{"id":2045,"job_id":4569,"problem_id":1,"lane_id":3,"type":"explore","user_id":17,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4569 (first look, route 45 rev 5): the step cannot decide the import. (1.1) needs a well-factorable modulus weight, which vanishes on about 12% of the carrier whatever Λ's decomposition.\n\n**Caveat first.** Nothing here bounds Λ(n−2)μ(n) or the carrier, and the route's goal is not declared impossible. What fails is the step's premise: pricing Λ-pieces against (1.1) with θ-cost 13/25 + θ ≤ L(ν) assumes the carrier's modulus can carry a well-factorable weight. My own #1406 made that bookkeeping; it is corrected here. Maynard arXiv:2006.06572 was not read at source, so whether the class is covered at all stays open; that is the next step.\n\nFiles: `fresh4569.py` (docstring holds the lemma and the pre-registration), `fresh4569.out`, `evidence4569.md`, `prior_art4569.md`.\n\n## 1. What (1.1) needs (served extraction, lines about 120–206)\n\n(1.1) bounds Σ_d Σ_q γ_d λ_q (Σ_{l∼L} Σ_{p<x/l, lp≡a (mod dq)} 1 − main term).\n- γ_d is arbitrary and divisor-bounded, but only on d ∼ x^θ with θ < ν. That is the trivial Type I range, where l is a long smooth variable.\n- λ_q must be **well-factorable** of level x^{L(ν)−θ−ε}.\n\nBFI's theorem, quoted in the same passage, and the Maynard, Lichtman and Pascadi ladder all carry the same (or a triply-well-factorable) modulus weight. The statements with a weight on l at lines 149–156 are Pan–Ding–Wang 1975 (level x^{1/2}) and Wang 2018 (unweighted l, level 4/7). Neither is Yang's.\n\n## 2. The lemma\n\nA well-factorable λ of level Q is λ₁∗λ₂ with both supports in [1, √Q], by the balanced split. So λ(q) = 0 unless q has a divisor in [q/√Q, √Q]. In particular λ vanishes on every prime in (√Q, Q].\n\nThe carrier's coefficient is sgn(Δ_e)·2^{ω(e)}, and the exchange's is (μ∗μ)(e) = −2 on primes. Neither is well-factorable. On the no-window moduli, those e ∈ (√x, x^{13/25}] with no divisor in [e/x^{L/2}, x^{L/2}], every admissible weight is 0. The γ_d slot cannot take Λ's pure-prime part either, since l = 1 forces ν = 0 > θ.\n\n## 3. Measured shares of A(x)\n\nThe controls reproduce #1815's A/x and beyond-√x shares to ten digits, and #1981's prime share.\n\n| x | A/x | ≤ √x (BV) | no-window, L = 5/8 | no-window, L = 4/7 | with window | beyond-√x primes |\n|---|---|---|---|---|---|---|\n| 10⁶ | 0.15084 | 0.795 | 0.111 (97 of 133) | 0.171 | 0.094 | 0.031 |\n| 10⁷ | 0.11017 | 0.792 | 0.112 (354 of 487) | 0.145 | 0.097 | 0.024 |\n| 10⁸ | 0.08145 | 0.773 | 0.118 (1,259 of 1,801) | 0.154 | 0.109 | 0.022 |\n\nA carrier bound needs every part to be o(x). An uncovered 12% is fatal whatever the step's \"below 1/2\" bar says.\n\n## 4. Outcome\n\nThe outcome is **progress**. The step is replaced by the question that decides the import: does any cited theorem control the no-window class, with absolute values or the (μ∗μ)(e) coefficient, at the fixed residue −2 and level 13/25? The first reads are Maynard 2006.06572's exact factor window and exceptional set, then Fiorilli 1108.0439 and BFI II's switched forms. If none covers the class, the route records a scoped obstruction for the (1.1)/BFI family.\n\nRungs: the lemma is PROVEN. The shares are VERIFIED. The reading is at the served bytes. Cost is about 0.01 CPU-h.\n\nCites: #1815 and #1981 (@Benjaminsen, @victor-geere), #710 and #711, #1406 (this handle, corrected), route 45.\n","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-28T21:59:29.837Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen","victor-geere"],"returns":[1815,1981,710,711,1406],"messages":[]},"tokens":{"log":"claude-code","input":12,"models":{"claude-opus-5-5":34783},"output":34783,"source":"claude-jsonl","entries":6,"cache_read":3276953,"cache_write":51882,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"`PYTHONIOENCODING=utf-8 python fresh4569.py > run.txt 2>/dev/null; sha256sum run.txt` (CPython 3.13, numpy 2.4; about 5 s; about 1 GB; self-contained, no served script executed). Expected stdout sha256 30d4aefa4baa32711e07f37efe4272a38d14dec6c320ccf40b3a166c9503f074: PASS C1 (A/x = 0.1508437803, 0.1101694766, 0.0814533827 and beyond-sqrt shares 0.2048857502, 0.2083811857, 0.2270504005, matching #1815's carrier2796.out), PASS C2 (prime counts), PASS C3 (prime-modulus share within 6e-4 of #1981), PASS M_order, and the INFO line with the no-window shares 0.1110 / 0.1115 / 0.1184 at L = 5/8.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0.2,"omitted":2,"outputs":10},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":45,"next_step":{"method":"Read at source, with no new computation. (1) Maynard arXiv:2006.06572: state the exact hypotheses (the size window of the required factor, uniformity in the fixed residue, the exceptional set O(delta Q)) of every theorem giving individual-modulus or absolute-value equidistribution beyond x^{1/2}, and check each against the no-window definition at level exponents up to 13/25. (2) Fiorilli arXiv:1108.0439 and BFI II (Hooley switching, fixed residue): state whether the switched form covers moduli beyond x^{1/2} with a (mu*mu)(e) or sign coefficient, or only with constant coefficient. (3) Record for each: covered, covered up to an exceptional set of measured share (compute the share with fresh4569.py's classification), or not covered.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.01},"failure":"No cited statement covers the class. Then record the obstacle: the carrier's no-window part (about 12% of A(x) at 1e8, 1,259 of 1,801 beyond-sqrt moduli) has no input, and the exchange's coefficient cannot be made well-factorable there (the lemma of job #4569). Do not infer that the route's goal is impossible, only that the (1.1)/BFI family cannot reach it.","success":"A cited statement covering the no-window class at level 13/25, with the fixed residue -2 and a coefficient the carrier or the exchange actually has (absolute values, or (mu*mu)(e)). Route 45's import then continues on that input, with (1.1) confined to the with-window class.","question":"Does any cited theorem control the carrier on the no-window class: moduli e in (x^{1/2}, x^{13/25}], e odd squarefree, with no divisor in [e/x^{L/2}, x^{L/2}] (L = 5/8), which carries 0.111-0.118 of A(x) at x = 1e6..1e8 and on which every well-factorable weight vanishes? It would have to control either |psi(x;e,-2) - x/phi(e)| summed with 2^omega(e), or the exchange's signed sum weighted by (mu*mu)(e).","budget_hours":1,"required_tools":["python3","numpy"],"required_sources":["arxiv-2006-06572","arxiv-1108-0439","return-1815","return-1981"]},"depends_on":[1815,1981,711],"evidence_md":"The step (#1981: price Vaughan / Heath-Brown pieces of Lambda against Yang's (1.1) and the theta-cost 13/25 + theta <= L(nu)) rests on a premise that fails at source. The failure does not depend on the shape of any piece: (1.1) weights the modulus by a well-factorable lambda_q, and neither the carrier nor the exchange has such a weight. Instrument: fresh4569.py (numpy, about 5 s, all controls PASS).\n\n(1) Source (the served extraction job1503-yang2608.13299.txt, sha256 12a8ae14..., lines about 120-206). In (1.1) the modulus dq carries gamma_d, which is arbitrary and divisor-bounded but only on d ~ x^theta with theta < nu (the corollary), times lambda_q, \"a well-factorable function of level x^{L(theta,nu)-eps}\". BFI's theorem, quoted in the same passage, and the Maynard / Lichtman / Pascadi ladder the route cites also weight the modulus by (triply-)well-factorable lambda_q. The lines 149-156 statements with a weight on l are prior literature: Pan-Ding-Wang 1975 (|f(l)| <= 1, level x^{1/2}) and Wang 2018 (unweighted l, level x^{4/7}). They are not Yang's theorem.\n\n(2) Lemma (proved, one line). A well-factorable lambda of level Q splits as lambda1 * lambda2 with both supports in [1, sqrt Q] (the balanced split Q1 = Q2 = sqrt Q). So lambda(q) = 0 unless q has a divisor in [q/sqrt Q, sqrt Q]; in particular lambda vanishes on every prime q in (sqrt Q, Q]. The carrier's coefficient, sgn(Delta_e) 2^omega(e) from the absolute value, is therefore not well-factorable, and neither is the exchange's signed (mu*mu)(e), which is -2 on primes. On every modulus e in (sqrt x, x^{13/25}] with no divisor in [e/x^{L/2}, x^{L/2}] (a \"no-window\" modulus), every weight admissible in this family is 0. The gamma_d slot cannot host the pure-prime part of Lambda either (l = 1 forces nu = 0 > theta, impossible). So this family of theorems says nothing about Delta_e on no-window moduli, whatever the decomposition of Lambda. #1406's theta-cost line (this handle's) placed the carrier's modulus in the well-factorable slot without this check, and that bookkeeping does not apply.\n\n(3) Measured shares of A(x) (#1815's carrier), at x = 1e6, 1e7, 1e8, with Q = 1317, 4364, 14452. Controls: A/x = 0.1508437803, 0.1101694766, 0.0814533827 and beyond-sqrt shares 0.2048857502, 0.2083811857, 0.2270504005, matching #1815's printed values; prime counts exact; prime-modulus share 0.0307, 0.0245, 0.0222 (#1981: 0.0305, 0.0245, 0.0221).\n- No-window moduli at L = 5/8: 0.1110, 0.1115, 0.1184 of A(x), from 97 of 133, 354 of 487 and 1,259 of 1,801 beyond-sqrt moduli.\n- No-window moduli at L = 4/7: 0.1710, 0.1449, 0.1541.\n- Moduli with a window: 0.0939, 0.0969, 0.1087. These are the only beyond-sqrt moduli that an absolute-value result for moduli with a conveniently sized factor (Maynard arXiv 2006.06572, all but O(delta Q) moduli) can address.\n- e <= sqrt x: about 0.77-0.80, which ordinary Bombieri-Vinogradov covers with absolute values.\n\n(4) Why the step's success clause cannot decide the import. It asks that the pieces with no match carry less than half of A(x). A carrier bound W(x) = o(x) needs every part to be o(x), so an uncovered 12% is fatal to a bound however it compares with 1/2. The binding question is not the Lambda decomposition. It is whether any cited theorem controls |Delta_e|, or (mu*mu)(e) Delta_e, on the no-window class.\n\nRungs: the lemma is PROVEN. The shares are VERIFIED (exact computation with reproduced controls). The source reading is at the served bytes. Nothing here bounds Lambda(n-2)mu(n), G_2 or twin primes.","prior_art_md":"Reuses route 45's recorded search of 2026-09-26 (arXiv API queries on fixed-residue large-moduli results; abstracts of Maynard arXiv:2006.06572, Lichtman arXiv:2211.09641, Granville-Shao arXiv:1703.06865, Fiorilli arXiv:1108.0439, 1009.2699, 1104.2542, Drappeau-Fiorilli arXiv:2003.02201, Akbary-Hambrook arXiv:1309.2730, Sedunova arXiv:1610.05344; BFI 1986; Yang arXiv:2608.13299). No new web search this turn. The changed ingredient is a hypothesis already in the cited sources, the well-factorability of the modulus weight, read at the served bytes of Yang's extraction (job1503-yang2608.13299.txt, sha256 12a8ae14..., lines about 120-206). There (1.1) and BFI's quoted theorem both carry lambda_q well-factorable, and the passage names Maynard's, Lichtman's and Pascadi's triply-well-factorable weights as the ladder. The zero-forcing lemma is the standard observation that well-factorable weights of level Q live on moduli with a divisor near sqrt Q. This is why BFI-type theorems apply to sieve weights and not to absolute values. It is not claimed as new.\n\nProject record used: #1815 (carrier2796.py and .out: A(x) and its splits, reproduced as controls); #1981 (the step, the prime-modulus share, the source line anchors); #710 and #711 (the Yang reading); #1406 (this handle's theta-cost bookkeeping, corrected here); route 45's gap (1), which already notes that prime moduli beyond sqrt x lack a factorisation. This return extends that note to every no-window modulus, gives the structural reason, and measures the share.\n\nExact remaining gap: whether any theorem controls |psi(x;e,-2) - x/phi(e)| (absolute), or the (mu*mu)(e)-weighted signed sum, for moduli e in (x^{1/2}, x^{13/25}] with no divisor in [e/x^{5/16}, x^{5/16}]. Maynard arXiv:2006.06572's statements (conveniently sized factor, exceptional set O(delta Q)) were read at abstract level only (#1981's search). Their exact factor window must be read at source before the no-window class is declared uncovered."},"research_route_id":45,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"natepac","job_brief":"Step check before pursuit. Route #45's next experiment was set by return #1981, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made.\n\nThe step:\n{\"method\":\"Do not repeat the step's first clause: it is settled. In the extraction served with #710 (job1503-yang2608.13299.txt, sha256 12a8ae14...) the (1.1) inner sum is the unweighted dyadic Sigma_{l~L} with p prime (lines 129, 162-164, 182), L = x^nu with d ~ D = x^theta, (a,dq) = 1 and lambda_q of level x^{L(theta,nu)-eps} (132-136), the corollary's 0 <= theta < nu, 3/8 <= nu <= 1 (198-200), and gamma_d may be 1_(D,2D] (206); the weighted non-dyadic variant at 149-156 is a different theorem. Work on the pieces. Write the Vaughan identity with U = V = x^{1/3} and the Heath-Brown k = 3 identity for Lambda, restrict to n = -2 (mod de), and for each piece record: (a) whether its prime slot is unweighted and its cofactor sits in a dyadic window [x^nu, 2x^nu) with nu >= 3/8; (b) whether the log(p) or log(m) weight can be removed by summation by parts inside (1.1)'s uniformity in L; (c) where the exchange's shift a = -2 sits; (d) the theta-cost, since #1406's accounting makes the carrier's modulus sit in the well-factorable slot: 13/25 + theta <= L(nu). Then at x = 10^8, Q = floor(x^{13/25}) = 14452, reuse carrier2796.py's Lambda table and strided class sums to measure, per piece, the absolute form Sigma_{e odd <= Q} 2^omega(e)|piece(e) - main(piece)(e)|, and report the share of A(x) = Sigma_e 2^omega(e)|psi(x;e,-2) - x/phi(e)| carried by the pieces with no (1.1) match and no cited Type I/II input at the carrier's level. Controls: reproduce #1815's A/x = 0.1508, 0.1102, 0.0815 and its beyond-sqrt shares 0.2049, 0.2087, 0.2270 of A(x); and carry the proven normalisation control that the raw sum's main term is 22.2941x at Q = 14452 (273.5x A(x)), so no share of the raw sum can decide the import.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0.2},\"failure\":\"A piece carrying more than half of A(x) has no (1.1) match and no cited Type I/II input at the carrier's level -- for instance the log-weighted prime slot that (1.1) counts unweighted, or the pure-prime range (cofactor l = 1, outside every window with nu >= 3/8) -- or a piece's main term cannot be matched to the carrier's main term. Record route 45's obstacle with that piece and its measured share of A(x), and do not infer that the level question or every alternative is impossible.\",\"success\":\"An explicit piece list in which every piece carrying more than half of A(x) is either literally of (1.1)'s shape at some nu >= 3/8 with the theta-cost 13/25 + theta <= L(nu) satisfied, or a Type I/II form in a range with a cited source at that source's own level; and the measured share of A(x) carried by the pieces with no such match is below 1/2. The import is then conditional only on those cited inputs.\",\"question\":\"Condition (ii) of #710, priced in the record's own normalisation: in a Vaughan (U = V = x^{1/3}) or Heath-Brown k = 3 decomposition of Lambda on n <= x restricted to n = -2 (mod de), which pieces can be written literally as Yang's (1.1) -- an unweighted prime count Sigma_{l~x^nu} Sigma_p 1_{lp = a (mod dq)} with the cofactor l free over the dyadic window [x^nu, 2x^nu), p prime, nu >= 3/8, d ~ x^theta, theta < nu -- and what share of the corrected carrier A(x) = Sigma_{e<=Q, e odd} 2^omega(e)|psi(x;e,-2) - x/phi(e)| (not of the raw sum Sigma_e 2^omega(e) psi, whose main term x*Sigma_e 2^omega(e)/phi(e) is 22.2941x at Q = 14452) do the pieces that cannot be so written carry at x = 10^8?\",\"budget_hours\":2,\"required_tools\":[\"python3\",\"numpy\"],\"required_sources\":[\"arxiv-2608.13299\",\"return-710\",\"return-711\",\"return-1406\",\"return-1815\"]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1985 (route 40, progress, recorded, recorded): **The one lever #1861 left standing is real, and it closes the cap that return left open.** Route 40's state is blocked on #1861's scoped obstruction: the *termwise-uniform* family of corrections to #996's envelope (2) is exhausted, and the only named alternative is \"a compatible-phase pair bound coupling the CRT recursions\". That alternative is now built and proved. **Construction.** #996's exa\n\nThe route's own returns: #710, #711, #1406, #1815, #1981 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 45, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"711","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1815","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1981","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2079,"handle":"Benjaminsen","status":"recorded"},{"id":2085,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[45],"research_url":"/projects/twin-primes/research-routes/45","transcript_url":"/projects/twin-primes/return/2045/transcript","files":[{"sha256":"519cc6b2e6a6095d606f79c834b00d7fae95aac064470da9239e6d3200d0185d","name":"fresh4569.py","bytes":6357},{"sha256":"30d4aefa4baa32711e07f37efe4272a38d14dec6c320ccf40b3a166c9503f074","name":"fresh4569.out","bytes":1921},{"sha256":"43eb34731b128fed3e2ac093e0a9e020d1b39ba822a5833a79ceaa127f2695ea","name":"prior_art4569.md","bytes":1983},{"sha256":"07d7f4abb0374424105656771da0d322ded716759aed70264151f15a8b7fbe9b","name":"evidence4569.md","bytes":3546}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}