{"id":1831,"job_id":2070,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2070 (pursuit, route 89): at fixed V the cutoff family is a Bombieri-Vinogradov main term, and its signed aggregate drifts to -infinity like -0.0015 x log^2 x. At the consumer's own cutoffs the reviewed (4.1) already makes every member O(x/log^A x)\n\n**Caveats first.**\n- Nothing here bounds P(1,e_1), (H_B), (4.1)'s complement B, (16) or the twin-prime statement. All of them stay **OPEN**.\n- The asymptotic in section 2 is a **heuristic** derivation. The steps are those of the reviewed section 4.1 of fixed-endpoint-discrepancy.md, with the Mobius-mean step replaced by an exact finite sum; I did not write out the tail bounds again. Its leading constant agrees with a fit to the measured main term to 3%. That agreement is **measured**, not proved.\n- The finite tables are **measured**. The scanner reproduces 74 published members (#1097 j=12..20, #1103 j=21..23, and the port anchor at 2^12 and 2^14) to relative 2.3e-15, so the gate is **verified**.\n\n## 1. The object and what changed\nRoute 89's family is tI(U) = sum_e sum_{m in I_e,(m,e)=1} log m Lambda(em-2) TI_U(m), with TI_U(m) = sum_{r|m, r<=UV} c_U(r), at **fixed** V = 32 and U <= 32 (served (2.6); #1097 family()). The consumer note proves (4.1), T_I^low = O(x log^-A x), only for U = V = floor(x^(eps'/3)). Its section 5 (2026-09-17) records that fixed-cutoff readings \"lie outside its hypothesis\". Its section 2.5 Remark adds that the cutoff movement of T_I and B \"carries no information about (H_B)\".\n\nFor fixed (U,V), each member is a Type I sum over prime classes -2 mod e·r with r <= UV <= 1024 and e < x^(1/2-1/60). So BV applies (with the coprimality divisor truncated as in section 4.1), and\n\n  tI(U) = MT(U) + o(x),   MT(U) = sum_e W_e A_e(U),   W_e = sum_{m in I_e,(m,e)=1} log m Lambda(em-2),   A_e(U) = sum_{r odd,(r,e)=1} c_U(r)/phi(r).\n\nAt fixed cutoffs the Mobius mean that kills this term in (4.1) is a fixed nonzero number.\n\n## 2. Leading term\nW_e ~ (x/2e)(log(x/e) - 0.3069), and sum over odd squarefree e <= E coprime to r of 1/e ~ (4/pi^2) prod_{p|r} p/(p+1) log E. Together these give\n\n  MT(U)/x = (2/pi^2) K(U) (lambda - lambda^2/2) log^2 x + O(log x),   lambda = 1/2 - 1/60,   K(U) = sum_{r odd} c_U(r)/phi(r) prod_{p|r} p/(p+1).\n\nAt V = 32: K(1) = +0.012417 and K(32) = -0.007308. The log^2 x coefficient of Tx = (tI(32)-tI(1))/x is therefore **-0.001465** (asym2070.json). A least-squares fit a + b log x + c log^2 x to the exact-W main term over j = 12..30 gives c = **-0.001423**, with b = +0.0229. The positive log x term dominates at reachable scales. That is the \"decay\" #1103 read at 2^21..2^23 (it regressed on ln j = ln log2 x, not on ln x). Tx changes sign near j = 26 and tends to -infinity.\n\n## 3. Measurements (fam2070.c, V = 32, chain {1,2,4,8,16,32}; ladder2070.jsonl)\n| j | Tx | MT diff/x | Tx - MT diff | Vx | max_U abs(tI-MT)/x |\n|---|---|---|---|---|---|\n| 12 | +0.4772 | +0.1315 | +0.3457 | 0.477 | 0.3710 |\n| 16 | +0.4269 | +0.1245 | +0.3024 | 0.484 | 0.1684 |\n| 20 | +0.0951 | +0.0915 | +0.0036 | 0.399 | 0.0509 |\n| 23 | +0.0564 | +0.0494 | +0.0071 | 0.495 | 0.0065 |\n| 24 | -0.0176 | +0.0357 | -0.0533 | 0.484 | 0.0426 |\n| 25 | +0.0036 | +0.0165 | -0.0129 | 0.549 | 0.0089 |\n| 26 | +0.0263 | -0.0028 | +0.0291 | 0.590 | 0.0162 |\n| 28 | -0.0488 | -0.0447 | -0.0041 | 0.638 | 0.0029 |\n| 30 | -0.0838 | -0.0929 | +0.0092 | 0.712 | 0.0056 |\n\nPre-registered verdicts (PREREG-2070.md, written before any run; verdict2070.json):\n- **S1** (Tx >= 0.08 at 2^24 and 2^25): did not fire (Tx = -0.018, +0.004). The fit of ln Tx that the route asked for is undefined because Tx < 0 at j = 24, 27..30. abs(Tx) first exceeds 0.08 at j = 30, on the negative side, as the main term predicts.\n- **S2**: did not fire. The dominant (2,4] window has 0% of its mass in r = dk > x^(1/3) (its r = 3k <= 96). The short-divisor support is definitional: c_U lives on r <= UV = 1024. It is not a measured mechanism.\n- **S3** (V = 64 probe at 2^24): **fired**. Tx(V=64) = -0.1512 against -0.0176 at V = 32, and MT predicts -0.1550.\n- **M1/M2** (main term explains members and Tx at j >= 20): hold. Members are within 0.006x of MT for j >= 27.\n- Controls at every scale: 0 violations of mu = TII - TI, window-identity residual 0, cutoff-invariance relative error <= 9.3e-10.\n\n## 4. What this changes for route 89\nAt the route's scope (fixed V), the signed telescoped aggregate is not bounded. It is an explicit main term ~ -0.0015 x log^2 x, and the absolute pad Vx grows too (0.40 -> 0.71). At the consumer's scope (growing cutoffs), the reviewed (4.1) already gives every member, hence any finite-chain variation, O(x/log^A x). The invariant T_I+B does not move with the cutoffs at all (Remark, section 2.5). Either way the family's variation cannot bound P(1,e_1): it only measures how the split moves. Route 49's obstacle stands in its own words: the repair must be \"a statement about the invariant, not a parameter choice\". Proposed outcome: blocked, claim_refuted, scoped to the fixed-V family. No next step.\n\n## Sources\n- research/fixed-endpoint-discrepancy.md, served, sha256 f68588601afedd32abf594421081aec49dbad9a2b69a65bdaa445af42151eae5: (2.5)-(2.9), the section 2.5 Remark, section 4.1 (4.1), section 5 (2026-09-17 addition).\n- Returns #1097 (window_increments.py, window-increments.json), #1103 (ladder-ext.json), #1088 (split_census.py port), and route 49's obstacle (GET /research-routes/49).\n- This job: fam2070.c, gate2070.py, asym2070.py, verdict2070.py, PREREG-2070.md and their outputs (hashes in `hashes`).\n- Required-source names: \"vaughan-identity\" is served (2.4)/(V); \"large-sieve-level-of-distribution\" is BV as used in section 4.1. Both were found on the record; nothing was rebuilt except the new scanner.\n\n36 returns wait for a verdict.\n\nTranscript: removed the API token, session/account identifiers, local absolute paths outside the working folder, and lines not belonging to this assignment. At startup the department also reconciled delayed usage for its previous return (#1830) and reran its readiness selftest.\n","patch":null,"cpu_hours":0.05,"hashes":{"asym2070.json":"6f86645fbadbcc6129901fd39b5d5476efc301d79730a1a755ea708166596cd9","gate2070.json":"1b0ec7a0643adeaac465f480e54e9f0ce984150c7ce191d1c5e2ad3eb1037a4c","ladder2070.jsonl":"f903591a605dc8117a90c1987f7ff249e6a3ef5cbf0a1f5252b4009a7cfc1af8","verdict2070.json":"9895945d4d63d9f46dfd6da42c872e36af3320d3ee0bf16c1b591d4283f0f8a5"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T13:52:08.849Z","repo_url":null,"commit":null,"cites":{"files":["b2ab3c2ded943a134a34e312e8675426d07cb6f8bbca8f4f0f762dc7c9168e61","5a69a06e69124adb4b17183a3863ffec67dfbe7cc714dd52e658c7cda2587027","10cbd9c9afaa0d7adf3614cbcc9e9e891e71bf0a757eb490abcc06e7e1e28e10","a3686dd172283266c15dd852504acbf6042652d5a3865c228c55b9050181faba","0f17cbc1d12f1fbb714660957708a4aef85b5d5a74a1f2a8a888cdd4931e67b7","f903591a605dc8117a90c1987f7ff249e6a3ef5cbf0a1f5252b4009a7cfc1af8","6f86645fbadbcc6129901fd39b5d5476efc301d79730a1a755ea708166596cd9","9895945d4d63d9f46dfd6da42c872e36af3320d3ee0bf16c1b591d4283f0f8a5","1b0ec7a0643adeaac465f480e54e9f0ce984150c7ce191d1c5e2ad3eb1037a4c"],"handles":[],"returns":[1097,1103,1088],"messages":[]},"tokens":{"log":"claude-code","input":116,"models":{"claude-opus-5-5":66521},"output":66521,"source":"claude-jsonl","entries":58,"cache_read":6499636,"cache_write":180207,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Files at <project base>/files/<sha256> (names in `files`). Needs: cc (C99, -lm), python3 >= 3.9 stdlib. For the gate, place #1097's window-increments.json at job2050/ and #1103's ladder-ext.json as r1103-ladder-ext.json.\n    cc -O2 -o fam2070 fam2070.c -lm\n    mkdir -p out gate\n    for j in $(seq 12 30); do ./fam2070 $j 32 1 2 4 8 16 32 > out/v32_$j.json; done\n    ./fam2070 24 64 1 2 4 8 16 32 64 > out/v64_24.json\n    for j in 12 14; do ./fam2070 $j 3 1 2 3 > gate/port$j.json; done; for j in $(seq 12 23); do cp out/v32_$j.json gate/; done\n    python3 gate2070.py > gate2070.json          # expect pass_1e-9 true, 74 compared\n    cat out/v32_{12..30}.json out/v64_24.json > ladder2070.jsonl\n    python3 verdict2070.py > verdict2070.json\n    python3 asym2070.py > asym2070.json\nExpected sha256: ladder2070.jsonl f903591a605dc8117a90c1987f7ff249e6a3ef5cbf0a1f5252b4009a7cfc1af8, verdict2070.json 9895945d4d63d9f46dfd6da42c872e36af3320d3ee0bf16c1b591d4283f0f8a5, asym2070.json 6f86645fbadbcc6129901fd39b5d5476efc301d79730a1a755ea708166596cd9, gate2070.json 1b0ec7a0643adeaac465f480e54e9f0ce984150c7ce191d1c5e2ad3eb1037a4c. Bytes may differ on another libm; then compare numbers to 1e-9 relative. Run time about 2.5 min on 1 core (j = 30 takes about 80 s and 5 GB RAM). Timing goes to stderr.\nCheapest check: j = 12..25 only (under 10 s). Confirm that MT tracks the members and that Tx(V=64) - Tx(V=32) at j = 24 is about -0.134. Then check K(1), K(32) and c in asym2070.json by hand.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.03225806451612903,"omitted":2,"outputs":62},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T13:53:30.180Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"claim_refuted","evidence":"fam2070.c -> ladder2070.jsonl (f903591a...), verdict2070.json (9895945d...), asym2070.json (6f86645f...), gate2070.json (1b0ec7a0...); fixed-endpoint-discrepancy.md (f6858860...) section 4.1, section 2.5 Remark, section 5; route 49 obstacle.","statement":"Route 89's premise that the signed (or absolute) variation of the cutoff family U -> tI(U) at fixed V stays within a constant pad of the 0.08x allowance is false. The members are a Bombieri-Vinogradov main term sum_e W_e A_e(U) plus o(x), and the signed aggregate grows like (2/pi^2)(K(32)-K(1))(lambda-lambda^2/2) x log^2 x = -0.0015 x log^2 x. At growing cutoffs the reviewed (4.1) already makes the whole family o(x) without bounding P(1,e_1).","assumptions":"Family as in #1097 (served (2.6), e < floor(x^(1/2-1/60)), V fixed at 32 or 64, U <= V). The asymptotic constant is a heuristic derivation following section 4.1 with the Mobius mean replaced by the exact finite sum; its agreement with the fit is measured on j = 12..30. Nothing is said about B, (H_B) or P(1,e_1).","revisit_when":"A reformulation in which the cutoff family carries information about the invariant T_I + B itself (e.g. a family whose members' main terms cancel identically at fixed cutoffs), or a signed lower bound for B with its actual coefficients, which is route 49's revisit condition."},"route_id":89,"depends_on":[1097],"evidence_md":"(1) At fixed V = 32 the family members are a BV main term plus o(x): MT(U) = sum_e W_e A_e(U) with A_e(U) = sum_{r odd,(r,e)=1} c_U(r)/phi(r). Measured: members track MT to within 0.006x for j = 27..30 (fam2070.c; gated on 74 published values to 2.3e-15).\n(2) The signed aggregate Tx = (tI(32)-tI(1))/x has main term (2/pi^2)(K(32)-K(1))(lambda-lambda^2/2) log^2 x + O(log x), with coefficient -0.001465 (heuristic derivation). The fitted coefficient over j = 12..30 is -0.001423 (measured). Tx = +0.121, +0.073, +0.056, -0.018, +0.004, ..., -0.049, -0.068, -0.084 at j = 21..25, 28..30. The small values at 2^21..2^23 were a +0.023 log x term being overtaken by the negative log^2 x term, not decay. abs(Tx) > 0.08 by j = 30, and Tx -> -infinity.\n(3) Route falsifiers: S1 did not fire at 2^24/2^25; S2 did not fire (the dominant window's legs are r = 3k <= 96; short support is definitional since r <= UV); S3 fired (V = 64: Tx = -0.151 vs -0.018, MT predicts -0.155).\n(4) At the consumer's cutoffs U = V = floor(x^(eps'/3)), the reviewed (4.1) already makes each member O(x/log^A x), and the Remark in section 2.5 says the cutoff movement carries no information about (H_B). The family's variation cannot bound the invariant at either scope. Route 49's obstacle stands.","prior_art_md":"Search date 2026-09-26. This reuses #1097's and #1103's record (Tao 254A Notes 3; Granville's alternative to Vaughan's identity; the 2026 restricted-Goldbach and short-gaps preprints; the zero-relevant \"total variation cutoff\" query) and adds:\n(1) web: \"Vaughan identity Type I sum fixed cutoff main term Bombieri-Vinogradov shifted primes Mobius mu(p+2) cutoff dependence\". Returns expository BV/Vaughan sources (Tao's Vaughan-identity posts, EoM \"Vaughan identity\", arXiv:1309.2730 explicit BV, arXiv:2508.16400). None treats the movement of a cutoff family.\n(2) web: \"sum over primes of mu(p+2) Vaughan decomposition Type I main term does not vanish fixed parameters U V log^2 x\". Returns general Vaughan applications (Montgomery vol. II draft, arXiv:1806.02699, 1809.10008). No relevant source.\n(3) Project record, the decisive prior art: research/fixed-endpoint-discrepancy.md (sha f6858860...), section 4.1 proves T_I^low = O(x log^-A x) at U = V = floor(x^(eps'/3)). Its section 5 addition says fixed-cutoff censuses lie outside that hypothesis, with #787's fixed-cutoff values at 2^20, e.g. T_I/x = +6.69 at (3,3) and -0.19 at (32,32). The section 2.5 Remark says the cutoff movement of T_I and B carries no information about (H_B). Route 49's obstacle (GET /research-routes/49) is unchanged.\nIt is textbook (e.g. BV proofs via Vaughan's identity) that a Type I piece with fixed small cutoffs has a main term given by truncated Mobius means, nonzero for fixed U, V. No source was found that writes the constant K(U) for this shift-2 object. That constant and its log^2 x consequence are this job's derivation (heuristic rung) plus measurement.\nAccess gaps: the full texts of the arXiv hits were not read; only titles and snippets were seen.\nExact remaining gap: route 89's contribution is covered at the consumer's cutoffs by (4.1) and the Remark, and refuted at fixed V by the main term. The open obligation is unchanged: a signed lower bound for B (equivalently for P_low + P_band) with its actual coefficients."},"research_route_id":89,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_e86de683358913c4a80c0e78","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/89 and return #1103. Return the ordinary report and transcript plus research: {route_id: 89, 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>, 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":"1097","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/89","transcript_url":"/projects/twin-primes/return/1831/transcript","files":[{"sha256":"b2ab3c2ded943a134a34e312e8675426d07cb6f8bbca8f4f0f762dc7c9168e61","name":"fam2070.c","bytes":8709},{"sha256":"5a69a06e69124adb4b17183a3863ffec67dfbe7cc714dd52e658c7cda2587027","name":"gate2070.py","bytes":1156},{"sha256":"10cbd9c9afaa0d7adf3614cbcc9e9e891e71bf0a757eb490abcc06e7e1e28e10","name":"asym2070.py","bytes":2916},{"sha256":"a3686dd172283266c15dd852504acbf6042652d5a3865c228c55b9050181faba","name":"verdict2070.py","bytes":1734},{"sha256":"0f17cbc1d12f1fbb714660957708a4aef85b5d5a74a1f2a8a888cdd4931e67b7","name":"PREREG-2070.md","bytes":2625},{"sha256":"f903591a605dc8117a90c1987f7ff249e6a3ef5cbf0a1f5252b4009a7cfc1af8","name":"ladder2070.jsonl","bytes":23991},{"sha256":"6f86645fbadbcc6129901fd39b5d5476efc301d79730a1a755ea708166596cd9","name":"asym2070.json","bytes":691},{"sha256":"9895945d4d63d9f46dfd6da42c872e36af3320d3ee0bf16c1b591d4283f0f8a5","name":"verdict2070.json","bytes":6581},{"sha256":"1b0ec7a0643adeaac465f480e54e9f0ce984150c7ce191d1c5e2ad3eb1037a4c","name":"gate2070.json","bytes":78}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}