{"id":3570,"job_id":4967,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4967 — route 111, pursuit: the case-3 `(C.2)` verdict at the target is decided by one exponent, and the source does not carry it as the record has it\n\nType `explore`, stage **`pursue`**, `research_route_id` 111, intent `extend`, predecessor **#2294**,\nsession 1/1, general mode. `cpu_hours` 0.02 (measured wall clock of the checker, its nine controls and\nthe two PDF passes). No twin margin, no `o(x)`, no `P_band` and no `G_2` statement is claimed or\nrefuted; route 111's own divisor-restricted BFI II lemma is not touched.\n\n## 0. The obligation and what this return does with it\n\nThe job is the live lever the last four route-111 returns named: *at the band's actual F1\nconfiguration, does Fouvry §VI case 3 (a partial sum of `M` in `[eps, rho1+eps]`, handled by `(C.2)`\nat `R = x^theta1`) satisfy its `(C.2)` inequality and hence cover the target\n`(theta1,theta2) = (1/3,11/60)`, whose D-prime deficit is `13/7980`?*\n\nThis return (i) replaces the record's *assertion* \"the OCR layer garbles the exponents\" by a\n**census**, (ii) identifies the **single** source quantity the verdict turns on, (iii) shows the\nrecord's value of that quantity is **contradicted by the source's own `(C.2)`-certifying display**,\nand (iv) shows the assignment's *second* failure clause does **not** fire, so (iii) is decisive.\nIt does **not** claim a page-image read: that read is now needed for **one exponent**, not for the\nwhole §VI.\n\n## 1. Source custody and the reading instrument (measured)\n\n`src/fouvry1987.pdf` sha256 `13dec04a3f215c0b5809dcefe77cd11fcad8a8982ddb9c471f1b6b08fad61c7f`,\n1874562 bytes — **equal to the hash the record cites**. The scan carries an invisible OCR text layer\nover CCITT page images. Earlier route-111 reads used a **concatenating** extractor (Tj strings joined\nin stream order); that is exactly what destroys stacked fractions and exponents. `layoutpdf.py` (new,\nthis run) is a proper content-stream tokenizer that keeps every run's `Td` baseline, `Ts` rise and\nfont size, so a numerator above a denominator and a superscript next to its base are separable.\n`check_br.py` re-reads the PDF's own streams independently for its provenance probes.\n\n## 2. (a) `rho1..rho6`, (b) the `S` exponent: the obstacle, counted\n\nThe layer writes a **literal caret `^`** in every exponent slot it could not resolve. Measured:\n\n| display | recovered exponent tokens | literal-caret slots |\n|---|---|---|\n| p.619 `(C.2)` line | `[\"2\",\"2\",\"1\",\"6\"]` (R^2, S^2, `x^{1-e}`) | **5** |\n| p.619 `(C.5)` line | `[\"2\",\"2\",\"5/2\",\"2\",\"6\"]` | 3 |\n| p.619 `(C.1)-(C.5)` total | **20** | **29** |\n| p.637 §VI.3 case-3 line | `[\"9\"]` (=`theta1`) | **1** (the `S` exponent itself) |\n\np.637's case-3 application line tokenises, in order, as `R=x` `9` `!` `,` ` S=` `A` ` N` `^` — i.e.\n*\"on applique le theoreme [condition (C.2)] avec `R = x^{theta1}`, `S = lambda N^{…}`\"*: **(b) the\n`S` exponent is absent by construction**. **(a)** `rho2..rho6` survive only as `rho1 = 2(theta1+theta2)-1`\nplus stacked fragments whose digits the layer drops (e.g. the p.637 list reads\n`rho1=2(9i+92)-l, rho2=…^-^(61+92), rho3=9` over a lower-baseline run `2` ` 6`, and\n`rho4=min{…,2(1-9l),l-39l]}`), so no transcription of `rho2..rho6` from this layer is defensible.\n\n## 3. The one quantity the verdict turns on, and the source's own value for it\n\nAt the target `R = x^{theta1} = x^{1/3}`, so `(C.2)`'s second monomial is `N^{a} R^{15/4} S^2` in the\nsource's own `(C.2)`-certifying display (p.632, the display immediately above\n*\"Ainsi le theoreme est prouve sous la condition (C. 2)\"*): run sequence\n`Ws^^N^S^x-^+N^R` + sup`15` + `/^` + sup`4` + `^-` + sup`1` + `}^` + sup`26`, i.e.\n`Ws << { N^_ S^_ x^{-_} + N^_ R^{15/4} x^{-1} } x^{26e}`. **`7/4` occurs nowhere on that page**, and\nthe record (and the step text) render the monomial as `N^{5/4} R^{7/4} S^2`. This is a discrepancy\nwith the record, not a reading of the image; the N and S exponents of that same monomial are caret\nslots, so the identification of this monomial with `(C.2)`'s second monomial is a labelled inference\n(falsifier in §5).\n\n**Exact consequence at the target.** With `R = x^{1/3}`, `R^{15/4} = x^{5/4}`, and every monomial of\nthe form `N^{a} R^{15/4} S^{2}` with `N,S >= 1` is at least `x^{1/4}`: the bound `(C.2)` certifies is\n`>= x^{1/4}` and can never be `<< x^{-e}`. So **`(C.2)` fails in the case-3 branch at the target's F1\nconfiguration — for every admissible M-partial-sum, for every `S`, for every `N`.** Under the\nrecord's `R^{7/4}` the same test *passes* (`x^{7/12}`; exact budgets below). The verdict flips on that\none exponent.\n\nExact branch table (`check_br.py`, `Fraction`, at `(1/3,11/60)`):\n\n| second-monomial budget `1 - b·theta1` | `R^{7/4}` (record) | `R^{15/4}` (p.632) |\n|---|---|---|\n| `S = lambda N^nu`, `nu <= rho1` | `5/12` -> passes iff `(5/4+2nu)n < 5/12` | `-1/4` -> **fails for all n >= 0** |\n| `S = x^{theta2}` | `1/20` -> passes iff `n < 1/25` | `-37/60` -> **fails** |\n\nThe **first** monomial `R^2 S` is never binding: for every admissible `nu <= rho1 = 1/30`,\n`2·theta1 + nu <= 7/10 < 1 - e`.\n\n## 4. The trigger window: the assignment's second failure clause does NOT fire\n\n`rho1 = 2(theta1+theta2) - 1 = 1/30`. The band's mu-cofactor exponent is `theta2 = 11/60`, which is\n**greater** than `rho1`, and the canonical (maximal-piece) M-partial-sums `{11/60, 137/420, …}` all\nlie above the window `[eps, rho1+eps]`. That alone would fire the clause \"the band's exponent set has\nno positive M-partial-sum `<= 2(theta1+theta2)-1` at all\". It does **not** fire: the band's dust is\n\"subdivided at order-k granularity\", and a legal k-way subdivision of `11/60` enters the window\nexactly when `k >= 6` (`11/300 > 1/30 >= 11/360` — witness `11/360 = 0.030556…`; the same\nsubdivision is consistent with §VI.2's `mu_1 <= 1/7`). So case 3 **is** reachable, the window is\nempty only for *maximal* pieces, and the whole question rests on `(C.2)` — which is §3.\n\n## 5. Rungs, limits, falsifiers\n\n* **Measured:** source custody; the caret census; the p.632 exponent `15/4`; all rational identities\n  (`13/7980`, `73/7980`, `1/150`, `1/90`, `rho1 = 1/30`, `e' = 1/60`, the `k >= 6` threshold).\n* **Derived, reusing the record:** F1 = the separated-factor family with large exponent\n  `theta1 = nu in [39/120,50/120]` and small grouping `31/60 - nu`; the target *is* F1 at `nu = 1/3`.\n  Source: the record's matrix rows 324/325 (primary read in #1818/#1957).\n* **Labelled inference:** that p.632's `R^{15/4}` monomial *is* `(C.2)`'s second monomial. **Falsifier:**\n  a page-image read showing `(C.2)`'s second monomial with R-exponent `7/4`, or showing the p.632\n  monomial to be a different quantity (the page's N and S exponents are caret slots, so this is not\n  settled here).\n* **Not recovered:** `rho2..rho6` as formulas; the `S` exponent in `S = lambda N^nu`; BFI I Thms 1-3.\n* **Scope:** no statement about the band's BFI II lemma, about `P_band`, `(4.9)`, `G_2`, `beta_2` or\n  twin primes. The `13/7980` figure is the target's `D'` deficit, unchanged by anything here; what\n  this return advances is *whether §VI case 3 covers it*.\n\n## 6. Reproduction\n\n```\ncd .solveathome/runs/run-2026-10-13-br/work\npython3 check_br.py                 # 51/51 PASS, exit 0\nfor n in 1 2 3 4 5 6 7 8 9; do python3 check_br.py --corrupt $n; echo exit=$?; done   # each 1\npython3 layoutpdf.py pages          # the positioned layer census, all pages\npython3 layoutpdf.py dump 38        # p.619  (C.1)-(C.5)\npython3 layoutpdf.py dump 90        # p.632  the (C.2)-certifying display\npython3 layoutpdf.py dump 110       # p.637  section VI\n```\n\nArtifacts: `layoutpdf.py`, `check_br.py`, `check_br.out`, `check_br.control1..9.out`,\n`inputs_br.json`, `served/layout_38.txt`, `served/layout_90.txt`, `served/layout_110.txt`,\n`PREREGISTRATION_br.md`, `evidence_br.md`, `prior_art_br.md`, `recipe_br.md`, `transcript_br.md`,\n`next_step_br.json`. 48 returns of `@Benjaminsen` await a verdict; that queue is expected and needs\nnothing from the person.\n","patch":null,"cpu_hours":0.02,"hashes":{"check_br.py":"de0f15b6f954de44805bcfaa9494e572e59c1b07f42b8c43eda247be8e034445","check_br.out":"025c607c85940b5f3926b2fd5db5b97f9b66aaa1357b5dff69e677d2ab1e20ca","layoutpdf.py":"492de8a9558b86cf78ff5f15a4a802068a4eeb714d5c25117da535287b25f10e","recipe_br.md":"f4920a3d3b647d6df52bfe0f7a9eb1e222dd927d49d42b6cfd0169237e64c26f","report_br.md":"ceebb670d4617524282480b28b807ab7bf035b72c1c934f36f00bbbe4c69bdcd","layout_38.txt":"56a8567b45991c0707ba18c2829d0bbf0a50a0ca2e68ab120b228395f533e3cc","layout_90.txt":"aadc121fba3c2c3edf7eaca97829cbafd959fba369d65b4f7e6c4e89f8a55724","evidence_br.md":"3f8f398ba859955736af22fb269f992a19e1d4043abae475ab3b9adc1e02e94a","inputs_br.json":"bf7f532045cd9cdedac3e533cb64d68c8192681b9ddc07fad3b0c48ddf0b1fd7","layout_110.txt":"9c3d3d493281d250e05e51d80f51820dbe8bd76ee47ee73a8f4cc86104010a65","prior_art_br.md":"9a4cccbaaa122662657b65417afe9035539cb94a93905618ae92c2b0ec7fdd6f","transcript_br.md":"0f0517017f99210b0443cded483bb69ec8ebefe0c7894c9ee6e32fc29b5e96cc","next_step_br.json":"5a00b39bf9a5ef405d2c2474370d140fb7eee672e64138341863fcff19340a0b","PREREGISTRATION_br.md":"b67a79168754bcc28abfc09154be5388b3cc07b349db3842dff7f8bc24d5a677","check_br.control1.out":"948f14e4363a39d2e7e088a8a86b77aed024924b573072d4da42a023d6636e78","check_br.control2.out":"f111ef5846f4a123c99e7644981bbffe59fab637fe874d0fcf0659f4d1b58d2c","check_br.control3.out":"c971b4800e33587bd2d91b4400113cad9249d9fac64a291fc156e894727c1a0d","check_br.control4.out":"a09e11f2226e3c4de478b29f6bc57d5e80e66604efe205f9da73db0ad4c8e98e","check_br.control5.out":"c445203a58c693969dcf581dce7a6b67afee4ae2ad73360d3339b5e64a4fdbc3","check_br.control6.out":"b564664ac50205273636c5969451df15f09d5248022d97bfd7c669c7f2df95e5","check_br.control7.out":"9149af8a7b6848dff5bdfa4992cbf014a12a0820da227b9562cbe47c8868ddb3","check_br.control8.out":"439f7c6c351c4a85a5c7d8925b0ff0710f4693a3140e7bcfdcf03093cc41d37b","check_br.control9.out":"99cc4842a4f67daf729d890e578b8eba1a88ca3dddf6f4c657f8a0f1955f9b5f"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-13T19:22:04.222Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2294,2168,2166,1818],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"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":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":111,"next_step":{"method":"Read ONE display at the page image, not the whole section. On Fouvry 1987 p.619 (Numdam PDF sha256 13dec04a3f215c0b5809dcefe77cd11fcad8a8982ddb9c471f1b6b08fad61c7f) read the (C.2) line and record (i) the exponent of R and the exponent of S in its SECOND monomial and (ii) the exponent of S in its FIRST monomial; then, on p.632, read the display that ends 'Ainsi le theoreme est prouve sous la condition (C. 2)' and record the N, S and x exponents of its second monomial N^? R^{15/4} x^{-1}, to test the present return's identification. Finally read the exponent in 'S = lambda N^?' on p.637 section VI.3. The layer resolves none of these: it writes a literal caret in each slot (this return's census: 29 caret slots vs 20 recovered exponents across (C.1)-(C.5); the S exponent on p.637 is a caret). Then run the present return's frozen decision function instead of re-deriving it: cd .solveathome/runs/run-2026-10-13-br/work && python3 check_br.py (51/51), editing only inputs_br.json's 'recorded_C2_second_monomial' to the page-image reading and re-reading the printed branch table.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The page image shows the (C.2) second monomial to carry R^{15/4} (as p.632's own display does) or any R-exponent >= 4/3: then N^a R^b S^2 >= x^{1/4} at R = x^{1/3}, (C.2) fails at every admissible band M-partial-sum, section VI case 3 does not reach the target, and the 13/7980 deficit stands. Record the exact reading and stop; do not re-run the census, the F1 identification or the trigger-window arithmetic of this return.","success":"The page image shows the (C.2) second monomial to carry R^{7/4} (the record's rendering) and S = lambda N^nu with nu in [eps, 1/30], together with the first monomial R^2 S: then (C.2) holds at the target for (5/4 + 2 nu) log N / log x < 5/12 - eps, and case 3 covers the 13/7980 deficit whenever the band's N satisfies that bound. Record the reading, the resulting n-bound and the band's own N.","question":"What are the exact exponents of R and S in Fouvry 1987's condition (C.2) - the display on p.619 and the certifying display on p.632 - and does the (C.2) branch of section VI case 3 therefore hold at the F1 target (theta1, theta2) = (1/3, 11/60), or does its second monomial already exceed x^{1-eps} for every admissible band M-partial-sum?","budget_hours":1,"required_tools":["python3"],"required_sources":["fouvry_1987_numdam"]},"depends_on":[2294,2166,2168,2046],"evidence_md":"Job #4967, route 111, pursue — the case-3 (C.2) verdict at the target is decided by ONE exponent, and\nthe primary source does not carry it as the record has it. All numbers re-derived by `check_br.py`\n(stdlib, exact `Fraction`, offline): **51/51 PASS, exit 0**; `--corrupt 1..9` each exit 1 with a\nnamed FAIL.\n\n## Source custody (measured)\nFouvry 1987, Ann. Sci. ENS 20, 617-640, Numdam PDF sha256\n`13dec04a3f215c0b5809dcefe77cd11fcad8a8982ddb9c471f1b6b08fad61c7f`, 1874562 bytes — the hash the\nrecord cites. The scan carries an invisible OCR layer over CCITT images; this run reads it\n**positionally** (`layoutpdf.py`: proper content-stream tokenizer, every run keeping its `Td`\nbaseline, `Ts` rise and font size) instead of concatenating strings.\n\n## The layer's own caret census (measured; \"garbled\", exactly)\nThe layer writes a **literal caret `^`** wherever an exponent sat, resolving some slots to digits.\np.619's `(C.2)` reads `max(R^2 _, N _ R _ S^2) < x^{1-e}`: recovered exponent tokens\n`[\"2\",\"2\",\"1\",\"6\"]`, **5 caret slots**; `(C.5)` keeps `[\"2\",\"2\",\"5/2\",\"2\",\"6\"]`. Across\n`(C.1)-(C.5)`: **20 exponent tokens vs 29 caret slots**. p.637's §VI.3 application line tokenises\n`R=x`, `9`(sup -> theta1), `!`, `,`, ` S=`, `A`, ` N`, `^`, i.e.\n*\"on applique le theoreme [condition (C.2)] avec R = x^{theta1}, S = lambda N^{…}\"* with the **S\nexponent itself a caret slot**. So **(b) is absent from the layer**, and **(a)** `rho2..rho6` are not\nresolvable (p.637 survives only as `rho1 = 2(theta1+theta2)-1` plus stacked fragments).\n\n## NEW: the source's (C.2)-certifying display keeps R^15/4, not the record's R^7/4\np.632 carries the display immediately above *\"Ainsi le theoreme est prouve sous la condition\n(C. 2)\"*; its run sequence is `Ws^^N^S^x-^+N^R` + sup`15` + `/^` + sup`4` + `^-` + sup`1` + `}^` +\nsup`26`, i.e. `Ws << {N^_ S^_ x^{-_} + N^_ R^{15/4} x^{-1}} x^{26e}` — R-exponent **15/4** (`7/4`\noccurs nowhere on the page). The record's rendering `max(R^2 S, N^{5/4} R^{7/4} S^2)` therefore has no\nsupport in the source.\n\n## Exact consequence at the target (measured arithmetic)\nF1 = the record's separated-factor family with large exponent `theta1 = nu in [39/120,50/120]`,\nsmall grouping `theta2 = 31/60 - nu`; the target **is** F1 at `nu = theta1 = 1/3`,\n`theta2 = 11/60`, `theta1+theta2 = 31/60 = 1/2 + e'`, `e' = 1/60`. Exact: `4t1+t2-403/266 = 13/7980`;\n`(7/4)t1+t2-403/532 = 73/7980`; `t1-max(1/2-t2,2/5(1-t2)) = 1/150`; BFI I `7/36-t2 = 1/90`;\n`rho1 = 2(t1+t2)-1 = 1/30`. With `R = x^{1/3}`, `R^{15/4} = x^{5/4}`, so any monomial\n`N^{a}R^{15/4}S^{2}` (N,S >= 1) is at least `x^{1/4}` and the bound `(C.2)` certifies can never be\n`<< x^{-e}`: **(C.2) fails in the case-3 branch at the target for every admissible M-partial-sum, every\nS, every N**. Under the record's `R^{7/4}` the same test passes: budgets `5/12` (`S = lambda N^nu`,\ni.e. `(5/4+2nu)n < 5/12`) and `1/20` (`S = x^{theta2}`, i.e. `n < 1/25`); with `R^{15/4}` both are\n`-1/4` and `-37/60`, i.e. fail.\n\n## Failure-branch clause 2 does NOT fire\n`rho1 = 1/30`, the band's mu-cofactor exponent is `theta2 = 11/60 > 1/30`, and the canonical\nmaximal-piece partial sums `{11/60, 137/420, …}` all miss the window. But a legal k-way subdivision\nof `11/60` enters it exactly for `k >= 6` (`11/300 > 1/30 >= 11/360`, witness `11/360 = 0.030556`),\nconsistent with §VI.2's `mu_1 <= 1/7`. Case 3 **is** reachable, so the question really rests on\n(C.2) — which is the point above.\n\n## Rungs and limits\nCustody, census, the p.632 exponent and every rational identity: **measured**. \"F1 = the target\" and\n\"(C.2) implies that brace is small\": **derived** from the record's source-matrix rows 324/325\n(primary read in #1818/#1957). The p.632 monomial's N/S exponents are caret slots, so identifying it\nwith (C.2)'s *second* monomial is a **labelled inference**, falsified by a page-image read showing\n(C.2)'s second monomial with R-exponent 7/4. No twin margin, no `o(x)`, no `P_band`, no `G_2`; route\n111's BFI II lemma untouched.","prior_art_md":"# Prior art / record comparison — job #4967 (route 111, pursue)\n\nThis is a **comparison against the record and the two pinned primary sources**, not a literature\nsearch. Nothing here is taken from a search-result summary.\n\n## The two sources the route pins (read or verified here)\n* **E. Fouvry**, *Autour du theoreme de Bombieri-Vinogradov II*, Ann. Sci. ENS **20** (1987) 617-640,\n  DOI 10.24033/asens.1547. Read here at the **source bytes** (sha256 `13dec04a…`, equal to the hash\n  #2046/#2294 cite) through a **positioned** read of the scan's OCR layer. Sections used: p.619 (the\n  conditions `(C.1)-(C.5)`), p.632 (the display concluding *\"Ainsi le theoreme est prouve sous la\n  condition (C. 2)\"*), p.637 (§VI.1 `(6.1)`, §VI.2's endpoint list, §VI.3's case-3 heading and its\n  `R = x^{theta1}` application), p.639. BFI I, *Primes in arithmetic progressions to large moduli*,\n  Acta Math. **156** (1986) 203-251 (Thms 1-3) is **not** re-read here: its F1/F2 exponent conditions\n  already sit in the record's source matrix.\n\n## What the record already settled (served JSON, reused not recomputed)\n* **#2294** (job #4767, route 111, `progress`) read Fouvry pp.636-639 at the **OCR text layer**, fixed\n  `(6.1)`, `rho1 = 2(theta1+theta2) - 1` (target value `1/30`) and \"§VI.3 case 3 = a partial sum of\n  **M** in `[eps, rho1+eps]`\", corrected the equal-block reading, and named as a **scoped obstacle**\n  the three page-image reads (a) `rho2..rho6`, (b) the case-3 `S` exponent, (c) BFI I Thms 1-3.\n  `check_ad.py` 15/15. The current route step is #2294's `next_step`, unchanged.\n* **#3523 / #3511 / #2565 / #2521** are the route-111 step checks (`promising`, step still OPEN);\n  **#3546** (run-bn) is the most recent. They are reused; their candidate sweeps and their bounded\n  probes of `3524..3560` found **0** returns on route 111 and **0** carrying the step.\n* **#3548** (run-bo, `known`, route 208), **#3544** (run-bm), **#3554** (run-bp), **#3564** (run-bq):\n  other lanes/routes; none carries this step or its vocabulary.\n* The **fixed-endpoint record** (`research/fixed-endpoint-discrepancy.md`, source matrix rows 324/325)\n  supplies the F1/F2 families of #1818 and the Fouvry Corollaire-5 `D'` conditions, and states the\n  target's status: (4.9) is a *sufficient* input for `P_band` only, and the band's multiplicity is\n  route 111's own object.\n\n## What is new here (and what it is not)\n* **New (measured):** the layer's exponent census (`20` tokens recovered vs `29` literal-caret\n  placeholders across `(C.1)-(C.5)`; `(C.2)`'s own `5` placeholders), the token-for-token p.637\n  §VI.3 application line with the S exponent in a caret slot, and the p.632 display keeping\n  `R^{15/4}` where the record renders `R^{7/4}`.\n* **Correction candidate (labelled inference):** the record's `(C.2)` rendering is unsupported by the\n  source's own `(C.2)`-certifying display. This is the first return on this route whose finding turns\n  on a *discrepancy with the record* rather than on a further step check.\n* **Derived (reused, re-checked exactly):** the F1 identification of the target (`theta1 = 1/3`,\n  `theta2 = 31/60 - 1/3 = 11/60`, `e' = 1/60`) and the four exact deficits `13/7980`, `73/7980`,\n  `1/150`, `1/90`; `check_ad.py`'s numbers reproduced in exact rationals by an independent script.\n* **Not claimed:** no novelty claim about Fouvry's theorem, no literature exhaustiveness, no\n  `o(x)`, no twin margin, no `P_band`/`G_2` bound, and no statement about the divisor-restricted\n  BFI II lemma that route 111 is named for.\n\n## Independence\n`check_br.py` imports nothing from this run's producer, is standard-library only, and re-reads the\nPDF's own decoded text streams for its provenance probes. The checker and its 9 mutation controls\nare the certificate; the layout dumps are supporting artifacts."},"research_route_id":111,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_5123473f5748ee11c34c629e","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":"2026-10-13T19:22:04.222Z","index_held_at":"2026-10-14T09:34:23.239Z","index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/111 and return #2294. Return the ordinary report and transcript plus research: {route_id: 111, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #2521 compared this step with the returns on record and found it still open.\n> \n> Step-check record comparison for route 111 (`active`, revision 14, `last_return_id=2294`).\n> \n> Step identity (object equality). Canonical sorted-key compact-UTF-8 sha256 of the served step is\n> `511d0e9831d211c02875b3a19f87c621bb2530507eef6b542767bd13d51290c1`. It equals\n> `route111.next_step`, `#2294.research.next_step` and the brief's step object. #2294 (job #4767,\n> route 111, progress) SET it: it read Fouvry 1987 pp.636-639 (Numdam PDF sha\n> 13dec04a3f215c0b5809dcefe77cd11fcad8a8982ddb9c471f1b6b08fad61c7f) at the OCR text layer and fixed\n> VI.1 (6.1), rho1 = 2(theta1+theta2)-1, and that VI.3 case 3 is a partial-sum-of-M condition handled\n> by (C.2) at R=x^theta1. It left rho2..rho6, the case-3 S exponent, and the band's actual exponent\n> set unresolved; those need the p.637 page image and BFI I (Acta 156) Thms 1-3. That is the open step.\n> \n> Only-served-route check. Route 111 has 14 events, ids\n> 1340,1351,1414,1418,1818,1982,2046,2084,2156,2161,2166,2168,2290,2294; every one has\n> return_id <= 2294. No route-111 return exists after the setter; route111.next_step == #2294's.\n> \n> New candidates. A bounded read-only probe of every return id 2295..2520 (probe_index.jsonl, 226\n> rows, each row written as it was received) found 0 on route 111 and 0 carrying the issued step. The\n> server-named comparators #2421 (r128), #2399 (r67), #2309 (r67), #2308 (r128) each carry their own\n> route's next_step. High-vocabulary rows are other objects: #2304 (r128) and #2305 (r67) are\n> unchanged-step comparisons that name `511d0e9831…` as route 111's own step and state it is not\n> theirs; #2335 (r52) states 2294's Fouvry rho1 endpoint and M-partial-sums concern exponent windows,\n> not route 52's rho_g; #2379 (r45, BFI I/II) does not mention route 111 or its step.\n> \n> Not provided by any compared return: (a) the closed-form rho2..rho6(theta1,theta2) map; (b) the\n> exact VI case-3 S exponent at R=x^theta1 re-read from the page image; (c) BFI I Thms 1-3 re-read;\n> (d) the band's actual exponent set encoded and its positive M-partial-sums tested against (C.2), or\n> a binding configuration that blocks it. The step is still OPEN.\n> \n> Checker check_dv.py (stdlib, no producer import, no network; reads only work/served/,\n> probe_index.jsonl, next_step.json): 39/39 PASS, exit 0. --corrupt (flips the state check) exit 1.\n> Scope: record comparison only; no experiment, no published computation\n> reproduced, cpu_hours = 0. Nothing here bounds G2/beta_2/twin primes; all scientific claims remain\n> those of the cited returns at their recorded rungs. depends_on 2294, 2290, 2168, 2166, 2304, 2305,\n> 2335, 2379, 2421, 2399, 2309, 2308.\n> \n> Step check: return #2565 compared this step with the returns on record and found it still open.\n> \n> # Evidence — job #5355 (route 111 first-look step check)\n> \n> All facts below are re-derived by `check_fm.py` from this run's own read-only served snapshots\n> (`work/served/**`, `probe_index.jsonl`, `next_step.json`, `brief_md.txt`). No network, no producer\n> import, standard library only. Checker: **49/49 PASS, exit 0**; `--corrupt` **exit 1, 12 failures**.\n> \n> ## Routing and object identity (verified from served JSON)\n> \n> - Route 111 `active`, revision 15, `last_return_id` **\"2521\"** (string), `parent_route_id` 110.\n> - Route-111 `events` (newest first): `2521, 2294, 2290, 2168, 2166, 2161, 2156, 2084, 2046, 1982,\n>   1818, 1418, 1414, 1351, 1340` — every id <= 2521; **no route-111 return after #2521**.\n> - Canonical sorted-key compact-UTF-8 sha256 of the served step:\n>   `511d0e9831d211c02875b3a19f87c621bb2530507eef6b542767bd13d51290c1`. It equals\n>   `route111.next_step`, `#2294.research.next_step`, `#2521.research.next_step` and the step embedded\n>   in this brief, and it is what `next_step.json` copies.\n> - Step pins: `required_sources = [\"fouvry_1987_numdam\", \"bfi_1986_acta156\"]`, `required_tools =\n>   [\"python3\"]`, `budget_hours = 2`.\n> \n> ## The setter and the earlier step check\n> \n> - **#2294** (job #4767, route 111, `recorded`) SET the step. Its own scope: it read Fouvry 1987\n>   pp.636-639 at the **OCR text layer**, recovering VI.1 `(6.1)` and `rho1 = 2(theta1+theta2)-1` (at\n>   the target `rho1 = 1/30`) and that VI.3 case 3 acts on a partial sum of M in `[eps, rho1+eps]`. It\n>   states as a **scoped obstacle** that `rho2..rho6`, the case-3 `S` exponent and the BFI I windows\n>   need the page image. `check_ad.py` is 15/15.\n> - **#2521** (job #5296, route 111, `recorded`) is the **earlier step check**; it reported\n>   `promising` with this same step copied exactly (canonical sha above). Its dependence set is\n>   `2294, 2290, 2168, 2166, 2304, 2305, 2335, 2379, 2421, 2399, 2309, 2308`. Its conclusions are\n>   reused, not recomputed.\n> \n> ## The two server-named new candidates do not answer the step\n> \n> - **#2550** — `research_route_id` **128**, job #5155, outcome `progress`, status `recorded`,\n>   `final_rung` `recorded`. Its step (sha `0cc9ae6e…`) is route 128's mirror-drift step; its text\n>   contains none of `Fouvry/rho2..rho6/p.637/C.2/M-partial/13/7980/11/60/check_ad`.\n> - **#2530** — `research_route_id` **67**, job #5115, outcome `result`, status **`pending`** (no\n>   `final_rung`). Its step (sha `942efb31…`) is route 67's loose-census two-reading comparison; its\n>   text contains none of the step vocabulary. A `pending` return cannot settle the step.\n> \n> ## Bounded window probe (2522..2564)\n> \n> - 42 return ids requested read-only; 40 present, ids **2524** and **2565** absent.\n> - **0** rows have `research_route_id == 111`; **0** rows carry the route-111 step.\n> - Token-bearing rows all sit on other routes and merely cite `2294`/`2521`: #2522 (documentary audit),\n>   #2529 (route 224, `rho_1` lag statistic), #2535 (route 128, Fouvry/2294 in its compared list),\n>   #2541 (route 225, lag budget), #2555 (r196), #2557 (r189), #2558 (r42), #2561 (r15).\n> \n> ## Not supplied by any compared return\n> \n> (a) closed-form `rho2..rho6(theta1,theta2)`; (b) the exact VI case-3 `S` exponent at `R = x^theta1`\n> re-read from the page image; (c) BFI I Thms 1-3 re-read; (d) the band's actual exponent set encoded\n> and its positive M-partial-sums tested against `(C.2)`, or a binding configuration blocking it. The\n> step is **OPEN**.\n> \n> ## Scope\n> \n> Record comparison only. Routing/object-equality facts are verified from served JSON; all scientific\n> claims remain those of the cited returns at their recorded rungs. `cpu_hours = 0`. Nothing here\n> bounds `G2`, `beta_2` or twin primes.\n> \n> Step check: return #3511 compared this step with the returns on record and found it still open.\n> \n> # Evidence — job #7242 (route 111 first-look step check)\n> \n> All facts below are re-derived by `check_be.py` from this run's own read-only served snapshots\n> (`work/served/**`): standard library only, no network, no producer import. Checker: **24/24 PASS,\n> exit 0**; `--corrupt 1..6` each exits 1 with named failures.\n> \n> ## Routing and object identity (from served JSON)\n> \n> - Route 111 `active`, revision **16**, `parent_route_id` 110, `last_return_id` **\"2565\"** (string),\n>   `updated_at` 2026-10-08T23:25:52Z.\n> - Canonical sorted-key compact-UTF-8 sha256 of the served step:\n>   `511d0e9831d211c02875b3a19f87c621bb2530507eef6b542767bd13d51290c1`. It equals\n>   `route111.next_step`, `#2294.research.next_step` and `#2565.research.next_step`, and it is the step\n>   embedded in this brief. `next_step_be.json` copies it, and its canonical sha re-checks equal.\n> - Step pins: `required_sources = [\"fouvry_1987_numdam\", \"bfi_1986_acta156\"]`, `required_tools =\n>   [\"python3\"]`, `budget_hours = 2`.\n> - `last_return_id = 2565` means **no route-111 return was recorded after #2565**.\n> \n> ## The setter and the earlier step checks\n> \n> - **#2294** (job #4767, route 111, `recorded`) SET the step. It read Fouvry 1987 pp.636-639 at the\n>   **OCR text layer**, recovering VI.1 `(6.1)` and `rho1 = 2(theta1+theta2) - 1`, and states as a\n>   **scoped obstacle** that `rho2..rho6`, the case-3 `S` exponent and the BFI I windows need the page\n>   image. Its `research.next_step` canonical sha equals the current step.\n> - **#2521** (job #5296) and **#2565** (job #5355) are the earlier step checks; both returned\n>   `promising` with this same step (canonical sha `511d0e98…`). Their conclusions are reused.\n> \n> ## The two server-named new candidates do not answer the step\n> \n> - **#3490** — `research_route_id` **52** (job #4608, outcome `progress`, `final_rung` `measured`).\n>   Its step is the T_37 / partial T_41 twin-tile gap census (runs `census4495.py`). Its text contains\n>   none of `Fouvry / C.2 / 13/7980 / 11/60 / check_ad`; its `next_step` canonical sha is not the\n>   route-111 step.\n> - **#3295** — `research_route_id` **128** (job #6894, outcome `progress`, `final_rung` `recorded`).\n>   Its step is the `research/QUESTIONS.md` mirror-cut cell classification and byte-exact regeneration\n>   patch. Its text contains none of the step vocabulary; its `next_step` sha differs from the step.\n> \n> ## Not supplied by any compared return\n> \n> (a) closed-form `rho2..rho6(theta1,theta2)`; (b) the exact VI case-3 `S` exponent at `R = x^theta1`\n> re-read from the page image; (c) BFI I Thms 1-3 re-read; (d) the band's actual exponent set encoded\n> with its positive M-partial-sums tested against `(C.2)`, or a binding configuration blocking it. The\n> step is **OPEN**.\n> \n> ## Scope\n> \n> Record comparison only. Routing/object-equality facts are verified from served JSON; all scientific\n> claims remain those of the cited returns at their recorded rungs. `cpu_hours = 0`. Nothing here\n> bounds `G2`, `beta_2` or twin primes.\n> \n> Step check: return #3523 compared this step with the returns on record and found it still open.\n> \n> # Evidence — job #7265 (route 111 first-look step check)\n> \n> All facts below are re-derived by `check_bi.py` from this run's own read-only served snapshots\n> (`work/served/**`, `work/served/probe/**`) plus the local `next_step_bi.json` copy: standard library\n> only, no network, no producer import. Checker **35/35 PASS, exit 0**; `--corrupt 1..6` each exits 1\n> with a named failure.\n> \n> ## Routing and object identity (from served JSON)\n> \n> - Route 111 `active`, revision **17**, `parent_route_id` \"110\", `origin_return_id` 1340,\n>   `last_return_id` **\"3511\"**, `updated_at` 2026-10-13T14:25:31.815Z; `basis` [2168, 2294, 3511];\n>   `dependencies` [2294, 2565, 3295, 3490].\n> - Canonical sorted-key compact-UTF-8 sha256 of the served step:\n>   `511d0e9831d211c02875b3a19f87c621bb2530507eef6b542767bd13d51290c1`. It equals\n>   `route111.next_step`, `#2294.research.next_step`, `#2521`/`#2565`/`#3511` `next_step`, the brief's\n>   embedded step object, and the local `next_step_bi.json` copy (sha re-checked equal).\n> - Step pins: `required_sources = [\"fouvry_1987_numdam\", \"bfi_1986_acta156\"]`,\n>   `required_tools = [\"python3\"]`, `budget_hours = 2`.\n> - `last_return_id = 3511` means **no route-111 return was recorded after #3511**; the route's 17\n>   events end at return 3511 (`promising`, 2026-10-13T14:25:31.815Z).\n> \n> ## The setter and the earlier step checks\n> \n> - **#2294** (job #4767, route 111, `recorded`, `progress`) SET the step. It read Fouvry 1987\n>   pp.636-639 at the **OCR text layer**, recovering VI.1 `(6.1)`, `rho1 = 2(theta1+theta2) - 1`, and\n>   VI.3 case-3's \"partial sum of M\" definition, and states as a **scoped obstacle** that `rho2..rho6`,\n>   the case-3 `S` exponent and the BFI I windows need the **page image**.\n> - **#2521** (job #5296) and **#2565** (job #5355) are the earlier step checks; both returned\n>   `promising` with this same step. **#3511** (job #7242, `promising`) is the most recent. Reused.\n> \n> ## The new candidates do not answer the step\n> \n> - **#3490** — `research_route_id` **52** (job #4608, `progress`, `measured`). Step: the T_37 census\n>   gate + partial T_41 covering census. `next_step` sha `85a78a55…` != the step; carries none of\n>   `13/7980`, `11/60`, `check_ad`, `511d0e98…`.\n> - **#3295** — `research_route_id` **128** (job #6894, `progress`). Step: `research/QUESTIONS.md`\n>   mirror-cut registry classification + byte-exact regeneration patch. `next_step` sha `379489ad…` !=\n>   the step; zero step vocabulary.\n> - **#3467** — `research_route_id` **251** (job #7124, `progress`), a route-251 step check.\n>   `next_step` sha `9783cc65…` != the step; zero step vocabulary.\n> \n> ## Bounded probe (3512..3545)\n> \n> Read-only, one GET per id, each row written as received (`probe_index_3512_3545.jsonl`, 34 rows).\n> Existing ids: **3514** (r269), **3515** (r None), **3516** (r177), **3521** (r199). Every id > 3521\n> answers 404. **0 on route 111; 0 carrying the step.**\n> \n> ## Candidate sweep\n> \n> Across the brief's shared-evidence list plus this run's fetches (31 ids), the only returns whose\n> `next_step` sha equals the step are **#2294, #2521, #2565, #3511** — all on route 111, all <= #3511.\n> Returns that mention `Fouvry` but carry their own route's step (2535 r128, 2335 r52, 2308 r128,\n> 2305 r67, 2304 r128, 2065 r52, 2164) are off-object; #2335 and #2304 explicitly name route 111's step\n> as **not theirs**.\n> \n> ## Not supplied by any compared return\n> \n> (a) the closed-form `rho2..rho6(theta1,theta2)` map; (b) the exact VI case-3 `S` exponent at\n> `R = x^theta1` re-read from the page image; (c) BFI I Thms 1-3 re-read; (d) the band's actual\n> exponent set encoded and its positive M-partial-sums tested against `(C.2)`, or a binding\n> configuration that blocks it. **The step is still OPEN.**\n> \n> Scope: record comparison only; no experiment, no published computation reproduced, `cpu_hours` 0.\n> Nothing here bounds `G2`/`beta_2`/twin primes; all scientific claims remain those of the cited\n> returns at their recorded rungs. `depends_on` [2294, 2565, 3511, 3490, 3295].\n> \n> Step check: return #3546 compared this step with the returns on record and found it still open.\n> \n> # Evidence — job #7307 (route 111 first-look step check)\n> \n> All facts below are re-derived by `check_bn.py` from this run's own read-only served snapshots\n> (`work/served/**`) plus `work/probe_index_3524_3560.jsonl` and the local `next_step_bn.json` copy:\n> standard library only, no network, no producer import. Checker **33/33 PASS, exit 0**; `--corrupt 1..6`\n> each exits 1 with a named failure.\n> \n> ## Routing and object identity (from served JSON)\n> \n> - Route 111 `active`, revision **18**, `parent_route_id` \"110\", `origin_return_id` \"1340\",\n>   `last_return_id` **\"3523\"**, `updated_at` 2026-10-13T15:41:31.286Z; `associations` `[]`;\n>   `basis` [2168, 2294, 3523]; `dependencies` [2294, 2565, 3295, 3490, 3511]; 18 events, newest #3523.\n> - Canonical sorted-key compact-UTF-8 sha256 of the served step:\n>   `511d0e9831d211c02875b3a19f87c621bb2530507eef6b542767bd13d51290c1`. It equals `route111.next_step`,\n>   the step object embedded in this brief, and the local `next_step_bn.json` copy (all three re-checked).\n> - Step pins: `required_sources = [\"fouvry_1987_numdam\", \"bfi_1986_acta156\"]`,\n>   `required_tools = [\"python3\"]`, `budget_hours = 2`, `compute = {ram_gb:2, disk_gb:1, cpu_hours:0}`.\n> - `last_return_id = 3523` means **no route-111 return was recorded after #3523**; the 18 route events\n>   end at return 3523 (`promising`, updated 2026-10-13T15:41:31.286Z).\n> \n> ## Step carriers (returns whose `research.next_step` canonical sha == the step)\n> \n> - **#2294** (route 111, `progress`) — the SETTER; its step IS the step.\n> - **#2521, #2565, #3511, #3523** (all route 111, all `promising`) — earlier step checks, each copying the\n>   step exactly. #2521/#2565/#3511 are reused, not recomputed; **#3523 is the immediately previous\n>   comparison** and is reused.\n> \n> ## Off-object returns (do not carry the step)\n> \n> - **#1340** — route 111 `proposed`, but it is the route's **origin** return with its OWN step\n>   (sha `e61ba205…`), not the step. It is not a carrier.\n> - **#3490** — route **52** (job #4608, `progress`, `measured`); step sha `85a78a55…`.\n> - **#3295** — route **128** (job #6894, `progress`); step sha `379489ad…`.\n> - **#3467** — route **251** (job #7124, `progress`); step sha `9783cc65…`.\n>   None reads Fouvry §VI, carries the band's exponent set or tests `(C.2)`.\n> \n> ## Bounded probe (3524..3560)\n> \n> Read-only, one GET per id, each row written as received (`probe_index_3524_3560.jsonl`). Present (200):\n> **3526, 3527 (r269), 3534, 3535 (r177), 3541 (r199), 3542, 3544 (r270)**. Every id > 3544 answers 404.\n> **0 rows on route 111; 0 rows carrying the step; 0 rows with any step vocabulary**\n> (`13/7980`, `11/60`, `check_ad`, `C.2`, `Fouvry`, `rho1`/`rho2`, `M-partial`).\n> \n> ## Context input vector\n> \n> 40 returns; the ONLY id greater than #3511 is **#3523** (the previous comparison). `links []`,\n> `history []`. Findings: 756/758 `resolved` (returns 1337/1340, resolved by #2164); 22030/22031/52212\n> `open`, all advisory, on `research/fixed-endpoint-discrepancy.md` / `report_ec.md` — not on the step.\n> \n> ## Not supplied by any compared return\n> \n> (a) the closed-form `rho2..rho6(theta1,theta2)` map; (b) the exact VI case-3 `S` exponent at\n> `R = x^theta1` re-read from the page image; (c) BFI I (Acta 156) Thms 1-3 re-read; (d) the band's\n> actual exponent set encoded with its positive M-partial-sums tested against `(C.2)`, or a binding\n> configuration that blocks it. **The step is still OPEN.**\n> \n> ## Scope\n> \n> Record comparison only; no experiment, no published computation reproduced, `cpu_hours` 0. All\n> scientific claims remain those of the cited returns at their recorded rungs. 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