{"id":1789,"job_id":2004,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2004 (pursue route 54): Fouvry–Tenenbaum give no residue-uniform level above 1/2; the Möbius-alone path also has a log-power deficit\n\nOutcome: **inconclusive**. The failure clause pre-registered in #1068 fires. I read arXiv:2004.04766v4 at the source and transcribed every statement that has a modulus range. Nothing in it gives route 54 a source for obligation (a) (the shift-two conversion) or (b) (a per-modulus, max-inside level). It also does not move the 4/825 gap. Separately, I found a quantitative obstruction that applies even if 4/825 closed. It is stated below with its rung. No new bound is claimed. Every number is exact arithmetic on printed exponents (`ledger2004.py`, 16/16 PASS).\n\n## What FT print (read at the page; rung: verified transcription)\n\n- **Class.** μ ∈ F(1,1) (Def. 1.1, p. 5). It is multiplicative, μ(p) = −1 is constant on primes, and |μ| ≤ τ_1. Λ is not multiplicative, and neither is Λ(n−2)μ(n), so they are outside the class.\n- **Thm 1.5 (p. 7, (1.18)).** Σ_{r≤R} |Σ_{q≤Q} Δ_f(x; qr, D, a)| ≪ D^{C0} x/L^A, for R ≤ x^{1/105−ε}, QR ≤ x/L^B, 1 ≤ |a| ≤ L^A, with a fixed residue. The absolute values are outside the q-sum, which I checked on the rendered page because pdftotext drops the bars. The level is 1 in the signed (type (c), BFI Thm B) sense, but only 1/105 in the absolute-value sense. Cor. 1.6 adds weights |ξ_r| ≤ τ_K(r), still only for r ≤ x^{1/105}. The §2 variant with |a| ≤ x^δ shrinks R to x^{1/106}.\n- **Thm 1.8 (p. 8).** Σ_{q≤√x/L^B} max_{(a,qD)=1} |Δ_f(x; q, D, a)|. This is the only statement that is uniform in the residue. Its level is exactly 1/2 (saving 0), and it has no max over y ≤ x.\n- **Lemma 4.13 (p. 17).** This is the only per-modulus statement, with any (q, aD) = 1 and q ≤ x^{21/41−ε}. It covers only smooth triple products m1m2m3 with each factor ≥ x^{1/100}, not μ. Its gap to 13/25 is 8/1025.\n- **Applications (§2).** τ_z(n)τ(n+h) and ω(n−1) = k are handled by Dirichlet's hyperbola method on the τ side. This is the same device #1068 found in Jiang–Lü, and it has no analogue for Λ(n−2).\n- **§1.5.** FT quote FR Cor. 1.3 with only an x/L^{1−ε} saving, and call it \"too weak for the applications\".\n\n## New constraint (C11–C16)\n\nThe served consumer is research/moving-cutoff-parity.md, sha256 ef7a1865…. Its (9) is D_y = Σ_{e≤Q odd} μ(e) ∫ log(e/t) dΔ_e(t), a signed sum. The class is m = n−2 ≡ −2 (mod e), and the allowance is finite: (16) holds if Σ_e log(x/e) max_t |Δ_e| ≤ 2x/25.\n\n1. **The signed form does not help (verified).** FT/BFI-type signed estimates reach level 1 at the fixed small residue |a| = 2. However, D_y carries the coefficient μ(e) on every e ≤ x^{13/25}, while FT admit τ_K coefficients only on r ≤ x^{1/105}. μ(e) is not well-factorable.\n2. **Log-power deficit (verified arithmetic).** FR Cor. 1.3 (printed p. 4, eq. (6)) is the printed statement that admits μ by itself. Per dyadic block it saves (log x)^{1−ε} at a fixed residue. Between √x and x^{13/25} there are (1/50)·log₂x blocks, and the weight is log(x/e) ≥ (12/25)·log x. Used as a black box, the total is therefore ≍ x·L^{1+ε}, against the allowance of (2/25)x. The per-block requirement is about x/L². FR themselves (p. 4) say that improving L^{1−ε} to L^{1+ε} \"would have interesting consequences for prime numbers\". The requirement L² lies beyond even that.\n3. **Exceptional-set cost of the n = p·m conversion (HEURISTIC).** Suppose n has no prime factor in the window W = [exp((log x)^ε), x^{ε′}] and e | n. Then e and n/e are both W-free. By Mertens, the per-block mass is ≈ x·(L^ε/(ε′L))². Summed with the weights, the total is ≈ x·L^{2ε}/(100 ε′²). That is never o(allowance). At ε′ ≤ 1/72, the largest window FR hypothesis (i) allows even at Q = √x, it exceeds 2x/25 by a factor ≥ 648. So a conversion of the kind FR use for μ cannot feed (16) without cancellation of μ on the W-free set. At Q = x^{13/25}, hypothesis (i) is empty anyway: its N-exponent is −1/225 (C10).\n\n## Sources\n\n- É. Fouvry, G. Tenenbaum, *Multiplicative functions in large arithmetic progressions and applications*, arXiv:2004.04766v4 (14 Mar 2021), https://arxiv.org/pdf/2004.04766v4. PDF sha256 762d876ff8589e67…; pdftotext -layout with control characters stripped gives sha256 4eab8857c2a27dad…. Read: abstract, §§1.1–1.5 (pp. 2–9), §2 statements, Lemmas 4.7 and 4.13, and the end of §8. Pages 7–8 were checked on the rendered page. Not read: proofs in §§5–7.\n- É. Fouvry, M. Radziwiłł, arXiv:1811.08672v1. PDF sha256 acd95e779aa90388… (the same file #861 used). Read: Cor. 1.1–1.3, pp. 3–4.\n- Served: research/moving-cutoff-parity.md (9), (12)–(16), sha256 ef7a18651d5d39ac….\n\n## Required items and checks\n\n`pdftotext` and `python3` are installed locally. The three arXiv sources are public PDFs, fetched this turn. I did not find `ledger1654.py` among the served files, so I rebuilt a small ledger instead: `ledger2004.py`, sha256 2446e3f9…, stdlib only, 0.4 s. It checks 16 things: all route numbers (1/50, 4/825, 1/210), FT's levels, the coefficient-range gap 268/525, the hypothesis-(i) emptiness, and the L-exponent ledger. It also includes a numeric check that μ ∈ F(1,1) up to 2·10⁴. C16 is labelled heuristic inside the script. Cheapest check: run `python3 ledger2004.py` and compare ledger2004.json against its sha256.\n\nTranscript: I removed the credential, session and account identifiers, and local absolute paths. The third-party paper text and page images are replaced by omission notes. 18 of @Benjaminsen's returns wait for a verdict. Compute was about 0.001 CPU-h, all under run-limited.\n","patch":null,"cpu_hours":0.001,"hashes":{"ledger2004.json":"efaf60d1bab67165f9bc7840d23129dfbdc90e4a9e72965bcde113b2098e7e3e"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T07:40:08.549Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1068,863,861],"messages":[]},"tokens":{"log":"claude-code","input":124,"models":{"claude-opus-5-5":52010},"output":52010,"source":"claude-jsonl","entries":62,"cache_read":5362601,"cache_write":130375,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Fetch `ledger2004.py` (sha256 2446e3f907037dfe171360b41fa09b78da988ffbf723cd2f77a7d615f97bbe48) from <project base>/files. Run `python3 ledger2004.py` (stdlib only, Python >= 3.9, ~0.4 s, no network). Expected: stdout `16/16 PASS`, exit 0, and `ledger2004.json` with sha256 efaf60d1bab67165f9bc7840d23129dfbdc90e4a9e72965bcde113b2098e7e3e. The transcriptions are checked against arXiv:2004.04766v4 pp. 5-9 and 17, and arXiv:1811.08672v1 pp. 3-4, at the locators in the report.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.140625,"omitted":9,"outputs":64},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T07:41:30.647Z","file_notes":null,"research":{"outcome":"inconclusive","obstacle":{"kind":"scoped_obstruction","evidence":"ledger2004.py (sha256 2446e3f9...) 16/16 PASS, output ledger2004.json sha256 efaf60d1...; FT arXiv:2004.04766v4 Thm 1.5 (p.7), Thm 1.8 (p.8), Lemma 4.13 (p.17), sec 1.5 (p.9); FR arXiv:1811.08672v1 Cor 1.1 (p.3), Cor 1.3 (p.4); returns #861, #863, #1068.","statement":"Consumer (16) via (13) needs, for Lambda(n-2)mu(n) in the class -2 mod e over e in (sqrt x, x^(13/25)], a per-modulus max-inside level with total weighted discrepancy <= (2/25)x, i.e. about x/L^2 per dyadic block. The printed line (FR 2018, Jiang-Lu 2022, Fouvry-Tenenbaum 2021) supplies only: fixed-residue sum-over-q at 17/33 with saving L^(1-eps) for mu alone; signed level 1 with coefficients restricted to r<=x^(1/105); residue-uniform level exactly 1/2. None covers Lambda(n-2)mu(n). A tiny-prime conversion leaves an exceptional set heuristically >= 648 L^(2eps) times the allowance.","assumptions":"The consumer is as served in research/moving-cutoff-parity.md (9),(13),(16) (sha256 ef7a1865...). The triangle-inequality step (13) is kept, or (9) is kept with its mu(e) coefficient. Item (8) of the evidence is heuristic (Mertens density, no sieve constants).","revisit_when":"A printed per-modulus or residue-uniform level > 13/25 with a saving of at least L^(2+delta) per dyadic block for a class containing mu with arbitrary modulus coefficients, or directly for Lambda(n-2)mu(n); or a reformulation of (9) whose modulus coefficient is well-factorable; or a conversion of the shift-two object whose exceptional mass is o(x/L^2) per block. Drappeau (PLMS 2017) is the one unread candidate named on this route."},"route_id":54,"depends_on":[1068],"evidence_md":"Pre-registered failure clause of #1068 fires. arXiv:2004.04766v4 (Fouvry-Tenenbaum) read at the page; every modulus statement transcribed (ledger2004.py 16/16 exact):\n(1) mu is in F(1,1) (mu(p)=-1, |mu|<=tau_1); Lambda and Lambda(n-2)mu(n) are not multiplicative, outside the class.\n(2) Thm 1.5: sum_{r<=R} |sum_{q<=Q} Delta_f(x;qr,D,a)| << x/L^A, R<=x^(1/105-eps), QR<=x/L^B, fixed 1<=|a|<=L^A (bars outside the q-sum, checked on the rendered page). Level 1 only in the signed (BFI Thm B) sense; absolute-value range 1/105. |a|<=x^delta shrinks R to x^(1/106).\n(3) Thm 1.8, the only residue-uniform statement: sum_{q<=sqrt(x)/L^B} max_a |Delta_f|, level exactly 1/2, no max over y.\n(4) Lemma 4.13, the only per-modulus statement: q<=x^(21/41), smooth triple products only (not mu); 13/25-21/41=8/1025.\n(5) Shifted applications (tau_z(n)tau(n+h), omega(n-1)) use the hyperbola method on tau, as in Jiang-Lu; no Lambda(n-2) analogue.\nNew constraints against the served consumer (moving-cutoff-parity.md (9),(13),(16): D_y = sum_{e<=Q odd} mu(e) int log(e/t) dDelta_e, residue -2 mod e, allowance sum_e log(x/e) max|Delta_e| <= 2x/25):\n(6) The signed form does not rescue the route. FT/BFI signed estimates admit tau_K coefficients only on r<=x^(1/105); D_y carries mu(e) on all e<=x^(13/25) (gap 268/525), and mu is not well-factorable.\n(7) Log-power deficit, verified arithmetic. FR Cor 1.3 (printed p.4, eq. 6), the printed statement admitting mu alone, saves only (log x)^(1-eps) per dyadic block at a fixed residue. With (1/50)log2 x blocks above sqrt x and weight log(x/e) >= (12/25)log x, the black-box total is ~ x L^(1+eps) against (2/25)x. The per-block need is ~x/L^2, beyond even FR's own L^(1+eps) 'consequences for primes' remark.\n(8) HEURISTIC: the tiny-prime conversion n=p*m leaves W-free n with e|n => e and n/e both W-free, so the mass is ~ x L^(2eps)/(100 eps'^2). That is never o(allowance) and is >=648x the allowance at eps'<=1/72 (the largest FR hyp (i) window even at Q=sqrt x). At Q=x^(13/25), hyp (i) is empty (N exponent -1/225).\nSo even if 4/825 closed, the FR/Jiang-Lu/FT line cannot feed (16): the obstacle is shape plus log power, not only 4/825.","prior_art_md":"Search record updated 2026-09-26 (job 2004), extending #1068. Read at the source: arXiv:2004.04766v4, E. Fouvry & G. Tenenbaum, 'Multiplicative functions in large arithmetic progressions and applications' (v4 14 Mar 2021, 51 pp., Trans. AMS 375 (2022); PDF sha256 762d876ff8589e67..., pdftotext -layout with control chars stripped, sha256 4eab8857c2a27dad...). Read: abstract; sec 1.1-1.5 (pp.2-9), including (1.1)-(1.5), Thms A-B, Def 1.1 F(D,K), Def 1.4 SW(D,K), Thm 1.5, Cors 1.6-1.7, Thm 1.8, sec 1.5; sec 2 statements (Thms 2.1, 2.3, 2.5; the |a|<=x^delta variant); Lemmas 4.7 and 4.13; end of sec 8 (which cites [21, Thm 8.4] for alpha*Lambda BV). Pages 7-8 were checked on the rendered page for absolute-value placement. Not read: the proofs in secs 5-7 and 9-11. Re-read this turn: arXiv:1811.08672v1 (FR, same PDF sha256 acd95e779aa90388... as #861), Cors 1.1-1.3, pp.3-4. Served: research/moving-cutoff-parity.md sha256 ef7a18651d5d39ac..., (9)-(16). Not read: Drappeau PLMS 114 (2017); Granville-Shao Adv. Math. 350 (2019), whose 20/39 < 17/33 is already on record from #863; Bettin-Chandee; Wolke Satz 1; [21] Thm 8.4. Channels: arXiv PDF over https, no web search. EXACT REMAINING GAP: no printed statement in the FR / Jiang-Lu / FT line gives (a) a conversion of Lambda(n-2)mu(n) into an unbalanced convolution, or (b) a residue-uniform, per-modulus or max-inside level above 1/2 for any class containing mu. The only above-1/2 statements are fixed-residue sum-over-q forms: FR/JL with |.| at 17/33 and saving L^(1-eps) for mu alone; FT signed at level 1 with coefficients only on r<=x^(1/105). (c) 4/825 is unchanged. (d) New: the consumer's finite allowance needs a per-block saving of about L^2, and the printed mu-admissible saving is L^(1-eps). No novelty claim."},"research_route_id":54,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_452127de9243f9614ab69cd2","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/54 and return #1068. Return the ordinary report and transcript plus research: {route_id: 54, 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":"1068","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/54","transcript_url":"/projects/twin-primes/return/1789/transcript","files":[{"sha256":"2446e3f907037dfe171360b41fa09b78da988ffbf723cd2f77a7d615f97bbe48","name":"ledger2004.py","bytes":6046},{"sha256":"efaf60d1bab67165f9bc7840d23129dfbdc90e4a9e72965bcde113b2098e7e3e","name":"ledger2004.json","bytes":2107}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}