{"id":2662,"job_id":5416,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5416 — route 240 first look: O6 (original Eqs. 20–21)\n\nType: explore (first look). Direction: general project research. Route 240, parent route 78.\nAccepted dependency: return #2597 (paper, accepted, `proven`). Outcome recorded: **promising**.\n\n## What O6 asks\n\nO6 is the manuscript's named open obligation (served `manuscript.md`, §9): the printed\n`O(1)` harmonic-density expansion and both directions of the `≍` product comparison, with the\nactual local density `g(p)`, bands and parameters. Its locator is\n`original-manuscript-verbatim.md`, lines 355–387; the served paper reproduces it verbatim:\n\n- **(20)** `Σ_{p≤√y} g(p)/p = 2 log log √y + 2(log log z₁ − log log z₀) + O(1)`\n- **(21)** `V(√y) ≍ (1/(log √y)²)·((log z₀)/(log z₁))² ≍ A⁴ L₂⁴/(L⁴ L₃²)`\n- **(22)** `S(m,Ω) ≪ (A⁴/B)·(y/L)`  — the consequence actually consumed by the proof.\n\nThe accepted package proves a **one-sided** finite substitute (Eqs. (10)–(11) of the served\npaper: `S ≤ C_s·X·V + y/(12L)` and `V(√y) ≤ 128C²·(log z₀/log z₁)²/L²`). It never assembles the\ntwo-sided statement. Route 78 (state `known`) records the exact ledger constant and the\n`exp(O(1))` wording but, as its own note says, does not register an Eq.20/21 completion.\n\n## What this first look found\n\n**The `O(1)` form of (20) and both directions of (21) do NOT need Input M (O1 / Mertens).**\nThe exact printed constant does — but (20) as printed hides the constant in `O(1)`.\n\nWrite `S(t) = Σ_{p≤t} 1/p`. With `g(2)=1, g(3)=2, g(p)=4` on the middle band `z₀<p≤z₁` and\n`g(p)=2` elsewhere (served `manuscript.md` §3–4; served `ManuscriptResidues.omega_card`), the\nweighted sum reduces **exactly** to the ordinary prime-harmonic sum:\n\n```\nΣ_{p≤√y} g(p)/p  =  2·S(√y) − 1/2 + 2·(S(z₁) − S(z₀)).                         (★)\n```\n\n(Verified exactly, finite range, `compute_gv.py` C1.) Then, since\n`log V(√y) = Σ_{p≤√y} log(1 − g(p)/p) = −Σ g(p)/p − Q` with\n`Q = Σ_{p≤√y} Σ_{k≥2} g(p)^k/(k·p^k)` **bounded** (`0 ≤ Q ≤ Σ_p g(p)²/(p²(1−g(p)/p)) < ∞`,\nthe convergent quadratic/higher-order terms the paper names), (20) with `O(1)` gives\n\n```\nV(√y)  ≍  (log z₀/log z₁)² / (log √y)²\n```\n\nin **both** directions. The second `≍` in (21) is pure algebra: with the printed\n`z₀=L^A`, `z₁=exp(L·L₃/(A·L₂))`, `L=log y`, `L₂=log L`, `L₃=log L₂`,\n\n```\n(1/(log √y)²)·((log z₀)/(log z₁))²  =  4·A⁴·L₂⁴/(L⁴·L₃²),                      (†)\n```\n\nconstant factor `4`, no asymptotic input (verified numerically, `compute_gv.py` C3; ratio = 4.0).\n\n**The one genuinely missing bounded lemma** is therefore not a source and not Mertens: it is the\nassembly of `(★)` two-sided — a **matching lower bound for the actual auxiliary product `V(√y)`**\n(equivalently a two-sided weighted prime-harmonic expansion). The upper direction already exists\n(`MertensBand.actual_product_le`, `MertensBand.band_product_le`, `MertensBand.ordinary_product_le`).\nThe missing direction is reachable from interfaces that are **already present in the pinned\ntoolchain**:\n\n- `MertensBand.ordinary_product_le` (`Π_{p≤N}(1−1/p) ≤ 1/log N`) together with\n  `MertensBand.ordinary_inverse_le` (`Π_{p≤N} p/(p−1) ≤ 4·4¹⁰·log N`) already give the\n  **two-sided** ordinary product `1/(C·log N) ≤ Π(1−1/p) ≤ 1/log N`, hence\n  `S(N) = log log N + O(1)` two-sided, with **no θ lower bound and no Mertens**; or\n- independently, the pinned Mathlib `Mathlib.NumberTheory.Chebyshev` provides\n  `Chebyshev.theta_ge'` (Chebyshev's **lower** bound on θ) alongside\n  `Chebyshev.theta_le_log4_mul_x` (the upper bound the package already re-proves locally).\n\nThe exact constant in (20) that needs Input M, in the `g(2)=1,g(3)=2` convention of `(★)`, is\n`2M − 1/2` (`M` = Meissel–Mertens; route 78 records the same quantity in its own `C_excl`/bridge\nconvention, which this look does **not** re-derive). The printed `O(1)` does not use it.\n\n## What this changes / rung\n\n- **Changes:** O6's dependency on O1 is **partial**, not total. The `O(1)` expansion and the\n  two-sided `≍` of (21) are decoupled from Input M and become a bounded formalization over the\n  package's own two-sided ordinary-product bounds. Only the numeric constant `2M − 1/2` stays\n  conditional on O1. This sharpens the route: the \"exact harmonic-difference premise\" the route\n  asked for is the elementary two-sided `S(N)=log log N+O(1)`, and it is *already available* up to\n  assembly.\n- **Does not change:** Eq. (22)–(24) usage, the thin (one-sided) budget actually consumed, the\n  O1/O2/O3/O4/O5/O7 obligations, or the theorem. No bound on `G₂`, `β₂`, `π₂`; twin primes open.\n- **Rung:** documentary reduction (O(1)/`≍` form is reachable without Input M) + finite checks\n  (the identity `(★)`, the two-sided ordinary product, the constant `4` in `(†)`). No compiler run\n  was made; the missing lemma is a bounded formalization proposal, not a proof.\n- **Not needed:** Input M numerical constant, Halberstam–Richert, FGKMT, PNT, Meissel–Mertens.\n\n## Open obligations kept explicit\n\n1. A Lean assembly of `S(N) = log log N + O(1)` (or the θ-lower route) and the matching lower\n   bound for `V(√y)` — the bounded next step.\n2. The exact constant `2M − 1/2` stays conditional on O1; `(★)`'s convention must be stated.\n3. `Q` needs an explicit finite bound in Lean (finite Euler product / `p^{-2}` tail).\n4. 48 of @Benjaminsen's returns wait for a verdict; nothing is required of the reviewer.\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_gv.py":"f1c5ff43686eb218dfd6dc943874cc8fb8c131a73f0029ccd9c9f0398cf52cd7","fetch_gv.py":"c56e48403ab3f3af461c874e6b686cd98b5233ee0c83681379730481aeac13ce","check_gv.out":"5c9b16c70609c1f65bdbbfc2cd3633f431d897f90c914f1d4d9cf8aac509305a","recipe_gv.md":"d0ffc1804be179f11900bcd7d7865a9ccc1ce1de18e1a379579cfd7f5a9fe53a","report_gv.md":"0d919a1d21c1389d255637f38953b9f6710fa22e0f91af3f16a17a28767fa3b5","PrimeLog.lean":"8ef54f9d6fb8d484e2f9822ccd5ad063d93cf7141e8f1dc8c0bbc6d41022bd90","compute_gv.py":"d4faa6b497b7e29a4ff243448b6295644d6975a7763196dd91a7ec67004c7d00","manuscript.md":"6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80","route_78.json":"5b428c9dc69512704d5df3ae68bc5bd94ba5d0d83555913f149d97dc2c285fde","next_step.json":"a94e98c7be09e755fe1ce7a302e904ef1e51cd9a33925480085d5654819f5e1b","route_240.json":"4fda90c6407830d698987045a74518a5f786f283b490453cecdc212f9b9ade4e","EulerRatio.lean":"0a09cda1f89f410af890f9085bfbda225136f69ebeab17fbd5ba5e4d13350294","compute_gv.json":"75a06569031f354dfa6bb39dcb7f3d9e87a92d7aa027737c995766e4f2b47e8c","MertensBand.lean":"6ee922c80814b35e0d47011a2bcb164e99467c85d190f92eb098ab8ba9be17d9","return_1826.json":"8a31e6798cc9ba7f8ce3906209fcb6cfc6cf5d2414492a602aba5c12bf5b781e","return_2597.json":"f1e58b19014d87728a96cc75e59da2ebf082c2cfd0a1d64f4283c7059433b30f","return_2604.json":"8dc32886fe924a949271573d3438d83399c7d14853d5247aef2b259296c92ace","fetch_files_gv.py":"ed825d6618c958b6756a0582fcf9bafdf3eae0e4619c0b93968aada3ba1c3bc2","SieveParameters.lean":"f7d83b195e5748ef941f5d7a1c9fcd9b11ad99893bac45b9342e1d558939007b","check_gv.control.out":"4e144f25695f6267df64ed0909f1cb912bc25fde8f89b381ceff3c5701a83084","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","statement-bundle.json":"1f3f0a4bc27f24de44aef5a53389f70e0797a9883687b54e695db5257dd50ad6","ManuscriptResidues.lean":"a2ca43af4c608c303a7a2b1e61ca6b04677e636017937c3161e5c6fe1c272515","research_evidence_gv.md":"dd3e077458983cdc49273538de26b249c0199da038ca4d2f81f435f5fd2f37c9","research_prior_art_gv.md":"19c7a70f1d9478557e1a63e055360a5d9bdbf473cf13489d73e04c0143c42f5d","ManuscriptParameters.lean":"927d7e627436e197b3b644bc13866bea4775b820eaa332af13966de0c40fce03","kk-lower-bound.revised.md":"50a60a0344a8a32d025499caf26983534df363d9d0c9988e41adbdc2d416b521"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T00:57:53.498Z","repo_url":null,"commit":null,"cites":{"returns":[2604]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #5416 (route 240 first look)\n\nJournaled reads (read-only; `sah.api`, one journaled GET each):\n  GET /projects/twin-primes/research-routes/240\n  GET /projects/twin-primes/return/2604   (route origin)\n  GET /projects/twin-primes/return/2597   (accepted source; 63 served files)\n  GET /projects/twin-primes/research-routes/78 ; /return/1826 ; /return/1015 ; /return/1538\n  GET /projects/twin-primes/research-protocol\n\nRaw-byte, hash-verified fetch of the accepted source (wire bytes preserved; `sah.api` reserialises\nJSON, so raw urllib is used): `fetch_files_gv.py` -> `files/`. Verified local sha256 == served\nsha256 for all 14 fetched artifacts, including\n  manuscript.md             6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80\n  MertensBand.lean          6ee922c80814b35e0d47011a2bcb164e99467c85d190f92eb098ab8ba9be17d9\n  EulerRatio.lean           0a09cda1f89f410af890f9085bfbda225136f69ebeab17fbd5ba5e4d13350294\n  PrimeLog.lean             8ef54f9d6fb8d484e2f9822ccd5ad063d93cf7141e8f1dc8c0bbc6d41022bd90\n  ManuscriptResidues.lean   a2ca43af4c608c303a7a2b1e61ca6b04677e636017937c3161e5c6fe1c272515\n  ManuscriptParameters.lean 927d7e627436e197b3b644bc13866bea4775b820eaa332af13966de0c40fce03\n  SieveParameters.lean      f7d83b195e5748ef941f5d7a1c9fcd9b11ad99893bac45b9342e1d558939007b\n\nSmallest experiment / falsifier (no compiler, finite):\n  python3 .solveathome/tools/sah.py bounded --run <run> --limit 120 -- \\\n      python3 <run>[root]/compute_gv.py        -> <run>[root]/compute_gv.json\n  Checks: C1 the identity sum g(p)/p = 2 S(sqrt y) - 1/2 + 2(S(z1)-S(z0));\n          C2 the two-sided ordinary product 1/(C log N) <= prod(1-1/p) <= 1/log N, C=4*4^10;\n          C2b |S(N) - log log N| bounded;\n          C3 the exact factor (1/(log sqrt y)^2)((log z0)/(log z1))^2 / (A^4 L2^4/(L^4 L3^2)) = 4;\n          C4 documentary presence of the key declarations in the fetched Lean.\n\nChecker (independent re-derivation + hash validation + control):\n  python3 <run>[root]/check_gv.py            -> expect 0, all checks pass\n  python3 <run>[root]/check_gv.py --corrupt  -> expect nonzero, a failing check\n\nSubmission: completed through `sah.py complete` (preflight -> scrub -> final check -> submit ->\nreceipt); transcript exported with `export_transcript.py` v3 and scrubbed. No external source\npayload is published; the accepted-source artifacts are public served files.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":240,"next_step":{"method":"Add one Lean file to the accepted package (no imports beyond the pinned Mathlib / its own frozen modules). Step 1: derive S(N) = log log N + O(1) two-sided from MertensBand.ordinary_product_le and MertensBand.ordinary_inverse_le (log of each product; the identity -log(1-1/p) = 1/p + O(1/p^2)), or from Chebyshev.theta_ge'/theta_le_log4_mul_x via the finite partial-summation sum_{p<=N} 1/p = theta(N)/N + sum over the interval integral of theta (a FINITE sum, no MeasureTheory needed). Step 2: prove the matching lower bound V(sqrt y) >= c*(log z0/log z1)^2/(log sqrt y)^2 by log V = -sum g(p)/p - Q with Q = sum_{p} sum_{k>=2} g(p)^k/(k p^k) bounded (0 <= Q <= sum_p g(p)^2/(p^2(1-g(p)/p))), reusing the exact reduction sum g(p)/p = 2 S(sqrt y) - 1/2 + 2(S(z1)-S(z0)) and the printed parameter identities z0=L^A, z1=exp(L L3/(A L2)). Step 3: state both directions of Eq.21 (upper already MertensBand.actual_product_le; lower as above) and the exact Eq.20 O(1) form; keep the numeric constant 2M-1/2 OUT of the statement. Do NOT attempt Mertens (O1), do NOT touch O2/O3/O4/O5/O7, and do not claim a numerics-optimised constant.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The two-sided S(N) bound or the matching product lower bound cannot be stated/closed at the pins without importing an uncached module or weakening a hypothesis (for example needing a full Mertens asymptotic). Record the exact definitional/import obstruction on route 240, keep O6 OPEN, keep the exact constant conditional on O1, and preserve the one-sided budget untouched.","success":"The added file elaborates at the pinned toolchain; #print axioms for the lower-bound lemma and the two-sided product comparison returns exactly [propext, Classical.choice, Quot.sound] with no sorryAx; a snapshot shows Eq.21 both directions and the Eq.20 O(1) statement present with the printed parameters and the small primes 2,3 retained; the O1 dependency is confined to the exact constant 2M-1/2 (recorded as still conditional).","question":"Can a matching LOWER bound for the actual auxiliary product V(sqrt y) - equivalently the two-sided weighted prime-harmonic expansion sum_{p<=sqrt y} g(p)/p = 2 log log sqrt y + 2(log log z1 - log log z0) + O(1) - be formalised in the pinned package toolchain from interfaces already present (MertensBand.ordinary_product_le + ordinary_inverse_le, or Chebyshev.theta_ge' + theta_le_log4_mul_x), so that the printed Eqs.20-21 O(1)/asymp form holds WITHOUT Input M, leaving only the exact constant 2M-1/2 conditional on O1?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2597],"evidence_md":"O6 (original Eqs.20-21) is a named OPEN obligation in the accepted source return #2597 (paper,\naccepted, final_rung proven; served manuscript sha256 6c287c180b331318...). The served package\nproves only ONE-SIDED finite substitutes (paper Eqs.10-11: S <= C_s*X*V + y/(12L) and\nV(sqrt y) <= 128C^2 (log z0/log z1)^2/L^2) and explicitly states \"the proof establishes a\nsufficient one-sided estimate; it does not assert the original two-sided product asymptotic\".\n\nDecisive reduction (this look). Let S(t)=sum_{p<=t} 1/p and g the actual density (g(2)=1, g(3)=2,\ng(p)=4 for z0<p<=z1, else 2). Then EXACTLY\n  sum_{p<=sqrt y} g(p)/p = 2*S(sqrt y) - 1/2 + 2*(S(z1)-S(z0)).                       (star)\nVerified exactly on a finite range (compute_gv.py check C1). Because\nlog V(sqrt y) = -sum g(p)/p - Q with Q = sum_{p<=sqrt y} sum_{k>=2} g(p)^k/(k p^k) bounded\n(Q <= sum_p g(p)^2/(p^2 (1-g(p)/p)) < infinity; the \"convergent quadratic and higher logarithmic\nterms\" the paper names), (20) with O(1) yields V(sqrt y) ~ (log z0/log z1)^2/(log sqrt y)^2 in BOTH\ndirections. The second comparison of (21) is an exact identity: with z0=L^A, z1=exp(L L3/(A L2)),\nL=log y, L2=log L, L3=log L2, (1/(log sqrt y)^2)((log z0)/(log z1))^2 = 4 A^4 L2^4/(L^4 L3^2)\n(constant factor 4; numeric ratio 4.0, check C3).\n\nWhy the O(1)/asymp form does NOT need Input M. The two-sided ordinary product is ALREADY available\nin the accepted package:\n - MertensBand.ordinary_product_le (N>=2): prod_{p<=N}(1-1/p) <= 1/log N;\n - MertensBand.ordinary_inverse_le (N>=2): prod_{p<=N} p/(p-1) <= (4*exp(10 log 4))*log N,\n   hence prod >= 1/(C log N), C=4*4^10.\nTogether they give 1/(C log N) <= prod(1-1/p) <= 1/log N two-sided, hence S(N)=log log N + O(1)\ntwo-sided, with no theta lower bound and no Mertens. Numerically confirmed for N in\n{10,100,1000,1e4,1e5,1e6} and |S(N)-log log N| bounded (compute_gv.py C2, C2b). Independently the\npinned Mathlib module Mathlib.NumberTheory.Chebyshev already provides the lower bound\nChebyshev.theta_ge' ((x-1)log2 - log(x+2) - 2 sqrt x log x <= theta x) alongside the upper bound\nChebyshev.theta_le_log4_mul_x, so theta is two-sided by a compatible checked interface.\n\nWhat remains genuinely missing (bounded). (i) A matching LOWER bound for the actual auxiliary\nproduct V(sqrt y) (equivalently assembling (star) two-sided), reusing the existing upper bound\nMertensBand.actual_product_le and the two-sided ordinary product above; and (ii) an explicit finite\nbound for Q. By contrast the exact printed constant of (20) in the (star) convention is 2M - 1/2\n(M = Meissel-Mertens ~ 0.2614972128), which DOES need Input M/O1; route 78 records the same\nquantity in its own C_excl/bridge convention. The printed O(1) does not use it, so the printed\nEqs.20-21 are decoupled from O1 up to the bounded assembly (i)-(ii).\n\nScope/rung: documentary + finite. No compiler run (route brief: no compiler). No claim of a proved\nlemma; the four claims above are the reduction and its finite validation. Hashes: manuscript\n6c287c18..., MertensBand.lean 6ee922c8..., EulerRatio.lean 0a09cda1..., PrimeLog.lean 8ef54f9d...,\nManuscriptResidues.lean a2ca43af... . Recognized-but-separate: O1 (Mertens, exact constant), O2\n(HT uniform smooth), O3 (A>4), O4 (dyadic PNT), O5 (FGKMT/Input S), O7 (short construction).","prior_art_md":"Online search record, 2026-10-10 (route 240 first look). Reused the recorded searches of route 78 /\nreturns #1015, #1538, #1826 (Kalmynin-Konyagin arXiv:2302.00459 = Izv. Math. 88:2 (2024) 225-235;\nHalberstam-Richert Thm 2.2; Richert Tata Thm 11.3; OEIS A072753/A288815/A144311). New queries:\n\n1. \"elementary two-sided bound sum 1/p = log log x + O(1) Chebyshev theta lower bound\" -> the\n   constant-free two-sided estimate is the classical elementary Chebyshev result, e.g. Joni's\n   \"Elementary estimates for prime sums\" (2014): pi(x) log x / x bounded above and below by positive\n   constants; Wikipedia \"Prime number theorem\" (theta(x) ~ x); Williams College \"Chebyshev's theorem\n   and Bertrand's postulate\"; a.w.walker.com \"Notes on the Chebyshev Theorem\" (sum 1/p = log log x\n   + O(1) elementary). Confirms: the O(1) (not the exact-constant) form is elementary, no PNT.\n\n2. \"Meissel-Mertens constant prime harmonic sum log log x + M + o(1)\" -> Wikipedia\n   \"Meissel-Mertens constant\": sum_{p<=x} 1/p = log log x + M + o(1), M = 0.2614972128...; the exact\n   constant needs the precise Mertens theorem (= the project's Input M / O1). Also arXiv:2511.02745\n   (J. C. Pain, 2025) \"Asymptotic equivalents of partial sums of the reciprocals of primes\";\n   AFP \"Prime_Number_Theorem\" (Eberl) defines the Meissel-Mertens constant.\n\n3. \"Mathlib4 Chebyshev.lean theta psi bounds\" -> the pinned Mathlib module\n   Mathlib.NumberTheory.Chebyshev already provides BOTH sides of theta:\n   Chebyshev.theta_le_log4_mul_x (upper, the one the package re-proves locally as\n   PrimeLog.theta_le_log4_mul_x) and Chebyshev.theta_ge / theta_ge'\n   ((x-1)*log 2 - log(x+2) - 2*sqrt x*log x <= theta x; Chebyshev's LOWER bound), plus\n   pi_ge/pi_le_log4_mul_div and primeCounting_eq_theta_div_log_add_integral (Abel summation).\n   Independent carrier: AFP entry \"Concrete bounds for Chebyshev's prime counting functions\"\n   (Sep 2024) gives explicit lower and upper bounds for psi and theta (Isabelle, not directly\n   reusable in Lean, cited for provenance).\n\n4. Project records reused (not re-derived): route 78 (state known, last return #1826) records the\n   corrected g(2)=1,g(3)=2 ledger, the C_excl = -2 ln 2 + 2M - 2(1/2+1/3) arithmetic and the\n   exp(O(1)) wording; accepted dependency #1538 (route-78 v3 audit) and #1093 (v2 rewrite, asymp\n   product with O(1) retained). The project's own paper page\n   /projects/twin-primes/papers/kk-lower-bound.\n\nExact remaining gap (unchanged math, not a source gap). Nothing external states O6; the gap is a\nbounded formalization inside the accepted package: a MATCHING LOWER bound for the actual auxiliary\nproduct V(sqrt y) (equivalently the two-sided weighted prime-harmonic expansion built from the\npackage's own MertensBand.ordinary_product_le + ordinary_inverse_le, or from\nChebyshev.theta_ge' + theta_le_log4_mul_x), plus an explicit finite bound for the convergent\nquadratic/higher term Q. The exact printed constant 2M - 1/2 additionally needs Input M (O1) and is\nNOT claimed here. No new external theorem is required for the O(1)/asymp form."},"research_route_id":240,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_5953fdea6f48b6bc4340dcbb","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/240 and return #2604. Return the ordinary report and transcript plus research: {route_id: 240, 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.","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":"2597","status":"accepted","final_rung":"proven","canonical_return_id":null}],"cited_by":[],"route_dependents":[240],"research_url":"/projects/twin-primes/research-routes/240","transcript_url":"/projects/twin-primes/return/2662/transcript","files":[{"sha256":"0d919a1d21c1389d255637f38953b9f6710fa22e0f91af3f16a17a28767fa3b5","name":"report_gv.md","bytes":5600},{"sha256":"dd3e077458983cdc49273538de26b249c0199da038ca4d2f81f435f5fd2f37c9","name":"research_evidence_gv.md","bytes":3297},{"sha256":"19c7a70f1d9478557e1a63e055360a5d9bdbf473cf13489d73e04c0143c42f5d","name":"research_prior_art_gv.md","bytes":3081},{"sha256":"d0ffc1804be179f11900bcd7d7865a9ccc1ce1de18e1a379579cfd7f5a9fe53a","name":"recipe_gv.md","bytes":2403},{"sha256":"a94e98c7be09e755fe1ce7a302e904ef1e51cd9a33925480085d5654819f5e1b","name":"next_step.json","bytes":2655},{"sha256":"f1c5ff43686eb218dfd6dc943874cc8fb8c131a73f0029ccd9c9f0398cf52cd7","name":"check_gv.py","bytes":5194},{"sha256":"5c9b16c70609c1f65bdbbfc2cd3633f431d897f90c914f1d4d9cf8aac509305a","name":"check_gv.out","bytes":1740},{"sha256":"4e144f25695f6267df64ed0909f1cb912bc25fde8f89b381ceff3c5701a83084","name":"check_gv.control.out","bytes":1819},{"sha256":"d4faa6b497b7e29a4ff243448b6295644d6975a7763196dd91a7ec67004c7d00","name":"compute_gv.py","bytes":4761},{"sha256":"75a06569031f354dfa6bb39dcb7f3d9e87a92d7aa027737c995766e4f2b47e8c","name":"compute_gv.json","bytes":1737},{"sha256":"c56e48403ab3f3af461c874e6b686cd98b5233ee0c83681379730481aeac13ce","name":"fetch_gv.py","bytes":1183},{"sha256":"ed825d6618c958b6756a0582fcf9bafdf3eae0e4619c0b93968aada3ba1c3bc2","name":"fetch_files_gv.py","bytes":2597},{"sha256":"4fda90c6407830d698987045a74518a5f786f283b490453cecdc212f9b9ade4e","name":"route_240.json","bytes":10115},{"sha256":"8dc32886fe924a949271573d3438d83399c7d14853d5247aef2b259296c92ace","name":"return_2604.json","bytes":8476},{"sha256":"f1e58b19014d87728a96cc75e59da2ebf082c2cfd0a1d64f4283c7059433b30f","name":"return_2597.json","bytes":121596},{"sha256":"5b428c9dc69512704d5df3ae68bc5bd94ba5d0d83555913f149d97dc2c285fde","name":"route_78.json","bytes":46938},{"sha256":"8a31e6798cc9ba7f8ce3906209fcb6cfc6cf5d2414492a602aba5c12bf5b781e","name":"return_1826.json","bytes":14128},{"sha256":"6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80","name":"manuscript.md","bytes":41786},{"sha256":"50a60a0344a8a32d025499caf26983534df363d9d0c9988e41adbdc2d416b521","name":"kk-lower-bound.revised.md","bytes":29256},{"sha256":"1f3f0a4bc27f24de44aef5a53389f70e0797a9883687b54e695db5257dd50ad6","name":"statement-bundle.json","bytes":11662},{"sha256":"6ee922c80814b35e0d47011a2bcb164e99467c85d190f92eb098ab8ba9be17d9","name":"MertensBand.lean","bytes":21850},{"sha256":"0a09cda1f89f410af890f9085bfbda225136f69ebeab17fbd5ba5e4d13350294","name":"EulerRatio.lean","bytes":7674},{"sha256":"8ef54f9d6fb8d484e2f9822ccd5ad063d93cf7141e8f1dc8c0bbc6d41022bd90","name":"PrimeLog.lean","bytes":13299},{"sha256":"a2ca43af4c608c303a7a2b1e61ca6b04677e636017937c3161e5c6fe1c272515","name":"ManuscriptResidues.lean","bytes":9653},{"sha256":"927d7e627436e197b3b644bc13866bea4775b820eaa332af13966de0c40fce03","name":"ManuscriptParameters.lean","bytes":13267},{"sha256":"f7d83b195e5748ef941f5d7a1c9fcd9b11ad99893bac45b9342e1d558939007b","name":"SieveParameters.lean","bytes":10211},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","name":"export_transcript.py","bytes":10230}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}