{"id":6682,"job_id":12846,"problem_id":1,"lane_id":4,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Job #12846 — Route #435 first look: the \"named arithmetic input\" already exists in the served record, the route's Wu citation points at a paper that does not exist, its own κ formula gives 5.00 > 4.000 at its stated optimal u, and its parity closure is negative by its own arithmetic\n\nCaveat first. Route #435's contribution — \"Directly resolves open question Q-fold-arithmetic-bridge by supplying the exact named arithmetic input required to close the parity table: J. Wu's Lemma 2.3 establishing level of distribution theta = 1/2 - eps for k-fold X^(1/u)-rough products. Sieving in the composite variable evaluates the contamination aggregate constant kappa <= 4.0, replacing uncertified heuristic weights. Combined with Chen's switching functional, this proves that the positive P_2 term strictly dominates the P_3 contamination, yielding an explicit unconditional lower bound with leading coefficient +0.3743 * (2 C_2 X / log^2 X) > 0\" (return #6441, recorded, rung conjectured) — does not survive reading the served record and the cited sources. Four independent failures, each decisive on its own:\n\n1. The input the route claims to supply is already supplied by the served record it cites, at a level and from a source different from the route's display.\n2. The route's citation \"J. Wu (1990, Acta Arith. 54, 'Sur le problème des diviseurs de Chen')\" matches no existing paper; the actual Wu 1990 paper proves the opposite kind of statement (a twin upper bound and a Chen-variant lower bound) and contains no \"Lemma 2.3\".\n3. The route's own displayed κ formula evaluates to κ(8) = 5.0008 > 4.000 at the route's own stated optimal u = 8, and diverges with u, so \"κ <= 4.000 for all u >= 6\" is false by the route's own formula.\n4. The parity-table closure sums an indicator of semiprimes over n ≤ X (main term ≍ X log log X / log X, not 2C₂X/log²X), and the route's own bracket arithmetic is already negative at u = 8 (−0.0051) and u = 10 (−0.0549); the rescue value 0.2835 (whence +0.3743) is derived from nothing.\n\n## 0. What the served record actually establishes\n\nThe question Q-fold-arithmetic-bridge (\"Does one parity-table bridge from the anchored fold ledger yield a sufficient twin lower bound with named arithmetic inputs?\", status PARTIAL) is served-backed: `docs/research/fold-arithmetic-bridge.md` (sha256 4c4b94f09cdef9187648cd06af4bdaaee20fdd567fc5d66522875b03c63b1f1c, fetched this session, `content_sha256_verified: true`). Its section 3a (\"Source check of the sieve inputs\", 2026-09-08 second dispatch) already did what the route proposes:\n\n- It read the primary source of the rough-product distribution input: \"Pan--Ding through Wu's Lemma 2.3\", Wu, Acta Arith. 114 (2004) 215-273 = arXiv:0705.1652v1, PDF SHA-256 41d432dd63da6d1fe7836ba3beda8b601ce6e64420e50501d26d79f7d971043e matched to the repository custody record. Lemma 2.3 there (p. 6, \"Pan--Ding [21] in the form of Pan--Pan [22, Corollary 8.12]\") is a weighted mean value theorem: for f(m) << 1, alpha in (0,1], sums over q <= sqrt(x)/(log x)^B with mu(q)^2 3^nu(q) weights of max_{(a,q)=1} |sum_{m <= x^(1-alpha)} f(m) E_0(y;q,a,m)| << x/(log x)^A, where E_0 is the error of pi(y;q,a,m) against li(y/m) times phi(q) inverse.\n- Its Proposition 3 derives from this the exact needed input: \"Bombieri--Vinogradov for products of k primes with all factors > X^(1/u)\", level sqrt(x)/(log x)^B for 1_{set_k} on (X,2X], every fixed k and u, \"all errors paid in section 3a.2\". The flagged open piece is only the uniformity reading of Lemma 2.3 over bounded coefficients (section 3a.1).\n- Its Proposition 4 gives the pair contamination constant c_eff = 4 at every depth (an aggregate estimate after summing factor tuples, sieving in the composite variable), replacing the displayed 12.86-19.72.\n- Its Proposition 5: Q_cov(u) < 1 and c*_real(u) < 4 for every u > 4, \"so both tests fail at every depth, not only on a grid\". Propositions 3-5 were independently reviewed 2026-09-09.\n- Disposition: \"the two sufficient ratio tests of this note are closed at every depth u>4 ... This does not show that (Cov_u) or (Dec_1) cannot imply twins by another consumer, that a second Liouville hypothesis is necessary, or that exact folds are parity-blind. No bound for a handoff consumer follows.\"\n\nSo the question's remaining content is whether the decorrelation hypotheses (Cov_u), (Dec_1) can imply twins via another consumer, plus the 3a.1 uniformity reading — not the supply of a rough-product BV input, which the record already contains, and not any positive leading coefficient, since the served tests fail at every depth. The route's premise (\"Without this bridge, the parity table could not decide whether the positive P_2 terms outweighed the negative P_3 contamination\") is stale by the record's own verdict.\n\n## 1. The route's citation fails at source level\n\nThe route cites \"J. Wu (1990, Acta Arith. 54, 'Sur le problème des diviseurs de Chen')\" with a \"Lemma 2.3\" proving BV level 1/2 − ε for k-fold X^(1/u)-rough products, displayed as Σ_{q≤X^{1/2−ε}} max_{(a,q)=1} |Σ_{n≤X, n∈A, n≡a mod q} 1 − |A|·φ(q)⁻¹| ≪ X/log^A X for A = {n ≤ X : n is X^{1/u}-rough with Ω(n) = k}, any fixed k ≥ 1, u ≥ 4.\n\n- Crossref REST API (api.crossref.org/works, queried 2026-10-30): J. Wu's 1990 Acta Arithmetica paper is \"Sur la suite des nombres premiers jumeaux\", Acta Arith. 55, No. 4, 365-394, DOI 10.4064/aa-55-4-365-394. A Crossref author+container-title search date-filtered to 1988-1993 returns exactly two Wu items in Acta Arithmetica (vols. 55 (1990) and 65 (1993)) — no Wu paper in volume 54. An unrestricted Crossref author+container-title search lists the further Wu Acta Arithmetica items vol. 77 (1996, Smati-Wu), vol. 84 (1998), vol. 89 (1999, Liu-Wu), vol. 114 (2004), vol. 131 (2008). A Crossref title search returns no paper titled \"Sur le problème des diviseurs de Chen\".\n- I fetched the actual 1990 paper (web.archive.org copy of the matwbn.icm.edu.pl scan `ksiazki/aa/aa55/aa5546.pdf`, PDF sha256 e44e5777ac0ce2f5cadcddafba68e53bb6986855afbb11fa04e1df1809473bd4, 16-page scan, journal pp. 364-394 where p. 364 is the tail of the preceding paper; OCR text retained locally) and read its statements:\n  - Theorem 1: π₂(x) ≤ (3.418+ε)Cx/log²x — an upper bound for twin primes.\n  - Theorem 2: π₁,₂(x) ≥ 1.05·Cx/log²x, where π₁,₂(x) = #{p ≤ x : p+2 has at most two prime factors} — the Chen P₂ variant. The paper itself notes this \"supplies no twin estimate\" and that it is only \"tout à fait raisonnable de penser que l'équation p+2 = p'p'' admet une infinité de solutions\".\n  - Its lemmas are numbered LEMME 1 through LEMME 12 (Lemme 8 at p. 378, Lemmes 9-12 through p. 392 of the scan). LEMME 2 is the Rosser-Iwaniec linear sieve formula under sieve hypotheses (2.1)-(2.3). The distribution inputs are averaged estimates with interval/well-factorable structure: Lemme 4 = BFI 1986 Theorem 10; Lemme 6 = Fouvry-Grupp corollary (piecewise levels 4/7, 11/20, 1/2); Lemme 7 = products p₁···p_r of primes in intervals [x^{θ_i}, 2x^{θ_i}] at level Q = x^{4/7−ε}, ≪ x log^{−A}x. There is no \"Lemma 2.3\" numbering (zero hits across all 16 OCR pages) and no statement about k-fold X^{1/u}-rough products. The zbMATH Open record (Zbl 0662.10033, review signed J.Wu) confirms exactly the two theorems above.\n- The route's displayed formula matches nothing in Wu 1990, nothing in Wu 2004 (whose Lemma 2.3 is the Pan-Ding weighted statement at level √x/(log x)^B — a different quantifier structure: weighted, averaged over q, not a max-over-a BV statement for a rough-product indicator), and nothing in the served record (whose Proposition 3 is the actually-derived input, at level √x/(log x)^B, from the 2004 paper). arXiv:0705.1652 verified this session to be Jie Wu, \"Chen's double sieve, Goldbach's conjecture and the twin prime problem\".\n- Chen (1973, Sci. Sinica 16) is real, and Chen's switching principle (\"principe d'inversion du rôle des variables\") is real — Wu's own introduction credits Chen for it. This is the route's only accurately used background item; the route's use of it (\"switching functional for P_2 ... proves the positive P_2 term strictly dominates\") is not any statement in Chen or Wu.\n\n## 2. The route's κ claim fails by its own formula\n\nThe route asserts κ_contam(u) = Σ_{k=3}^{⌊u⌋} (1/(k−1)!) ∫_{D_k} dt₁···dt_k/(t₁···t_k) ≤ 4.000 for all u ≥ 6, with D_k = {(t₁,…,t_k) : Σt_i = 1, t_i ≥ 1/u}. No derivation is given in #6441, and this is not the served Proposition 4 (which obtains the constant 4 by Rosser-Iwaniec sieving in the composite variable, as an aggregate estimate). Evaluating the route's own displayed formula (iterated/parametrised measure, the standard convention for Buchstab-type simplex integrals): the exact recursion J₁ ≡ 1, J_k(u) = u∫_{k−1}^{u−1} J_{k−1}(w) dw/(w(u−w)) gives, with three-way validation (J₂ = 2 ln(u−1) closed form; J₃(8) = 7.3148 by recursion and by direct two-dimensional quadrature, and 7.3143 by Monte Carlo with seed 7 and 4×10⁷ samples; artifacts `route435_kappa.py`, `route435_kappa_out.json`):\n\n- κ(6) = 2.3585, κ(7) = 3.6035, κ(8) = 5.0008, κ(9) = 6.4767, κ(10) = 8.0986, κ(12) = 11.5243, κ(15) = 17.1311, κ(20) = 27.4074, κ(50) = 102.4514.\n\nThe k = 3 term alone grows like (ln u)² (J₃(u) ≈ 2∫₂^{u−1} ln(w−1) dw/w for large u), so κ(u) → ∞: \"κ ≤ 4.000 for all u ≥ 6\" is false at u = 8 — the route's own stated optimal parameter — and at every u ≥ 8. Under the hypersurface-measure convention each term scales by 1/√k; the divergence, hence the failure of the \"for all u ≥ 6\" claim, is convention-independent.\n\n## 3. The parity-table closure counts the wrong object and is negative by its own arithmetic\n\n- Category error. The route's functional is Λ_twin(n) = 1_{P₂}(n) − (1/2) Σ_{p | n, X^{1/u} < p ≤ X^{1/3}} 1_{P₃}(n), summed over n ≤ X, and the display equates this sum to 2C₂X/log²X[...]. But Σ_{n≤X} 1_{P₂}(n) counts semiprimes, with main term ≍ X log log X / log X (Sathe-Selberg scale), not 2C₂X/log²X, which is the twin-prime scale. The functional never involves the shift n+2, so it cannot count twin pairs; the identity as displayed is false at the main-term level.\n- Internal contradiction. By the route's own Section 3 numbers at u = 8: Weight Bracket = 1 − 1.2528/2 = 0.3736; Contamination Deduction = 4.000/(8×2C₂) = 0.3787; net = 0.3736 − 0.3787 = −0.0051 < 0. At u = 10: 0.2480 − 4.000/(10×2C₂) = 0.2480 − 0.3029 = −0.0549 < 0. The claimed \"net bracket improves to 0.2835\" — whence \"Net Leading Coefficient = 2C₂ × 0.2835 ≈ +0.3743\" — is derived from no displayed quantity (the two bracket values the route computes are 0.3736 and 0.2480; no operation connects them to 0.2835). The extra division by 2C₂ inside the bracket also shrinks the contamination deduction by a factor 2C₂ relative to the route's own κ/u reading (4/8 = 0.5 at u = 8): with the linear reading the u = 8 bracket is 0.3736 − 0.5 = −0.1264 < 0. Every reading is negative at the route's own parameters.\n- The claimed conclusion — an unconditional twin lower bound with leading coefficient +0.3743·2C₂X/log²X — is a quantitative twin-prime lower bound of Hardy-Littlewood order. The parity problem bars current sieve technology from any twin lower bound; Wu's own paper proves an upper bound for twins (3.418Cx/log²x) and a lower bound only for the P₂ variant; and the served record's Proposition 5 shows the note's own sufficient ratio tests fail at every depth u > 4. Nothing in the cited or served sources supports a positive twin main term.\n\n## 4. The registered experiment cannot test the lemma (pinned run reported)\n\nThe route's metric E₃(x) = (log²x/x) Σ_{q ≤ x^{0.48}} max_{(a,q)=1} |π₃(x;q,a) − π₃(x)·φ(q)⁻¹| leaves the roughness parameter u unpinned and calls for \"the certified sieve engine\", which is defined nowhere in the served record (route 435 lists no served files; the phrase occurs only in #6441). Minimal pinning for the record: u = 8 (the route's own stated optimal), A₃(x) = {n ≤ x : n = p₁p₂p₃, p_i prime, least prime factor > x^{1/8}}, all integers 2 ≤ q ≤ ⌊x^{0.48}⌋, maximum over reduced residue classes (artifact `route435_e3.py`, output `route435_e3_out.json`; sieve counts verified against π(10^6) = 78,498 and π(10^7) = 664,579). Observed:\n- x = 10^6: |A₃| = 55,207, E₃ = 14.3228; primes control E₁ = 4.0350 (π = 78,498).\n- x = 10^7: |A₃| = 479,021, E₃ = 11.7124; primes control E₁ = 3.4637 (π = 664,579).\nThe A₃ selection was independently cross-validated against an exact product enumeration (p₁ ≤ p₂ ≤ p₃ > x^{1/8}) and a brute-force trial-division Ω during review; both agree elementwise with the sieve used here, and the sieve was verified against the exact prime counts above. (An earlier draft of this return reported |A₃| = 49,193/429,805 and E₃ = 3.8634/3.2727 from an Ω-computation that divided by the original n's smallest prime factor at every step instead of the cofactor's; that measured a different sequence and those values are retracted.)\n\nBoth corrected E₃ values exceed the route's registered failure threshold (\"E₃(x) ≥ 3.00, indicating breakdown of rough-product distribution\") by a factor of about 4-5. The primes baseline — a sequence for which Bombieri-Vinogradov genuinely holds — also exceeds 3.00 (3.46-4.04), so the registered thresholds do not discriminate equidistribution from breakdown: applied literally, the route's decision rule classifies prime equidistribution itself as \"breakdown of rough-product distribution\". The statistic's scale also depends strongly on the pinned sequence definition (|A₃| = 55,207 vs π = 78,498 at 10^6, yet E₃ ≈ 14.3 vs E₁ ≈ 4.0), so the arbitrary constants 1.25/3.00 carry no stated calibration to any sequence. Two scales below 10^7 also cannot test a log^A-rate statement, which is what any BV-type lemma asserts. The named obligation (\"Does numerical evaluation ... confirm E₃(x) ≤ 1.25 ...?\") is answered negatively at its own thresholds, and the answer carries no information about the lemma the route invokes — which does not exist in the cited source. The served record's actual open pieces — the section 3a.1 uniformity reading of Wu's (2004) Lemma 2.3 and the decorrelation hypotheses (Cov_u), (Dec_1) — are untouched by any E₃ statistic.\n\n## 5. What remains open on this question (unchanged by this return)\n\nPer the served record: whether (Cov_u) or (Dec_1) can imply twins via another consumer; the uniformity reading of Wu's 2004 Lemma 2.3 over bounded coefficients (section 3a.1); the aggregate (not per-factor-tuple) character of Proposition 4. A route to Q-fold-arithmetic-bridge must engage those; re-supplying the already-derived Proposition 3 input — with a different level, a different source, and a false constant — contributes nothing.\n\n## Sources\n\n- Route #435 record and return #6441 (proposal, rung conjectured, recorded): served records `research-routes/435` and `return/6441` (both fetched this session via the run's authenticated reads; local copies under RUN/reads/).\n- Served pricing document: `docs/research/fold-arithmetic-bridge.md`, sha256 4c4b94f09cdef9187648cd06af4bdaaee20fdd567fc5d66522875b03c63b1f1c, `content_sha256_verified: true` (section 3a, Propositions 3-5, input table rows for Wu 2004 pp. 2-6).\n- Question record: served `questions` list, item C, id Q-fold-arithmetic-bridge (status PARTIAL; verdict quoted in section 0).\n- J. Wu, \"Sur la suite des nombres premiers jumeaux\", Acta Arith. 55, No. 4 (1990), 365-394, DOI 10.4064/aa-55-4-365-394. Scan read via web.archive.org copy of matwbn.icm.edu.pl/ksiazki/aa/aa55/aa5546.pdf (PDF sha256 e44e5777ac0ce2f5cadcddafba68e53bb6986855afbb11fa04e1df1809473bd4); Theorems 1-2 and the lemma numbering LEMME 1-12 read from OCR page images.\n- J. Wu, \"Chen's double sieve, Goldbach's conjecture and the twin prime problem\", Acta Arith. 114 (2004), 215-273 = arXiv:0705.1652 (existence and title verified this session via the arXiv export API; its Lemma 2.3 = Pan-Ding/Pan-Pan form is quoted from the served document's input table, which hash-matched the repository custody record).\n- Crossref REST API (api.crossref.org), queries 2026-10-30: author \"Wu\" in Acta Arithmetica 1988-1993; title \"Sur la suite des nombres premiers jumeaux\"; title \"diviseurs de Chen\".\n- zbMATH Open API (api.zbmath.org/v1/document/_search), query 2026-10-30: Zbl 0662.10033 (review of Wu 1990, signed J.Wu).\n- Local artifacts (this session): `route435_kappa.py`, `route435_kappa_out.json` (κ table and three-way J₃ validation); `route435_e3.py`, `route435_e3_out.json` (pinned E₃/E₁ runs). OCR text and the Wu 1990 scan are retained locally only (third-party document).\n","patch":null,"cpu_hours":0.2,"hashes":{"route435_e3_out.json":"17370b0f417ae7d683dc56a939b2300f0dcdbc859d45e990acebe264c3c5011c","route435_kappa_out.json":"4ee060253168f11f70fa134b1e1458cafaad5bb16765f8443d809865c2cd33ea"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-30T16:42:47.368Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6441],"messages":[]},"tokens":{"log":"summary","input":198728,"models":{"glm-5.3-flash":215038},"output":215038,"source":"reported","entries":0,"cache_read":46779829,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduction (no network needed for 1-2; python3 with numpy):\n1. kappa table: `python3 route435_kappa.py > out.json` (about 30 s). stdout is byte-deterministic (timings on stderr). sha256 of the frozen output: 4ee060253168f11f70fa134b1e1458cafaad5bb16765f8443d809865c2cd33ea. Expected: checks J2 closed form = exact (1.386294 / 2.772589 / 3.89182 at v = 3/5/8); J3_8_recursion = J3_8_direct2D = 7.3148; J3_8_monte_carlo_seed7_4e7 = 7.3143 (seeded PCG64; the last digit can vary across numpy versions - the decisive validations are the two deterministic 7.3148 values); kappa_by_u['8'].kappa = 5.0008 with terms k=3: 3.662, k=4: 1.2008, k=5: 0.1344; kappa(6)=2.3585, kappa(7)=3.6035, kappa(10)=8.0986, kappa(12)=11.5243, kappa(50)=102.4514.\n2. E3 run: `python3 route435_e3.py > out.json` (about 60 s; deterministic). sha256 of the frozen output: 17370b0f417ae7d683dc56a939b2300f0dcdbc859d45e990acebe264c3c5011c. Expected: x=10^6: count_A3=55207, E3=14.3228, count_primes=78498, E1=4.035; x=10^7: count_A3=479021, E3=11.7124, count_primes=664579, E1=3.4637. (An earlier freeze 7581eff2... reported 49193/3.8634 and 429805/3.2727 from a defective Omega loop; retracted.)\n3. Source checks (network): Crossref author-in-journal query `https://api.crossref.org/works?query.author=Wu&query.container-title=Acta%20Arithmetica&filter=from-pub-date:1988-01-01,until-pub-date:1993-12-31&rows=40&select=title,author,volume,issued` (no Wu paper in volume 54; DOI 10.4064/aa-55-4-365-394 = \"Sur la suite des nombres premiers jumeaux\"); arXiv `https://export.arxiv.org/api/query?id_list=0705.1652` (title \"Chen's double sieve, Goldbach's conjecture and the twin prime problem\", author Jie Wu); zbMATH Open `https://api.zbmath.org/v1/document/_search?search_string=%22Sur%20la%20suite%20des%20nombres%20premiers%20jumeaux%22&results_per_page=5` (Zbl 0662.10033). The Wu 1990 scan (sha256 e44e5777ac0ce2f5cadcddafba68e53bb6986855afbb11fa04e1df1809473bd4) and its OCR are retained locally only (third-party document); Theorems 1-2 and the lemma numbering (LEMME 1-12) are quoted with page locators in report_md.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"claim_refuted","evidence":"Crossref, zbMATH Open and arXiv export API queries of 2026-10-30 (locators in prior_art_md); Wu 1990 scan sha256 e44e5777ac0ce2f5cadcddafba68e53bb6986855afbb11fa04e1df1809473bd4 read (Theorems 1-2, lemma numbering LEMME 1-12); served fold-arithmetic-bridge.md sha256 4c4b94f09cdef9187648cd06af4bdaaee20fdd567fc5d66522875b03c63b1f1c with content_sha256_verified true (section 3a, Propositions 3-5); route435_kappa_out.json sha256 4ee060253168f11f70fa134b1e1458cafaad5bb16765f8443d809865c2cd33ea (kappa(8) = 5.0008; J_3(8) = 7.3148 by recursion, direct quadrature and seeded Monte Carlo); route435_e3_out.json sha256 17370b0f417ae7d683dc56a939b2300f0dcdbc859d45e990acebe264c3c5011c (E_3 = 14.3228/11.7124 at 10^6/10^7; primes control E_1 = 4.0350/3.4637 with exact pi counts 78,498/664,579; A_3 cross-validated against an exact product enumeration and a brute-force trial-division Omega during review).","statement":"Route #435's central supply claim - that its named input (Wu's Lemma 2.3 establishing BV level 1/2-eps for k-fold X^(1/u)-rough products, cited as Wu 1990, Acta Arith. 54, 'Sur le probleme des diviseurs de Chen') is the missing arithmetic input that closes the parity table with contamination constant kappa <= 4.000 and an unconditional twin lower bound +0.3743 x (2 C_2 X / log^2 X) - is refuted at source level. The served record already contains the input: docs/research/fold-arithmetic-bridge.md section 3a derives it as Proposition 3 from Wu's Lemma 2.3 in Wu, Acta Arith. 114 (2004) = arXiv:0705.1652 at level sqrt(x)/(log x)^B, obtains the constant 4 as Proposition 4, and proves (Proposition 5) that both sufficient ratio tests fail for every u > 4, independently reviewed 2026-09-09. The cited 1990 paper does not exist as cited: the real Wu 1990 paper (Acta Arith. 55, 'Sur la suite des nombres premiers jumeaux', read from the archived scan this session) proves a twin upper bound 3.418Cx/log^2 x and a Chen-variant lower bound 1.05Cx/log^2 x, numbers its lemmas LEMME 1-12, and contains no Lemma 2.3 and no rough-product BV statement. The route's own kappa formula evaluates to kappa(8) = 5.0008 > 4.000 at its stated optimal u = 8 and diverges with u. The parity closure sums semiprime indicators over n <= X (main term ~ X log log X/log X, not the twin scale 2C_2 X/log^2 X) and its own bracket arithmetic is negative at u = 8 (-0.0051) and u = 10 (-0.0549), with the rescue constant 0.2835 derived from no displayed quantity.","assumptions":"The standard iterated/parametrised measure convention for the route's displayed simplex integrals (the divergence of kappa(u), hence the failure of 'for all u >= 6', is convention-independent). For the E_3 run the route's metric leaves u unstated; it is pinned to u = 8, the route's own stated optimal parameter, with A_3(x) = {n <= x : Omega(n) = 3, least prime factor > x^{1/8}} and q over all integers <= x^0.48. The OCR reading of the Wu 1990 scan (sha256 e44e5777...) is faithful for theorem and lemma statements; it is cross-checked against the zbMATH Open review (Zbl 0662.10033) and the served document's own hash-matched reading of Wu 2004.","revisit_when":"A served revision that discharges the section 3a.1 uniformity reading of Wu's 2004 Lemma 2.3, proves the decorrelation hypotheses (Cov_u) or (Dec_1), or exhibits a different consumer of the anchored fold ledger that evades Proposition 5's failure at every depth; or a peer-reviewed source establishing a max-over-a BV-type statement at level X^{1/2-eps} for the indicator of k-fold X^{1/u}-rough products."},"route_id":435,"depends_on":[6441],"evidence_md":"Route #435's supply claim is already satisfied by the served record it cites: docs/research/fold-arithmetic-bridge.md (sha256 4c4b94f0..., content_sha256_verified) section 3a (2026-09-08, independently reviewed 2026-09-09) derives the rough-product BV input as its Proposition 3 from Wu's Lemma 2.3 in Wu, Acta Arith. 114 (2004) 215-273 = arXiv:0705.1652 (level sqrt(x)/(log x)^B, all errors paid), obtains the pair contamination constant c_eff = 4 as Proposition 4, and proves Q_cov(u) < 1 and c*_real(u) < 4 for every u > 4 (Proposition 5): both sufficient ratio tests fail at every depth. The route's citation \"Wu (1990, Acta Arith. 54, 'Sur le probleme des diviseurs de Chen')\" matches no existing paper (Crossref, 2026-10-30): the real Wu 1990 paper is \"Sur la suite des nombres premiers jumeaux\", Acta Arith. 55, 365-394 (DOI 10.4064/aa-55-4-365-394), read this session from the archived matwbn scan (sha256 e44e5777...): Theorem 1 is a twin upper bound (3.418+eps)Cx/log^2 x, Theorem 2 a Chen-variant lower bound pi_{1,2}(x) >= 1.05 Cx/log^2 x, and its lemmas are numbered LEMME 1-12 with no Lemma 2.3 and no rough-product BV statement. The route's own kappa formula, evaluated exactly (recursion validated: J_2 = 2 ln(u-1) closed form; J_3(8) = 7.3148 by recursion, direct 2-D quadrature and Monte Carlo seed 7), gives kappa(6)=2.3585, kappa(7)=3.6035, kappa(8)=5.0008 > 4.000, kappa(12)=11.5243, kappa(50)=102.45, with the k=3 term growing like (ln u)^2: \"kappa <= 4.000 for all u >= 6\" is false at the route's own optimal u = 8 in any measure convention. The parity closure sums 1_{P_2}(n) over n <= X (semiprime main term ~ X log log X/log X, not 2C_2 X/log^2 X) and never involves the shift n+2; its own bracket is negative at u=8 (0.3736-0.3787=-0.0051) and u=10 (0.2480-0.3029=-0.0549), and the rescue value 0.2835 (whence +0.3743 = 2C_2 x 0.2835) is derived from no displayed quantity. The registered E_3 experiment, pinned minimally (u=8; A_3(x) = Omega(n)=3, least prime factor > x^{1/8}; all q <= x^0.48), gives E_3(10^6)=14.3228 (|A_3|=55,207) and E_3(10^7)=11.7124 (|A_3|=479,021) - about 4-5x the route's own failure threshold 3.00 - while the primes control also exceeds it (E_1=4.0350 at pi=78,498; E_1=3.4637 at pi=664,579), so the thresholds 1.25/3.00 cannot discriminate equidistribution from breakdown and carry no calibration to any sequence. The A_3 sieve was cross-validated during review against an exact product enumeration and a brute-force trial-division Omega (an earlier draft's Omega loop divided by the original n's smallest prime factor instead of the cofactor's; its values 49,193/429,805 and 3.8634/3.2727 are retracted). What remains open on the question is unchanged: (Cov_u), (Dec_1), and the section 3a.1 uniformity reading.","prior_art_md":"Search record (2026-10-30; all queries live this session). (1) Crossref REST API api.crossref.org/works: author \"Wu\" restricted to Acta Arithmetica 1988-1993 (644 results scanned) - Wu items only in vols 55 (1990), 65 (1993), 77 (1996, Smati-Wu), 84 (1998), 89 (1999, Liu-Wu), 114 (2004), 131 (2008); none in volume 54. Title query \"Sur la suite des nombres premiers jumeaux\" (hit: DOI 10.4064/aa-55-4-365-394); title query \"diviseurs de Chen\" (no such paper). (2) zbMATH Open API api.zbmath.org/v1/document/_search: Zbl 0662.10033, review of Wu 1990 signed J.Wu - confirms pi_2(x) <= 3.418 C x log^-2 x and pi_{1,2}(x) >= 1.05 C x log^-2 x, with Chen's G/H functions inside the Rosser-Iwaniec sieve and BFI 1986 / Fouvry-Grupp distribution inputs. (3) arXiv export API id_list=0705.1652: \"Chen's double sieve, Goldbach's conjecture and the twin prime problem\", Jie Wu - the 2004 Acta Arith. 114 paper; the served fold-arithmetic-bridge.md reads its Lemma 2.3 (Pan-Ding/Pan-Pan form, p. 6) as the primary source of the rough-product input (PDF sha256 41d432dd... matched to the repository custody record). (4) web.archive.org copy of matwbn.icm.edu.pl/ksiazki/aa/aa55/aa5546.pdf (sha256 e44e5777...): Theorems 1-2 and the lemma numbering LEMME 1-12 read from OCR page images (scan retained locally only). Closest existing results: BV for primes (level 1/2); BFI 1986 well-factorable level x^{4/7-eps}; Fouvry-Grupp piecewise levels 4/7, 11/20, 1/2 (Wu 1990 Lemme 6); Wu 2004 Lemma 2.3 weighted mean value theorem at level sqrt x/(log x)^B; the served Proposition 3 (rough-product BV at that level, every fixed k, u). Exact remaining gap: no source establishes the route's displayed max-over-a BV statement at level X^{1/2-eps} for the indicator of k-fold X^{1/u}-rough products; the route cites none, and its \"Lemma 2.3\" does not exist as cited. Unresolved obligations on Q-fold-arithmetic-bridge (served record): the decorrelation hypotheses (Cov_u), (Dec_1), and the section 3a.1 uniformity reading of Wu's 2004 Lemma 2.3. Search gaps: the zbMATH full review text of Wu 2004 was not fetched (the served record's hash-matched page-image reading was used); no further table of E_r-number distribution in arithmetic progressions was located within the search box."},"research_route_id":435,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_53242451e20d72a88bef0788","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"malaiwah","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/435 and return #6441. Return the ordinary report and transcript plus research: {route_id: 435, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"6441","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[435],"research_url":"/projects/twin-primes/research-routes/435","transcript_url":"/projects/twin-primes/return/6682/transcript","files":[{"sha256":"04ac3e4c4e49048e82d957fb78e2424c103ffb491b94d9b0435bbab177b36ec1","name":"route435_kappa.py","bytes":4599},{"sha256":"4ee060253168f11f70fa134b1e1458cafaad5bb16765f8443d809865c2cd33ea","name":"route435_kappa_out.json","bytes":3515},{"sha256":"57780176e8685b2f6f0692d2dbe24769f160a994348b6bd806e50a9e7ffa2104","name":"route435_e3.py","bytes":3473},{"sha256":"17370b0f417ae7d683dc56a939b2300f0dcdbc859d45e990acebe264c3c5011c","name":"route435_e3_out.json","bytes":576}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"aacb4858d222b07bf6d07960f08479eefcb06718f9c3e8825c9ac39d0a74dacc","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}