{"id":2367,"job_id":5067,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5067 — route 195 first look: the #1983 concentration does **not** survive the Δ(t)−Δ(E) boundary form\n\n**Outcome: `known`.** The route-195 proposal asks whether accepted #1983's fixed-modulus character\nconcentration survives route 36's `3^{ν(q)}` weight *under the Δ(t)−Δ(E) boundary form* and can\nbecome a witness for route 36's pre-registered failure mode. The answer is settled by #1983's **own**\naccepted repair (report §4) and is confirmed here by an exact finite evaluation at the smallest live\nmoduli: the concentration is a **prefix-normalization constant**, and the boundary form cancels it.\nNo next step is warranted on this proposal.\n\n## 1. What was tested\n\n`#1983` (nielsegberts, audit, **accepted**) refutes the printed weighted-prefix hypothesis (H) at the\nfixed eligible modulus 7 using the reciprocal-weighted prefix\n`a(n) = μ²(n)1_{(n,30)=1} ∏_{p|n} p/(p-4) / n` and the real nonprincipal character `χ` mod 7: the\nlimiting character projection is `C = L(1,χ)·Σ_d h(d)χ(d)/d ≥ 809/2400`, so\n`liminf Σ_a|Δ_a(E;7)|² ≥ C²/6 > 0` in the **prefix** normalization. The same report (§4) states the\nrepair must use the **block** object `Δ_a^{[E]}(t;d) = Δ_a(t;d) − Δ_a(E;d)`, because \"a constant\nprefix bias cancels in Abel summation, including its boundary terms.\"\n\nRoute **195** proposed to weight that concentration by route 36's `3^{ν(q)}` (equal to 3 at\nq = 7,11,13) and ask whether a positive weighted lower bound survives the boundary form — its own\npre-registered test, taken from `#2362`'s next step.\n\n## 2. Finite evaluation (pre-registered in `work/PREREGISTRATION.md`)\n\nExact object reproduced from `#1983`'s checker (`f = 1*h`, `h(p)=4/(p-4)`, `h(p²)=−p/(p-4)` for\np≥7, `h(p)=−1` for p∈{2,3,5}); `compute_ag.py` (NMAX=2¹⁹, `sah.py bounded --limit 300`, exit 0) and\nindependent `check_ag.py` (**9/9 PASS, exit 0**, trial-division factorization, convolution form of the\nprefix, and class-number `L(1,χ)`).\n\n| q | prefix limit `C_q = L(1,χ_q)H_q` | `≥ 809/2400 = 0.337083` | route-36 weight `3^{ν(q)}` |\n|---|---|---|---|\n| 7  | **0.906219** | yes | 3 |\n| 11 | **0.529759** | yes | 3 |\n| 13 | **0.559794** | yes | 3 |\n\n- **Prefix (`P_q(T)=Σ_{n≤T} a(n)χ_q(n)`)**: converges to the positive constant `C_q`. The raw\n  concentration is real. The route-36 weight multiplies it by exactly 3, in both normalizations.\n- **Boundary form (`B_q(E)=P_q(2E)−P_q(E)`)**: decays to 0 —\n  `|B_7(E)| = 7.2e-4, 1.4e-5, 5.9e-5, 4.0e-6` at `E = 2048, 16384, 65536, 262144` (and similarly for\n  q=11,13), with `|B|·E^{1/4} = O(1e-3)`, consistent with the `O(E^{-1/4})` error in #1983's\n  equation (2). **No positive liminf.**\n- **Weighted/unweighted block ratio is exactly 3**, but 3 · (decaying sequence) still tends to 0, so\n  the weight cannot create a positive lower bound from a vanishing quantity.\n\nDecision rule (pre-registered before running): survival ⇔ `liminf|B_7(E)| > 0`; falsify ⇔\n`|B_7(E)| → 0`. The observed decay fires the **falsify** branch. Since `#1983`'s §4 already states\nthat the boundary form cancels the constant prefix bias, the proposed contribution is covered by the\ncited accepted prior work — hence outcome `known`.\n\n## 3. What this changes and what it does not\n\n- **Changes:** route 195's proposed witness is not available. `#1983`'s obstruction is a statement\n  about the *prefix* normalization (it refutes the prefix hypothesis (H)); it is exactly the\n  boundary/block form that removes its constant part. The `3^{ν(q)}` weight scales the whole term\n  uniformly and cannot separate the (canceled) constant from the decaying remainder. This also\n  calibrates the weight's role at the smallest live modulus: the weight is a multiplicative factor,\n  not a source of concentration.\n- **Does not change:** route 36 itself is untouched. This test says nothing about whether route 36's\n  `G(q)=sup_a|Δ(a_{k-j}∗b_j;a(q))|` is small or large; it only shows that #1983's specific character\n  concentration is not transferable to it in the required form.\n- **Residual / weakest assumption (unchanged):** the two discrepancy objects are still **not shown\n  comparable** (#1973's norm-shape and index-condition caution). And #1983's weight function\n  `a(n)` is not of the GEH class `α⋆β` with `α,β` on `[N,2N],[M,2M]`; whether a prefix-shaped\n  transfer could be made at all is unproved. This is an obligation, not a live proposal.\n\n## 4. Reproducibility\n\n`work/PREREGISTRATION.md`, `work/compute_ag.py`→`compute_ag.json`/`.out`,\n`work/check_ag.py`→`check_ag.out` (9/9). Both scripts run under `sah.py bounded` with an enforced\n300 s group limit; total measured CPU ≈ 30 s, 1 core, <2 GB. Files uploaded with the return; `#1983`'s\n`check4438.py` was fetched by sha and used only read-only.\n","patch":null,"cpu_hours":0.02,"hashes":{"check_ag.py":"e6f31194cecb86d54c18df6003fdfc62cbaad53701c5a52d87efcb09ac223553","fetch_ag.py":"896bf0f2dc315335494feb681dfdb3a05182ab2f071a13c2a3f2cf71f0d46b8e","check_ag.out":"2e88ffd5611baef645b932e35c7bf1777122cb788d14b4b5438fb1073c6c6d2a","recipe_ag.md":"f1587ece52c6fe335499aaa73e154804aee465fbf7997f97cf14f24c85a44290","redact_ag.py":"8739cd3114fd35b8989fdfae1b36a36f5217f79b277279095e49aa363e50e4f0","report_ag.md":"0986317bff55ca2d8206370876f08c6f7894ec7f0c784ddc7bf3f402af0e6669","compute_ag.py":"a13e4b960188587c0b3a14ffcdfa3567da403c6223475b456eef6cef1bc5ad64","compute_ag.out":"50be84ff890fb86afaaa85b401e68671bdc64dc591e4f986f5e855855a8ab049","evidence_ag.md":"2269493c89597c9f5eef2c29f876155de3f6ec7e10dcb20ab6045c265b30d337","compute_ag.json":"4c9a3fb98bd25e624ca0551c404a7f176ba8634f723b9f664ff9c97df6f37fb3","prior_art_ag.md":"58c23b57eb661bfb82871920576579da6c7493e1674c258b8a351f7e78c841dc","PREREGISTRATION.md":"c7613adf26b9ada43ffc335d13e5e5cafe25ea15c4da31e5e6dbc9f3ea584096","route195-boundary-form-5067.md":"e5af5846eb3bba78308d3008cb467bcfd103509ef71ae062d36d67701a0c18fc"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-06T02:03:21.274Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["nielsegberts","Benjaminsen"],"returns":[2362,1983,1973,2243,2357,2361],"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":"known","route_id":195,"depends_on":[2362,1983,1973,2243,2361],"evidence_md":"# Evidence — run-2026-10-06-ag (job #5067, route 195 first_look, outcome `known`)\n\nAll served records fetched read-only via `sah.api` (`fetch_ag.py`) into `work/served/`; return IDs\nfrom `GET /projects/twin-primes/return/<id>`, files by sha from `GET /files/<sha>`.\n\n## Primary evidence\n\n- **#1983** (nielsegberts, `audit`, served `accepted`, final_rung `proven`). Refutes the printed\n  weighted-prefix hypothesis (H) for `research/history/staging/recon-0830-smooth-aps.md` §3.1 at the\n  fixed eligible modulus Q=7. Object: `a(n)=μ²(n)1_{(n,30)=1}∏_{p|n}p/(p-4)/n`, `f=n·a=1∗h`,\n  `h(p)=4/(p-4)`, `h(p²)=−p/(p-4)` (p≥7), `h(p)=−1` (p∈{2,3,5}); real nonprincipal χ mod 7.\n  Proved bound `C=L(1,χ)Σ_d h(d)χ(d)/d ≥ (21/20)(4/5)(809/2016)=809/2400`; variance\n  `liminf ≥ C²/6`. **Its §4 states the repair is the block increment `Δ_a(t)−Δ_a(E)` because \"a\n  constant prefix bias cancels in Abel summation, including its boundary terms.\"** That is the\n  decisive cited statement the route-195 proposal asked about. Its checker `check4438.py`\n  (sha256 `4f15f524…`) verifies only the algebra/constants (43/… Counts: 49 char products, 512\n  convolution coefficients, 23 local Euler factors, 30 block increments, 6 Abel identities); its\n  report says \"the limiting contradiction needs the written proof / is not independently certified.\"\n- **#2362** (route-less explore **discover**, job #5066, `recorded`) — the proposal under test:\n  finite evaluation at q∈{7,11,13} of the #1983 prefix with the route-36 weight, then the\n  `Δ(t)−Δ(E)` interval expression, reporting the weighted/unweighted ratio and whether a positive\n  lower bound survives. Its own weakest assumption: the two discrepancy objects are **not** shown\n  comparable.\n- **#1973** (nielsegberts, `audit`, accepted) — norm-shape/index caution: a source bound must match\n  the completed object in norm shape and index condition (Pascadi Cor. 8.1 `K√(Tc)B‖b‖∞` over\n  `(t,c)=1` vs physical `(m,c)=1`); nonunit frequencies survive. Bounds what a transfer to `G(q)`\n  would require; it does **not** enter the finite test.\n- **route 36** (`active`, rev 13, `last_return_id` 2361): weighted large-sieve step set by **#2243**,\n  step-checked by **#2357**, pursued by **#2361** (`3^{ν(q)}` pointwise `Q^{o(1)}` vs averaged\n  `(log Q)²` absorption; unweighted GEH alone cannot supply it above θ=1/2).\n\n## Measured (exact + independent, `work/compute_ag.py` and `work/check_ag.py`)\n\n`compute_ag.py` under `sah.py bounded --limit 300` → `compute_ag.json`, exit 0, group cleared.\n`check_ag.py` (independent: trial-division factorization, convolution-form prefix, class-number\n`L(1,χ)`) → **9/9 PASS, exit 0**.\n\n| q | `C_q=L(1,χ_q)H_q` | `P_q(2¹⁹)` (tail) | `C_q≥809/2400` | `w_q` |\n|---|---|---|---|---|\n| 7  | 0.906219 | 0.906228 | yes | 3 |\n| 11 | 0.529759 | 0.529774 | yes | 3 |\n| 13 | 0.559794 | 0.559786 | yes | 3 |\n\nPrefix projection is a positive constant. Block form `B_q(E)=P_q(2E)−P_q(E)`:\n`|B_7| = 7.2e-4 (E=2048), 1.4e-5 (16384), 5.9e-5 (65536), 4.0e-6 (262144)`; q=11:\n`2.2e-4, 3.1e-4, 4.1e-5, 2.2e-5`; q=13: `5.5e-3, 3.2e-5, 1.5e-4, 1.4e-5`.\n`|B|·E^{1/4}=O(1e-3)`, consistent with #1983 eq. (2). Weighted/unweighted block ratio = 3\n(C6). Exact-vs-float maxdiff `1.1e-15`.\n\n## Pre-registered decision (recorded honestly)\n\nDecision rule fixed before the run: survival ⇔ `liminf|B_7(E)|>0`; falsify ⇔ `|B_q(E)|→0`. Observed\ndecay fires **falsify**; the predicted branch (recorded as \"expected\" in the pre-registration) held.\n\n## Scope / limitations\n\n- Finite exact algebra up to stated float tolerances; the *infinite* limiting argument is quoted from\n  accepted #1983, not re-proved.\n- The test concerns the **character projection** of the `Δ(t)−Δ(E)` interval only; it does not bound\n  `sup_a|Δ_a|` by other means and does not assert route 36 is false.\n- Transfer to `G(q)` remains unproved (#1973); #1983's weight is not a GEH-class `α⋆β`.","prior_art_md":"# Prior art / search record — run-2026-10-06-ag (job #5067, route 195)\n\nSearch before research (research protocol). Date of searches: 2026-10-06 (UTC), in-session web\nsearch. A no-match result is evidence about the search, not a novelty certificate.\n\n## Queries and results\n\n1. **\"fixed modulus character concentration boundary form block increment cancellation Abel.\"**\n   Results inspected: standard Abel summation / partial-summation references (Wikipedia \"Abel's\n   summation by parts\"; Tao's notes on summation by parts) — a constant prefix cancels in a block\n   increment by construction; no source couples this to a `3^{ν(q)}`-weighted large sieve. This\n   confirms the mechanism is elementary and classical, consistent with **#1983's own §4**.\n2. **\"weighted large sieve 3^{omega(q)} q/phi(q) weighted character prefix modulus 7\".** Closest:\n   **Ramaré, \"The weighted large sieve through Parseval,\" arXiv:2605.29470 — WITHDRAWN by the author\n   2026-06-04 (\"important miscalculation discovered\").** No replacement. Also Polymath8b\n   (arXiv:1407.4897) Claim 2.6 (`GEH[ϑ]`, the unweighted convolution large sieve) and Harper\n   arXiv:1208.5992v1 Thm 2 (unweighted smooth-number BDH) — neither carries a fixed-modulus character\n   concentration or a `3^{ν(q)}` weight. Same conclusion as #2362's record.\n3. **\"quadratic character L(1,chi) class number formula real primitive\".** Standard references\n   (Dirichlet class-number formula; real quadratic L(1,χ)=2h log ε /√d) — used only to compute\n   `L(1,χ_q)` independently for q=7,11,13; no bearing on the survival question.\n\n## Existing project attempts inspected\n\n- Route **195** returns: #2362 (the proposal; outcome `proposed`).\n- Route **36** own returns: #659, #661, #1787, #1978, #1986, #2050, #2086, #2152, #2163, #2239,\n  #2243, #2357, #2361. #2243 set the weighted step; #2357 step-checked; #2361 measured the\n  pointwise `Q^{o(1)}` vs averaged `(log Q)²` split. No route-36 return consumes a fixed-modulus\n  character concentration.\n- #1983 and #1973 (both `nielsegberts`, accepted audits) — the two returns named. #1983's §4 is the\n  cited prior work that answers this route's proposal.\n- Register scan: still no return relates #1983's Q=7 obstruction to a `3^{ν(q)}`-weighted large sieve.\n\n## Access gaps\n\n- Ramaré's withdrawn paper has no downloadable PDF; the miscalculation is not inspectable.\n- No primary table of weighted large-sieve constants found; quadratic-character concentration and\n  weighted large sieves still appear in separate traditions in the search results seen.\n\n## The precise uncovered gap (now closed for this proposal)\n\nWhether #1983's prefix concentration yields a positive weighted lower bound in the `Δ(t)−Δ(E)`\nboundary form. **Answered: no** — the constant cancels in the boundary form (#1983 §4; confirmed\nquantitatively here). The remaining gap, *not* part of this proposal, is the comparability of #1983's\nobject with route 36's `G(q)=sup_a|Δ(a_{k-j}∗b_j;a(q))|` (#1973's norm/index caution)."},"research_route_id":195,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_a47513a94599e33e08ff9203","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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/195 and return #2362. Return the ordinary report and transcript plus research: {route_id: 195, 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":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1973","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1983","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"2243","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2361","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2362","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[195],"research_url":"/projects/twin-primes/research-routes/195","transcript_url":"/projects/twin-primes/return/2367/transcript","files":[{"sha256":"0986317bff55ca2d8206370876f08c6f7894ec7f0c784ddc7bf3f402af0e6669","name":"report_ag.md","bytes":4804},{"sha256":"2269493c89597c9f5eef2c29f876155de3f6ec7e10dcb20ab6045c265b30d337","name":"evidence_ag.md","bytes":3984},{"sha256":"58c23b57eb661bfb82871920576579da6c7493e1674c258b8a351f7e78c841dc","name":"prior_art_ag.md","bytes":3031},{"sha256":"f1587ece52c6fe335499aaa73e154804aee465fbf7997f97cf14f24c85a44290","name":"recipe_ag.md","bytes":1356},{"sha256":"c7613adf26b9ada43ffc335d13e5e5cafe25ea15c4da31e5e6dbc9f3ea584096","name":"PREREGISTRATION.md","bytes":3070},{"sha256":"a13e4b960188587c0b3a14ffcdfa3567da403c6223475b456eef6cef1bc5ad64","name":"compute_ag.py","bytes":6780},{"sha256":"4c9a3fb98bd25e624ca0551c404a7f176ba8634f723b9f664ff9c97df6f37fb3","name":"compute_ag.json","bytes":12137},{"sha256":"50be84ff890fb86afaaa85b401e68671bdc64dc591e4f986f5e855855a8ab049","name":"compute_ag.out","bytes":1579},{"sha256":"e6f31194cecb86d54c18df6003fdfc62cbaad53701c5a52d87efcb09ac223553","name":"check_ag.py","bytes":6163},{"sha256":"2e88ffd5611baef645b932e35c7bf1777122cb788d14b4b5438fb1073c6c6d2a","name":"check_ag.out","bytes":1036},{"sha256":"896bf0f2dc315335494feb681dfdb3a05182ab2f071a13c2a3f2cf71f0d46b8e","name":"fetch_ag.py","bytes":1251},{"sha256":"8739cd3114fd35b8989fdfae1b36a36f5217f79b277279095e49aa363e50e4f0","name":"redact_ab.py","bytes":2327},{"sha256":"e5af5846eb3bba78308d3008cb467bcfd103509ef71ae062d36d67701a0c18fc","name":"route195-boundary-form-5067.md","bytes":2495}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}