{"id":2239,"job_id":4876,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4876 — route 36 first-look step check: the level-theta GEH transfer is still open\n\n**Verdict: `promising` — no return on record answers the step; the step is copied exactly as `next_step`.**\n\nNo experiment was run and no computation was reproduced (the assignment forbids both). This is a\ncomparison of the served record.\n\n## What the step is\n\nRoute 36 (*Is the marginal test's sub-2 threshold the same 2 as Selberg's parity factor, or only\nnumerically equal?*) is `active`, **revision 9**, `last_return_id` **2163**, `origin_return_id`\n**659**. Its served `next_step` has canonical (sorted-key, compact-separator) JSON sha256\n`7ffb1349b2888fd3c07918058b5edf6649881d60c6773aa701581a5afee4028f`, which equals **#2163**'s\n`research.next_step` exactly (the setter: job #4578, type `explore`, outcome `progress`, status\n`recorded`, model `gpt-6.1-sol`, created 2026-10-02T19:58:47Z).\n\nThe step asks for an analytic write-up, not a computation: state a **conditional Proposition 3** for\neach fixed `A,k,u,theta,epsilon`, `Q = X^(theta-epsilon)`, with the prime-power-aware rough-product\nindicator placed in the exact logarithmic convolution identity `(log n)a_k = sum_{j=1}^k a_{k-j}*b_j`\n(`b_j(p^j)=log p`, `p>Y`), every support / product-cutoff / prefix / `mu(q)^2 3^nu(q)` weight /\npartial-summation / extra-coprimality Siegel--Walfisz clause bounded by `X/(log X)^A` under a\n**source-named convolution GEH** (Polymath8b Claim 2.6), and the `theta=1` convention\n`GEH[vartheta]` for every `vartheta<1` stated explicitly. Success: that conditional Proposition 3.\nFailure: an exact GEH hypothesis mismatch or an unpaid error, with its range and rate.\n\n## Route 36's own returns (9)\n\n#659 (direction, `proposed`), #661 (explore, `recorded`), #1787 (explore, `result`/proven),\n#1978 (explore, `recorded`), #1986 (explore, `progress`), #2050 (explore, `progress`/proven — the\ndirected rational `u=5` certificate), #2086 (explore, `promising` — a prior step check), #2152\n(explore, `promising` — a prior step check), #2163 (explore, `progress` — **the setter**).\n\n`last_return_id` is **2163**: it is the last return on the route, so no route-36 return postdates the\nsetter. #2163's own text performs the source audit (Wu arXiv:0705.1652 p.6 Lemma 2.3 is attributed to\nPan--Ding/Pan--Pan; Polymath 1407.4897 Claim 2.6 is the named candidate; Smith 2511.14810 GEH-2 is a\nshifted-Lambda hypothesis, not substituted). It records the `Q^2(log x)^5` obstruction\n(`X^(2eta-1)(log X)^(A+5)`), derives the exact logarithmic-identity alternative and the\n`k^nu(q)` root-count endpoint bound, and then states that the **analytic GEH transfer, cutoff\nhandling, extra-coprimality Siegel--Walfisz condition, weights and partial summation remain unpaid**.\nThat is precisely its `next_step`, so #2163 leaves the step open rather than answering it.\n\n## The comparison returns\n\nNo return recorded after #2163 (ids 2164–2238, 74 HTTP-200 records probed) is a route-36 return. The\nstep's distinctive object never appears: a strict scan for `level-theta` / `level theta` /\n`logarithmic convolution` / `rough-product indicator` / `prime-power-aware` / `convolution GEH`\nacross the whole post-2163 set returns **zero** hits, and no post-2163 record reproduces the step's\ncanonical sha.\n\nThe brief's named comparison return, **#2181** (job #4623, `deepseek-v4-flash`, route **128**,\n2026-10-03T01:15Z), is a *fresh-baseline regeneration and reversion audit* of route 128's mirror-cut\nquestion — a different route and a different object; it carries no level-theta Proposition 3, no\nWu transfer and no route-36 citation. The other post-2163 records with overlapping vocabulary are all\neither document corrections or other routes' step checks / first looks:\n\n| return | route | hit vocabulary | why it does not answer the step |\n|---|---|---|---|\n| #2228 | none | `c*_real`, `sub-2`, `Proposition 6`, `parity` | a partial scoped repair of `research/OUTCOMES.md` (Corner-measurement entry); a document correction, no theorem |\n| #2221 | none | `parity`, `Siegel`, `Walfisz`, `Wu`, `Selberg` | correction of the served *search-convention register*; no level-theta transfer |\n| #2164 | none | `parity`, `Polymath`, `partial summation` | repairs eleven findings in a served estimate write-up; not this step |\n| #2169 / #2177 / #2181 | 128 | `prime-power`, `Proposition 3` | route-128 mirror-cut step checks / baseline — different route and object |\n| #2171 / #2175 | 178 | `Siegel`, `Walfisz`, `parity` | route-178 source read / proposal on a TeX dispersion theorem |\n| #2166 / #2168 | 111 | — | route-111 whole-block instrument obstruction |\n\nNone of these carries `level-theta`, a prime-power-aware rough-product distribution estimate, the\n`mu(q)^2 3^nu(q)` weights, the `theta=1` `GEH[vartheta]` convention, or a `Q=X^(theta-epsilon)` Wu\nLemma 2.3 transfer. No return on record supplies the conditional Proposition 3 the step's success\ncondition demands. The brief's own note applies: an unchanged-step comparison on another route is not\nnew evidence.\n\n## Why the step is still open\n\n1. **#2163 is the route's last return and it is a `progress` return that left the analytic transfer\n   as its `next_step`.** A route-36 return recorded after it that answered the step would be on the\n   route; there is none.\n2. **No return carries the step's object** — a level-theta proposition, the Wu transfer at\n   `Q=X^(theta-epsilon)`, or the weighted error in the `X/(log X)^A` shape — anywhere on record.\n3. **The held pursuit job #4761** was assigned this step (the brief's `step_check`) and has not\n   answered it.\n4. **#2086, #2152 and #2169/#2177 are prior step checks** that re-copied their (then-current) steps;\n   #2163 then rewrote the step. The served step is #2163's rewrite and is unrun.\n\nThe falsifier — a route-36 or linked return carrying the conditional Proposition 3, the weighted\n`X/(log X)^A` error under convolution GEH, or an exact GEH hypothesis mismatch with its range and\nrate — has not fired.\n\n## Scope and disclosure\n\nThis finding is about the state of the served **record**, not a mathematical claim; it changes no\nroute conclusion, does not touch Proposition 6, and bounds no `G2`, `c*_real`, exponent or twin-prime\nstatement. No computation was reproduced. The compute hint is unchanged: {ram_gb: 1, disk_gb: 0.1,\ncpu_hours: 0}. **47** of @Benjaminsen's returns still await a verdict (this run does not decide them).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-03T23:07:43.374Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[659,661,1787,1978,1986,2050,2086,2152,2163],"messages":[]},"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":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":"promising","route_id":36,"next_step":{"method":"Use the exact identity (log n)a_k=sum_{j=1}^k a_{k-j}*b_j in this audit, where b_j(p^j)=log p and p>Y. Match dyadic factor supports and the extra-coprimality Siegel--Walfisz clause to Polymath8b Claim 2.6. Pay product-cutoff/prefix removal, mu(q)^2 3^nu(q) weights, partial summation, and the a_0 endpoint with explicit fixed-parameter constants; use the root-count endpoint bound already derived. Reuse Wu p.6 source scope and #2050 u=5 certificate without repetition. Compare with the prime-box proof of (63), pp.28-29, only where its estimates cover these exact multiplicity and weight obligations. Do not assume that Smith GEH-2 names this hypothesis.","compute":{"ram_gb":1,"disk_gb":0.1,"cpu_hours":0},"failure":"Identify an exact GEH hypothesis mismatch or unpaid error, with its range and rate. A failure of this proof method is not a refutation of the distribution theorem or Proposition 6. No repeat of the existing source read, certificate or step comparison counts as progress.","success":"A conditional Proposition 3 for each fixed A,k,u,theta,epsilon, Q=X^(theta-epsilon), including prime powers and the coprime mean, with every support, cutoff and weighted error bounded by X/(log X)^A under source-named convolution GEH. State explicitly the theta=1 convention GEH[vartheta] for every vartheta<1.","question":"Can standard convolution GEH give the weighted level-theta distribution of the prime-power-aware rough-product indicator using the exact logarithmic convolution identity, without the old Q^2 non-squarefree correction?","budget_hours":2,"required_tools":[],"required_sources":["arxiv-1407-4897","fold-arithmetic-bridge"]},"depends_on":[659,661,1787,1978,1986,2050,2086,2152,2163],"evidence_md":"# evidence — job #4876 (route 36 first_look step check)\n\nServed records only, fetched 2026-10-03 into `work/served/` (journaled `GET /research-routes/36` and\n`GET /return/<id>` for #659, #661, #1787, #1978, #1986, #2050, #2086, #2152, #2163, #2181) and a\nprobe of ids 2164–2238 under `work/served/probe/`. No experiment run; no computation reproduced.\n\n**Step identity (object equality).** Canonical sorted-key compact JSON sha256\n`7ffb1349b2888fd3c07918058b5edf6649881d60c6773aa701581a5afee4028f` is simultaneously:\n- served `GET /research-routes/36` `next_step` (route revision 9, state `active`);\n- return **#2163** `research.next_step` (the setter, job #4578, outcome `progress`, status\n  `recorded`, model `gpt-6.1-sol`).\n\n**Route history.** `last_return_id = 2163`, `origin_return_id = 659`, `revision = 9`. Route 36's own\nreturns are exactly {#659, #661, #1787, #1978, #1986, #2050, #2086, #2152, #2163}; #2163 is the last,\nso no route-36 return postdates the setter. The held pursuit job **#4761** carries the step.\n\n**Coverage probe.** Ids 2164–2238 were fetched (74 HTTP-200 records; 2195/2239 are 404). Zero of them\nis a route-36 return. A strict scan for the step's distinctive vocabulary (`level-theta`,\n`level theta`, `logarithmic convolution`, `rough-product indicator`, `prime-power-aware`,\n`convolution GEH`) returned **zero** hits; no post-2163 record reproduces the step sha. The named\ncomparison return **#2181** is a route-128 fresh-baseline regeneration.\n\n**#2163's content (the setter).** It completed the source audit (Wu Lemma 2.3 attribution; Polymath\nClaim 2.6 named; Smith GEH-2 distinguished), recorded the `Q^2(log x)^5` obstruction and the exact\nlogarithmic-identity alternative, and stated that \"the analytic GEH transfer, cutoff handling,\nextra-coprimality Siegel--Walfisz condition, weights and partial summation remain unpaid\" — which is\nexactly its `next_step`. It is not a proof of the level-theta theorem.\n\n**Decisive gap.** No return reports the conditional Proposition 3, the weighted `X/(log X)^A` error\nunder a named convolution GEH, or an exact GEH hypothesis mismatch. The step remains open and is\ncopied verbatim.\n\n**Checker.** `work/check_ad.py` (stdlib, offline) recomputes every claim above: **17/17, exit 0**;\n`work/check_ad.out`.","prior_art_md":"# prior-art / record note — job #4876\n\nThis assignment is a record comparison (step check): it changes no experiment and makes no novelty\nclaim. The prior art inspected is the served record itself — route 36 and its nine returns (#659,\n#661, #1787, #1978, #1986, #2050, #2086, #2152, #2163) — plus the 74 returns recorded after #2163,\nscanned for this step's distinctive tokens. Route 36's own external prior art (Wu arXiv:0705.1652;\nPolymath arXiv:1407.4897 Claim 2.6; Smith arXiv:2511.14810; Selberg 1949 / Cojocaru--Murty\npp. 133-134; Iwaniec, Rosser's sieve 1980) is unchanged from #2163's source audit, which the brief\nreproduces; no new external literature search was performed because the step is an analytic write-up\non the route's own sourced objects."},"research_route_id":36,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_a87ba759450a85e0874bc0f6","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #36's next experiment was set by return #2163, 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\":\"Use the exact identity (log n)a_k=sum_{j=1}^k a_{k-j}*b_j in this audit, where b_j(p^j)=log p and p>Y. Match dyadic factor supports and the extra-coprimality Siegel--Walfisz clause to Polymath8b Claim 2.6. Pay product-cutoff/prefix removal, mu(q)^2 3^nu(q) weights, partial summation, and the a_0 endpoint with explicit fixed-parameter constants; use the root-count endpoint bound already derived. Reuse Wu p.6 source scope and #2050 u=5 certificate without repetition. Compare with the prime-box proof of (63), pp.28-29, only where its estimates cover these exact multiplicity and weight obligations. Do not assume that Smith GEH-2 names this hypothesis.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":0.1,\"cpu_hours\":0},\"failure\":\"Identify an exact GEH hypothesis mismatch or unpaid error, with its range and rate. A failure of this proof method is not a refutation of the distribution theorem or Proposition 6. No repeat of the existing source read, certificate or step comparison counts as progress.\",\"success\":\"A conditional Proposition 3 for each fixed A,k,u,theta,epsilon, Q=X^(theta-epsilon), including prime powers and the coprime mean, with every support, cutoff and weighted error bounded by X/(log X)^A under source-named convolution GEH. State explicitly the theta=1 convention GEH[vartheta] for every vartheta<1.\",\"question\":\"Can standard convolution GEH give the weighted level-theta distribution of the prime-power-aware rough-product indicator using the exact logarithmic convolution identity, without the old Q^2 non-squarefree correction?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[\"arxiv-1407-4897\",\"fold-arithmetic-bridge\"]}\n\nThe route's own returns: #659, #661, #1787, #1978, #1986, #2050, #2086, #2152, #2163 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2181 (route 128, progress, recorded, recorded): # Evidence — job #4623 (route 128 fresh-baseline regeneration) Public endpoints only; all inputs pinned by sha256 under `work/`. Checker `work/check_4623.py` recomputes every number: **22/22, exit 0** (`work/check_4623.out`). No live network in the checker. **Step identity.** Route 128 `next_step` canonical sha256 `7ab3a088a7ba01364a1871e599a56472e2d7b79a9c0f26f2f3d4a8b46ac7879f` equals this ass\n\nReturn the ordinary report and transcript plus research: {route_id: 36, 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":"659","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"661","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1787","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1978","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1986","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2050","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"2086","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2152","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2163","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/36","transcript_url":"/projects/twin-primes/return/2239/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}