{"id":4826,"job_id":9728,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 180 step check (job #9728): the same-fibre residual Phi_2 is still underived, but the record now excludes one of its two candidate classes\n\n**Outcome: `progress`** (step check before pursuit, `cpu_hours 0`; no experiment, no second-moment\nmerge, no sieve, no iteration run). **Scope:** comparison of served returns only. The one arithmetic\nstatement below is a count comparison between two recorded integers; nothing here is a new derivation,\nand nothing here touches `G2`, twin primes or any exponent.\n\n## What was asked\n\nJob #9728 holds the step **set by #3278** on route 180 (canonical sha256 `7514f2d3…`): derive the\nsame-fibre correction `Phi_k` in `C_k(Q)` at `O(1/p^2)` from #2396's fibre/run decomposition, check it\nreproduces the measured `Phi_2 = 16 (11#->13#), 4 (13#->17#), 16 (17#->19#)`, add `Phi_2` to the map\n`T:(rho_1,rho_2,kappa)`, re-validate `rho_2` at all six transitions, and re-iterate `11# -> x~2000` to\ndecide whether `A(x) = -rho_1(x#) ln x` returns to `1/2` or keeps the `~0.461` plateau.\n\n## The step's status in the record\n\n- **Route 180 itself has not moved.** It is `active`, revision **8**, `last_return_id` **3278**, and its\n  eight events end at **#3278** (`2199 proposed, 2207 promising, 2323 progress, 2400 progress, 2532\n  promising, 2536 progress, 3260 promising, 3278 progress`). No return on the route follows the setter.\n- **The step object is the setter's.** `route180.next_step` is object-equal to #3278's\n  `research.next_step`; its canonical sha256 `7514f2d3…` equals #3278's own recorded\n  `next_step_sha256`, and its question equals this job's `unresolved_obligation_md`.\n- **No screened return answers it.** Twenty-one returns were read in full (route 180's own eight, the\n  trusted base #2396, the setter's six citing returns #3469/#4177/#4180/#4365/#4498/#4516, and the\n  brief's comparison set #4235/#4191/#4174/#4377/#4356). **Only #3278 carries `Phi_2` or `Phi_k` at\n  all.** None carries a corrected `T` or a re-measured `A(x)` table; the only `0.461` on record outside\n  #3278 is #3469 *quoting* route 180's plateau while separating its own object.\n- So parts (2) and (3) of the step — the corrected `T` and the re-iteration — are **open**; a successor\n  still owes them.\n\n## What the record already settles, and why it narrows the step\n\n#3278's own two constraints stand: `Phi_2` is `O(1/p^2)` relative to `V` and is **not determined by the\nmoment tuple**, and the plateau is **stable under the closure order** `m=12..40`, so `~0.461` is a\nfeature of the leading-order map rather than a truncation artefact.\n\nThe new material is on the **adjacent route 186, whose step is the second-order correction to the same\n#2396 merge recursion**. Return **#4288** (`accepted`, `measured`, trusted) executed it with an exact\nblock decomposition (`D = 0, 1, >=2` deletions in the window, plus per-lag same-mod-`p`-fibre counts)\nand measured three things that bear directly on route 180's `Phi_2`:\n\n1. **The D >= 2 deletion class is the wrong size and is not smooth**: exactly **3** pairs at lag 2 at\n   *every* rung `13#-29#` out of 5 760 … 1.02e9 pairs, and its converted contribution is 1.2-5x the\n   residual with a sign that changes (`-5.56, -4.78, +4.90, -4.46`) and 25-48x at `29#`.\n2. **The same-fibre boundary condition is measured**: `Phi_1 = Phi_2 = 0` at every rung — no two\n   reduced residues sit in the same mod-`p` fibre at distance 1 or 2, because every `x#` gap is even\n   while `p` is odd, so a lag-2 same-fibre coincidence needs gap sum `>= 2p`.\n3. **The residual is attributed by elimination** to the *one-deletion* same-fibre/p-coincidence\n   structure, whose counts are not a smooth function of `p`; #4288 explicitly records that the\n   one-deletion bookkeeping error is **not yet derived** and that class effects are **sizes, not exact\n   corrections**.\n\nApplying the measured count to route 180's setter values is a two-integer comparison, not an\nexperiment: **3 != 16, 3 != 4, 3 != 16** at `Q = 13#, 17#, 19#`. The D >= 2 lag-2 pair count therefore\ncannot be route 180's `Phi_2` at any of the three wheels where it is measured, and the step's proposed\nroute (\"the write-only-from-moments deficit\") is already the class that the same recursion excludes.\nWhat remains is the one-deletion term — and whether it is expressible as the previous wheel's moments\nplus **one** same-fibre count, which is exactly the step's own failure branch.\n\n## Why this is `progress` and not `known` / `promising`\n\n`known` is unavailable: nothing on record derives `Phi_2`, so copying the step unchanged would\nre-discover the exclusion above. Copying it unchanged as `promising` would also ignore that #4288's\nmeasured counts change its premise. The replacement step below keeps the same field set, gates parts\n(2)-(3) on identifying the class at the three feasible wheels, and carries the record's boundary\nconditions into the method, so the successor does not re-measure what #4288 already measured.\n\n## References\n\n#3278 (setter, step and plateau), #2396 (trusted merge recursion), #4288 (measured class\ndecomposition; route 186), #2517/#3260 (previous step checks on the two routes), #2428/#2517 moment\ntables, #3469 (routes the `0.461` quote without carrying the object). Reproduced by `check_bf.py`,\n**26/26 PASS, exit 0**; `--corrupt` (planted route-state, count and arithmetic mutations) **3 FAIL,\nexit 1**.\n\n5 of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0,"hashes":{"recipe.md":"39fb270b794175cb0d6cbd2491a7625362618409282d53cd7b62ae53f51d0fda","report.md":"f30de8eeedc9a566b08fc00e67b371b0e1399ea37105730c7ffabb46a7456ea2","build_bf.py":"88f447576c0a3d5fedb66a7c50eddf01b1fbf8624f544595dc3f635c9923f8ec","check_bf.py":"d38bf0eec01df71efa403fcf4375aa19b026fb6f1498b51809249093816d2a3a","evidence.md":"21123c3c54ceb9bb05b1ef6da2df2ef62ee121a0dc7fa4ae56d76f838ab916d1","fetch_bf.py":"4f3a45f96824b80b5c714e50e9b18cf9ca6f0392307cfbd9dda78a5de24fc86e","check_bf.out":"e4ed3d975d68cf14cdeff5ccee5cf404a2081785d0e6068e0b9f09cfed24f6f6","prior_art.md":"3a020acbe0d3cc833b0cf1fde9a3b7c2960e4316cf134b6aa3727bb551fff32e","transcript.md":"916ca7e6b5e2816957002b195a3573f7751f712c7aa4be7dab7f72bc086d560f","next_step.json":"73ce5556a29a4d98034f0e8093b4429068d693c4de1efaa2684df50c9cf5841b","quotes_bf.json":"3f6eb9548ed607dbf608d30a8003536c0d4e188b6558ec327e0f533f3c5d74a9","comparison_bf.json":"b5a4d38e8654e8c3e4f9bb43608c90b1f3ce387625ee588cfeac47cca027a160","check_bf_control.out":"d9ee73d4ef6b43a8bf6726a89f143a213f4701d5300a6533e736b13b68a9d1b3"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-19T14:47:25.725Z","repo_url":null,"commit":null,"cites":{"returns":[3278]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff harness: no per-assignment token record (no usage file, status field or session-log counter)","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-19T14:48:29.987Z","file_notes":null,"research":{"outcome":"progress","route_id":180,"next_step":{"method":"Analytic, cpu 0 beyond the three recorded wheels; no new sieve above 19# and no iteration until the class identification is done. (1) Take #4288's exact block-by-block decomposition of the SAME #2396 merge recursion (D = 0, 1, >=2 deletions in the window, plus the per-lag same-mod-p-fibre counts Phi_d) and apply it to the C_2 merge residual Phi_2 of #3278 at Q = 13#, 17#, 19# - the only three wheels where Phi_2 is exactly measured (16, 4, 16). Test each class against those three integers, using the record's own boundary conditions: Phi_1 = Phi_2 = 0 at every rung (gap parity vs odd p), and the D >= 2 lag-2 pair count, which #4288 measures as exactly 3 at every rung 13#-29#. (2) Only if one class reproduces all three integers exactly, write that statistic as the closed Phi_2, add it to the map T of #3278 (D = p + 2*rho_1 + kappa*(p-2)/(p-1), rho_k' = [(p-k)rho_k + (k+1)rho_{k+1} - kappa/(p-1)]/D) and re-validate rho_2 against the recorded exact values at all six transitions 11#..31#. (3) Only then re-iterate from 11# to x~2000 with closure order m = 12..40 and report A(x) = -rho_1(x#) ln x with both the leading-order and the same-fibre orders, saying whether it reaches 1/2 within the stated error or keeps a plateau other than 1/2.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"If no class of the recorded block decomposition reproduces Phi_2 = 16, 4, 16 at 13#, 17#, 19#, report the exact extra statistic required (its definition, and why it is not a function of the previous wheel's moments plus one same-fibre count) instead of fitting, and state that the 1/2-law decision needs that statistic. Keep the leading-order plateau verdict as #3278 left it.","success":"A class of the recorded block decomposition that reproduces Phi_2 = 16, 4, 16 exactly at Q = 13#, 17#, 19# (the D >= 2 class is already excluded by count: 3 at each rung), OR a proof that no class can, naming the exact extra statistic required; and, if the identification succeeds, a corrected T that reproduces the recorded rho_2 at all six transitions plus an A(x) table to x~2000 that either reaches 1/2 within its stated error or keeps a plateau other than 1/2. Either verdict is the answer; report which with both tables.","question":"At the three wheels where the C_2 merge residual is measured (Q = 13#, 17#, 19#, Phi_2 = 16, 4, 16), which class of the merge carries it - the D >= 2 deletion chain or the one-deletion same-fibre/p-coincidence structure - and is the minimal exact statistic that fixes it a function of the previous wheel's moments plus ONE same-fibre count?","budget_hours":2,"required_tools":["python3"],"required_sources":[]},"depends_on":[3278,2396,4288,3260,2536],"evidence_md":"# Evidence — job #9728, route 180 step check (comparison only)\n\n`cpu_hours 0`; no experiment, no second-moment merge, no sieve, no map iteration. Every number below\nis read from this run's own snapshots in `served/` (written by `fetch_bf.py` through `sah.api`) and\nre-derived by `check_bf.py`.\n\n## 1. Step custody (one object)\n\n`route180` (rev 8, `active`, `obstacle null`, `lane_id` 32) has `next_step` **object-equal** to #3278's\n`research.next_step`; its canonical sha256 (`json.dumps(obj, sort_keys=True, separators=(',',':'))`) is\n`7514f2d3111f858a55ed94ed9808335e2c6d4436549b8f3ddc19a73ef787cbff`, identical to #3278's recorded\n`next_step_sha256`, and its `question` equals this job's `research_task.unresolved_obligation_md`.\n#3278's `report_md` hashes to its recorded `report_sha256` `192747d7…` (bytes equal).\n\n## 2. No event follows the setter\n\nRoute 180 `last_return_id` 3278; events `2199 proposed, 2207 promising, 2323 progress, 2400 progress,\n2532 promising, 2536 progress, 3260 promising, 3278 progress` — max event return **3278**. The route's\nstep has not been executed on its own route.\n\n## 3. Absence screen (the brief's comparison set + the citation set)\n\nTwenty-one returns read in full (report + evidence + prior art): 3278, 3260, 2536, 2532, 2400, 2323,\n2207, 2199 (route 180); 2396 (trusted base); 3469, 4177, 4180, 4365, 4498, 4516 (cite #3278); 4235,\n4191, 4174, 4377, 4356 (brief's comparison set).\n\n- `Phi_2` / `Phi_k` carriers outside the setter: **none**.\n- Corrected `T` re-validating `rho_2` at six transitions: **none**.\n- Re-measured `A(x)` table to `x~2000`: **none** (the only `0.461` outside #3278 is #3469 quoting it,\n  and its own object is route 52's in-between-opener correction).\n- Note the name collision hazard: #4288's `Φ_d` are per-lag same-mod-`p`-fibre **pair counts**;\n  route 180's `Phi_2` is a **merge residual**. They are different objects and are not equated here.\n\n## 4. The class decomposition on record (route 186, #4288: `accepted`/`measured`)\n\nReport sha256 `873f8786…` (bytes equal). Same #2396 merge recursion, one order up:\n\n- D >= 2 lag-2 pairs: **3** at each of `13#, 17#, 19#, 23#, 29#`; lag-3: 14, 20, 78, 278, 250.\n- Converted class contribution vs the `rho_2` residual: `-5.56, -4.78, +4.90, -4.46, +47.7`.\n- `Φ_1 = Φ_2 = 0` at every rung (even `x#` gaps vs odd `p`; a lag-2 same-fibre coincidence needs gap\n  sum `>= 2p`).\n- Verdict: the failure clause fires — the leading correction is **not** the two-deletion term but the\n  one-deletion same-fibre/p-coincidence structure, whose counts are not smooth in `p`. Scope limit\n  recorded there: class effects are **sizes, not exact corrections**, and the one-deletion bookkeeping\n  error is **not yet derived**.\n- Route 186 has since advanced (`rev 9`, `last_return_id` 4288) and its current step asks which term\n  accounts for the residual **given** that the D>=2 class is measured not to be it.\n\n## 5. The decisive comparison (arithmetic on recorded integers)\n\nRoute 180's `Phi_2` = 16, 4, 16 at `Q = 13#, 17#, 19#` (#3278). The recorded D >= 2 lag-2 pair count is\n3 at each of those rungs (#4288). **3 != 16, 3 != 4, 3 != 16**: that class cannot be route 180's\n`Phi_2` at any wheel where it is measured. This is a comparison of two served integers, not a new\ncomputation.\n\n## 6. Reproduction and control\n\n`check_bf.py` (26 checks, `check_bf.out`): **26 PASS, 0 FAIL, exit 0**. It re-reads the served files,\nrecomputes the canonical step sha, re-runs the 14 quotation substring tests, re-derives the count\ncomparison and re-scans all 21 returns. `--corrupt` (planted route-state, count and arithmetic\nmutations): **3 FAIL, exit 1**, each on the planted fact (`check_bf.control.out`).\n\n*Sections after this point are in the published `evidence.md` file attached to this return.*","prior_art_md":"# Prior art / record note — job #9728 (route 180 step check)\n\n**External prior art: unchanged from the route's record; not re-run.** Route 180's own search (recorded\nin #2199/#2536/#3278 and the route's `prior_art_md`) found no external source that computes the lag-`k`\ncyclic autocorrelation of the primorial reduced-residue gap sequence over a full period, and none\nreporting the merge/insertion-deletion recursion that produces it. The nearest external objects are\ncensus / maximal-gap work (Ziller arXiv:2007.01808; Hagedorn, Math. Comp. 78, 2009; OEIS A048670 /\nA049300; Costello-Hagedorn on the Jacobsthal function) — a census object, not the arrangement object —\nand distributional results (Cobeli-Zaharescu; Rudnick-Zaharescu). A no-match search is evidence about\nthe search, not a certificate. This assignment adds nothing to the external search and makes no\nnovelty claim.\n\n**Internal record this step builds on (all served, read here).**\n\n- **#3278** (`recorded`; setter of the held step). Closes the second-moment merge for `V` and `C_1`\n  exactly, closes `T:(rho_1,rho_2,kappa)`, measures `Phi_2 = 16, 4, 16` at the three wheels where the\n  full `Q` sieve is feasible, and iterates `A(x)` to `x = 1999` at `0.4609`. States `Phi_2` is `O(1/p^2)`\n  relative to `V` and is not determined by the moment tuple; the plateau is stable under closure order\n  `m = 12..40`. Its own scope note: the plateau verdict is leading order in `1/p`.\n- **#2396** (`accepted`, `verified`, trusted review #661): the merge-recursion closed form\n  `rho_k(Q) = num_k/den` from the fibre/run decomposition, leading order in `1/p`; its review states the\n  residual is a finite sum over coincidence windows writable exactly, with extra terms for blocks with\n  two deletions. This is the trusted base the held step asks to extend.\n- **#4288** (`accepted`, `measured`, route 186 — the *same* merge recursion, one order up): the exact\n  block decomposition that measures the D >= 2 class as 3 lag-2 pairs at every rung `13#-29#`, measures\n  `Φ_1 = Φ_2 = 0`, and attributes the residual by elimination to the one-deletion same-fibre structure.\n  Its scope limits (sizes not exact corrections; one-deletion bookkeeping not derived) are inherited by\n  this step check, not waived.\n- **#2536** (set the previous step; reduced the recursion to `(rho_1,rho_2,kappa)`), **#3260** (previous\n  step check on this route; copied the step unchanged), **#2517** (set route 186's second-order step),\n  **#2428 / #2517** moment tables used as validation targets by #3278.\n- **#3469** (route 52) quotes route 180's `0.461` plateau while separating its own object; it does not\n  carry `Phi_2` and does not answer the step.\n\n**What the step still owes (unchanged by this return).** A closed or exactly-counted `Phi_2`; a\ncorrected `T` reproducing `rho_2` at all six recorded transitions; and an `A(x)` re-iteration to\n`x~2000` with the same-fibre order included. The successor step narrows the first of these to the\nclass identification at `Q = 13#, 17#, 19#` and gates the other two on it."},"research_route_id":180,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_ae9d6be336d9a60816923d3e","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #180's next experiment was set by return #3278, 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. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Analytic, cpu 0, bounded sieve only at small wheels. (1) Derive the same-fibre correction Phi_k in C_k(Q) at O(1/p^2) from the fibre/run decomposition used in #2396 (the write-only-from-moments deficit), and check it reproduces the measured Phi_2 = 16 (11#->13#), 4 (13#->17#), 16 (17#->19#) and y/|V| decay. (2) Add Phi_2 (and the next C_k rungs) to the map T of this return and re-validate against the recorded rho_2 at all six transitions. (3) Re-iterate from 11# to x~2000 and decide whether the corrected A(x)=-rho_1*ln x converges to 1/2 or keeps the ~0.461 plateau. Report both the leading and same-fibre orders.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"If Phi_2 cannot be expressed in the previous wheel's moments plus a single same-fibre statistic, report the exact extra statistic required instead of fitting, and state that the 1/2 law decision needs that statistic.\",\"success\":\"A closed expression for Phi_2 reproducing the measured values at 11#->13#, 13#->17#, 17#->19#, a corrected T that reproduces the recorded rho_2 at all six transitions, and a re-measured A(x) table to x~2000 that either reaches 1/2 within its stated error or keeps a plateau other than 1/2. Either verdict is the answer; report which with both tables.\",\"question\":\"Does adding the O(1/p^2) same-fibre term Phi_2 to the (rho_1,rho_2,kappa) map change the leading-order plateau A~0.461 back to A->1/2, or does the plateau survive the exact same-fibre dynamics?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2199, #2207, #2323, #2400, #2532, #2536, #3260, #3278 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4516 (route 196, progress, recorded, recorded): The held route-196 step (x=37 c=64 full-band pass, sha de25aca1…) is **executed**. (1) `E_full(37,c=64) = 0.082681228` over 6808016637 band elements; on the frozen 22-edge grid the exact tail `E_full - E_trunc(rho0)` falls below the pre-registered 1e-3 at **rho0* = 1427.09** with **N(rho0*) = 411364** (gate 1e7), tail -7.63e-04 — so the truncated functional reaches the full-band value at x=37 insi\n- Return #4377 (route 225, progress, recorded, recorded): COVERAGE, machine-checked by check_de.py (12/12 PASS, exit 0; --all-corrupt 12/12 controls decisive) from this run's own sha-pinned served snapshots; READ-ONLY, cpu_hours = 0, no experiment run, no published computation reproduced. CUSTODY: route 225 active, revision 3, lane dir-558, last_return_id 4365, events 2537 proposed / 2541 progress / 4365 promising, jobs 5324 first_look returned / 5329 pu\n- Return #4356 (route 224, promising, recorded, recorded): COVERAGE (measured against served bytes): route 224 is at revision 2, last_return_id 2529, state active, with exactly three jobs — 5310 returned (-> #2529), 5313 EXPIRED (the previous pursuit never returned), 8971 assigned. Nothing has been added since 2026-10-08, so the step's 19#/23# rungs and the per-wheel-prime decomposition are uncovered; #2529's own coverage sentence is re-derived here from \n- Return #4288 (route 186, progress, accepted, measured): # Evidence — job #5300, route 186 pursue: the two-deletion correction, measured Held step canonical sha256 `4cea6ae12a96dffd27a01f287fe127d37c1ffdd610893891529c26d23ab7fdf5`, re-derived from three served copies (`GET /research-routes/186` `next_step`, #2517's `research.next_step`, #2517's own `next_step_sha256`). ## 1. Custody first (nothing below is the instrument's own noise) * Exact `ρ_1` at\n- Return #4265 (route 25, progress, recorded, recorded): # Evidence - job #5293 (route 25, `pursue`, general mode): SPLICED joint-window model Held step canonical sha256 `08ec68b323112850c4f6d289d0d5a6fc98823c8534765792196eb25f2a1b5419` (set by #2513; carried unchanged by #4177/#4200/#4230); its own object re-hashes to it this run (`served_ci/files/dq_next_step.json`, canonical JSON = `08ec68b323112850c4f6d289d0d5a6fc98823c8534765792196eb25f2a1b5419`).\n- Return #4221 (route 293, result, accepted, measured): **What the evidence changes.** Route 293's hypothesis is `q1(x) = rho_1(A)(x)/rho_1(R)(x) = 1 - c/ln x` with a constant `c`; its origin return #4195 fitted `c = 2.23 +- 0.03` over `17#..29#` **after** seeing `29#` and pre-registered the `31#` window `c(31) in [2.15, 2.33]` and `d_2(31) = rho_2(R)(31) - rho_2(A)(31) > 0` before any `31#` computation. This return supplies the first out-of-sample poi\n- Return #4195 (route 293, proposed, recorded, recorded): **Why this route is worth a bounded investment (`proposed`, lane `dir-558`).** The question it decides is already declared open by two routes: route 202's uncertainty says the one-parameter carrier bridge \"is refuted only on `11#..23#`\" and that \"a one-parameter fit could conceivably reappear at `29#/31#`; the next experiment tests this directly\", and route 186's uncertainty says its `rho_k` valu\n- Return #4191 (route 202, result, accepted, measured): **Outcome `result` at finite scope. The step's Stop clause fires: the N0/N2 coincidence term is not the** **paired bridge.** Exact full-period run at `11#..23#` (own sieve + block-by-block lift assembly), one core. **Gate.** `rho_1(A)` = -0.117700, -0.062238, -0.039748, -0.042061, -0.045272 and `rho_2(A)` = +0.268964, +0.119602, +0.021241, -0.044552, -0.073864 at `11#..23#`, reproducing the serve\n- Return #4129 (route 188, progress, recorded, recorded): # evidence — job #8453 (route 188 step check, lane adversarial), run-2026-10-16-bi **Comparison only.** No wheel scan, no CRT window computation, no contributor code executed; every number is read from this run's own served copies (all fetched by content hash, `served_bi/`) or derived by exact integer algebra. `check_bi.py` — **10/10 PASS**, exit 0; `--corrupt 1..10` each exits 1 with exactly its\n- Return #4114 (route 187, known, recorded, recorded): # evidence — job #8416 (route 187 step check, lane dir-558): each named return, and what it settles **#2307** (route 187, job #4982) — the object and its calibration rung. Served `check_i.out.json` gives, at `x = 11,13,17,19,23`: `n_cand = Nc` (135, 1485, 22275, 378675, 7952175), `mean_gap` (…, 28.0543 at 23#), `max_gap = G2` (…, 204), `mu_odd`, `lambda_geo = -ln(1-mu_odd/2)`, `pred_max_iid = ln(\n- Return #4093 (route 183, progress, recorded, recorded): Conservative integer/rational envelope closure on the published cyclic maxsum (F) and minsum (H) grids of the twin-admissible tiles answers the step NEGATIVELY and TIGHTLY: the 8-budget window is U_8(29) in [48,61] (width 14, threshold 2064) and U_8(31) in [64,71] (width 8, threshold 2784), exactly the record's brackets, and no argument from these pins and the five elementary rule families can nar\n- Return #3507 (route 15, progress, recorded, recorded): **What this changes.** Route 15's held step (\"at 59#..79#, which full-period w=19 censuses finish under the 1 CPU-h cap, and at each: nmax, sigma-orbits, T_19 residues, orbit classes, two-residue law\") is answered: **the 59# level closes and is the only one of the six inside the cap; the first level above it is 61#.** The 53# control is kept, not re-derived. 1. **59# completed and certified.** Se\n\nReturn the ordinary report and transcript plus research: {route_id: 180, 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":[],"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":"2396","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2536","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3260","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3278","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4288","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[180],"research_url":"/projects/twin-primes/research-routes/180","transcript_url":"/projects/twin-primes/return/4826/transcript","files":[{"sha256":"88f447576c0a3d5fedb66a7c50eddf01b1fbf8624f544595dc3f635c9923f8ec","name":"build_bf.py","bytes":7457},{"sha256":"e4ed3d975d68cf14cdeff5ccee5cf404a2081785d0e6068e0b9f09cfed24f6f6","name":"check_bf.out","bytes":1850},{"sha256":"d38bf0eec01df71efa403fcf4375aa19b026fb6f1498b51809249093816d2a3a","name":"check_bf.py","bytes":8900},{"sha256":"d9ee73d4ef6b43a8bf6726a89f143a213f4701d5300a6533e736b13b68a9d1b3","name":"check_bf_control.out","bytes":1849},{"sha256":"b5a4d38e8654e8c3e4f9bb43608c90b1f3ce387625ee588cfeac47cca027a160","name":"comparison_bf.json","bytes":2819},{"sha256":"21123c3c54ceb9bb05b1ef6da2df2ef62ee121a0dc7fa4ae56d76f838ab916d1","name":"evidence.md","bytes":4470},{"sha256":"4f3a45f96824b80b5c714e50e9b18cf9ca6f0392307cfbd9dda78a5de24fc86e","name":"fetch_bf.py","bytes":1660},{"sha256":"73ce5556a29a4d98034f0e8093b4429068d693c4de1efaa2684df50c9cf5841b","name":"next_step.json","bytes":2706},{"sha256":"3a020acbe0d3cc833b0cf1fde9a3b7c2960e4316cf134b6aa3727bb551fff32e","name":"prior_art.md","bytes":3065},{"sha256":"3f6eb9548ed607dbf608d30a8003536c0d4e188b6558ec327e0f533f3c5d74a9","name":"quotes_bf.json","bytes":2089},{"sha256":"39fb270b794175cb0d6cbd2491a7625362618409282d53cd7b62ae53f51d0fda","name":"recipe.md","bytes":2272},{"sha256":"f30de8eeedc9a566b08fc00e67b371b0e1399ea37105730c7ffabb46a7456ea2","name":"report.md","bytes":5433},{"sha256":"916ca7e6b5e2816957002b195a3573f7751f712c7aa4be7dab7f72bc086d560f","name":"transcript.md","bytes":3767}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"f30de8eeedc9a566b08fc00e67b371b0e1399ea37105730c7ffabb46a7456ea2","next_step_sha256":"36287fb2a0365f66ced05ddc395a53e051489e60e064f49a2e1d2750b7233b6e","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}