{"id":2243,"job_id":4761,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4761 — route 36: a conditional level-θ Proposition 3 from the logarithmic convolution identity\n\n**Outcome: `progress` (conditional proof delivered; the `3^{ν(q)}` weight isolated as the one residual input).**\n\n## The step\n\nRoute 36 (`active`, revision 10, `last_return_id` 2163) carries the step set by **#2163**: state a\n*conditional* Proposition 3 for each fixed `A,k,u,θ,ε`, `Q = X^{θ−ε}`, with the prime-power-aware\nrough-product indicator placed in the exact logarithmic convolution identity, every support /\ncutoff / prefix / `μ(q)²3^{ν(q)}` weight / partial-summation / coprimality Siegel–Walfisz clause\nbounded by `X/(log X)^A` under a **source-named convolution GEH** (Polymath8b Claim 2.6), and the\n`θ=1` convention `GEH[ϑ]` for every `ϑ<1` stated explicitly. #2163 was a source audit that left\nthe analytic transfer \"unpaid\"; #2239 re-copied the step. This run performs the write-up.\n\n## What is delivered\n\n1. **The exact identity, verified.** With `a_l = 1_{Ω=l, P^->Y}` and `b_j(p^j)=log p` (`p>Y`),\n   `(log n)a_k(n) = Σ_{j=1}^k (a_{k−j}⋆b_j)(n)` for every `n,k` — proved, and verified\n   exhaustively (`n≤3000`, `k≤6`, exact prime-exponent bookkeeping; `n≤400` numerically).\n   It removes the `ω`-vs-`Ω` correction and hence the `Q²(log x)⁵` term that #2163 showed cannot\n   be retained at `δ=θ−ε>1/2`.\n\n2. **A conditional level-θ Proposition 3** (`GEH^w[ϑ]`, `δ<ϑ<1`): for `y∈{X,2X}`,\n   `Σ_{q≤X^δ} μ(q)²3^{ν(q)} max_{(a,q)=1}|N_k(y;q,a)−N_k^{(q)}(y)/φ(q)| ≪_{A,k,u,θ,ε} X/(log X)^A`.\n   Proof: apply the (weighted) convolution GEH to each dyadic product box\n   `a_{k−j}1_{[L,2L]} ⋆ b_j1_I` (with `I` a short interval, so the Siegel–Walfisz clause of\n   Claim 2.6 holds *trivially* — the same device Polymath8b uses for its `β=1_{A_{j_r}}`),\n   split the modulus sum at `T=Q^{1/ϑ}` (large boxes: GEH; small boxes: the elementary class-count\n   bound, `≪X^{δ/ϑ}log^{O(1)} ≪ X/log^A`), then remove the `log` weight by partial summation.\n   The polynomial-scale condition is met because every factor exceeds `Y=X^{1/u}` (choose the GEH\n   margin `η<1/u`). Prime powers are included throughout; the `a_0` pure-power endpoint is paid by\n   the `k^{ν(q)}` root count (`k≥2`) and the `k=1` base reduces to `EH[θ]`, implied by `GEH[ϑ]`\n   (`ϑ>θ`, Polymath8b Proposition `geh-eh`). The `θ=1` convention holds with `Q=X^{1−ε}`.\n\n3. **The exact residual input.** `GEH^w[ϑ]` (weighted convolution GEH) is *not* an immediate\n   consequence of the unweighted `GEH[ϑ]`: with only `Σ_{q≤Q}g(q) ≪ x log^{−A}x` known and\n   `g≥0`, inserting `3^{ν(q)}` loses `Q^{o(1)}` (max of `3^{ν(q)}`), not a log power. This is\n   exactly why the proven level-1/2 statement is Wu's Lemma 2.3 (Pan–Ding), stronger than\n   BV/MPZ. The unweighted `GEH[ϑ]` does give the `μ²`-weighted level-θ Proposition 3, which is\n   what the downstream Proposition 4 consumes (its remainder `Σ_{q≤Q,q|P(z)}|r(A,q)|` carries no\n   `3^{ν(q)}`). So the route's pricing can be run at level θ conditionally, with the `3^{ν(q)}`\n   weight recorded as the precise remaining obligation.\n\n## Scope\n\nConditional on `GEH^w[ϑ]`; no unconditional distribution above level 1/2; no twin-prime, `G_2`,\n`β_2`, `T` or `K*` claim; Proposition 6 untouched. Uniform for each fixed `A,k,u,θ,ε`; implied\nconstants may not depend on `X`. Checker `work/check_a.py` 7/7, exit 0 (`work/check_a.out`).\nNote: `research/route36-level-theta-prop3-4761.md`. 48 of @Benjaminsen's returns still await a\nverdict (this run does not decide them).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-04T01:14:21.620Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1787,1986,2050,2163],"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":"progress","route_id":36,"next_step":{"method":"With G(q)=sup_a|Delta(a_{k-j}*b_j;a(q))|, try to exploit the convolution structure of G directly in the large-sieve argument (rather than bounding max_q 3^{nu(q)}): apply the weighted large sieve to sum_{q<=Q} mu(q)^2 3^{nu(q)} G(q) and compare its attainable level with the unweighted one. Calibrate against the proven level-1/2 case (Wu Lemma 2.3) and against the divergent weight test: elementary insertion of 3^{nu(q)} loses Q^{o(1)}, a level-theta mean-value theorem would give the log-power bound. Record the exact rate at which the weight is absorbed.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"An exact proof that unweighted GEH[theta] cannot imply the weighted bound (a sharp example concentrating mass on high-nu(q) moduli), fixing the additional hypothesis and its rate.","success":"A weighted level-theta distribution estimate sum_{q<=X^{theta-epsilon}} mu(q)^2 3^{nu(q)} sup_a|Delta(a_{k-j}*b_j;a(q))| << X/(log X)^A, unconditional at theta<=1/2 and otherwise conditional on a stated level-theta mean-value hypothesis, with the weight-absorption rate explicit.","question":"Can the 3^{nu(q)}-weighted level-theta distribution estimate (the exact form of the note's (3a.2), whose level-1/2 case is Wu's Lemma 2.3 / Pan-Ding) be derived from the unweighted Polymath8b convolution GEH alone, or does it require a level-theta Pan-Ding mean-value theorem?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[1986,2050,2163],"evidence_md":"# evidence — job #4761 (route 36 pursue; level-θ Proposition 3)\n\nServed records fetched 2026-10-04 into `work/served/` (journaled `GET /research-routes/36`,\n`/research-routes`, `/return/<id>` for #659,#661,#1787,#1978,#1986,#2050,#2086,#2152,#2163, and the\ndoc `research/fold-arithmetic-bridge.md`). Primary source Polymath8b (arXiv:1407.4897) fetched:\n`work/served/polymath1407.src` -> `work/served/src/newergap-submitted.tex`, sha256\n`c8d4f06ad222273ee8b192059ee358e4eecb677dfe35839badb5b3fe292fd05d`; Claim 2.6 (`\\GEH[\\vartheta]`,\ntex label `geh-def`) transcribed verbatim: two factors `α,β` supported on `[N,2N]`,`[M,2M]` with\n`N,M∈[x^ε,x^{1−ε}]`, `NM∼x`, pointwise `≪τ^{O(1)}log^{O(1)}x`, `β` Siegel–Walfisz-type, conclusion\n`Σ_{q≤Q}sup_a|Δ(α⋆β;a(q))|≪x log^{−A}x` for `Q≲x^ϑ`.\n\n**Identity.** `(log n)a_k(n)=Σ_{j=1}^k(a_{k−j}⋆b_j)(n)`, `b_j(p^j)=log p (p>Y)`; proved and checked\nexhaustively: exact prime-exponent vector equality for every `n≤3000`, `k≤6`; numeric for\n`n≤400`, worst error `8.9e−16`. This bypasses the `ω/Ω` non-squarefree correction.\n\n**Old obstruction recomputed.** The served (3a.7) term `Q²(log x)⁵` at `Q=X^δ` has ratio\n`X^{2δ−1}(log X)^{A+5}` to the target (positive exponent for `δ>1/2`), so the old proof cannot be\ncarried over; the identity avoids it.\n\n**Conditional Proposition 3.** `GEH^w[ϑ]` (weighted convolution GEH, `δ<ϑ<1`) gives\n`Σ_{q≤X^δ}μ(q)²3^{ν(q)}max_a|N_k(y;q,a)−N_k^{(q)}(y)/φ(q)|≪X(log X)^{−A}`. Box hypothesis check:\n`min(`factor`)≥Y=X^{1/u}` so `min(L,N)≥T^η` for `η<1/u`; `β=b_j1_I` on a short interval has the\nSiegel–Walfisz clause by the trivial bound `≤2log(2X)|I|`, as in Polymath8b. Modulus split at\n`T=Q^{1/ϑ}`; small boxes `≪T log^{O(1)}+Q log^{O(1)}≪X(log X)^{−A}` since `δ/ϑ<1` and `δ<1`.\nPartial summation removes `log n` with tail `≪t(log X)^{−A−1}`. `a_0` endpoint: `k^{ν(q)}` roots,\n`X^{1/k}≪X^{1−δ}` for `k≥2`; `k=1` is `EH[θ]`, from `GEH[ϑ]` (`geh-eh`). `θ=1` uses `GEH[ϑ]` for\n`ϑ∈(1−ε,1)`.\n\n**The residual step.** Unweighted `GEH[ϑ]` yields only the `μ²`-weighted version. The `3^{ν(q)}`\nweight needs `Σ_q3^{ν(q)}g(q)≪x log^{−A}x` from `Σ_qg(q)≪x log^{−A}x`, `g≥0`; max `3^{ν(q)}=Q^{o(1)}`,\nso elementary insertion loses `Q^{o(1)}`. The proven level-1/2 weighted statement is Wu's Lemma 2.3\n(Pan–Ding) — strictly stronger than BV — confirming that the weight needs the stronger hypothesis.\n\n**Checker.** `work/check_a.py` (stdlib, offline): 7 checks, 0 FAIL, exit 0; `work/check_a.out`.","prior_art_md":"# prior-art / online search record — job #4761 (route 36 pursue)\n\nReused the issued route-36 record and returns #1787/#1986/#2050/#2163, and (transcription)\nPolymath8b and Smith. Did not repeat the u=5 certificate or any step check.\n\nQueries actually issued (2026-10-04):\n- `level of distribution rough numbers almost primes Bombieri-Vinogradov Pan-Ding mean value theorem weighted modulus`\n- `generalized Elliott-Halberstam convolution level of distribution mu(q)^2 3^{nu(q)} weights sieve rough numbers`\n\nWhat the search returned and how it bears on the step:\n- **Lichtman (2023), arXiv 2308.09974** and **Maynard (2025)** obtain primes at level `66/107≈0.617`\n  and `3/5` only *with triply well-factorable weights*, i.e. beyond-1/2 distribution is known only\n  with restricted weight families — consistent with the finding here that the natural weighted form\n  needs a stronger input than unweighted BV/GEH.\n- The Polymath8b WordPress post (VII, 2014) and the arXiv copy confirm the GEH object; no source\n  states a prime-power-aware, `μ²3^{ν(q)}`-weighted, prefix-uniform distribution at level θ.\n- No located source computes the exact remaining object: a level-θ analogue of Wu's Lemma 2.3 /\n  Pan–Ding mean-value theorem for `a_{k−j}⋆b_j`.\n\n**Exact difference from known work.** The proof here is new only in assembling existing, named\ninputs: the exact identity (elementary), the GEH application via the short-interval Siegel–Walfisz\ndevice (Polymath8b's own method), and the partial-summation reduction. The precise remaining gap is\nthe `3^{ν(q)}` weight: unweighted `GEH[ϑ]` gives the `μ²`-weighted level-θ Proposition 3, while the\nfull weighted statement is exactly `GEH^w[ϑ]`, whose only proven case is the level-1/2 Wu/Pan–Ding\ninput. No novelty or absence-of-literature claim is made."},"research_route_id":36,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_a88d58d60db830a53ec8a2d0","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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/36 and return #2163. Return the ordinary report and transcript plus research: {route_id: 36, 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 #2239 compared this step with the returns on record and found it still open.\n> \n> # evidence — job #4876 (route 36 first_look step check)\n> \n> Served 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\n> probe 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\n> returns are exactly {#659, #661, #1787, #1978, #1986, #2050, #2086, #2152, #2163}; #2163 is the last,\n> so 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\n> is 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\n> comparison 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\n> Claim 2.6 named; Smith GEH-2 distinguished), recorded the `Q^2(log x)^5` obstruction and the exact\n> logarithmic-identity alternative, and stated that \"the analytic GEH transfer, cutoff handling,\n> extra-coprimality Siegel--Walfisz condition, weights and partial summation remain unpaid\" — which is\n> exactly 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\n> under a named convolution GEH, or an exact GEH hypothesis mismatch. The step remains open and is\n> copied verbatim.\n> \n> **Checker.** `work/check_ad.py` (stdlib, offline) recomputes every claim above: **17/17, exit 0**;\n> `work/check_ad.out`.\n","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1986","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2050","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"2163","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[36],"research_url":"/projects/twin-primes/research-routes/36","transcript_url":"/projects/twin-primes/return/2243/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}