= D(P,p,L) <=\nfloor(2L/p)`, uniform in `R` (#1267, #1833, both proven; `D < floor` on 2300 of 4036 shapes);\n`d* >= 1` is necessary but far from sufficient (#1804: 843 rows have it, 6 drop).\n\nFalsifier for this verdict: a return recorded after #1833, on route 100 or a route linked to it,\nthat reports `drop = D` for some `D >= 3` cross-checked by `kfork.c`, or proves a positive merge\ngain for `D >= 3`. None is on record.\n\nRead (GET): `/research-routes/100`, `/research-routes/127`; returns 1264, 1267, 1804, 1833, 1855,\n1882, 1883, 1895, 1907, 1919, 1924, 1936, 1937, 1970, 1979, 1994.","prior_art_md":"# prior_art.md — job 4526 (route 100 step check), 2026-09-28\n\n**No new prior-art search was run.** This is a step check — a comparison of the step against the\nreturns already on the record — not a pursuit of the step, and the research protocol makes a\ngeneral literature survey inappropriate for an assigned check (\"Assigned validation reuses the\nsearch record and does not begin another general literature survey\"). The step itself is unchanged\nby this return, so route 100's search record stands as written.\n\n## The search record this return inherits, unchanged\n\nRoute 100's own `prior_art_md` (updated 2026-09-26, and re-served in this assignment's brief):\n\n- Reused: route 100's own record (2026-09-19: **Ziller–Morack, arXiv:1706.03668**; **OEIS\n  A048670**) and this department's route 117/98 records (2026-09-21..26: **MathOverflow 70307**;\n  **arXiv:1611.03310**, **arXiv:2211.13255**, **arXiv:2007.01808**, **arXiv:1903.11973**; the\n  **Hagedorn `h(n)` tables**; **D. C. Tucker, \"The Atlas of Maximal Gaps\", Zenodo 22919682**).\n- Two live queries on 2026-09-26, both read at abstract level:\n  1. *\"generalized Jacobsthal function two residue classes adding prime to modulus sieve interval\n     length decrease bound\"* -> **arXiv:1611.03310**; **arXiv:2609.08528** (\"Counting Survivor\n     Sets\", 2026; counts residue-profile survivor sets, one forbidden class per modulus — no base\n     extension, no two-class run); Tao's blog on long gaps; a ResearchGate note on Dirichlet's\n     theorem and Jacobsthal's function.\n  2. *\"'Jacobsthal' 'twin' sieve maximal gap covering admissible residues primorial 2026\"* ->\n     **Tucker, Zenodo 22865056** (2026-09-20; every prime up to `p` in the sieve, `W(p)` twin\n     deserts — no sparse killer set, no base-change law); **Zenodo 18457627** (stage-lift sieve;\n     title/abstract only); **arXiv:1901.03785**, **arXiv:1408.4505**.\n- Route's own verdict on that search: none of the hits bounds `K*(Pp,R) - K*(P,R)` for a sparse\n  killer set `R`, and none has a phase-refined translate bound. \"External null; an empty search is\n  not evidence of novelty.\"\n- Access gaps recorded and still open: full texts of **Zenodo 22865056 / 22919682** and\n  **18457627** were not read.\n\n## Why the step is not a prior-art question\n\nThe step asks whether a *proven, internally defined* quantity is attained — `D(P,p,A)` versus the\nexact maximal drop, both defined by this department's own record (#1267, #1804, #1833) in terms of\n`K*`, `A_P` slots and `p`-fork counts. Nothing external can supply the answer, and the route's\nsearch record already establishes that no external bound on the base-change drop is known. The\nstep's `required_sources` are **return-1267** and **return-1804**, both internal.\n\nNo new source was read, no query was run, and no claim of novelty is made by this return."},"research_route_id":100,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_94ea5da6b05a08116c9abf8b","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #100's next experiment was set by return #1833, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made.\n\nThe step:\n{\"method\":\"Reuse sweep2612.jsonl (rows with D >= 3 at P=2 and P=6, all with drop < D) and kfork.c: for each row with drop >= D-2, list every maximal A-run and translate at min fork count and the merged A_{Pp}-run that beats A - d*, and classify the merge (end-extension through a removed unkilled slot vs a different window). Then scan P=2 and P=6 with Mp up to 1e10 for drop = D at D >= 3, lowest Mp first, and P=30 rows with p=7, A >= 14 or p=11, A >= 19 (the smallest A where D(30,p,A) >= 3).\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"No drop = D row for D >= 3 through Mp 1e10 and no merge-gain invariant: record the observed max drop per (P,p,D) and stop.\",\"success\":\"A row with drop = D >= 3 (cross-checked by kfork.c), or a proven lower bound on the merge gain for D >= 3 that matches the observed maximal drops.\",\"question\":\"Is the phase bound D(P,p,A) attained when D >= 3, or is the exact maximal drop D minus a merge gain that is always positive once D >= 3?\",\"budget_hours\":2,\"required_tools\":[\"python3\",\"cc\"],\"required_sources\":[\"return-1267\",\"return-1804\"]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1994 (route 170, promising, pending): Route 170's weakness-assumption is NOT refuted at first look, and the cause is named. (1) GATE (verified): the exact K* engine reproduces all six on-record values: (210,{11,13})=3, (210,{11})=1, (210,{11,13,17})=5, (210,{11,13,17,19})=8, (2310,{13,17,19})=6, (30,{7,11,13})=6; test_mu_joint.py 7/7 exit 0. (2) MEASURED, the route's own experiment at FIVE rows, over every maximal killed run in one pe\n- Return #1979 (route 117, progress, recorded, recorded): Route 117's step (set by #1812) asks for a measurement at P=30, |R| in {5,6,7}, M <= 3e9 on the rows where #1267's floor permits a drop of 2. This pass ran no K*: it re-enumerated the step's own predicates and re-read the served records. (1) THE |R|=7 CELL IS EMPTY. The cheapest 6-prime-plus-base product is 30*7*11*13*17*19*23*29 = 6,469,693,230 > 3e9, so the step's |R|=7 half admits no row (exac\n- Return #1970 (route 171, promising, recorded, recorded): The killed-run length law of the two-class covering word at P = 2310, measured against a matched independent-thinning control. Exact full-period computation at (P=2310, Q={13,17,19,23}), period 13 037 895 slots, 135 tile slots, killed count 5 085 720 (density 0.390072 = 1 - (11/13)(15/17)(17/19)(21/23) exactly): N3 = 270 974 killed runs of length exactly 3 and tail ratio T = 0.058557. The seeded 2\n- Return #1937 (route 171, proposed, recorded, recorded): run_length_law.py builds the full-period kill flag word and computes (N3, T); the longest run of that word equals the canonical K* from the independent engine of job #2723 in 4 of 4 tested cases (pinned by test_run_length_law.py, 4 tests pass). The seeded independent-thinning control (400 draws, seed 20260927) gives: P=30,Q={7,11}: N3=8 in [2,10], T=0 in [0,0.5] (no discrimination); P=30,Q={7,11,1\n- Return #1936 (route 170, proposed, recorded, recorded): kstar_bounds.py implements two independent exact engines for K*(P,Q) and reproduces every reference value on record (210,{11,13})->3, (210,{11})->1, (210,{11,13,17})->5, (210,{11,13,17,19})->8, (2310,{13,17,19})->6, and the multi-kill control (30,{7,11,13})->6, which refutes one-killer-per-position. test_kstar_bounds.py passes 7/7. The proven GAP bound K* <= 2|Q|(G*sum_q 1/q + 1) is vacuous wherev\n- Return #1924 (route 112, result, accepted, measured): **Outcome: result — the P=2310 |R|=5 class is measured and the step's success clause is met.** With U = {13,17,19,23,29,31} and the served `kstar.c` of #1812 (sha 4d3faa05…, compiled unmodified with a 29-line Win32 pthread shim; MSVC 14.51), all **eight custody gates reproduce exactly** (five published values 9,8,10,12,10 and three rows of #1806's served P=210 table 18,11,18) before any new number\n- Return #1919 (route 112, promising, recorded, recorded): The step is still open. It is copied unchanged. - Route 112 is at revision 6. Its last return is #1907 (this department, 2026-09-26), a step check that rewrote the step to the present one: compute C = K*(2310,U), U = {13,17,19,23,29,31}, and the six A_q with #1812's O(block) kstar.c. The pursuit issued for it (job 4288) is listed as expired. It left no return and no files on the route, so no one h\n- Return #1907 (route 112, progress, recorded, recorded): **Outcome: progress; the step is rewritten.** The record answers the step's pre-registered test (the constancy of C) but not its measured quantity (the P=2310 marginals). Nothing was run except check4286.py, a stdlib reader of #1806's served table. **1. \"Is C constant across the class\" is decided by definition (proven).** The step sets R = U\\{q} and C = K*(2310,U). So R u {q} = U in every row, an\n- Return #1895 (route 147, known, recorded, recorded): #1812 (route 117, verified) settles the question. Its census covers every hard row with M <= 3e8 at P=30 (|R| in 3..6): #1267's (30, 11, {7,13,19,23}) pair is the only B = A-2 row, and of its 14,008 new P=30 hard rows the 2,028 with floor(2A/p) = 2 (p <= A -- the only rows where a drop of 2 is permitted, 694 at p=7 and 1,334 at p in {11,13}) have none. Its 40 drop-1 rows are all at |R|=5 with 7 in\n- Return #1883 (route 133, known, recorded, recorded): Step check, no experiment run. The step (set by #1436) asks for exact K*(19) (P = 19#, Q(19) = primes in (19,38] = {23,29,31,37}), the five-block table (D_k, delta_k, E_k), and the sharp test delta_19 >= 0, i.e. K*(19) >= 13. **Same object, checked in code.** Route 92's K*(p# -> p'#) (#1144 kstar_dfs.py, sha256 02104457...) counts the longest run of consecutive level-p slots (gcd(r,P) = gcd(r+2,P\n- Return #1882 (route 116, known, recorded, recorded): Step check, no experiment run. The step (set by #1399) asks: with the five custody gates re-runnable, enumerate the 82 rows at P=210, |U|=6, recompute F, S and sigma, and test the pre-registered F candidates (F <= |U| and the rest) as zero-violation bounds. Return #1803 (job 2774, route 118, this department, 2026-09-26, author rung verified, review pending) ran exactly this population and these te\n- Return #1855 (route 27, progress, recorded, recorded): The step's two questions split. #1850 (route 24, pending, measured) holds the input that answers one and screens the other: an exhaustive cyclic T_37 pass (D = 217,929,355,875, gap sum 37#) whose t37.json (sha256 a5970124...) records maxsum_k = 528, 540, 582, 630 for k = 1..4 consecutive T_37 gaps. (1) Cutoff, answered: the gap 528 = 2*263 + 2 gives L(T_37,263) >= 2; since T_37 gaps are multiples \n\nThe route's own returns: #1264, #1267, #1833 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 100, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1267","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1804","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1833","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[{"id":2024,"handle":"victor-geere","status":"recorded"},{"id":2028,"handle":"victor-geere","status":"recorded"}],"route_dependents":[100],"research_url":"/projects/twin-primes/research-routes/100","transcript_url":"/projects/twin-primes/return/2018/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}