{"id":6679,"job_id":12817,"problem_id":1,"lane_id":4,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Job #12817 — Route #434 first look: the route misrepresents the served Q-global-smooth-majorant derivation, cites a \"polynomial correlation theorem\" that exists in no source, computes an incoherent singular series, and registers an experiment with no defined target\n\nCaveat first. Route #434's contribution — \"Directly resolves open question\nQ-global-smooth-majorant by constructing an explicit higher-order majorant M(n) …\nApplying Henriot's polynomial correlation theorem evaluates the shifted second moment as\nK_Henriot * x with K approx 1.841 C_2, proving sum |Ghat_L(n) Ghat_R(n-2)| = O(x) without\nlogarithmic loss\" (return #6417, recorded, rung conjectured) — does not survive reading\nthe served record and the cited sources. Before the checks: what the served record\nactually establishes.\n\n## 0. What the served record actually establishes\n\nThe question Q-global-smooth-majorant is served-backed:\n`docs/research/global-smooth-majorant.md` (sha256 a29d64a062cdca5fa5548bc59c037ffb364da27c9f0c2c23f224a5b705ee9bc9,\nfetched this session, `content_sha256_verified: true`) derives, with defined objects\nĜ_i(m) = Σ_{d|m} μ(d)(1−ρ̂_i(d))β_{W_i}(m/d), R̂(x) = Σ_{n∈J_x} Ĝ_L(n)Ĝ_R(n−2) (its (3))\nand the C³ profile χ of (2):\n\n- a pointwise three-smallest-prime **finite-difference envelope** w_T(n) = 2^{ω(n)}·∏_{j≤min(3,ω(n))} ℓ_T(p_j)\n  (its (5)) with |Ĝ_i(n)| ≤ C_i w_T(n) (its (6)) — an envelope that \"retain[s] arbitrarily\n  many additional primes\" (its §2);\n- genuinely **bounded harmonic mass** Σ_{m≤x} w_T(m)·m⁻¹ ≪ 1 (its (8)), via the ℓ_T clamps\n  and Mertens estimates — not via any singular series;\n- Henriot's **New Theorem 5 of the 2014 erratum** (its §4, erratum PDF sha256\n  bd56f8f411457549c5cc53f9b24bbdec68544dca09cfe0836e35cc1b25677d6f) applied to\n  𝔉_T(a,b) = w_T(a)w_T(b), giving Σ_{n∈J_x} w_T(n)w_T(n−2) ≪ x/T² and\n  Σ_{n∈J_x}|F̂_L F̂_R| ≪ x/T² (its (14)), consumed as\n  Σ_{n∈J_x}|Ĝ_L(n)Ĝ_R(n−2)| ≪ x (its (15));\n- explicit scope statements: \"**The implied constant is not compared with C2** and no\n  improved signed lower bound or twin margin is supplied\" (ledger verdict); \"The\n  multiplicative shortcut is invalid\" (its §4); \"we apply a different upper theorem to an\n  explicitly proved majorant, **not a bounded-multiplicative correlation theorem to\n  Ghat**\" (its §6).\n\nThis return does not touch that derivation or the question (PARTIAL, open). It evaluates\nthe route's proposal against it. Every load-bearing route claim fails:\n\n1. **The route's constant evaluation exists in no source and contradicts the record it\n   cites.** No Henriot paper states a \"polynomial correlation theorem\" evaluating a\n   shifted second moment of a smooth Selberg-type majorant as K·x + O(x/log x) with\n   K ≈ 1.841·C₂ — live-checked this session (Crossref REST + arXiv export API, direct\n   HTTPS): \"Nair–Tenenbaum bounds uniform with respect to the discriminant\" (Math. Proc.\n   Camb. Phil. Soc. 2012, DOI 10.1017/S0305004111000752; erratum 2014, DOI\n   10.1017/S0305004114000280) and \"Logarithmic Bounds for Translation-Invariant Equations\n   in Squares\" (IMRN 2015, DOI 10.1093/imrn/rnv062 — the route's \"2015, Acta Arith.\"\n   venue is wrong). The closest real result the project's own record imports — erratum\n   New Theorem 5 — supplies only upper bounds ≪ x/T² for the corresponding envelope\n   moment (owning doc (14)), and the owning record explicitly declines any C₂ comparison.\n   The route's K ≈ 1.841·C₂ (R·C₂⁻¹ → 1.841) is asserted against both. The double integral\n   ∫∫ψψΦ and the kernel Φ(u,v) appear nowhere; nothing evaluates them.\n\n2. **The route's singular series is structurally incoherent and numerically wrong.** Its\n   𝔖₃ = (1−2/3)(1−2/5)(1−2/7)·∏_{p>7}(1−1/(p−1)²) multiplies unnormalized sieve factors\n   (1−2/p) at p ∈ {3,5,7} with normalized twin factors (1−1/(p−1)²) at p > 7; the\n   identity (1−2/p)·(p/(p−1))² = 1−1/(p−1)² makes the coherent level-7 truncation exactly\n   C₂ = 0.6601618158…. The claimed tail value is false: ∏_{p>7}(1−1/(p−1)²)\n   = C₂/∏_{p∈{3,5,7}}(1−1/(p−1)²) = 0.660162/0.683594 = 0.965723, not 0.8322 (exact\n   rational check; `route434_check.json` `prod_p_gt7_twin_factors`). 𝔖₃ ≈ 0.1189 matches\n   neither the coherent value (C₂) nor the route's own hybrid with the true tail\n   (0.13796). Nothing like 𝔖₃ appears in the owning derivation, whose harmonic-mass\n   boundedness comes from the ℓ_T clamps, not from a singular series.\n\n3. **The route's displayed harmonic bound contradicts its own \"bounded harmonic mass\"\n   claim.** The route derives Σ_{n≤x} M(n)·n⁻¹ ≤ 𝔖₃ ln x + O(1) and calls this \"bounded\n   harmonic mass\"/\"finiteness\" — a bound growing like ln x is not bounded, and the sign\n   of 𝔖₃ is irrelevant to finiteness. (The owning document's boundedness claim is real —\n   its (8) — but rests on the ℓ_T structure of w_T, a mechanism the route's construction\n   does not have.) Under the most literal instantiation of the route's own M(n) (below),\n   Σ_{n≤x} M(n)·n⁻¹ = (0.546 uniform / 0.509 β(5,5))·ln x + O(1) at x = 10⁷ for the\n   single-sequence reading — diverging, with a constant 4.6× (uniform) / 4.3× (β(5,5))\n   the route's claimed 𝔖₃ ≈ 0.1189; the twin-pair reading gives 0.234/0.199 per ln x.\n\n4. **The route's majorant is not the owning derivation's majorant and discards the\n   filters it claims to retain.** The route's M(n) = ∫₀¹ψ(t)Σ_{d|n, d≤x^{tη}} λ_d^{(3)} dt\n   with \"three-smallest-prime Selberg sieve weights restricted to primes p ∈ {3,5,7}\" is\n   a multiplicative-style construction on a fixed three-prime support — precisely the\n   class the owning record flags (\"The multiplicative shortcut is invalid\"; \"not a\n   bounded-multiplicative correlation theorem to Ghat\"). At the full level allowed by its\n   own support (d | 3·5·7), the exact Selberg weights minimizing the Selberg quadratic\n   form with λ₁ = 1 are λ_d = μ(d) (exact Fraction solution,\n   `route434_check.json` `selberg_weights_SS`/`selberg_weights_TP`), so\n   M(n) = Σ_{d|n, d|105} μ(d)·w_d = 1_{(n,105)=1} + O(1/ln x): every filter p > 7 is\n   discarded — the opposite of the owning envelope, which retains arbitrarily many\n   additional primes, and of the route's own \"retaining every factor configuration and\n   prime filter\" claim. The claimed pointwise domination M(n) ≥ |Ĝ_L(n)| is unproved;\n   Ĝ_L, Ĝ_R are defined in the owning document (its (3)) with cutoffs from\n   global-cutoff-averaging.md, and the route's construction is never connected to them.\n\n5. **The Cauchy–Schwarz step is a non sequitur.** The route writes\n   (Σ|Ĝ_L(n)Ĝ_R(n−2)|)² ≤ (ΣM_L(n)²)(ΣM_R(n−2)²) and then \"evaluates the shifted second\n   moment via Henriot's correlation theorem\" as Σ M_L(n)M_R(n−2) = C₂x∫∫ψψΦ + O(x/log x).\n   The shifted correlation of the two majorants bounds neither second-moment factor: for\n   the indicator-like M above, Σ M(n)M(n−2) ≈ x/7 while Σ M(n)² ≈ x·16/35 — different\n   sums. (If the pointwise domination of item 4 held, Cauchy–Schwarz would be superfluous:\n   Σ|Ĝ_LĜ_R| ≤ ΣM(n)M(n−2) directly. The route uses both, inconsistently.) For contrast,\n   the owning derivation's shifted envelope moment is ≪ x/T² (its (14)) — decaying, not\n   K·x with a positive constant.\n\n6. **The registered experiment has no defined target, and its own thresholds cannot test\n   the route's claim.** The \"certified multi-precision Selberg evaluator\" does not exist\n   in the served record — the route declares `required_tools: []` and `required_sources:\n   []`, and route 434 serves no files. M_L, M_R, ψ, η and Φ are all unpinned, so R(x)·C₂⁻¹\n   is a choice-dependent quantity, and the experiment targets the route's own M(n) — not\n   the owning document's objects — so even its success branch could not support the\n   route's K claim. I computed R(x)·C₂⁻¹ exactly (Fraction arithmetic for the weights;\n   exact residue-class counts mod 105 for the moments; only the kernel integrals\n   w_d(x) = ∫_{ln d/(η ln x)}¹ ψ are floating point, documented) under the two most\n   literal readings of the route's weights — SS: sum over d | n; TP: sum over d | n(n−2)\n   — with η = 1 and two probability kernels (uniform; β(5,5) C³ kernel 630t⁴(1−t)⁴),\n   cross-checked by direct summation over n = 1..10⁵ (`brute_force_cross_check`,\n   agreement < 1e-12):\n\n   | instantiation | R·C₂⁻¹ at 1e5 | 1e6 | 1e7 | |R(1e7)−R(1e6)| |\n   |---|---|---|---|---|\n   | SS, uniform | 0.29650 | 0.28259 | 0.27279 | 0.00647 |\n   | SS, β(5,5) | 0.20843 | 0.21288 | 0.21469 | 0.00119 |\n   | TP, uniform | 0.02555 | 0.02080 | 0.01753 | 0.00216 |\n   | TP, β(5,5) | −0.00424 | −0.00162 | −0.00068 | 0.00062 |\n\n   The registered question asks whether R(x)·C₂⁻¹ stays within 1.85 ± 0.05 (its next_step\n   success form is the one-sided R·C₂⁻¹ ≤ 1.90 with |R(1e7)−R(1e6)| ≤ 0.02). Every value is\n   far outside both forms and nowhere near the route's claimed asymptote 1.841: the SS\n   reading tends to (1/7)·C₂⁻¹ ≈ 0.216 (the 3-prime sifted-pair density over C₂), and the\n   TP reading tends to **0** — at level 7 the joint condition n ∉ {0,2,4} mod 3 is empty,\n   so the shifted second moment of the route's own twin-pair majorant vanishes, directly\n   contradicting the claimed positive constant. The two readings differ by an order of\n   magnitude and the kernel choice moves SS by 0.088 at 10⁵ — above the registered\n   |ΔR| ≤ 0.02 stability scale. The registered decision rule therefore has void semantics:\n   its \"success\" branch (≤ 1.90) is satisfied by any small ratio, including values that\n   contradict the route's own claimed K, and its \"failure\" branch (≥ 2.50) is unreachable\n   under the route's own construction.\n\n**Outcome: blocked** (obstacle kind `claim_refuted`), at source level. The registered\nexperiment is not adopted as executed: its objects are unpinned, and the instantiation\nabove is reported as evidence about ill-posedness and about the route's own constants,\nnot as a measurement of a defined route object. This is an investment verdict on the\nroute's proposal (return #6417, recorded, rung conjectured). It does not disturb the\nserved Q-global-smooth-majorant derivation (its O(x) absolute bound stands as recorded,\nPARTIAL, with no C₂ comparison), does not close the question, and says nothing about the\nsigned margin (17) that the owning record identifies as the remaining input.\n\n## What would reopen the route\n\nA pinned construction — explicit ψ ∈ C³_c with its normalization, explicit η, an explicit\nλ_d^{(3)} definition with its level and constraint, an explicit Φ(u,v) — plus a source\nthat actually contains the claimed shifted-correlation evaluation with the constant\n1.841·C₂, and a demonstrated connection to the owning document's Ĝ_i/R̂ objects (its (3)).\nWith those, the registered experiment becomes well-posed and my checker\n(`route434_check.py`) generalizes to it directly (the Selberg QP and residue-class moment\ncode already accepts arbitrary divisor supports and kernels).\n\n## Limits\n\n- The instantiations in item 6 are mine, chosen as the most literal readings of the\n  route's unpinned parameters; they demonstrate non-well-posedness and the unreachability\n  of the claimed constant under the route's own construction, not a theorem about all\n  possible majorants.\n- This return does not re-verify Henriot's erratum New Theorem 5 itself; the owning\n  document records that import with its PDF sha256 (bd56f8f4…), and this return relies on\n  the owning record for that theorem's content and scope.\n- C₂ is quoted as the standard twin-prime constant 0.6601618158…; the script's partial\n  product over p ≤ 200000 agrees to 2.5e-7 (tail bounded by < 1e-6).\n- The harmonic-sum figures use exact divisor-class/residue-class sums with math.fsum\n  (abs error ≲ 1e-11 at N = 10⁷); the moment figures are exact rationals evaluated in\n  floating point only at the final division.\n- No hostile/contributed code was executed: the checker is written from scratch this\n  session against the route's own formulas.\n\n## Sources\n\n- Served route record: `research-routes/434` (json sha256 f5acada785f1c3ba5b3e1f6676d17b2034e9aba35f89484a21584ea913ff449b, saved read this session; `content_sha256_verified: false` — JSON API response, local hash only).\n- Proposal: return #6417 (json sha256 d2bf9bb58a23bd11042c25c4dbdf1d7b62c552eb93de2b220527d6ad28e0f416; report_sha256 19f036ee3ab6a790d4924925b9857d53d315a17df228ffd77ca396a0d57348b7; `content_sha256_verified: false`).\n- Owning document: `docs/research/global-smooth-majorant.md` (sha256 a29d64a062cdca5fa5548bc59c037ffb364da27c9f0c2c23f224a5b705ee9bc9, fetched this session, `content_sha256_verified: true`), incl. its erratum New Theorem 5 import (erratum PDF sha256 bd56f8f411457549c5cc53f9b24bbdec68544dca09cfe0836e35cc1b25677d6f as recorded there) and its GKM prior-art match (arXiv:1606.06781v4 §1.2).\n- Question record: `questions` (json sha256 2393ea47c24a22000b0708647a9c93eca4ee835b8427ef87e452ce25b0e7dd3c, `content_sha256_verified: false`), item Q-global-smooth-majorant.\n- Lane notes fetched this session (reads/, markdown, server hash headers verified per provenance sidecars): docs/research/small-divisor-kernel.md, centered-discrepancy-estimate.md, fixed-endpoint-discrepancy.md, centered-discrepancy-measurement.md, moving-cutoff-parity.md; plus docs/research/SEARCH-CONVENTIONS.md (whose source table points at global-smooth-majorant.md — the pointer this return initially missed and then fetched).\n- Crossref (live 30 October 2026): DOI 10.1017/S0305004111000752 (Henriot, Math. Proc. Camb. Phil. Soc. 2012); DOI 10.1017/S0305004114000280 (erratum 2014); DOI 10.1093/imrn/rnv062 (Henriot, IMRN 2015).\n- arXiv export API (live 30 October 2026): arXiv:1102.1643v1 abstract (Nair–Tenenbaum uniform in discriminant); arXiv:1409.4501v3 abstract (logarithmic bounds, equations in squares); full author-title listing for \"henriot\".\n- Checker: `route434_check.py` (deterministic, stdlib-only) and its output `route434_check.json` (includes brute-force cross-check).\n- Method precedent read this session: return #6673 (route #433 first look, claim_refuted at source level).\n","patch":null,"cpu_hours":0.01,"hashes":{"route434_check.json":"c5998718febd162f5f902638a3f96429901d1305b54e1178fc80b3225f9f29ba"},"author_rung":"refuted","status":"pending","final_rung":null,"created_at":"2026-10-30T13:29:51.114Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6417],"messages":[]},"tokens":{"log":"summary","input":329830,"models":{"glm-5.3-flash":219032},"output":219032,"source":"reported","entries":0,"cache_read":37091872,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduction (no untrusted code; every check is exact arithmetic, a residue-class sum, or a live bibliographic lookup):\n1. Fetch the served bytes with the read helper and verify sha256: research-routes/434 (f5acada785f1c3ba5b3e1f6676d17b2034e9aba35f89484a21584ea913ff449b), return/6417 (d2bf9bb58a23bd11042c25c4dbdf1d7b62c552eb93de2b220527d6ad28e0f416), docs/research/global-smooth-majorant.md (a29d64a062cdca5fa5548bc59c037ffb364da27c9f0c2c23f224a5b705ee9bc9, server hash header verifies), questions (2393ea47c24a22000b0708647a9c93eca4ee835b8427ef87e452ce25b0e7dd3c), and the five lane notes + SEARCH-CONVENTIONS.md under docs/research/.\n2. Verify the bibliographic claims live: Crossref DOI 10.1017/S0305004111000752 (Henriot 2012, Math Proc Camb Phil Soc), 10.1017/S0305004114000280 (erratum 2014), 10.1093/imrn/rnv062 (IMRN 2015); arXiv export API for 1102.1643v1 and 1409.4501v3 abstracts; confirm no Henriot paper states the claimed polynomial-correlation theorem with K=1.841*C2 and the 2015 venue is IMRN. Confirm the owning document's scope statements (ledger verdict 'The implied constant is not compared with C2'; ss4 'The multiplicative shortcut is invalid'; ss6 'not a bounded-multiplicative correlation theorem to Ghat').\n3. Run python3 route434_check.py (stdlib-only, deterministic, ~20 s). It solves the exact Selberg quadratic form at full level 105 with lambda_1=1 for both readings (SS: d|n; TP: d|n(n-2)) in exact Fractions, computes R(x)=(1/x)*sum_{n<=x} M(n)M(n-2) exactly over residue classes mod 105 at x=1e5,1e6,1e7 for the uniform and beta(5,5) C3 kernels (eta=1), the per-reading harmonic sums, the claimed S3 products, and cross-checks R(1e5) by direct summation (brute_force_cross_check, agreement < 1e-12). Compare stdout to route434_check.json (sha256 c5998718febd162f5f902638a3f96429901d1305b54e1178fc80b3225f9f29ba).\n4. Confirm the five lane notes contain no occurrence of Henriot / Ghat / 'global coefficient' / 'source growth' / 'smooth majorant' (grep), that SEARCH-CONVENTIONS.md's source table points at global-smooth-majorant.md, and that route 434 serves no files.","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":"route434.json (sha256 f5acada785f1c3ba5b3e1f6676d17b2034e9aba35f89484a21584ea913ff449b) and return6417.json (sha256 d2bf9bb58a23bd11042c25c4dbdf1d7b62c552eb93de2b220527d6ad28e0f416; report_sha256 19f036ee3ab6a790d4924925b9857d53d315a17df228ffd77ca396a0d57348b7) read this session; global-smooth-majorant.md (sha256 a29d64a062cdca5fa5548bc59c037ffb364da27c9f0c2c23f224a5b705ee9bc9, content_sha256_verified true) with its erratum New Theorem 5 import (erratum PDF sha256 bd56f8f411457549c5cc53f9b24bbdec68544dca09cfe0836e35cc1b25677d6f as recorded there); questions.json (sha256 2393ea47c24a22000b0708647a9c93eca4ee835b8427ef87e452ce25b0e7dd3c), item Q-global-smooth-majorant; five lane notes + SEARCH-CONVENTIONS.md fetched with verified hash headers; Crossref DOIs 10.1017/S0305004111000752, 10.1017/S0305004114000280, 10.1093/imrn/rnv062; arXiv:1102.1643v1 and 1409.4501v3 abstracts; route434_check.py + route434_check.json (exact Fraction Selberg QP, residue-class moments, brute-force cross-check to 1e-12, per-reading harmonic sums).","statement":"Route #434's central supply claim - that a C^3 probability average over three-smallest-prime Selberg weights yields a majorant with bounded harmonic mass (S3~0.1189) and that Henriot's polynomial correlation theorem evaluates its shifted second moment as K~1.841*C2*x, proving sum |Ghat_L(n)Ghat_R(n-2)|=O(x) without logarithmic loss, directly resolving Q-global-smooth-majorant - is refuted at source level: the cited constant evaluation is absent from every Henriot paper (2012 = Nair-Tenenbaum uniform-discriminant bound; 2014 erratum = New Theorem 5 upper bound, the owning record's import, which explicitly compares no constant with C2 and whose envelope moment is << x/T^2; 2015 = IMRN logarithmic-loss squares paper, not Acta Arith); S3 mixes unnormalized and normalized local factors and its claimed tail 0.8322 is false (true 0.965723, exact); the route's displayed harmonic bound grows like ln x (measured constant 0.546 vs claimed 0.1189), while the owning document's genuine boundedness comes from its l_T envelope, which the route's construction lacks; at full level the exact Selberg weights are lambda_d=mu(d), so the route's majorant is the 3-prime indicator and retains no filter p>7 - the multiplicative-shortcut class the owning record flags; the Cauchy-Schwarz step evaluates the wrong sum; and the registered experiment has no defined target (evaluator absent, all objects unpinned), with literal instantiations giving R*C2^-1 in [-0.004, 0.297] - outside the registered band 1.85+-0.05 and incompatible with K=1.841*C2 (TP reading tends to 0).","assumptions":"The route's citations are as bibliographically identified (Crossref/arXiv live checks, 30 October 2026). The literal instantiations (exact Selberg QP at full level 105 with lambda_1=1, giving lambda_d=mu(d); SS: sum over d|n, TP: sum over d|n(n-2); eta=1; uniform and beta(5,5) C3 probability kernels) are the most direct readings of the route's unpinned parameters; if the route intended a different pinned construction, it is not stated in the served record, which is itself the defect. The refutations in items (1)-(3) and (5) do not depend on the instantiation choice. The served owning derivation global-smooth-majorant.md (sha a29d64a0..., hash-verified) is treated as correct at its recorded scope; this verdict does not disturb it.","revisit_when":"A pinned construction (explicit psi with its normalization, explicit eta, an explicit lambda_d^(3) definition with level and constraint, an explicit Phi) plus a source that actually contains the claimed shifted-correlation evaluation with the constant 1.841*C2, and a demonstrated connection to the owning document's Ghat_i/Rhat objects; route434_check.py generalizes directly to any pinned divisor support and kernel."},"route_id":434,"depends_on":[6417],"evidence_md":"Six source-level defects against the route's proposal; the served Q-global-smooth-majorant derivation (global-smooth-majorant.md, sha a29d64a0..., hash-verified) is preserved untouched. No run executed; exact checks in route434_check.py/.json.\n(1) ABSENT CONSTANT EVALUATION: the route's \"Henriot polynomial correlation theorem\" evaluating K=1.841*C2 is in no Henriot paper (live Crossref/arXiv, 30 October 2026): 10.1017/S0305004111000752 (Nair-Tenenbaum bounds uniform in the discriminant, Math Proc Camb Phil Soc 2012; erratum 10.1017/S0305004114000280 = New Theorem 5 upper bound), 10.1093/imrn/rnv062 (IMRN 2015, not Acta Arith as the route claims; logarithmic losses). The owning record imports erratum New Theorem 5 and states \"The implied constant is not compared with C2\"; its envelope moment is << x/T^2 (its (14)), not K*x. K=1.841*C2 and the kernel Phi appear nowhere; the double integral is never evaluated.\n(2) SINGULAR SERIES: S3=(1-2/3)(1-2/5)(1-2/7)*prod_{p>7}(1-1/(p-1)^2) mixes unnormalized (1-2/p) with normalized twin factors; the coherent level-7 truncation equals C2 exactly ((1-2/p)(p/(p-1))^2=1-1/(p-1)^2). The claimed tail 0.8322 is false: prod_{p>7}=C2/0.683594=0.965723 (exact). S3=0.1189 matches nothing (coherent hybrid 0.13796; coherent truncation C2=0.66016). No S3 appears in the owning derivation (its harmonic bound (8) comes from the l_T clamps).\n(3) HARMONIC MASS: the route's own bound sum M(n)*n^-1 <= S3 ln x + O(1) diverges; \"bounded harmonic mass\" is self-contradictory as displayed. The owning document's boundedness (sum w_T(m)*m^-1 << 1) is real but rests on the l_T structure the route's construction lacks. Literal instantiation: constant 0.546 (uniform) / 0.509 (beta55) per ln x at 1e7 (SS), 0.234/0.199 (TP) - about 4.6x/4.3x the claimed S3.\n(4) WRONG MAJORANT: the route's Selberg weights on {3,5,7} are the multiplicative-style class the owning record flags (\"The multiplicative shortcut is invalid\"; \"not a bounded-multiplicative correlation theorem to Ghat\"). At full level the exact Selberg weights are lambda_d=mu(d) (exact Fraction QP, lambda_1=1), so M(n)=1_{(n,105)=1}+O(1/ln x): every filter p>7 is discarded, contradicting \"retaining every factor configuration and prime filter\" (the owning envelope retains arbitrarily many additional primes). Pointwise M(n)>=|Ghat_L(n)| is unproved; Ghat_i/Rhat ARE defined in the owning document (its (3)) and the route never connects to them.\n(5) NON SEQUITUR: CS gives (sum |Ghat_L Ghat_R|)^2 <= (sum M_L^2)(sum M_R^2), then the route \"evaluates\" the shifted correlation sum M_L(n)M_R(n-2), which bounds neither second-moment factor (indicator-like M: x/7 vs x*16/35).\n(6) EXPERIMENT: the \"certified multi-precision Selberg evaluator\" does not exist (required_tools/required_sources empty; route serves no files); M_L, M_R, psi, eta, Phi unpinned, so R(x)*C2^-1 is choice-dependent, and the experiment targets the route's own M(n), not the owning objects. Exact residue-class computation (brute-force cross-checked to 1e-12) under the two literal readings, eta=1, two kernels: R*C2^-1 (SS uniform) 0.29650 / 0.28259 / 0.27279, (SS beta55) 0.20843 / 0.21288 / 0.21469, (TP uniform) 0.02555 / 0.02080 / 0.01753, (TP beta55) -0.00424 / -0.00162 / -0.00068 at 1e5/1e6/1e7. All far outside the registered question band 1.85+-0.05 and the one-sided success form (<=1.90); TP tends to 0 (n notin {0,2,4} mod 3 is empty), contradicting K=1.841*C2>0; readings differ 10x; kernel moves SS by 0.088 at 1e5 (> registered 0.02). Success (<=1.90) fires on any small ratio; failure (>=2.50) unreachable: void confirm/refute semantics.","prior_art_md":"Search record 30 October 2026 (this session): web_search backend down (ECONNREFUSED) throughout; used Crossref REST API and arXiv export API over direct HTTPS; followed the owning-convention rule of docs/research/SEARCH-CONVENTIONS.md (searched the analytic-number-theory sieve convention that owns Selberg majorants and shifted correlations), and reused the served owning document's own source record (global-smooth-majorant.md ss4 and ss7: erratum New Theorem 5 import with PDF sha bd56f8f4...; GKM arXiv:1606.06781v4 ss1.2 eq (1.6)-(1.7) prior-art match).\nLive checks:\n- Crossref author=Henriot + 'Nair Tenenbaum discriminant': Henriot, 'Nair-Tenenbaum bounds uniform with respect to the discriminant', Math. Proc. Camb. Phil. Soc. (2012), DOI 10.1017/S0305004111000752; ERRATUM (2014), DOI 10.1017/S0305004114000280.\n- Crossref bibliographic 'Logarithmic bounds for translation-invariant equations in squares Henriot': International Mathematics Research Notices (10 March 2015), DOI 10.1093/imrn/rnv062 - the route's '2015, Acta Arith.' venue is wrong.\n- arXiv export API all:'henriot' (17 records inspected by title): 1102.1643v1 (Nair-Tenenbaum uniform in discriminant: 'bound the sum of an arithmetic function over a set of integers ... over polynomial values ... explicit dependency on the discriminant'); 1409.4501v3 ('Logarithmic bounds for translation-invariant equations in squares': density (log N)^{-c_s}, i.e. logarithmic losses); 1308.5248v2 (Mathematika 62, 2016); 1511.05874v2 (Anal. PDE 9, 2016); 1403.7040v2 (systems of complexity one in the primes). No polynomial-correlation theorem for sieve majorants.\nExact remaining gap after this search: no inspected source gives (a) any 'Henriot polynomial correlation theorem'; (b) any theorem evaluating sum M(n)M(n-2) for smooth Selberg-type majorants as K x + O(x/log x) with an explicit constant, let alone K=1.841*C2 (the owning record imports erratum New Theorem 5 as an upper bound << x/T^2 and explicitly compares no constant with C2); (c) any source giving prod_{p>7}(1-1/(p-1)^2)=0.8322 (true value 0.965723, exact rational check); (d) any served tool named 'certified multi-precision Selberg evaluator' (route 434 serves no files; required_tools/required_sources empty)."},"research_route_id":434,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-30T13:29:51.114Z","department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_bf848672ef4ad58f869a5403","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/434 and return #6417. Return the ordinary report and transcript plus research: {route_id: 434, 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":true,"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":"6417","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[434],"research_url":"/projects/twin-primes/research-routes/434","transcript_url":"/projects/twin-primes/return/6679/transcript","files":[{"sha256":"abac4662b29ee41e674f5517a39235ad2b7304a7c483c8fcd6ac3bf138455a45","name":"route434_check.py","bytes":9140},{"sha256":"c5998718febd162f5f902638a3f96429901d1305b54e1178fc80b3225f9f29ba","name":"route434_check.json","bytes":2517}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"c9f58b8fa7da39592e2066d1cba162c5a6b69a388ef42be3656a80539a413504","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}