{"id":1813,"job_id":4153,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4153 (first look, route 165): a dyadic union over t cannot remove consumer (16)'s 1/50. The max over t spans one dyadic block, the 13/25 is the modulus range, and the sequence is not mu\n\n**Caveats first.**\n- This refutes the route's proposed mechanism only. It says nothing new about whether (16) holds. (16) stays **OPEN**, and nothing here bears on twin primes, G2 or beta_2.\n- The one literature claim (\"no printed equidistribution statement for Lambda(n-2)mu(n) in progressions\") is **heuristic**, scoped to the searches listed in research.prior_art_md and to #1789's record.\n- Rungs: the exponent ledger (ledger4153.py, 6/6 checks) is **proven** exact arithmetic. The mesh bound in (c) is an elementary two-line bound, **proven**. The sequence mismatch in (b) is read from the served consumer, so it is **verified** against the document.\n\n## The consumer, as served\nresearch/moving-cutoff-parity.md (sha256 ef7a1865...), (3): x = 2^j, y = ceil(x^(12/25)), Q = floor(x/y), f(n) = Lambda(n-2)mu(n). §4 defines Delta_e(t) for odd e <= Q and **x/2 <= t <= x** as the discrepancy of f on n = 0 mod e. (13) prices D_y by 2 sum_e log(x/e) max_t |Delta_e(t)|, and (16) is met if that sum is <= (2/25)x.\n\n## Why the route's success branch is unreachable\n**(a) There is nothing to split.** The max in (13) runs over t in (x/2, x]. That is exactly one dyadic block (C2, checked for j = 10..60). A union over t-blocks has one term, so the loss factor is 1 and nothing changes. The \"O(log x) blocks above sqrt x\" in the route's method are **modulus** blocks e in (sqrt x, x^(13/25)], about (1/50)·log2 x of them. Splitting those is #1789's pricing: each block must reach about (25/3)(ln 2)·x/(ln x)^2 (C5). That is a level statement at 13/25 in every block, not a way around one.\n\n**(b) The 13/25 is set by the moduli, not by t.** Here Q = x/y = x^(13/25) (C1). Any cutoff split has y·Q = x, so the levels needed by T1 (d <= y) and T2 (e <= x/y) are theta_y and 1 - theta_y. Their maximum is >= 1/2, with equality only at theta_y = 1/2 (C3). No manipulation of t touches that. Also, even the blockwise input the route names is for the wrong sequence. Granville-Shao Thm 1.1(b) is BV for **mu**. Delta_e is for **Lambda(n-2)mu(n)**, and the served document's §5 says so (\"ordinary BV concerns a different sequence\"). So there is no published 1/2-o(1) input to apply blockwise at **any** level, and #1789 already recorded this for the same consumer.\n\n**(c) The max over t is already free.** Take a mesh on (x/2, x] with spacing h = x/L^K. Per modulus e, moving from a mesh point to t costs at most (h/e + 1)·log x, since |f| <= log x. Summed over e <= x^(13/25), that is x·L^(2-K) + x^(13/25)·log x = o(x/L^A) (C4). This is the device of shifted-prime-decomposition.md §2 and of mobius-bv-derivation.md §1 (\"the added maximum is free\"). So a **fixed-t** estimate at moduli up to x^(13/25) would already give (13). Such an estimate is exactly as unavailable as the max form. The max-over-y clause was never what costs the 1/50.\n\n## What changes\n- Route 165's premise (\"if the union loss fits, the 1/50 disappears\") is false as an implication. The union loss is trivially affordable, and afterwards the requirement is unchanged: a residue-uniform level-13/25 bound for Lambda(n-2)mu(n) with saving about L^2 per modulus block.\n- #1183's statement (\"no printed max-form mu level >= 1/2\") is not the binding obstacle for this consumer. The binding one is #1789's: no printed equidistribution in progressions for Lambda(n-2)mu(n) at any level.\n- Proposed outcome: blocked, obstacle claim_refuted, scoped to the dyadic-union mechanism. No next step on this route. The live alternatives are the ones in #1789's revisit condition.\n\n## Sources\n- research/moving-cutoff-parity.md, served, sha256 ef7a18651d5d39ac45bb7f96727c0b2d6e16f39f620f359d63d220c6a7f2cd9d: (3), §4 (9)-(13), (16), §5.\n- research/mobius-bv-derivation.md, served, sha256 f6a966ed725fc9cd38bc610d54b568a3bb44516f80eaf85f51e430569df9b904: §1 statement (M) and the note on the max over y.\n- research/shifted-prime-decomposition.md, served, §2 mesh paragraph.\n- Returns #1811 (origin of the route), #1183 (with review 420), #1171, #1789 (route 54, same consumer), #4.\n- ledger4153.py (e72a23f4...), output ledger4153.json (55bb7d2b...). python3 stdlib, 0.4 s.\n- Online search: see research.prior_art_md.\n\n29 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. At startup the department also reconciled delayed usage for its previous return (#1812) and reran its readiness selftest.\n","patch":null,"cpu_hours":0,"hashes":{"ledger4153.json":"55bb7d2b2fa150ea632ce1f39c78d5dff65559653e9fd55ad6cbf6e5123272ba"},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T11:11:32.502Z","repo_url":null,"commit":null,"cites":{"files":["e72a23f4b8244289e26fabfab1c0dc372393171f0fa172ea16f3f2793e18147c","55bb7d2b2fa150ea632ce1f39c78d5dff65559653e9fd55ad6cbf6e5123272ba"],"handles":[],"returns":[1811,1183,1171,1789,4],"messages":[]},"tokens":{"log":"claude-code","input":90,"models":{"claude-opus-5-5":32302},"output":32302,"source":"claude-jsonl","entries":45,"cache_read":3815848,"cache_write":119919,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"File at <project base>/files/e72a23f4b8244289e26fabfab1c0dc372393171f0fa172ea16f3f2793e18147c (ledger4153.py), python3 >= 3.9, stdlib only.\n    python3 ledger4153.py > ledger4153.json\nExpected sha256 of ledger4153.json: 55bb7d2b2fa150ea632ce1f39c78d5dff65559653e9fd55ad6cbf6e5123272ba. Judgment: \"all_pass\": true, with checks C1-C6 all true. Run time under 1 s. Deterministic (sort_keys JSON, no timing).\nThe mathematical content is (a)-(c) of the report: checking them means reading moving-cutoff-parity.md (3), §4 and (13) at sha256 ef7a1865... and confirming (i) t ranges over (x/2, x], (ii) Q = x/y, (iii) f(n) = Lambda(n-2)mu(n).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.043478260869565216,"omitted":2,"outputs":46},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T11:13:05.663Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"claim_refuted","evidence":"ledger4153.py (e72a23f4...) -> ledger4153.json (55bb7d2b...), checks C1-C6 all pass. Served consumer §4-§5; mobius-bv-derivation.md §1; shifted-prime-decomposition.md §2 mesh; #1789 (route 54, same consumer).","statement":"The implication 'if the dyadic-union loss over y-blocks fits the (2/25)x allowance, consumer (16)'s 1/50 disappears' is false. The max over t in (13) spans one dyadic block and can be removed at o(x) cost by a mesh, while the level 13/25 is the modulus range e <= x/y and is untouched by any split in t. The blockwise input named (mu-BV at 1/2-o(1)) is for mu, not for Delta_e's sequence Lambda(n-2)mu(n).","assumptions":"The consumer is as served in research/moving-cutoff-parity.md (3), (9), (13), (16) at sha256 ef7a18651d5d39ac45bb7f96727c0b2d6e16f39f620f359d63d220c6a7f2cd9d, with the fixed cutoff y = x^(12/25). The scope is this mechanism only; the broader question of bounding D_y is not closed here.","revisit_when":"A reformulation of (9) in which the modulus range depends on t (so that a split in t lowers the level on most blocks), or a printed residue-uniform equidistribution bound for Lambda(n-2)mu(n) (or a class containing it) at any level, in which case #1789's revisit condition applies."},"route_id":165,"depends_on":[],"evidence_md":"Refutes the route's mechanism (exact arithmetic plus a two-line bound; ledger4153.py 6/6, output 55bb7d2b...), against the served consumer research/moving-cutoff-parity.md (sha256 ef7a1865...).\n(a) The max in (13) runs over t in (x/2, x], one dyadic block (C2). A dyadic union over t has one term and loss factor 1. The route's \"O(log x) blocks above sqrt x\" are modulus blocks e in (sqrt x, x^(13/25)], about (1/50) log2 x of them. Splitting those only restates #1789's requirement of about (25/3)(ln 2) x/(ln x)^2 per block at level 13/25 (C5).\n(b) The 13/25 is Q = x/y with y = x^(12/25) (C1). For any cutoff, y·Q = x, so the larger of the two levels needed is >= 1/2, with equality only at y = sqrt x (C3). The published 1/2-o(1) input (Granville-Shao Thm 1.1(b)) is for mu, while Delta_e is for f(n) = Lambda(n-2)mu(n) (served §5: \"ordinary BV concerns a different sequence\"). So there is no published input to apply blockwise at any level.\n(c) A mesh of spacing x/L^K removes the max over t at cost x·L^(2-K) + x^(13/25) log x = o(x/L^A) (C4; same device as shifted-prime-decomposition.md §2). A fixed-t estimate at moduli <= x^(13/25) would already suffice, and it is equally unavailable.\nConclusion: the union loss fits trivially, and the 1/50 (indeed the whole shift-two equidistribution) is still required. (16) stays OPEN. Rungs: (a), (c) and the ledger are proven; (b) is verified against the served text; \"no printed statement for Lambda(n-2)mu(n)\" is heuristic, scoped to the searches in prior_art_md.","prior_art_md":"Search date 2026-09-26. This reuses the route's recorded search (Granville-Shao arXiv:1706.05710v1 Thm 1.1(b)+2.1; Shao-Teravainen arXiv:2006.05954v2 Rem. 1.8; Tao 254A Notes 3 Ex. 21-22, read by #1171/#1183, review 420) and #1789's route-54 record for the same consumer (Fouvry-Tenenbaum arXiv:2004.04766v4 Thm 1.5, 1.8, Lemma 4.13; Fouvry-Radziwill arXiv:1811.08672v1 Cor 1.1, 1.3). New queries this job (web search): (1) \"Bombieri-Vinogradov max over y removed fixed endpoint equivalent mesh argument level of distribution\": standard statements with and without the max over y at the same level (Kedlaya, Notes ch. 17; MathWorld; P. S. Park's BV notes). Removing or adding the max over y never changes the level. This is the textbook situation and matches the corpus's own mesh (shifted-prime-decomposition.md §2). (2) \"Mobius function shifted primes arithmetic progressions level of distribution mu(p+2) Bombieri-Vinogradov\": Carella, arXiv:2206.12956v3 (abstract read via the arXiv API). It claims sum_{p<=x} mu(p+a) = O(x/(log x)^c). It is unrefereed and would break the parity barrier, so it is not relied on, and in any case it is an unweighted mean, not an equidistribution in progressions. Murty-Vatwani, Math. Z. (2018), \"Variants of equidistribution in arithmetic progression and the twin prime conjecture\": title and venue seen in results, full text not re-read (the consumer already cites its Props 3.2-3.3). It is the conditional framework, not an unconditional input. Access gaps: Drappeau (PLMS 2017), named in #1789, still unread. Exact remaining gap (unchanged by this route): a residue-uniform bound sum_{e<=x^(13/25), e odd} log(x/e) max_t|Delta_e(t)| <= (2/25)x for f(n) = Lambda(n-2)mu(n). The max over t is not part of the gap."},"research_route_id":165,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_c143a3a5092ca52323d71930","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/165 and return #1811. Return the ordinary report and transcript plus research: {route_id: 165, 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":[],"research_url":"/projects/twin-primes/research-routes/165","transcript_url":"/projects/twin-primes/return/1813/transcript","files":[{"sha256":"e72a23f4b8244289e26fabfab1c0dc372393171f0fa172ea16f3f2793e18147c","name":"ledger4153.py","bytes":3852},{"sha256":"55bb7d2b2fa150ea632ce1f39c78d5dff65559653e9fd55ad6cbf6e5123272ba","name":"ledger4153.json","bytes":1561}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}