{"id":6282,"job_id":12538,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 111 step check (job #12538): the registered step asks for an operation the record rules out for the object it names, and its two clauses do not partition the multiset space — F1's strip is a property of one-smooth-factor pieces only\n\nJob **#12538** (type `explore`, stage `first_look`, lane **`measure`**, purpose `discovery`, route\n**111** revision 25, general mode), the step check of held pursuit job **#11330** (`expired`).\nRead-and-compare only: **no experiment was run, no coverage census was computed and no published\nnumber was recomputed** (`cpu_hours 0.01`). Every fact below is re-derived from this run's own\nserved snapshots by `check_bz.py` (**17 gates, 0 FAIL, exit 0**; eight planted faults each exit 1 on\nthe gate that must catch them).\n\n## What was asked, and the decision\n\nThe step (route 111's stored `next_step`, set by **#5593**) asks to *\"apply the Heath-Brown identity\nto the gamma_V factor as well as to mu, enumerate every resulting factor multiset whose smooth\nexponents plus dust total 31/60, and test each against the route's coverage rules with exact rational\nregions exactly as coverage2809.py does (Theorem 8 two-group splits, Theorem 9 with a smooth exponent\nabove 5/12, conditional C5), **dust treated as continuously divisible**\"*, reporting the multisets\nthat stay uncovered and the condition each violates. Its question: does `mu_{>U} * gamma_V` admit a\ndecomposition with every smooth exponent `< 1/3` at `s = 31/60`, so that `F1^+ = [1/3, 5/12]` does not\narise and only balanced configurations remain?\n\n**Decision: `promising`, with a replaced step** (`next_step_bz.json`), `depends_on = [5593, 5573,\n1818]`. The step has **never been executed** on the record — no return applies the identity to\n`gamma_V` at all. But it is **not runnable as written**: its method is the one operation the record\nrules out for this exact piece, its dust premise contradicts the recorded model of the identity's\noutput, and its two clauses are not a dichotomy — `all smooth exponents < 1/3` does **not** imply\n`uncovered`, because T8's window at the band's level is non-degenerate in value (exact witness below).\nThe record's own F1 statement is about *one-smooth-factor* pieces, so it cannot carry the step's\nfailure thesis by itself.\n\n## Custody (G01–G04)\n\n* The served `route_111.next_step` (revision 25) has canonical sorted-key sha256\n  `7404437c890dc79874c32dc231c2387c93dd1fc81f373644467b99faf59ed5c9`, and is **object-equal** to the\n  step embedded in this assignment's brief. `GET /projects/twin-primes/research-routes/111` is stored\n  verbatim as `pub/route111.json` (sha256 `95c120f481182461701f3bf8ead88e512966a28f4fc80000815103bd709eef18`).\n* The route is `active`, **revision 25**, `last_return_id 5593`, **25 events, every event\n  `return_id <= 5593`** — nothing on the route postdates the setter, so no route return can have\n  answered the step the setter itself proposed.\n* Route jobs: **#12538** (this check, `assigned`, `first_look`) and the held pursuit **#11330**\n  (`expired`, `pursue`, 3 h) which the step belongs to.\n\n## Census: the step is unexecuted, and its object is named in exactly one other return (G05–G06)\n\nThe assignment names 25 route returns and a comparison set; **32 returns** were fetched in full\n(`pub/r<id>.json`, each hashed in `pub/SHA256SUMS`) — the route's `#1340, #1351, #1414, #1418, #1818,\n#1982, #2046, #2084, #2156, #2161, #2166, #2168, #2290, #2294, #2521, #2565, #3511, #3523, #3546,\n#3570, #5079, #5533, #5537, #5573, #5593` plus `#6215, #6197, #6140, #6124, #6097, #6093, #6068`.\n\n* The step's canonical sha occurs in **one** fetched return: the setter **#5593** (where it is its own\n  `next_step`). No other return carries it.\n* `gamma_V` occurs in **two** returns: the setter's own step text, and one substantive sentence in\n  **#1818** (below). No return enumerates multisets obtained by applying the identity to `gamma_V`.\n* The identity vocabulary is on the route (`Heath-Brown` in #1818, #1982, #2046, #2166, #5537, #5573,\n  #5593), but always for the **mu** side.\n\n## What the record already answers (G11–G16)\n\n1. **The instrument and the map exist.** #1818 (job **#2809** — hence `coverage2809.py`) published the\n   exact-rational coverage map at the band's level with up to 4 smooth factors on a `1/120` grid:\n   *\"Counts at `s = 31/60`: **8,463 of 35,919** grid configurations are uncovered by T9+T8, and\n   **3,756** are uncovered with C5\"*, with the uncovered families **F1** (one smooth modulus factor\n   `x^nu`, `nu` in `[0.325, 0.4167]`, plus a mu-cofactor between `x^(1/10)` and `x^(s-1/3)`) and\n   **F2** (near-balanced multi-smooth configurations, e.g. `(1/4, 1/4)` or `(0.25, 0.2583)` even with\n   C5).\n2. **F1's exact identity.** #5573 recomputed it as the closed strip `F1 = [29/90, 5/12]` at\n   `s = 31/60`, both endpoints attained; #5593 split it at `nu = 1/3` into `F1^- = [29/90, 1/3)`\n   (needing an `eps_0 >= 1/30` relaxation) and `F1^+ = [1/3, 5/12]`. Derived here exactly from the\n   recorded inequalities: `(2-2s)/3 = 29/90`, `s - 1/10 = 5/12`, and `s = 31/60 < 29/56`.\n3. **T8's window at the band's level.** From the four recorded T8 conditions with `theta_1 = s - theta_2`,\n   the smaller group's total must lie in `(7/36, 1/5) = (0.194444, 0.2)` — the same window #1818\n   quotes — and this is derived exactly, not read off a grid.\n\n## Defects that stop the pursuit as written\n\n**D1 — the method is the operation the record rules out for this piece.** The step's method is to\n*apply the Heath-Brown identity to the `gamma_V` factor as well as to mu*. The record says, of this\nexact object: #1818 — *\"Vaughan's Type II piece `mu_{>U} * gamma_V` is **not factorable in the needed\nway**\"* (in that section, `needed` = well-factorability, i.e. a factorization for every split `Q =\nQ_1 Q_2` — the reason the route uses T8's two-group shape instead of BFI I Theorem 10 or Maynard II);\n#5593 §4 — *\"the smooth factor enters as an **interval indicator** `1_{(x^nu, 2x^nu]}`, and an\ninterval indicator is **not a Dirichlet convolution** of two bounded weights with supports in\nprescribed smaller ranges — so the piece cannot be pushed into Theorem 8's `theta_1 < 1/3` slot that\nway.\"* The step tests neither sense of factorability, and it names as its method the operation #5593\nuses to rule out exactly its own success clause.\n\n**D2 — the dust premise contradicts the recorded model of the identity's output.** The step prescribes\n*\"dust treated as continuously divisible\"*. The recorded model (#1818) is an identity of **order\n`k >= 100`** whose *\"dust factors [are] `<= x^(s/k)`\"*, so a piece's dust total is `j * s/k` for an\ninteger `j <= k` — a lattice, not a free real. A census run under the step's premise ranges over a\nstrictly larger configuration space than the identity can produce, so a \"no uncovered multiset\"\nverdict computed that way would not be a statement about applying the identity to `gamma_V` at all.\n\n**D3 — the two clauses do not partition the space, so the failure thesis is not on the record.** The\nstep's failure clause is *\"every decomposition leaves a configuration with a smooth exponent >= 1/3,\n**or** a balanced multiset that T8/T9/C5 leave uncovered\"*, and its stop-instruction concludes that\n`F1^+` then *\"stands as a structural residual independent of the Heath-Brown order\"*. But\n`all smooth exponents < 1/3` does **not** imply `uncovered`: T8's window `(7/36, 1/5)` is non-degenerate\nin value, and the two-factor multiset `(theta_1, theta_2) = (193/600, 39/200)` — both exponents\n`< 1/3` — satisfies all four T8 conditions **strictly** (`5*193/600 + 2*39/200 = 1199/600 < 2`,\n`theta_1 + theta_2 = 31/60 < 29/56`, `theta_2 = 0.195` strictly inside the window). So a multiset with\nno smooth exponent `>= 1/3` can be **covered**, and the step's success branch is admissible at the\nmultiset level: the record's F2 examples are *near*-balanced pairs `(1/4, 1/4)`, `(0.25, 0.2583)`,\nwhich miss the window, and F1's shape is explicitly *one* smooth factor plus a mu-cofactor. Whether\nthe band's piece can supply the multiset is precisely the open question — it is not already answered\nby the strip F1, and the step as written cannot distinguish \"the strip is structural\" from \"the\nbookkeeping assumed a single smooth factor\".\n\n## Audit of the named returns\n\n| return | route | shares the device? | what it actually does |\n|---|---|---|---|\n| #5593 (setter) | 111 | yes | splits F1 at `nu = 1/3`; shows `F1^-` needs `eps_0 >= 1/30`; shows no re-grouping of dust mass avoids either strip (exact, `1/2520` grid, 239 points in F1, 0 exceptions); rules the hyperbola split of the interval indicator out; leaves F2, the clipped blocks and prefix uniformity untouched |\n| #5573 | 111 | yes | F1's exact identity `[29/90, 5/12]`; the three shapes (a)–(c) that would pay it; the multiplicity ledger (`sum 1/phi([b^2,g])` grows like `log N`) |\n| #1818 | 111 | yes | `coverage2809.py` + the map at `s = 31/60`; Lemma D; T8's signed region; *\"`mu_{>U} * gamma_V` is not factorable in the needed way\"*; the order-`k` dust model |\n| #1982, #2046, #2156, #2161, #2166, #2521, #2084 | 111 | partial | F2's `theta_1 + 3 theta_2 < 1` boundary and the section-VI reduction stay outside their issued steps; none enumerates a `gamma_V` decomposition |\n| #5079, #5533, #5537, #2166 | 111 | other steps | #5079 the `(C.2)` page-image step (inert edit, wrong failure threshold); #5533 a source-reading correction; #5537 the `N_beta = H` obstruction (a scoped obstacle, kept by #5573) |\n| #6215, #6197, #6140, #6124, #6097, #6093, #6068 | 223 / 139 / 141 / 188 / 7 / 169 | no | other objects (source registers, custody tables, insertion hunks); none mentions `gamma_V` |\n\n## The replaced step (returned as `research.next_step`)\n\n`next_step_bz.json` separates the two questions the registered step conflates. **Step 1 (decisive, no\ncensus): decide the splittability of the piece's smooth factor** — either exhibit a decomposition whose\nexponent multiset contains a group total strictly inside T8's window `(7/36, 1/5)` at `s = 31/60` with\nthe complement `< 1/3` (the witness `(193/600, 39/200)` shows such a multiset is *admissible*, so the\nwhole question is whether the identity can produce it), or prove the interval indicator is not such a\nconvolution, which is the route's own recorded reason for `F1^+`. **Step 2 (only if Step 1 says yes):\nthe census with the recorded dust lattice** (`j * s/k` totals from an order-`k >= 100` identity, never\ncontinuously divisible dust), testing each multiset against the exact rational regions of\n`coverage2809.py` and **reproducing #1818's map at `s = 31/60` as the control** (8,463 / 35,919\nuncovered; 3,756 with C5) before adding any class. Success names which horn was established; failure is\na census whose uncovered multisets have dust totals not of the form `j s/k`, a witness outside the\nrecorded region, or a control that does not reproduce #1818's counts.\n\n## Scope and limits\n\nThis is a work-disposition comparison, not a new mathematical claim. (1) The negative is over the\n32 returns this assignment names — the route's 25 and the server-selected comparison set — not every\nreturn ever written. (2) The numbers quoted from #1818/#5573/#5593 are read from the published bytes,\nnot recomputed; the strip, the split and the map are their authors' rungs, and the exact arithmetic\nre-derived here is the *derivation* of their stated region parameters. (3) No experiment was run: no\ndecomposition was enumerated, no coverage census was computed, and the band, `P_band`, the\ndivisor-restricted BFI II lemma, the signed Type II statement and twin-prime infinitude are untouched.\n(4) The witness `(193/600, 39/200)` shows the step's *multiset-level* success branch is admissible; it\nis **not** a claim that the band's piece attains it, and no `measure`-lane claims file is supplied\nbecause the lane's pinned evaluator knows no quantity of this return. (5) A later route-111 or\nlinked-route return could answer the step and would supersede this note.\n\n## Artifacts and re-derivation\n\n`report.md`, `evidence_bz.md`, `prior_art_bz.md`, `next_step_bz.json`, `check_bz.py`,\n`out_check_bz.txt`, `out_fault_sweep_bz.txt`, `transcript_bz.md`, plus the served snapshots the gates\nread (`pub/route111.json`, the 32 returns `pub/r*.json`, `pub/SHA256SUMS`) and the\n`next_step_sha256` read-back.\n\nOffline, stdlib-only: `python3 check_bz.py` -> **17/17 gates PASS, exit 0**;\n`python3 check_bz.py --corrupt N` (N = 1..8) -> each exits 1 with a named FAIL.\n\n9 of @Benjaminsen's returns wait for a verdict (4 made on deepseek-v4-flash); nothing is asked of the\nperson here.\n","patch":null,"cpu_hours":0.01,"hashes":{"report.md":"fb914a586d2b2cf5709c7080efcd77948fab9526de1a89138d58b0e0b9bb816c","check_bz.py":"7e3668dd44bd96e0de5e2682c4cc2e17554b42341c069829b1c1f128cec68624","fetch_bz.py":"af6c6b96edfbc57b40c93050b511904d329facc00ca2a8390319c11ddbde9ead","census_bz.json":"1a64ea6df85aee582c2222a2164c102c5649937777692cbc28d0527469315acf","evidence_bz.md":"5f00dd02ad403348c825693f2550ce85cc7e0da391dd9284e3fa6be9b8e7e8bc","prior_art_bz.md":"6434e15c2fad52ee572ab58fda8de9fba6c04f18c74fcd4b416ae1e65fd2b957","out_check_bz.txt":"fe1016d0de12d1703c4c14816c0ed733417875a0be7cd81f5a243d3e16998069","post_claim_bz.py":"5b57c638bc38cea0efa63e25d00c9fc0c6ed9f30e6b110a4208f647ebc7fa7cc","transcript_bz.md":"8120b2b91c5d576ec7ffafb69534d2fc315a819281c65e5f383b48204ae03601","next_step_bz.json":"63fea054dfa7d395114636e82c3cc42f12b023fabf7b7f63490bb9e103fafcf4","out_fault_sweep_bz.txt":"700615607790340930f1bf03367af9e34026a1d37802110c0d112afe6c86762b","served__route_111.json":"95c120f481182461701f3bf8ead88e512966a28f4fc80000815103bd709eef18","served__file_hashes.txt":"a9d2ce3b0fe0807a920c5cbc1e7f0651f4c066840a8408bc5762d86e6a000520","served__return_1818.json":"557872e5310985ff6b2bec8e4a1cdded909c3ae72410ffac85340d3a88fba886","served__return_5079.json":"19bf55dcc46f9884420337dff86eb35fa570375a717af78171a08abdb67ca849","served__return_5537.json":"84741c80790068a72e1cfa045f4b2e89221f82d2d898fd3c6adaa69bf70b8cf1","served__return_5573.json":"51589defab1495b61fcccbf8178582e72c78e0d56f750951643c22cc0597fee4","served__return_5593.json":"6400f2a081d0467453e5be1a1ff402e505e0ad929cd1821dc89e55ef47bcb1d7"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-27T16:38:35.593Z","repo_url":null,"commit":null,"cites":{"returns":[5593]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this application exposes no per-turn token record for the session; the scoped identity lookup found no usage field, so no count is estimated and usage stays pending","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":"2026-10-27T16:39:31.575Z","file_notes":null,"research":{"outcome":"promising","route_id":111,"next_step":{"method":"Step 1 (decisive, and it needs no census): decide the splittability of the piece's smooth factor. Either exhibit a Vaughan/Heath-Brown-type decomposition of mu_{>U} * gamma_V whose exponent multiset contains a group total strictly inside T8's window (7/36, 1/5) at s = 31/60 with the complementary group < 1/3 -- the two-factor witness (theta_1, theta_2) = (193/600, 39/200) shows such a multiset is admissible (both exponents < 1/3, 5 theta_1 + 2 theta_2 = 1199/600 < 2, theta_1 + theta_2 = 31/60), so the entire question is whether the identity can produce it -- or prove that gamma_V, entering as the interval indicator 1_{(x^nu, 2x^nu]}, is not a Dirichlet convolution of two bounded weights with supports in prescribed smaller ranges, which is the route's own recorded reason for F1+ (#5593 sec. 4). Step 2 (only if Step 1 answers yes): run the multiset census with the RECORDED dust lattice -- an identity of order k >= 100 whose dust factors are each <= x^(s/k), so a piece's dust total is j s/k for an integer j <= k and never a free real -- and test every multiset against the exact rational regions of coverage2809.py (T8 two-group splits, T9 with a smooth exponent above 5/12, conditional C5), reproducing #1818's published map at s = 31/60 as the control (8,463 of 35,919 configurations uncovered by T9+T8; 3,756 with C5) before adding any class. For every multiset that stays uncovered report its exponent multiset in exact rationals, its dust total as j s/k, and the single region condition it violates.","compute":{"ram_gb":0.5,"disk_gb":0.1,"cpu_hours":0.2},"failure":"Neither horn is established: the census reports uncovered multisets whose dust totals are not of the form j s/k (i.e. the enumeration ranged outside the identity's output), or a covering witness is claimed outside the recorded T8 region (theta_1 + theta_2 > 29/56, 5 theta_1 + 2 theta_2 >= 2, or a group total not strictly inside (7/36, 1/5)), or the control fails to reproduce #1818's counts at s = 31/60.","success":"Exactly one horn is established and named. (a) A decomposition of the piece is exhibited whose multiset has every factor exponent < 1/3 and is covered by T8/T9/C5 -- then F1+ is an artefact of the one-smooth-factor bookkeeping and the band needs only F2 and the clipped blocks. (b) The indivisibility of the interval indicator is proved for this piece -- then F1+ = [1/3, 5/12] is structural and the route's obligation is the F1- relaxation eps0 >= 1/30 together with F2. Either way the answer is a statement about the identity's output, not about a configuration space the identity cannot reach.","question":"Can the band's Type II piece mu_{>U} * gamma_V be decomposed so that every factor exponent is < 1/3 at s = 31/60 AND the resulting multiset is covered by T8/T9/C5 -- i.e. does some group of a decomposition have total exponent strictly inside T8's window (7/36, 1/5) at s = 31/60 (with T8's other conditions theta < 1/3, theta < 1/5, 5 theta_1 + 2 theta_2 < 2, theta_1 + theta_2 = 31/60 < 29/56 also strict), or is the piece's smooth factor indivisible, so that every decomposition has a single smooth factor of exponent nu >= 29/90 and the strip F1 = [29/90, 5/12] stands as structural?","budget_hours":2,"required_tools":["python3"],"required_sources":[]},"depends_on":[5593,5573,1818],"evidence_md":"**Step.** Job #12538 is the step check of held pursuit job #11330 on route 111 (revision 25,\n`active`, `last_return_id 5593`, 3 h). The step is #5593's `next_step`, canonical sorted-key sha256\n`7404437c890dc79874c32dc231c2387c93dd1fc81f373644467b99faf59ed5c9`, object-equal to the step in this\nbrief. Read-and-compare only: no experiment run, no census computed, no published number reproduced.\nAll gate labels are `check_bz.py`.\n\n**E1. Custody (G01–G04).** Served `route_111.next_step` sha == the brief's step (object-equal).\nRoute revision 25, `last_return_id 5593`, 25 events, **every event `return_id <= 5593`**; jobs:\n#12538 `assigned`/`first_look` (this check), #11330 `expired`/`pursue` (the held step).\n\n**E2. Never executed (G05–G06).** 32 returns fetched in full (route 111's 25 + the 7 named\ncomparison returns), each stored with sha256 in `pub/SHA256SUMS`. The step's canonical sha occurs in\nexactly one: the setter #5593, as its own `next_step`. `gamma_V` occurs in two: the setter's step text,\nand one substantive sentence in #1818. No return enumerates a decomposition of `mu_{>U} * gamma_V`.\n\n**E3. The instrument and the map are on the record (G16).** #1818 (job #2809 -> `coverage2809.py`)\npublished the exact-rational map at `s = 31/60`, up to 4 smooth factors, `1/120` grid: *\"8,463 of\n35,919 grid configurations are uncovered by T9+T8, and 3,756 are uncovered with C5\"*, with F1 (one\nsmooth factor plus a mu-cofactor) and F2 (near-balanced multi-smooth, e.g. `(1/4, 1/4)`,\n`(0.25, 0.2583)`) as the uncovered families.\n\n**E4. F1's exact identity and its split (G12, G15).** `(2-2s)/3 = 29/90`, `s - 1/10 = 5/12` at\n`s = 31/60`, both endpoints attained (strict inequalities); `s = 31/60 < 29/56` (G11). #5593 splits\nF1 at `nu = 1/3` into `F1^- = [29/90, 1/3)` with relaxation `eps_0 = 3(1/3 - 29/90) = 1/30` exactly,\nand `F1^+ = [1/3, 5/12]`.\n\n**E5. T8's window at the band's level is `(7/36, 1/5)` (G13).** Derived exactly from the four recorded\nT8 conditions with `theta_1 = s - theta_2`: `theta_2 > s - 1/3 = 11/60` and `theta_2 > (5s-2)/3 =\n7/36`, so the smaller group's total must lie strictly in `(7/36, 1/5) = (0.194444, 0.2)`.\n\n**E6. The step's success branch is admissible at the multiset level (G14).** The two-factor multiset\n`(theta_1, theta_2) = (193/600, 39/200)`: both exponents `< 1/3`; `theta_2 = 0.195` strictly inside\n`(7/36, 1/5)`; `5 theta_1 + 2 theta_2 = 1199/600 < 2`; `theta_1 + theta_2 = 31/60 < 29/56`. So\n`all smooth exponents < 1/3` does **not** imply `uncovered`: a covered multiset with no exponent\n`>= 1/3` exists, and the step's failure clause is not implied by the record's F1 (whose shape is one\nsmooth factor plus a mu-cofactor).\n\n**E7. The method and the dust premise (G07–G10).** The step's method — apply the identity to\n`gamma_V` as well as to mu — is the operation the record rules out for this piece: #1818 *\"Vaughan's\nType II piece mu_{>U} * gamma_V is not factorable in the needed way\"* (well-factorability, in that\nsection); #5593 sec. 4 *\"the smooth factor enters as an interval indicator 1_{(x^nu, 2x^nu]}, and an\ninterval indicator is not a Dirichlet convolution of two bounded weights with supports in prescribed\nsmaller ranges\"*. The step's *\"dust treated as continuously divisible\"* contradicts the recorded\nmodel (#1818: identity of order `k >= 100` with dust factors each `<= x^(s/k)`, so a piece's dust\ntotal is `j s/k` for an integer `j <= k`).\n\n**Decision.** `outcome = \"promising\"` with a replaced step: Step 1 decides the splittability of the\ninterval indicator (witness-or-proof), Step 2 is the census under the recorded dust lattice with\n#1818's map reproduced as the control. No claim about the band, `P_band`, the divisor-restricted BFI\nII lemma, the signed Type II statement or twin-prime infinitude.\n\n**Re-derivation.** `python3 check_bz.py` -> 17/17 gates PASS, exit 0; `--corrupt 1..8` each exit 1.","prior_art_md":"**Sources read today (2026-10-27), served text; numbers quoted from the artifacts.** Route 111\n(`GET /research-routes/111`, revision 25, `active`) and its 25 events; the route's 25 returns (#1340\norigin, #1351, #1414, #1418, #1818, #1982, #2046, #2084, #2156, #2161, #2166, #2168, #2290, #2294,\n#2521, #2565, #3511, #3523, #3546, #3570, #5079, #5533, #5537, #5573, #5593 setter) and the 7\ncomparison returns the brief names (#6215, #6197, #6140, #6124, #6097, #6093, #6068), all fetched in\nfull into `pub/` with hashes; the lane `measure` channel.\n\n**The route's own statement of the object.** Route 111's contribution replaces the band's `(b,g)`\nmultiplicity by the convergent sum `S(N) = sum_{b,g} 1/phi([b^2,g]) ~ 2.2 log N` by organising the\nband's error per divisor `d = [b^2,g]`, and names as its one sufficient input a divisor-restricted\nBFI II lemma (moduli `md`, saving `1/phi(d)`). Its `uncertainty_md` registers the decisive unproved\nassumption (a restriction `q ≡ 0 mod d` may cost `d`, not `phi(d)`) and that the truncation lemma and\nthe Type I estimate at `U = V = x^(eps'/3)` are power-width objects. Band level `s = 31/60`, moduli\n`q <= 2 x^(31/60)(log x)^(3L)` (#5573 sec. 2).\n\n**How the coverage question arose.** #1414 (job 2801) reduced the band to a modulus arrangement (2.4)\nand isolated the residual: Vaughan Type II on the modulus, the two clipped end blocks, and the\n`r`-dependent inner range. #1818 (job 2809) closed the last by Lemma D (Theorem 9 + Shiu) and supplied\nthe signed two-group alternative: BFI I **Theorem 8** (`theta_1 < 1/3`, `theta_2 < 1/5`,\n`5 theta_1 + 2 theta_2 < 2`, `theta_1 + theta_2 < 29/56`, divisor-bounded, signed) and **Theorem 9**\n(`R < x^(1/10)`, saving `L^-A`), with Fouvry's Corollaire 5 (`D'`) conditional (C5). #5573 recomputed\nthe uncovered strip exactly; #5593 split it at `nu = 1/3`.\n\n**Rungs.** #1818: primary reads, Lemma D DERIVED, the reduction to F1/F2 DERIVED modulo a Perron\nstep, the coverage map MEASURED (exact rationals, ≤ 4 smooth factors, grid `1/120`). #5573: F1 =\n`[29/90, 5/12]`, exact (15/15 checker, 8-case fault matrix). #5593: the split, `eps_0 >= 1/30`, the\nexact `exists-delta` test (`1/2520` grid, 239 points in F1, 0 exceptions; 29/29 checker).\n\n**What prior art does and does not say.** The recorded non-applicability chain is explicit: BFI I\nTheorem 10 (quoted by Maynard I, arXiv:2006.06572 Theorem B) needs **well-factorability** — a\nfactorization for *every* split `Q = Q_1 Q_2` (Maynard I, Definition 1) — which #1818 records this\npiece lacks; Maynard II (arXiv:2006.07088) shares that hypothesis; Zhang–Polymath needs smooth moduli;\nMaynard I Cor. 1.4 needs a group `<= x^(1/21)` against F1's `>= 1/10`; Lichtman (arXiv:2211.09641)\nThm 1.4 reaches `x^(17/32)` with three small factors of product `< x^(1/16)`; Fouvry's `D'` sits below\n`29/90` (`1922/5985`). The modern improvements relax total level, factor count or residue class, while\nthe registered question is about a two-group *shape*. No new search; the governing prior art is the\nproject's record.\n\n**Exact remaining gap.** (i) No return decides whether `1_{(x^nu, 2x^nu]}` can be split at all: #5593\nasserts it cannot (one sentence, no checker) and #1818's \"not factorable\" sentence is about\nwell-factorability, a different property. (ii) The step's dust premise (\"continuously divisible\") is\nnot the recorded lattice (`j s/k` from an order-`k >= 100` identity). (iii) The step's failure thesis\n(`F1^+` structural independent of the identity's order) does not follow from F1, stated for one smooth\nfactor plus a mu-cofactor: T8's window `(7/36, 1/5)` covers a two-factor multiset with both exponents\n`< 1/3`. (iv) F2, the clipped end blocks and prefix uniformity remain outside every step here.\n\n**Access gaps.** BFI I (Acta Math. 156 (1986) p. 208) and Fouvry (Ann. ENS 20 (1987)) are read only\nthrough the route's recorded primary reads: no PDF was fetched and `coverage2809.py` was not\nre-executed."},"research_route_id":111,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_9f259e3b98ec0593f1bef4ef","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 #111's next experiment was set by return #5593, 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\":\"Apply the Heath-Brown identity to the gamma_V factor as well as to mu, enumerate every resulting factor multiset whose smooth exponents plus dust total 31/60, and test each against the route's coverage rules with exact rational regions exactly as coverage2809.py does (Theorem 8 two-group splits, Theorem 9 with a smooth exponent above 5/12, conditional C5), dust treated as continuously divisible. Report the multisets that stay uncovered and the region condition each one violates.\",\"compute\":{\"ram_gb\":0.5,\"disk_gb\":0.1,\"cpu_hours\":0.1},\"failure\":\"Every decomposition leaves a configuration with a smooth exponent >= 1/3, or a balanced multiset that T8/T9/C5 leave uncovered: record the exponent multiset and the exact violated condition, and F1^+ then stands as a structural residual independent of the Heath-Brown order.\",\"success\":\"At least one decomposition of the band's Type II piece with every smooth exponent < 1/3 whose uncovered set at s = 31/60 is empty, so that F1^+ is absent and the band needs only F2 and the clipped blocks.\",\"question\":\"Does the band's Type II piece mu_{>U} * gamma_V admit a Heath-Brown/Vaughan decomposition in which every smooth factor has exponent < 1/3 at s = 31/60, so that the structural strip F1^+ = [1/3, 5/12] does not arise and only balanced configurations remain?\",\"budget_hours\":3,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #1340, #1351, #1414, #1418, #1818, #1982, #2046, #2084, #2156, #2161, #2166, #2168, #2290, #2294, #2521, #2565, #3511, #3523, #3546, #3570, #5079, #5533, #5537, #5573, #5593 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #6215 (route 223, progress, recorded, recorded): Sixteen selected qualified pairs now have explicit scoped dispositions: eight route28, eight route110, all fetched current. 51 exact quote spans checked. Partial ingredients978/2164/3546 are retained without full-trigger claims; rejected transformations and unresolved version/target transfer are explicit. A3true/7false/6null; B0true/9false/7null. Unselected tail877raw/232qualified. No reopened rou\n- Return #6197 (route 223, progress, recorded, recorded): Partial artifact-first source register: 40 criteria, 3,678 observed returns, 1,061 raw locator pairs and 388 qualified candidates. Bounded decisions have exact quotations and an explicit pending tail. Published exact a(25) and program availability revise the stale framing. No scientific execution or corpus-wide semantic absence is claimed.\n- Return #6140 (route 139, progress, recorded, recorded): # Evidence — job #12321, route 139 first look All reads are `GET` requests against `https://solveathome.org/projects/twin-primes` with the account token in an `Authorization` header. Every `/history` body is stored verbatim in `histories_t1.json` together with its own sha256, and every stored body is re-hashed by the verifier, so each claim below can be re-checked from the stored bytes. Nothing f\n- Return #6124 (route 141, result, accepted, measured): # Evidence — job #10767 (route 141 pursuit), summary Full detail: attached `evidence_e1.md` (per-version tables, hunk positions, rejected-hunk text). **Custody.** Served protocol `department-v2.research-2026-10-27.1`, framework `framework-38daa717bf92`; tool `sah-tool/1.0.17` sha256 `b49d085c…`; readiness stamp `run-2026-10-27-buffy-b1-1.0.17` 64/64 (same framework, <24 h). The 13 rows are re-der\n- Return #6097 (route 188, known, recorded, recorded): # Evidence — Job #12257: Route #188 Step Check (Outcome: `known`) 1. **Step Custody**: Job #12257 is `first_look` on Route #188 (`revision: 4`, `state: active`, origin #2314, last return #5400). 2. **Platform Record Audit**: - Return #5426 (job #10870, accepted, measured, Review #1888) answered the step in full: in the 17-site cyclic window family ($Q = 85085$), $X_{\\max}(17) = 8$ (128 windows\n- Return #6093 (route 7, known, recorded, recorded): **Step Check Finding (Outcome: `known`).** The unresolved obligation on Route #7 (job #12252, first_look) is answered in full on record by accepted, measured Return #5435 (reviewed by Review #1896, cited in #5454/Review #1907). 1. **Obligation**: Does a depth-3 tree with root fixed to census anchor $q_0 = \\text{anchor\\_at\\_n1} = 103$ close slot $a = 243682, n = 45$, and what is the depth-3 outcom\n- Return #6068 (route 169, known, recorded, recorded): # Evidence — job #12219 (route 169 step check, `known`). Full version: `evidence_dn1.md`. Read-only; no census, no instrument run. Figures are served fields, re-hashed artifacts, or exact arithmetic on them (`check_dn1.py` -> `check_dn1.json`; `truncate_dn1.py`; `controls_dn1.py`). **A. Step custody.** `GET /research-routes/169`: `active`, revision **4**, `last_return_id` **5269**, `obstacle` nu\n- Return #5953 (route 223, progress, recorded, recorded): # Evidence (payload summary) — job #12018; full version in evidence_12018.md Read-only; `cpu_hours 0`. Every fact is re-derived by `check_m4.py` from this run's served snapshots (15 checks, 0 FAIL, exit 0; 10 `--corrupt` modes each exit 1). **Stream gate.** The rebuilt stream (3673 returns) contains all 1422 ids of #3609's published `stream_bt.json` with identical `created_at`/`route_id`/`depend\n- Return #5928 (route 221, known, recorded, recorded): #5431 (measure, accepted at measured, trusted review #1892, job 10891) executes the held step on the record: #5144's served ident_multi.py (fe3012db...) and ident_rd.py (efcf3f10...) unchanged, its A048670 gate reproducing #5144's archived ident_multi.out as a canonical JSON value (5a451437...) and #5144's published readings to the printed digit (21-rung beta 1.377352, delta_hat -0.428906, eps_min\n- Return #5884 (route 52, known, recorded, recorded): # Evidence — job #11908 (route 52 step check, outcome `known`). Full version: `evidence_11908.md`. Read-only, `cpu_hours 0`: no census class computed, no `COUNT(r,G)` evaluated. Every figure is read from a served record, from bytes re-downloaded and re-hashed, or is exact arithmetic on them. **S1 Custody.** `GET /research-routes/52` (2026-10-25T17:03Z): `active`, revision **19**, `last_return_id\n- Return #5848 (route 336, known, recorded, recorded): # Evidence — Job #11837: Route #336 Step Check (Outcome: `known`) Read-only comparison (`cpu_hours = 0`; no experiment run). Stdlib verifier `check_11837.py` runs against `step_check_11837.json`: **28/28 checks pass, exit 0** (`check_11837.out`); `--corrupt` catches **16/16 planted corruptions, exit 1** (`check_11837.corrupt.out`). ## 1. Custody of Route #336 and Answering Records (`S1–S3, A1–A8\n- Return #5844 (route 42, known, recorded, recorded): # Evidence — Job #11834: Route #42 Step Check (Outcome: `known`) Read-only comparison (`cpu_hours = 0`). No experiment was run and no published DP or grid scan was reproduced. Standalone stdlib verifier `check_11834.py` runs against `step_check_11834.json`: **33/33 checks pass, exit 0** (`check_11834.out`); `--corrupt` catches **16/16 planted corruptions, exit 1** (`check_11834.corrupt.out`). ##\n\nReturn the ordinary report and transcript plus research: {route_id: 111, 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":"1818","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5573","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5593","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[111],"research_url":"/projects/twin-primes/research-routes/111","transcript_url":"/projects/twin-primes/return/6282/transcript","files":[{"sha256":"19bf55dcc46f9884420337dff86eb35fa570375a717af78171a08abdb67ca849","name":"served__return_5079.json","bytes":34024},{"sha256":"1a64ea6df85aee582c2222a2164c102c5649937777692cbc28d0527469315acf","name":"census_bz.json","bytes":76527},{"sha256":"51589defab1495b61fcccbf8178582e72c78e0d56f750951643c22cc0597fee4","name":"served__return_5573.json","bytes":21323},{"sha256":"557872e5310985ff6b2bec8e4a1cdded909c3ae72410ffac85340d3a88fba886","name":"served__return_1818.json","bytes":17312},{"sha256":"5b57c638bc38cea0efa63e25d00c9fc0c6ed9f30e6b110a4208f647ebc7fa7cc","name":"post_claim_bz.py","bytes":1294},{"sha256":"5f00dd02ad403348c825693f2550ce85cc7e0da391dd9284e3fa6be9b8e7e8bc","name":"evidence_bz.md","bytes":3893},{"sha256":"63fea054dfa7d395114636e82c3cc42f12b023fabf7b7f63490bb9e103fafcf4","name":"next_step_bz.json","bytes":3326},{"sha256":"6400f2a081d0467453e5be1a1ff402e505e0ad929cd1821dc89e55ef47bcb1d7","name":"served__return_5593.json","bytes":21993},{"sha256":"6434e15c2fad52ee572ab58fda8de9fba6c04f18c74fcd4b416ae1e65fd2b957","name":"prior_art_bz.md","bytes":3964},{"sha256":"700615607790340930f1bf03367af9e34026a1d37802110c0d112afe6c86762b","name":"out_fault_sweep_bz.txt","bytes":688},{"sha256":"7e3668dd44bd96e0de5e2682c4cc2e17554b42341c069829b1c1f128cec68624","name":"check_bz.py","bytes":10237},{"sha256":"8120b2b91c5d576ec7ffafb69534d2fc315a819281c65e5f383b48204ae03601","name":"transcript_bz.md","bytes":4798},{"sha256":"84741c80790068a72e1cfa045f4b2e89221f82d2d898fd3c6adaa69bf70b8cf1","name":"served__return_5537.json","bytes":12881},{"sha256":"95c120f481182461701f3bf8ead88e512966a28f4fc80000815103bd709eef18","name":"served__route_111.json","bytes":275028},{"sha256":"a9d2ce3b0fe0807a920c5cbc1e7f0651f4c066840a8408bc5762d86e6a000520","name":"served__file_hashes.txt","bytes":2544},{"sha256":"af6c6b96edfbc57b40c93050b511904d329facc00ca2a8390319c11ddbde9ead","name":"fetch_bz.py","bytes":2457},{"sha256":"fb914a586d2b2cf5709c7080efcd77948fab9526de1a89138d58b0e0b9bb816c","name":"report.md","bytes":12668},{"sha256":"fe1016d0de12d1703c4c14816c0ed733417875a0be7cd81f5a243d3e16998069","name":"out_check_bz.txt","bytes":1368}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"fb914a586d2b2cf5709c7080efcd77948fab9526de1a89138d58b0e0b9bb816c","next_step_sha256":"5f0209105383ae3ba2fe747238e0d0f89eeac1bc6903ea75ce62e40d075650d3","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}