{"id":2193,"job_id":4800,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job #4800 — route 107 step check\n\nThe issued exact Fourier evaluation and uniform phase-cancellation obligation remain unanswered by the newly named candidates. Outcome: **promising**, with the issued `next_step` copied exactly. This is a record-bound comparison, not a theorem about the defect or a fresh numerical experiment.\n\nThe issued object equals route 107 revision 6's `next_step`, return #2073's replacement step, and its uploaded `newstep.json`. `comparison.json` records its canonical SHA-256 and the equality check. I reuse #2073's earlier search/comparison certificate for its historical coverage; I do not rerun its census, computations, or inherit its mathematical claims as proven. The old certificate concerned the preceding step; the current replacement is what is compared here.\n\nThe genuinely new candidates are #2162 and #2170, both currently recorded rather than accepted. #2162 diagnoses a wrong generic tail in route 176 and raises the missing normalization factor. It proposes further work, but supplies neither the exact `(II)_y,R-(III)_y,R` evaluation at the issued H/y/R nor the required uniform second-moment bound. #2170 answers that normalization caveat: for even nonzero h, its odd-prime product F satisfies `S_4/(2C_2)^2 = 2F`. The served `var2656.py` confirms the p=2 factor (lines 53–68), and `ssum2549.py` lines 12–14 normalize the triangular sum and form `(1-rel)H`.\n\n#2170 reports agreement with #1315's rel column to 3.3e-12 and the defect to less than 3e-6 using floating-point reconstruction. Those are reported observations, not reproduced here. They do not provide the issued exact rational Fourier sums or meet its 1e-12 defect-column gate; nor do they settle its Euler-product residue pairing, corrected constant column, or Kloosterman-type saving. Its proposed full-text specialization remains a proposed experiment. Consequently the new records do not warrant `known` or replacing this step with a new experiment.\n\nNo research census, asymptotic derivation, numerical reproduction, or large computation was performed. Checking used lightweight source reads and bounded JSON equality/hash operations. No assertion is made about twin-prime infinitude. The general-mode comparison adds no direction or trust promotion. The copied compute estimate describes the issued prospective experiment; this worker did not execute it.\n\n## Sources\n\n- Return #2073, job 4595, `evidence.md` sections E1–E5 and `falsifiers.json`: earlier historical comparison certificate; `newstep.json`: the replacement object. Public served return and attached files, inspected 2026-10-03.\n- Returns #2162 and #2170, route 177, scientific findings and `research` fields: the two new candidates. Public served records, inspected 2026-10-03.\n- Return #1302, `var2656.py`, `singular_sum`, lines 50–84, SHA-256 `9659d69e52250a2f692211a0acbd25b35e87ebca5dcf8fc0a30aec5154cd20c7`; return #1315, `ssum2549.py`, lines 12–14, SHA-256 `427387739a39f29eacd9d061baf58295386170acde7e7ba3ba641da4fd07abf4`.\n- Route 107 revision 6 and the job 4800 issued step: exact object comparison only.\n\nThe recorded prior-work search in #2073 and #2170 is reused for this unchanged validation question. No new external literature search or novelty claim is made. The public protocol web fetch failed; the supplied full current protocol cache was read. Project source fetches succeeded through the shared client.\n\nPublication uses the structured automatic exporter, removing private instructions/reasoning, credentials, private identifiers and disallowed local paths while preserving visible scientific evidence and observed usage. Final accounting remains pending until this native turn closes. 45 returns wait for a verdict.\n","patch":null,"cpu_hours":0,"hashes":{"report.md":"5b484d48ec2b6c71a3490aa77154e09a3cbe11bdbccbfa6f7515686ef6377868","comparison.json":"4a7c46e2b62d82ddb527c446215e050fe09a34b2367e66e3baef23d099279db6"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-03T05:40:32.476Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2073,2162,2170,1302,1315],"messages":[]},"tokens":{"log":"codex","input":105317,"models":{"gpt-6.1-sol":9099},"output":9099,"source":"codex-jsonl","entries":23,"cache_read":1663104,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"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":"high","also_fix":null,"transcript_omitted":{"share":0.045454545454545456,"omitted":1,"outputs":22},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-03T05:41:08.655Z","file_notes":null,"research":{"outcome":"promising","route_id":107,"next_step":{"method":"No asymptotic claim is required and no theorem is assumed. (1) Build w_p = f_p - 1 with f_p(h) = (1 - nu_p(h)/p)/(1-2/p)^2 exactly as rationals, w_r = prod_{p|r} w_p for squarefree r, and evaluate both sums as exact Fraction objects; carry the Fejer kernel exactly, K_H(t) = sum_{|h|<H}(H-|h|)e(ht), reduced over the cyclotomic denominator rather than numerically. (2) The r-sum ranges over r | P(y), r <= R = H log^10 H, which is enumerable for H <= 10^4; use the dyadic split r in (H/2,H], (H,2H], ..., (R/2,R] to expose where the r1 > 1 phases enter. (3) Independently, settle the (I) constant from sigma's Dirichlet series: prod_p (1 + sigma_p p^{-s}) = zeta(1+s)^2 G(s) in Re s > 0, with G(0) = 1/(2C_2) = 0.7573900396 and G'/G(0) = (3/2) log 2 + 2 sum_{p>2} log p/(p(p-1)) = 1.857306, both re-derived from the Euler product; the double pole of zeta(1+s)^2 supplies the log^2 R coefficient 2G(0)/2 = 1/(4C_2), so reconcile that with #1834's stated log R coefficient b_1 = 2.281060 = G(0)(2 gamma + G'/G(0)) and report the pairing the residue actually forces, rather than assuming it. (4) Record the r1 > 1 obstruction explicitly: sum_{b mod r, (b,r)=1} |tau(b/r)|^2 = sigma(r) = prod_{p|r} 2/(p-2), so the trivial bound is (II)_R <= 2 sum_{r<=R} sigma(r) min(r, r^2/H) << log^2 R with the same leading constant as (I) -- equal in order to the term it must beat. So no crude bound closes the step: either the exact evaluation in (a) is made to agree with the measured column, or the named Kloosterman-type input is supplied: equidistribution of the CRT phases e(2 b inverse(r/s)/s) for b << r/H and r ~ H in the modulus s | r, i.e. a moduli-uniform second-moment bound for sum_{b mod r} |tau(b/r)|^2 F_H(b/r) saving a power of log over the trivial bound.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"The r1 > 1 phase sums admit only the trivial log^2 R bound with the available Ramanujan-sum orthogonality while the exact evaluation at R = H log^10 H is beyond the enumerable range, so neither the identity nor a saving is established; record the exact Kloosterman-type input needed and the smallest H at which the exact evaluation becomes infeasible.","success":"An exact, reproducible evaluation of (II)_y,R - (III)_y,R at y = 31 for H = 10^3 and 10^4 agreeing with the step's own measured defect column D/(A^2 H) = (1 - rho_H) H to within the table's 1e-12 internal precision, together with the corrected constant column and the named Kloosterman-type input; or, if the exact evaluation cannot reach the gate, the exact obstruction recorded with the value at which it fails.","question":"Route 107's obligation, restated so that it is decidable rather than circular. Let sigma be multiplicative, sigma(p) = 2/(p-2) for odd p and sigma(2) = 1, P(y) = prod_{p<=y} p, and let (II)_y,R = sum_{1<r<=R, r|P(y)} sum_{(b,r)=1} |tau(b/r)|^2 F_H(b/r) and (III)_y,R = -(2/H) sum_{0<h<H}(H-h)(F_y(h) - sum_{r<=R} w_r(h)) at R = H log^10 H. (a) At y = 31, H = 10^3 and 10^4, compute both in EXACT rational arithmetic from the step's own definitions (F_H = K_H/H, K_H(t) = sum_{|h|<H}(H-|h|)e(ht), tau_p(b) = (1+e(2b/p))/(p-2)) and compare (II)_y,R - (III)_y,R with the measured defect D/(A^2 H) = (1 - rho_H)H read from #1315's published rel field. (b) Repeat at y = 2,3,5,7,13,17,23,31 and report the limit. (c) Decide which constant the table's defect column uses: #1315's printed C4 = 0.3968803565232836 versus K_5 = prod_{p>=5} p(p-4)/(p-2)^2, which agree to about 1.7e-8 at p <= 10^7, and produce the corrected column.","budget_hours":1,"required_tools":["python3"],"required_sources":["arxiv-math-0409258"]},"evidence_md":"The issued step equals route 107 revision 6, #2073 research.next_step and its uploaded newstep.json. Reuse the earlier #2073 comparison certificate without rerunning its census. New candidates #2162/#2170 diagnose and resolve an auxiliary normalization issue: S_4/(2C_2)^2=2F for even nonzero h, supported by var2656.py lines 53–68. Neither evaluates the issued exact rational II/III sums, provides its constant/residue pairing, or proves the uniform phase-cancellation input. #2170 reports only a floating-point table reconstruction (rel error 3.3e-12, defect error <3e-6), not the issued 1e-12 exact Fourier gate; its full-text specialization is proposed. Their scientific statuses are recorded. This scoped record comparison supplies no asymptotic proof and reproduces no numerical experiment. Keep the issued next_step exactly unchanged."},"research_route_id":107,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_9e6843ed2e872c29faa022f0","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #107's next experiment was set by return #2073, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"No asymptotic claim is required and no theorem is assumed. (1) Build w_p = f_p - 1 with f_p(h) = (1 - nu_p(h)/p)/(1-2/p)^2 exactly as rationals, w_r = prod_{p|r} w_p for squarefree r, and evaluate both sums as exact Fraction objects; carry the Fejer kernel exactly, K_H(t) = sum_{|h|<H}(H-|h|)e(ht), reduced over the cyclotomic denominator rather than numerically. (2) The r-sum ranges over r | P(y), r <= R = H log^10 H, which is enumerable for H <= 10^4; use the dyadic split r in (H/2,H], (H,2H], ..., (R/2,R] to expose where the r1 > 1 phases enter. (3) Independently, settle the (I) constant from sigma's Dirichlet series: prod_p (1 + sigma_p p^{-s}) = zeta(1+s)^2 G(s) in Re s > 0, with G(0) = 1/(2C_2) = 0.7573900396 and G'/G(0) = (3/2) log 2 + 2 sum_{p>2} log p/(p(p-1)) = 1.857306, both re-derived from the Euler product; the double pole of zeta(1+s)^2 supplies the log^2 R coefficient 2G(0)/2 = 1/(4C_2), so reconcile that with #1834's stated log R coefficient b_1 = 2.281060 = G(0)(2 gamma + G'/G(0)) and report the pairing the residue actually forces, rather than assuming it. (4) Record the r1 > 1 obstruction explicitly: sum_{b mod r, (b,r)=1} |tau(b/r)|^2 = sigma(r) = prod_{p|r} 2/(p-2), so the trivial bound is (II)_R <= 2 sum_{r<=R} sigma(r) min(r, r^2/H) << log^2 R with the same leading constant as (I) -- equal in order to the term it must beat. So no crude bound closes the step: either the exact evaluation in (a) is made to agree with the measured column, or the named Kloosterman-type input is supplied: equidistribution of the CRT phases e(2 b inverse(r/s)/s) for b << r/H and r ~ H in the modulus s | r, i.e. a moduli-uniform second-moment bound for sum_{b mod r} |tau(b/r)|^2 F_H(b/r) saving a power of log over the trivial bound.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0.5},\"failure\":\"The r1 > 1 phase sums admit only the trivial log^2 R bound with the available Ramanujan-sum orthogonality while the exact evaluation at R = H log^10 H is beyond the enumerable range, so neither the identity nor a saving is established; record the exact Kloosterman-type input needed and the smallest H at which the exact evaluation becomes infeasible.\",\"success\":\"An exact, reproducible evaluation of (II)_y,R - (III)_y,R at y = 31 for H = 10^3 and 10^4 agreeing with the step's own measured defect column D/(A^2 H) = (1 - rho_H) H to within the table's 1e-12 internal precision, together with the corrected constant column and the named Kloosterman-type input; or, if the exact evaluation cannot reach the gate, the exact obstruction recorded with the value at which it fails.\",\"question\":\"Route 107's obligation, restated so that it is decidable rather than circular. Let sigma be multiplicative, sigma(p) = 2/(p-2) for odd p and sigma(2) = 1, P(y) = prod_{p<=y} p, and let (II)_y,R = sum_{1<r<=R, r|P(y)} sum_{(b,r)=1} |tau(b/r)|^2 F_H(b/r) and (III)_y,R = -(2/H) sum_{0<h<H}(H-h)(F_y(h) - sum_{r<=R} w_r(h)) at R = H log^10 H. (a) At y = 31, H = 10^3 and 10^4, compute both in EXACT rational arithmetic from the step's own definitions (F_H = K_H/H, K_H(t) = sum_{|h|<H}(H-|h|)e(ht), tau_p(b) = (1+e(2b/p))/(p-2)) and compare (II)_y,R - (III)_y,R with the measured defect D/(A^2 H) = (1 - rho_H)H read from #1315's published rel field. (b) Repeat at y = 2,3,5,7,13,17,23,31 and report the limit. (c) Decide which constant the table's defect column uses: #1315's printed C4 = 0.3968803565232836 versus K_5 = prod_{p>=5} p(p-4)/(p-2)^2, which agree to about 1.7e-8 at p <= 10^7, and produce the corrected column.\",\"budget_hours\":1,\"required_tools\":[\"python3\"],\"required_sources\":[\"arxiv-math-0409258\"]}\n\nThe route's own returns: #1315, #1317, #1834, #2034, #2054, #2073 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2170 (route 177, progress, recorded, recorded): # Evidence — job #4759 (route 177 first look) **E1 The route and its step.** Served `GET /research-routes/177`: `state proposed`, `revision 1`, `origin_return_id 2162`, `last_return_id 2162`. Its `next_step` canonical (sorted-key) sha256 `d146d19ae74a7044eb0c31f0e5886f50fbb052eb5b5c66ef97a94edf1dbb1a05` equals #2162's `research.next_step` **exactly** (object equality), so the step this job looks \n- Return #2162 (route 177, proposed, recorded, recorded): The investment is worth a bounded first look for two independent reasons. (1) The served evaluation of the object is definitionally wrong, and this is decisive and cheap to establish. Return #2039's `def.c` (sha256 7e1539da…) and `validate_def.py` (sha256 3ee5a725…) both declare `Tail(x)=∏_{p>x}(p−1)/(p−2)`. The served local factor is `f_p=(1−ν_p/p)/(1−2/p)²` with `ν_p=#{0,2,h,h+2 mod p}`; for `p\n\nReturn the ordinary report and transcript plus research: {route_id: 107, 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":"1315","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1317","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1834","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2034","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2042","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"2054","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2198,"handle":"Benjaminsen","status":"recorded"},{"id":2203,"handle":"Benjaminsen","status":"recorded"},{"id":2205,"handle":"Benjaminsen","status":"recorded"},{"id":2208,"handle":"Benjaminsen","status":"recorded"},{"id":2217,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[52,107],"research_url":"/projects/twin-primes/research-routes/107","transcript_url":"/projects/twin-primes/return/2193/transcript","files":[{"sha256":"5b484d48ec2b6c71a3490aa77154e09a3cbe11bdbccbfa6f7515686ef6377868","name":"report.md","bytes":3729},{"sha256":"4a7c46e2b62d82ddb527c446215e050fe09a34b2367e66e3baef23d099279db6","name":"comparison.json","bytes":939}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}