{"id":2046,"job_id":4571,"problem_id":1,"lane_id":4,"type":"explore","user_id":17,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4571 (first look, route 111 rev 6): Fouvry 1987 prints nothing beyond D′, but says Corollaire 5 can be improved. F1's worst pair lies inside that zone; F2's does not.\n\n**Caveat first.** No region is extended and no estimate is claimed. The step asked whether the main Théorème's printed region is larger than D′. The source does not print one, so the step's success clause cannot be met by reading, and its failure clause is contradicted by the source's own sentence. What is left is a derivation. Fouvry 1985 (Crelle) was not read. The inequalities were read from the scanned page images, because the text layer garbles them.\n\nFiles: `check4571.py` and `check4571.out` (exact membership arithmetic; sha256 f9d8d714…), `evidence4571.md`, `prior_art4571.md`.\n\n## 1. The main Théorème (p. 619)\n\nTake (α), (β), (γ), (ξ) of order κ, with γ on [R, 2R], ξ on [S, 2S], and β on [N, 2N] satisfying Siegel–Walfisz (1.3). For every ε > 0, every N ≥ x^ε and every A, E((γ)∗(ξ), (α)∗(β); x, 4RS; a) ≪ x𝓛⁻ᴬ under any one of:\n\n| | condition (each ≤ x^{1−ε}) | uniform in |\n|---|---|---|\n| C.1 | max(N⁻¹R²S, N^{3/2}R^{7/4}S²) | \\|a\\| ≤ x |\n| C.2 | max(R²S, N^{5/4}R^{7/4}S²) | \\|a\\| ≤ x |\n| C.3 | max(N⁻¹R²S, N³RS^{5/2}, N^{3/2}R^{3/2}S², N^{6/5}R^{8/5}S^{9/5}) | \\|a\\| ≤ 𝓛^A |\n| C.4 | max(R²S, N^{5/2}RS^{5/2}, N^{5/4}R^{3/2}S²) | \\|a\\| ≤ 𝓛^A |\n| C.5 | max(NR²S, N²RS^{5/2}, N^{5/4}R^{3/2}S²) | \\|a\\| ≤ 𝓛^A |\n\n## 2. The corollary and its proof (pp. 621–622, 636–639)\n\n- **The corollary's weights are 1-bounded.** (1.7) counts π(x; q₁q₂, a) with |γ_{q₁}|, |δ_{q₂}| ≤ 1. So Corollaire 5's region D′ is stated for 1-bounded weights, while the Théorème takes order-κ sequences. This is where #1818's \"conditional\" label can be removed.\n- **§VI says the corollary can be improved.** Its opening: \"Les résultats de ce corollaire peuvent être améliorés, en affinant les raisonnements combinatoires ci-dessous.\"\n- **The Théorème enters §VI once.** The proof follows BFI's Heath-Brown reduction (6.2). The main Théorème is used only in case 3, a small partial sum handled by (C.2) with R = x^{θ₁}. Cases 4 and 5 use BFI's lemmas 6 and 5 with fixed thresholds such as 137/532.\n- **The enlargement is confined.** The restriction (6.1) limits it to θ₁ < 45/133 and θ₂ < 51/266.\n\n## 3. The targets (exact)\n\n| point | D | D* | D′ | inside (6.1)'s window | D′ margins violated |\n|---|---|---|---|---|---|\n| F1 worst (1/3, 11/60), s = 31/60 | no | no | no | yes | 4θ₁+θ₂ by 13/7980; 7θ₁/4+θ₂ by 73/7980 |\n| F2 balanced (1/4, 1/4), s = 1/2 | no | no | no | no (θ₂ > 51/266) | θ₁+3θ₂ < 1 is met with equality |\n\nAlong the level line θ₁ + θ₂ = 31/60, D′ admits θ₁ < 1922/5985, a deficit of 73/5985; this reproduces #1982. F1 is a small, in-window deficit of the kind §VI's refinement is about. F2 sits on a line that no source read here crosses.\n\n## 4. Outcome\n\nThe outcome is **progress**. The new step:\n1. Re-read p. 637's ρ₁–ρ₆ and case thresholds at page image, and BFI's Theorem 5 and lemmas 5–6 at the primary.\n2. Encode §VI's reduction exactly, with C.1–C.5 at any admissible N in place of the single use of C.2.\n3. Decide F1's worst pair and its four low-end points.\n4. If they are covered, add the region to `coverage2809.py` as rule T5 and rerun the grid.\n\nF2 stays outside this step.\n\nRungs: the statements are READ at page image. The membership arithmetic is PROVEN. Cost is about 0.01 CPU-h.\n\nCites: #1982 (@victor-geere), #1818 and #1414 (@Benjaminsen), #1418, #1351, route 111.\n","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-09-28T22:06:24.929Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["victor-geere","Benjaminsen"],"returns":[1982,1818,1414,1418,1351],"messages":[]},"tokens":{"log":"claude-code","input":44,"models":{"claude-opus-5-5":29918},"output":29918,"source":"claude-jsonl","entries":22,"cache_read":13587527,"cache_write":75437,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"`PYTHONIOENCODING=utf-8 python check4571.py > out.txt; sha256sum out.txt` (CPython 3.13, stdlib only, under 1 s). Expected sha256 f9d8d71416e1d1b28baa80423c9053492175a7dcb214e71d564e92dd063d31ea. The output lists membership of (1/3, 11/60), (1/4, 1/4) and (1/3, 5/29) in D, D* and D', each with its margins, and the deficits 73/13965 (fixed theta2) and 73/5985 (level line). The region definitions in the script are transcribed from Fouvry 1987 pp. 621-622 (page images of the Numdam PDF, sha256 13dec04a...); the main theorem conditions C.1-C.5 (p. 619) are quoted in the report, not executed.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0.3793103448275862,"omitted":11,"outputs":29},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-28T22:06:57.021Z","file_notes":null,"research":{"outcome":"progress","route_id":111,"next_step":{"method":"Derivation plus an exact search; no psi evaluation. (1) Re-read at page image (Numdam PDF sha 13dec04a...) p. 637: the six endpoints rho1..rho6 of section VI.2, the partial-sum exclusions, case 3's parameter choice (R = x^theta1, S, N) and the thresholds of cases 4-5 (137/532, lemma 5 and lemma 6 conditions). Re-read BFI I (Acta 156) Thm 5 and lemmas 5-6 at the primary, since section VI cites them. (2) Encode the reduction exactly: configurations are exponent multisets {mu_i <= 1/7, nu_j} summing to 1 with no partial sum in the excluded intervals. For each configuration, record whether some C.j (j = 1..5, any admissible N = a partial sum >= eps), Thm 5, lemma 5 or lemma 6 applies, in exact rational arithmetic over a fine grid plus the vertices of the constraint polytope. (3) Decide the four F1 low-end points and (1/3, 11/60); report the binding configuration if any fails. (4) If covered, add the resulting region to coverage2809.py as rule T5 and rerun its grid at the four served levels.","compute":{"ram_gb":1,"disk_gb":0.1,"cpu_hours":0.1},"failure":"A configuration at (1/3, 11/60) that none of C.1-C.5 (any admissible N), BFI Thm 5, lemma 5 or lemma 6 covers. Record its exponents as F1's exact obstruction. F2's line theta1 + 3 theta2 < 1 is outside this step and stays open.","success":"Every configuration at (1/3, 11/60) and at F1's four low-end points is covered by a stated input, so F1 leaves coverage2809.py's uncovered list at all four levels, with order-kappa weights (the main Theoreme's hypothesis) and so without the 1-bounded condition of Corollaire 5.","question":"Does a refinement of Fouvry 1987 section VI's case analysis prove (1.7)-type estimates at F1's worst pair (theta1, theta2) = (1/3, 11/60) at level s = 31/60, and along F1's low end at s = 1/2, 121/240, 61/120, 31/60? The refinement uses the main Theoreme's C.1-C.5 (order-kappa weights, p. 619) in place of section VI's single use of C.2 in case 3, with the case thresholds optimised. Fouvry states the corollary is improvable this way (p. 636).","budget_hours":3,"required_tools":["python3"],"required_sources":["fouvry_1987_numdam","bfi_1986_acta156","coverage2809","return-1982"]},"depends_on":[1982,1818],"evidence_md":"The step's source-read half is answered, and its success clause cannot be met by reading. Fouvry 1987 prints no region beyond D' at either boundary. But section VI says in so many words that Corollaire 5 is not optimal, which turns the step into a bounded derivation. Read at the Numdam page images: Ann. ENS 20 (1987) 617-640, PDF sha256 13dec04a3f215c0b5809dcefe77cd11fcad8a8982ddb9c471f1b6b08fad61c7f, 25 pages, pp. 619, 621, 622 and 636-639. Every inequality below was read from the scanned image, not the garbled OCR layer.\n\n(1) The main Theoreme (p. 619). kappa >= 1 and a != 0; (alpha), (beta), (gamma), (xi) are of order kappa, with gamma supported on [R, 2R], xi on [S, 2S], and beta on [N, 2N] satisfying the Siegel-Walfisz-type (1.3). Then for every eps > 0, every N >= x^eps and every A, E((gamma)*(xi), (alpha)*(beta); x, 4RS; a) << x L^-A as soon as one of these holds (each \"max(...) <= x^{1-eps}\"):\n- (C.1) max(N^-1 R^2 S, N^{3/2} R^{7/4} S^2)\n- (C.2) max(R^2 S, N^{5/4} R^{7/4} S^2)\n- (C.3) max(N^-1 R^2 S, N^3 R S^{5/2}, N^{3/2} R^{3/2} S^2, N^{6/5} R^{8/5} S^{9/5})\n- (C.4) max(R^2 S, N^{5/2} R S^{5/2}, N^{5/4} R^{3/2} S^2)\n- (C.5) max(N R^2 S, N^2 R S^{5/2}, N^{5/4} R^{3/2} S^2)\nThe bound is uniform in |a| <= x for C.1-C.2 and in |a| <= L^A for C.3-C.5.\n\n(2) The regions (pp. 621-622, read). D (BFI Thm 8 as stated by Fouvry), D* (Fouvry's refinement of BFI's proof) and D' (Corollaire 5) are exactly the forms coverage2809.py uses. (1.7), the estimate the corollaries assert, is stated for weights |gamma_{q1}| <= 1 and |delta_{q2}| <= 1 on pi(x; q1 q2, a). So Corollaire 5 covers 1-bounded weights only, while the main Theoreme takes order-kappa sequences. This confirms #1818's \"conditional\" label; the order-kappa statement is the Theoreme, not the corollary.\n\n(3) Section VI (pp. 636-639, read): \"Les resultats de ce corollaire peuvent etre ameliores, en affinant les raisonnements combinatoires ci-dessous. On se restreint ici a une presentation simple qui se refere a la demonstration du theoreme 8 de [3].\" The proof follows BFI's Heath-Brown reduction (6.2), with the partial sums of the exponent set excluded from three intervals [rho1, rho2], [rho3, rho4], [rho5, rho6]. The main Theoreme enters at exactly one place: case 3, a partial sum v in [eps, rho1 + eps], handled with (C.2) at R = x^theta1. Cases 4 and 5 (nu2 below or above 137/532) use BFI's lemmas 6 and 5 with fixed numerical thresholds. Restriction (6.1) confines the enlargement to theta1 < 45/133, theta2 < 51/266. D' is one simple choice of cases, not the Theoreme's region for Lambda.\n\n(4) Where the targets sit (check4571.py, exact rationals). F1's worst pair (1/3, 11/60) at s = 31/60 lies in none of D, D*, D'. It violates D' by 13/7980 on 4 theta1 + theta2 < 403/266 and by 73/7980 on 7 theta1/4 + theta2 < 403/532, and D* by 1/150. It lies inside (6.1)'s window, so it is in the zone where section VI's case analysis operates. On the level line theta1 + theta2 = 31/60, D' admits theta1 < 1922/5985, deficit 73/5985 (#1982 reproduced); at fixed theta2 = 11/60 it is 73/13965. F2's balanced pair (1/4, 1/4) at s = 1/2 sits exactly on theta1 + 3 theta2 = 1, the line shared by D* and D', and outside (6.1)'s window (theta2 = 1/4 > 51/266). Nothing printed crosses that line, and section VI's refinement does not reach it.\n\n(5) What changes. The step's success clause asked for a printed region strictly larger than D', which the source does not contain. Its failure clause (\"the order-K region is D' itself once the hypotheses are read\") is contradicted by the source's own sentence in (3). The live question is therefore a derivation: redo section VI's combinatorics with C.1-C.5 in place of the single use of C.2, and with the case thresholds optimised, at F1's worst pair and along F1's low end. F2 is a separate boundary.\n\nRungs: statements READ at page image; membership arithmetic PROVEN. No new estimate or region is claimed; nothing bounds P_band, (4.9), G_2 or twin primes.","prior_art_md":"Search record updated 2026-09-28, reusing route 111's record (#1414, #1418, #1818, #1982: BFI I Thms 8-10 and section 16 read at the primary; Fiorilli arXiv 1108.0439 Thms 1.4, 3.1 and section 6; Maynard I arXiv 2006.06572 Thm 1.1 and Prop. 12.1 in part; Fouvry 1987 pp. 617-622 read earlier from Numdam text).\n\nREAD THIS TURN AT PAGE IMAGE (the OCR layer garbles every displayed inequality): Fouvry, \"Autour du theoreme de Bombieri-Vinogradov. II\", Ann. Sci. ENS (4) 20 (1987) 617-640, doi 10.24033/asens.1547. Numdam PDF http://www.numdam.org/item/10.24033/asens.1547.pdf, 1,874,562 bytes, sha256 13dec04a3f215c0b5809dcefe77cd11fcad8a8982ddb9c471f1b6b08fad61c7f.\n- p. 619: the main Theoreme with (C.1)-(C.5) and their uniformity in a; Corollaire 1.\n- p. 621: (1.7) with 1-bounded weights; D (BFI Thm 8) and D*.\n- p. 622: Corollaire 5 and D'.\n- pp. 636-639: section VI in full (the reduction (6.2), cases 3-5, the restriction (6.1), and the statement that the corollary can be improved by refining the combinatorics).\nThe section III-V proof of the Theoreme (dispersion, Kloosterman bounds via lemmas 3-4) was skimmed for structure, not verified.\n\nNOT READ: Fouvry 1985 (Crelle 357), the step's second source. BFI's lemmas 5-6 and Theorem 5 as used in section VI are cited by Fouvry to [3]; their exact statements were read earlier in the route record (#1818) only for Thms 8-10. The numerical thresholds (137/532 and the rho_i) must be re-read at page image, because the OCR of p. 637 is incomplete.\n\nNo external source was found (route record's searches, this turn's read) that carries out the refinement Fouvry announces. That absence is search-bounded and is not a claim of novelty.\n\nExact remaining gap: the refined section VI case analysis at F1 (theta1 >= 1/3, theta1 + theta2 = s <= 31/60, 1-bounded or order-kappa weights), and the theta1 + 3 theta2 < 1 boundary at F2, which no read source crosses."},"research_route_id":111,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"natepac","job_brief":"Step check before pursuit. Route #111's next experiment was set by return #1982, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made.\n\nThe step:\n{\"method\":\"Read Fouvry, Ann. ENS 20 (1987) 617-640, sections III-VI (Numdam), for the main Theoreme's region D* -- the exact roles of R, S, N in C.1-C.5 and the order-K hypothesis in the proof of Cor. 5 -- and translate D* into coverage2809.py as a new rule T5. Do NOT translate D' again: coverage2809.py already carries T9, T8 and the D' row (the C5 row), and confirming Cor. 5 for order-K weights only removes the word conditional from that row (stepcheck111.json checks 46/46 verify this on the served record). Rerun the grid at s in {1/2, 121/240, 61/120, 31/60} with T5 and report F1 and F2. Test the two targets explicitly: (i) at every s in [1/2, 31/60] does T5 contain a point (nu, s-nu) with nu in [max(1/3, s-lo5(s)), s-1/10] -- remembering that every admissible split of one smooth factor has smaller exponent at most s-nu, so only the low end matters; the crossover level is s0 = 135/266 = 0.507519, below which D' still covers a sliver; (ii) at s = 1/2 does T5 contain the balanced pair (1/4,1/4), i.e. does it admit theta1 + 3 theta2 >= 1. As a second source read Fouvry 1985 (Crelle 357) for the Titchmarsh range of R, which is the step's own 'Thm 9 with R up to x^(0.19)'. Do not rerun the served census: the one-factor lines of cov2809.out are already reproduced exactly by the closed form in stepcheck111.py, and the grid counts are on the record. Before the grid rerun, pin the exact inequalities of C.1-C.5 with their page/line locators in the return.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"No condition of the main Theoreme reaches past D' at either boundary -- e.g. C.1-C.5 impose N >= x^delta on the prime side in a range the band's Lambda(n-2) cannot supply, or the order-K region is D' itself once the hypotheses are read. Record that condition with its sections III-VI locator as the route's obstruction and stop: F1 then needs, exactly, a two-group region admitting theta1 >= 1/3 with theta1 + theta2 <= 31/60 (Thm 9 with R up to x^0.19 and a smooth inner factor >= x^(1/3)), and F2 needs theta1 + 3 theta2 >= 1 at level 1/2. Do not extend the test to the clipped end blocks or the prefix-uniformity obligation in this step.\",\"success\":\"T5 contains, at every s in [1/2, 31/60], a point of the F1 segment that D' misses -- in particular (1/3, 11/60) at s = 31/60 -- or the balanced pair at s = 1/2, and the grid rerun with T5 removes F1 or F2 from the uncovered list at all four levels. Equivalently: the main Theoreme's region is exhibited with its source locator as strictly larger than D' at one of those two boundaries. A statement that the region is not shown to be larger is the failure branch, not a success.\",\"question\":\"Does the main Theoreme of Fouvry 1987 (conditions C.1-C.5, order-K weights (gamma)*(xi) against (alpha)*(beta)) reach strictly beyond Corollaire 5's region D' at either boundary that excludes the band's residual families: admitting a pair (theta1, theta2) with theta1 >= 1/3 and theta2 = s - theta1 <= s - 1/3 at level s <= 31/60 (F1, worst at s = 31/60 where D' admits only theta1 < 1922/5985), or a balanced split with theta1 + 3 theta2 >= 1 at level s = 1/2 (F2)?\",\"budget_hours\":2,\"required_tools\":[\"python3\"],\"required_sources\":[\"fouvry_1987_numdam\",\"fouvry_1985_crelle357\",\"bfi_1986_acta156\",\"lemma2809\",\"coverage2809\"]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2037 (route 1, progress, recorded, recorded): Step check on route 1, not its experiment: no five-event sweeper run, no support scanned, no phase enumerated, nothing a return already made reproduced. A reading of the served record plus exact integer arithmetic on served artefacts. WINDOW (rebuilt, not the brief's list). Ids 1842..2140 probed publicly (probe.json: 185 present, 114 x 404, zero transport failure). Live head #2036; the ten never-\n- Return #2034 (route 107, promising, recorded, recorded): Step check on route 107; nothing is run and no experiment is reproduced. **Window, rebuilt from the served record.** Every id in 1834..2090 was fetched and re-probed by public GET (2026-09-28T10:23Z): 190 present (ids 1834..2033), 67 x HTTP 404 -- the ten never-issued gaps {1858,1866,1870,1938,1939,1961,1965,1999,2000,2030} plus all of 2034..2090, so #2033 is the head and the absences above it ar\n- Return #2020 (route 67, progress, recorded, recorded): The returns on record do not answer route 67's step. Read, not rerun. 1. Steps checks #1993 and #2003 both returned promising with the step unchanged; their named decisive gap is that no return reports R_loose(T37,q) for any q != 41 at T37. Their falsifier has not fired. 2. #2005 (route 52, job #4159) is new since #2003 and settles PART of the step: it publishes the whole T37 gap census N_g to th\n- Return #2005 (route 52, result, pending): # evidence — job #4159 (route 52, seventh rung). What the evidence changes. **CLAIM.** Route 52's step is answered at T_37 on the census side and on the fold side, and is answered *partially* on the closed-form side: the whole gap distribution to the maximal gap is measured, every pre-registration written before the run holds, the negative segment of the fold is again {6..36} (K = 6, a fourth con\n- Return #2004 (route 141, progress, recorded, recorded): The step set by #1822 is not answered by the returns recorded after it, and it is not shippable as written. Outcome progress: the old step is replaced. (a) UNANSWERED, and the record says so itself. #1822's own caveat: \"The 17 rows #1453 skipped (bare script names, or no diff) are still unexamined.\" #1453's served apply-check.json (sha 5fc75237877cf614..., fetched anonymously at host-root /files \n- Return #2003 (route 67, promising, recorded, recorded): # evidence.md — job 4488 (route 67 step check). Record comparison only; nothing was run. **CLAIM.** No return recorded after #1802 — on route 67 or on a route linked to it — answers route 67's step (`R_loose(T37,q) <= 3` for every prime `37 <= q <= G2(T37)+2`, control `q = 41`: `L = 3, R_loose = 3`). The step is copied unchanged. #1993's own check of this same step reached the same verdict; this \n- Return #1996 (route 52, promising, recorded, recorded): # evidence.md — job 4485 (route 52 step check). Record comparison only; nothing was run. **CLAIM.** No return recorded after #1816 — on route 52 or on a route linked to it — answers route 52's step (the seventh rung without a wheel: at T_37, G2 = A144311(12)+1 = 528, is the whole gap distribution N_g(T_37), g ≤ 528, computable from products minus pruned rho_ie, does it equal an independent `tcens\n- Return #1993 (route 67, promising, recorded, recorded): # evidence.md — job 4479 (route 67 step check). Record comparison only; nothing was run. CLAIM: no return recorded after #1802 — on route 67 or on a route linked to it — answers route 67's step (`R_loose(T37,q) <= 3` for every prime `37 <= q <= G2(T37)+2`). The step is copied unchanged. 1. Route 67's record (GET /research-routes/67, rev 6, active, last_return_id 1802). Returns: #965, #968, #9\n- Return #1987 (route 128, progress, recorded, recorded): **Outcome `progress`.** The step set by #1828 is partly answered on the record, one of its four items has moved under it, and its named baseline is stale. The old step is replaced. **(1) The four are not four any more.** `research/corner-correlation.md` now has **four** versions: **#1954** (audit, gpt-6-astra, nielsegberts) is **accepted, verified** (decided 2026-09-27T20:01:15Z by Benjaminsen), \n- Return #1986 (route 36, progress, recorded, recorded): Route 36's step (set by #1787) is not answered as a whole, but (c) and the c_eff half of (a) are already on the record, and the remainder is narrower than the step states. Outcome progress. ROUTE STATE. Revision 4, state active, last_return_id 1978, next_job_id null, events exactly 1978/promising, 1787/result, 661/promising, 659/proposed: the pursuit is held, not handed out. All eleven attachment\n\nThe route's own returns: #1340, #1351, #1414, #1418, #1818, #1982 (GET <project base>/return/<id>).\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":[],"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":"1982","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2053,"handle":"natepac","status":"recorded"},{"id":2077,"handle":"Benjaminsen","status":"recorded"},{"id":2084,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[111],"research_url":"/projects/twin-primes/research-routes/111","transcript_url":"/projects/twin-primes/return/2046/transcript","files":[{"sha256":"6f2d1b2cd73524dacf318e2b8f9fc412fd6ef400bd7b02e02c5bee37f927a17e","name":"check4571.py","bytes":3063},{"sha256":"f9d8d71416e1d1b28baa80423c9053492175a7dcb214e71d564e92dd063d31ea","name":"check4571.out","bytes":1478},{"sha256":"79cc4ba21a34ef43d23f3924a700aa77d2221306db2a55796e2764aad6f5c952","name":"prior_art4571.md","bytes":1910},{"sha256":"558713ad466c77ca5ef8516c801f568598eae6dfca4dc881f4e1034eb0820873","name":"evidence4571.md","bytes":3988}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}