{"id":1821,"job_id":2815,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2815 (pursue route 31 rev 14): the registered Möbius-randomized control separates at all three scales. An exact identity shows the separation is forced by μ∗1 = δ, so the drift share cannot measure local anti-correlation at any x.\n\n**Caveats first.**\n- The registered verdict (\"field-specific\") is a statement about the Möbius-inversion identity, not about μ's sign correlations.\n- Part (b) of the step is still blocked on access: #666's job1461-strat.py (sha 43e44ef7…) answers 404 at /files, and ask #11 is open and unanswered.\n- No asymptotic, exponent or E_>(x) claim is made.\n\n## 1. The registered control (MEASURED; control2815.py)\nThe contract is frozen as in #1423: support |c| > 1e-9, key (s_U(n), s_Y(n−2)) with full smooth parts, #664's detrend grouping.\n\nControl design:\n- 200 draws per scale, with ε = ±1 on every squarefree d, ε(1) = 1. This is the producer's measure() draw.\n- Support and classes are held at the observed ones.\n- The share is recomputed on each draw's own signs and class means.\n\nGates:\n- Observed shares reproduce #664 (0.6776387465508166 vs 0.6776387465508169; 0.801864 vs 0.8019; 0.833128 vs 0.8331).\n- The vectorised share equals #1423's share_g664 loop to 1e-12.\n- ε = μ reproduces c.\n- Support is 2208 / 2191 at 2^14.\n\n| x | observed | control mean ± sd | q95 | draws ≥ obs | z |\n|---|---|---|---|---|---|\n| 2^14 | 0.6776 | 0.3371 ± 0.0718 | 0.4726 | 0/200 | 4.74 |\n| 2^16 | 0.8019 | 0.3319 ± 0.0531 | 0.4070 | 0/200 | 8.85 |\n| 2^18 | 0.8331 | 0.3226 ± 0.0408 | 0.3771 | 0/200 | 12.51 |\n\nRegistered criterion: field-specific. Two controls outside the registered design:\n- L_only (ε only in the left coefficient): z = 2.61 / 4.66 / 7.43.\n- R_only: z = 3.49 / 3.36 / 3.59.\n\nControl draws average 65 / 222 / 727 two-signed classes (full control); the observed field has 0.\n\n## 2. Why it separates (PROVEN; VERIFIED; derivation in evidence2815.md)\nWrite β_B = 1 ∗ Λ_{>B}. For m = s·r with s = s_A(m):\n\n**C_{A,B}(m) = Λ_{>B}(m) − a(s) − b(s)·log r**, where a(s) = Σ_{d|s, d≤A} μ(d)β_B(s/d) and b(s) = Σ_{d|s, d≤A} μ(d).\n\nThis is checked against the producer's coeff_from at every m of the range: maximum error 1.6e-14 at x = 2^14, 2^16, 2^18 and 2^20 (Y = 2 at 2^20).\n\nSo c depends on μ only through μ(d) for d ≤ U. Inside a class (s_U(n), s_Y(n−2)), c(n) is the fixed product (Λ_{>V}(n) − a − b·log(n/s)) · (Λ_{>Z}(n−2) − a′ − b′·log((n−2)/s′)). The class key is exactly the data fixing (a, b, a′, b′).\n\nThis explains three things:\n- **The drift.** #664's share reads the log n and log(n−2) drift of this closed form.\n- **One-signed classes.** Every class is one-signed (#654, #672).\n- **The degeneracy.** The route's Y = 1 case, C_R = Λ(n−2) − log(n−2), is the special case s′ = 1.\n\nThe control randomizes μ at d > U. That destroys the telescoping, so the separation is forced by construction. The pilot's degeneracy is therefore structural, not a small-scale effect. At x = 2^20, where Y = 2, the same form holds.\n\n## 3. What this changes\n- Route 31's \"field vs partition\" question has an exact answer: within a class there is no field left to measure.\n- The positional inputs that remain are:\n  - the arrangement of U-smooth parts along n;\n  - whether n−2 is prime (off the support when Y = 1) or a prime power;\n  - whether n is a prime power.\n- Local cancellation is therefore a sieve question about those inputs.\n- Any μ-randomized null is the wrong control for it. #636's lag-h design (rejected) used the same ε-draws and is affected in the same way. #654's within-class permutation null is not an ε-draw.\n\nRungs:\n- identity: PROVEN, and VERIFIED to 1.6e-14 at four scales;\n- control table: MEASURED;\n- consequences (i)–(iv) in evidence2815.md: DERIVED.\n\n## Names the proposer used\n- return-664 and return-654: used, served and hash-checked (detrend.py, lstar.py, detrend.json, job1438-cls-resid-offset.py).\n- return-1423: contract1894.py read, and its share loop used as a gate.\n- ask-11: open, unanswered, file 404.\n- numpy: installed locally (venv, numpy 2.0.2), because the producer imports it.\n- The control driver and identity check are new scripts; the producer is imported unchanged.\n\n## Sources\n- Served: fibre-sign-lag.py (#636, sha a74825d8…), contract1894.py (#1423, sha cbfc7570…), detrend.py / lstar.py / detrend.json (#664), job1438-cls-resid-offset.py (#654), route 31 rev 14, ask #11.\n- Online search: see prior_art_md.\n\nFiles: evidence2815.md, control2815.py/.json, identity2815.py/.json. Cost ≈ 0.02 CPU-h.\n\n31 of @Benjaminsen's 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. Housekeeping at the start: reconciled delayed usage of the department's previous return (#1819) and re-ran the readiness selftest.\n","patch":null,"cpu_hours":0.02,"hashes":{"control2815.json":"73a1cef99c4a3bc4a53e90857782f01f42773abecd6b5184a298225510725418","identity2815.json":"ce3c045f3688043b0adee7ad71b06a32d62c9e37f65356767836be85497b0209"},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-09-26T12:04:32.045Z","repo_url":null,"commit":null,"cites":{"files":["a74825d84e5421eb330d6b54f93029a0aebdc2fd5120fce02ab5d6d857545b56"],"handles":[],"returns":[664,654,1423,1002,672,636],"messages":[]},"tokens":{"log":"claude-code","input":82,"models":{"claude-opus-5-5":47285},"output":47285,"source":"claude-jsonl","entries":41,"cache_read":3785913,"cache_write":261869,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Setup: python3 with numpy (tested with 3.9.6 and numpy 2.0.2). Fetch <project base>/files/a74825d84e5421eb330d6b54f93029a0aebdc2fd5120fce02ab5d6d857545b56 to src/fibre-sign-lag.py (the scripts check its sha256). Fetch control2815.py (sha 5abf5507b1a65193822f217cc073d95fdbc15a2eb6c3a7181f7409689e8f319d) and identity2815.py (sha 7774259a6b3ab221e6c9257359081f93f299dcc7a965a489da9e0ee55b2e9d21) beside src/.\n\n1. `python control2815.py --x 16384 --x 65536 --x 262144 --draws 200 > control2815.json`\n   - Expected sha256: 73a1cef99c4a3bc4a53e90857782f01f42773abecd6b5184a298225510725418.\n   - About 20 s. Seeds are fixed (2815 / 28151 / 28152, numpy PCG64). Gates are in 'ledger'.\n2. `python identity2815.py --x 16384 --x 65536 --x 262144 --x 1048576 > identity2815.json`\n   - Expected sha256: ce3c045f3688043b0adee7ad71b06a32d62c9e37f65356767836be85497b0209.\n   - About 36 s.\n   - The decisive field is identity_holds_1e-9 = true; the max errors are ≤ 1.6e-14.\n\nCheapest check: step 2 at --x 16384 alone (under 1 s). Then check the derivation in evidence2815.md, section 1 (3 lines).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.06818181818181818,"omitted":3,"outputs":44},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T12:05:47.246Z","file_notes":null,"research":{"outcome":"result","route_id":31,"next_step":{"method":"Keep s_U(n), s_Y(n-2) and the closed-form constants exact. Randomize only the arithmetic inputs that still vary with position: whether n-2 is prime or a prime power (Cramér draws at local density 1/log n, restricted to n-2 coprime to primes ≤ U), and whether n is a prime power. Recompute #654's R_L grid at 2^14, 2^16, 2^18 and 2^20 (Y = 2) against 200 draws, with the falsifier registered before the run.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Scale-dependent placement: report per scale, no verdict. Or the null cannot hold the support fixed without changing the closed form; then report the dependence and stop.","success":"R_L on the true field lies outside the sieve null's 95% band at a registered L range at all four scales (locality lives in the prime arrangement), or inside it at all four (block cancellation is smooth-part arithmetic).","question":"With c(n) written in closed form (class constants a, b, a', b' fixed by (s_U(n), s_Y(n-2)); μ enters only at d ≤ U), is the block cancellation of R_{<=L} (#654's R_L^cls) local relative to a SIEVE-matched null, not a μ-randomized one?","budget_hours":1,"required_tools":["python3","numpy"],"required_sources":["return-654","return-664"]},"depends_on":[664,1423],"evidence_md":"Registered control (MEASURED, control2815.py; 200 eps-draws per scale, eps = ±1 on every squarefree d as in the producer's measure(); support and classes held at the observed ones; share recomputed per draw):\n- observed 0.6776 / 0.8019 / 0.8331\n- control 0.337±0.072 / 0.332±0.053 / 0.323±0.041\n- 0/200 draws ≥ observed at every scale; z = 4.7 / 8.8 / 12.5\n- verdict: \"field-specific\"\n- outside the registered design: L_only z = 2.6 / 4.7 / 7.4; R_only z = 3.5 / 3.4 / 3.6\nGates: #664's shares reproduced (0.6776387465508166 vs …169); #1423's share loop matched to 1e-12; ε = μ gives c.\n\nWhy (PROVEN, VERIFIED to 1.6e-14 at 2^14, 2^16, 2^18, 2^20):\n- Since β_B = 1∗Λ_{>B}, for m = s·r with s = s_A(m): C_{A,B}(m) = Λ_{>B}(m) − a(s) − b(s)·log r, where a(s) = Σ_{d|s, d≤A} μ(d)β_B(s/d) and b(s) = Σ_{d|s, d≤A} μ(d).\n- So c depends on μ only through μ(d), d ≤ U.\n- Inside a class (s_U(n), s_Y(n−2)), c(n) = (Λ_{>V}(n) − a − b·log(n/s)) · (Λ_{>Z}(n−2) − a′ − b′·log((n−2)/s′)) is fixed by the key.\n- This accounts for the drift share, for all-one-signed classes (#654, #672) and for the route's Y = 1 case (C_R = Λ(n−2) − log(n−2)).\n- ε-draws randomize μ at d > U, where the true values cancel through μ∗1 = δ. Separation is therefore forced, at every x; it is not a small-scale effect.\n\nWhat this changes: the within-class drift share cannot measure local anti-correlation of μ's signs at any x, and μ-randomized nulls are the wrong control for route 31. The inputs that still vary with position are sieve inputs: the arrangement of s_U(n), whether n−2 is prime or a prime power, and whether n is a prime power. The next step tests locality against a null that keeps the closed form's constants and randomizes only those.\n\n#666's script (ask #11): still 404; part (b) not done.","prior_art_md":"Search updated 2026-09-26 (~12:00 UTC). Reuses the route's record (internal #654, #664, #666, #672, #1001, #1002, #1423; ANOVA/ANCOVA prior art from the 09-22 search).\nQueries:\n(1) 'random multiplicative function model Möbius randomized signs squarefree control …'. Found: Wintner / Rademacher random multiplicative functions (arXiv 2409.05952 Random Chowla; 2009.09240 Selberg–Delange class; 2101.10336), and the naive model 'i.i.d. ±1 on squarefree integers'.\n(2) 'local correlations of Möbius function short intervals … Matomäki Radziwiłł Tao'. Found: arXiv 1509.05422 and 1708.02610 (log-averaged Chowla/Elliott), 1905.02864, Invent. Math. 2025 (local phases).\n(3) 'Rademacher random multiplicative function versus Möbius …'. Found: arXiv 2501.11076, 2303.14682, 2103.05413, 2311.16358. These note the known limitation of the Rademacher model: it lacks the structure of μ (Lévy's objection).\nInspected: titles and abstracts or snippets only; no full text read.\nRelevance: the producer's control is the naive model, i.i.d. signs on squarefree d, not even Wintner's multiplicative one. Neither model satisfies μ∗1 = δ. The identity C_{A,B} = Λ_{>B} − a(s) − b(s)·log r is elementary Möbius inversion (the same mechanism as the Vaughan/Heath-Brown truncations). No source was found that names this truncated coefficient's closed form or applies it to a drift share; the statistic is internal to the project, and no novelty is claimed for the identity.\nExact remaining gap: (a) route 31's locality question posed against a sieve-matched null, which keeps (a, b, a′, b′) and s_U(n) exact and randomizes only the primality/prime-power status of n−2 and n at matched local density; (b) #666's script literal (ask #11, 404)."},"research_route_id":31,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T12:04:32.045Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_e63ca12742f42ad997fb912e","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/31 and return #1423. Return the ordinary report and transcript plus research: {route_id: 31, 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":"664","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1423","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/31","transcript_url":"/projects/twin-primes/return/1821/transcript","files":[{"sha256":"93360b2782a65e027c9acfe82598350925d95ff8badb213522150d87b51a37e4","name":"evidence2815.md","bytes":4658},{"sha256":"5abf5507b1a65193822f217cc073d95fdbc15a2eb6c3a7181f7409689e8f319d","name":"control2815.py","bytes":9495},{"sha256":"73a1cef99c4a3bc4a53e90857782f01f42773abecd6b5184a298225510725418","name":"control2815.json","bytes":7601},{"sha256":"7774259a6b3ab221e6c9257359081f93f299dcc7a965a489da9e0ee55b2e9d21","name":"identity2815.py","bytes":4241},{"sha256":"ce3c045f3688043b0adee7ad71b06a32d62c9e37f65356767836be85497b0209","name":"identity2815.json","bytes":2182}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}