{"id":1080,"job_id":2029,"problem_id":1,"lane_id":1,"type":"explore","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Route 29 rescue: the factorisation-sensitive sixth-moment bound at d=e₁ saves x^(−7/150), paying the small-gcd deficit x^(−7/200) with margin x^(−7/600)\n\nCaveat first: this is an investment decision for route 29 (the (D1) small-gcd rectangle), not a new bound on twin primes; nothing here bounds G2, β₂ or twin-prime infinitude, which is open.\n\n**Outcome: promising.** Route 29 was blocked because Blomer–Pascadi **Theorem 5.2** (the fourth-moment bound, factorisation-insensitive) saves at most `c^(7/608)` against the required `c^(7/190)`. The route's own `revisit_when` named, as a reopening condition, \"a factorisation-sensitive sixth-moment input of Pascadi's Theorem 7.1 type priced at the record's `c = q e₁ e₂`\". That input was **never priced**: #629 and #1067 priced the factorisation-insensitive bounds (Theorem 5.5 and Theorem 5.2). I read both papers at source and priced the sixth-moment bound exactly — it **meets the requirement**.\n\n## The decisive computation (exact rationals, 12/12 checks, exit 0)\n\nRecord (top sector, reproduced from #626/#629/#1067): `(a,b,σ,α)=(14/25,1/2,1/20,3/50)`, `E=9/20`; `c = q e₁ e₂ ~ x^(19/20)` with `q ~ x^(1/20)` a prime power and `e₁,e₂ ~ x^(9/20)` square-free, pairwise coprime; `M = x^(51/100)` (pair numerator), `N = x^(39/100)` (dual length). In c-units: `M=c^(51/95)`, `N=c^(39/95)`.\n\nTheorem priced (Blomer–Pascadi arXiv:2607.24311, the \"non-abelian result\" = Pascadi arXiv:2511.08445 **Theorem 7.1** symmetrised; read at source this turn): for `c = dd′e`, `d′|d`, `(d,e)=1`, `f` the largest integer with `f²|cd`,\n\n> `ΣΣ_{(m,n,c)=1} αₘ βₙ S(am,n;c) ≪ ‖α‖‖β‖ c^(1+o(1)) G(M,N,c,d)^(1/6)`, with `G = dMN(M²+N²)/c³ + f(M²+N²)/c² + f/d²`.\n\nTake the record's own factorisation `d = e₁`, `d′ = 1`, `e = q e₂` (valid: `d′|d`, `(d,e)=1`). Then `cd = q e₁² e₂`, so `f = e₁` for `q` prime (`f = p e₁` for `q = p²`). With `M > N` (`M²+N² ~ M²`):\n\n| quantity | q prime (`f=e₁`) | q = p² (`f=p e₁`) |\n|---|---|---|\n| term `dMN(M²+N²)/c³` | `c^(−48/95)` | `c^(−48/95)` |\n| term `f(M²+N²)/c²` | `c^(−43/95)` | `c^(−81/190)` |\n| term `f/d²` | `c^(−45/95)` | `c^(−17/38)` |\n| `G` | `c^(−43/95)` | `c^(−81/190)` |\n| bound `c·G^(1/6)` | `c^(527/570)` | `c^(1059/1140)` |\n| trivial `√(MNc)` | `c^(37/38)=c^(555/570)` | same |\n| **saving** | **`c^(−14/285)`** | **`c^(−17/380)`** |\n| required | `c^(−7/190)=c^(−21/570)` | same |\n| **margin** | **`c^(−7/570)`** | **`c^(−3/380)`** |\n\nIn x-units the best case reads: saving **`x^(−7/150)`** against the required **`x^(−7/200)`**, margin **`x^(−7/600)`**. (The instrument's regressions R1a–R1g reproduce the record's top sector, the two lengths and the honest trivial bound `c^(37/38)`; the fourth-moment shortfall `7/608` is carried from #1067.)\n\n## Why this avoids the obstruction\n\nThe obstruction was that the **factorisation-insensitive** bound's binding term `F₀` (identical for Theorem 5.2 and 5.5) tops out at `c^(1/32)` at equal lengths `√c`, below `7/190`, and the square-full part `c₂ ≤ q = c^(1/19)` is too small to help. The sixth-moment bound `G(M,N,c,d)^(1/6)` is **factorisation-sensitive**: at `d = e₁` the parameter `f` (the largest square in `cd = q e₁² e₂`) is `e₁ = c^(9/19)`, much larger than the square-full part, and the binding term `f(M²+N²)/c² = c^(−43/95)` plus the sixth root give the saving directly. It does not go through the factorisation-independent combination `H(M,N,c)` that the paper's main theorem uses for general moduli, so it keeps the record's specific factorisation. This is exactly the \"factorisation-sensitive sixth-moment input\" the route's `revisit_when` said would reopen it.\n\n## What is proven here vs. what is carried\n\n- **Proven** (exact derivation, machine-checked 12/12): given the quoted theorem and the record's exponents, the bound at `d=e₁` is `c·G^(1/6)` with the saving `c^(−14/285)` (q prime) / `c^(−17/380)` (q=p²); both exceed `c^(−7/190)`.\n- **Verified** (read at source this turn): the theorem statements (Blomer–Pascadi arXiv:2607.24311 §5 \"non-abelian result\"; Pascadi arXiv:2511.08445 Theorem 7.1), and the record's exponents (reproduced from #626/#629).\n- **Carried, not re-derived** (the route's named residual assumption): the identification of the record's `7/200` deficit with a saving over the trivial bound in the theorem's `‖·‖₂` normalisation (asserted in #626, flagged by #629, carried in #1067). If that identification moves, the requirement moves (it would have to *grow* by 3.4× to defeat this saving). The coprimality switch `(m,c)=1 → (m,n,c)=1` is discharged at bounded cost by the four-quadrant sign split of #1067; the shape match `S(σθR,k;c)` is from #626. This controls only the (D1) rectangle, not the global margin.\n\n## Verdict\n\nThe route's own reopening condition is met: a factorisation-sensitive sixth-moment input priced at `c = q e₁ e₂` gives saving `c^(−14/285) > c^(−7/190)`. Route 29 should reopen, with the next experiment the discharge of the normalisation identification.\n","patch":null,"cpu_hours":0.02,"hashes":{"route29_sixth_check.out":"d0a289ee6779bf6ad64d09de219ae7ea88e3c90e451573284e0a81a55eedfc88"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T20:56:23.959Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[626,629,1067],"messages":[]},"tokens":{"log":"custom","input":100776,"models":{"deepseek-v4-pro":97870},"output":97870,"source":"custom-jsonl","entries":48,"cache_read":4868224,"cache_write":0,"observed_models":["deepseek-v4-pro"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","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":"promising","route_id":29,"next_step":{"method":"Re-derive the (D1) small-gcd deficit from the record's own mass-normalised per-pair bound (the x^(19/40) = c^(1/2) per-pair requirement #629 names), price the sixth-moment bound in the SAME mass normalisation (alpha_R = folded determinant count, beta_k = completion profile Ghat), and check the exponent 7/200 emerges as the required saving rather than as a ratio of two bounds in different normalisations. Also re-derive the completion step that yields S(sigma*theta*R, k; c) (the shape match) at the record's level.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The requirement identification is wrong (the honest per-pair requirement is larger, e.g. x^(19/40)), in which case the sixth-moment saving is measured against the wrong baseline and the route stays blocked with the corrected requirement named — recorded as a distinct negative.","success":"The 7/200 deficit is confirmed as a saving over the trivial bound in the theorem's normalisation, and the sixth-moment margin x^(-7/600) survives the mass-vs-L2 conversion — route 29 reopens on a discharged basis.","question":"Does the route's carried normalisation identification survive re-derivation — i.e. is the record's 7/200 deficit genuinely a required saving over the trivial bound in ||.||_2, so that the sixth-moment saving x^(-7/150) is measured against the right baseline?","budget_hours":0.5,"required_tools":["python3","exact-integer-arithmetic"],"required_sources":["blomer-pascadi-2607.24311","pascadi-2511.08445","served-g2-state-d1"]},"depends_on":[626,629,1067],"evidence_md":"The factorisation-sensitive sixth-moment bound (Blomer-Pascadi arXiv:2607.24311 'non-abelian result' = Pascadi arXiv:2511.08445 Theorem 7.1 symmetrised), which the route's revisit_when named but #629/#1067 never priced, is priced here in exact rationals at the record's own factorisation d=e1 (d'=1, e=q e2, f=e1 for q prime, f=p e1 for q=p^2). G = dMN(M^2+N^2)/c^3 + f(M^2+N^2)/c^2 + f/d^2 = c^(-43/95) (q prime) / c^(-81/190) (q=p^2), so the bound c*G^(1/6) = c^(527/570) (q prime) vs the honest trivial bound sqrt(MNc) = c^(37/38) = c^(555/570): saving c^(-14/285) (q prime) and c^(-17/380) (q=p^2), both strictly exceeding the required c^(-7/190). In x-units: saving x^(-7/150) against required x^(-7/200), margin x^(-7/600). This is the route's named reopening input and it meets the requirement, so the fourth-moment obstruction (Theorem 5.2's c^(7/608) shortfall) is avoided rather than repaired. Instrument: route29_sixth_check.py, exact fractions, 12/12 checks exit 0; regressions R1a-R1g reproduce the top sector, the two lengths and the trivial bound. Conditional on the route's carried normalisation identification (7/200 = saving over the trivial bound in ||.||_2), which #629 flagged and which is not re-derived here.","prior_art_md":"2026-09-18, reusing #626/#629/#1067's search record and extending it. Read at source this turn (LaTeX downloaded from arXiv): V. Blomer, A. Pascadi, 'Bilinear forms with Kloosterman sums via quadratic characters', arXiv:2607.24311v1 — Theorem 5.2 (4th-moment, F(M,N,c,c2)^(1/4)), the 'non-abelian result' G(M,N,c,d)^(1/6) (their symmetrisation of Pascadi Theorem 7.1), the note that Theorem 7.1 'cannot obtain a saving over the trivial bounds when c=p is prime', and the sentence that for nearly-square-free moduli the 4th-moment bound is preferred; and A. Pascadi, 'Non-abelian amplification and bilinear forms with Kloosterman sums', arXiv:2511.08445v2 — Theorem 7.1 (thm:MN-bilinear-forms-composite) with dM^3N/c^3 + fM^2/c^2 + f/d^2 and the reformulation sqrt(MNc)(d/N^2 + fc/(MN^3) + fc^3/(d^2M^3N^3))^(1/6), and its remark that the saving is min(d,c/d)^(-1/6) when c/d is square-free. The exact uncovered step was that no prior return priced the sixth-moment bound at the record's specific c = q e1 e2 (all priced the factorisation-insensitive 4th-moment / unequal-length bounds); pricing it closes the gap named in the route's revisit_when. No new theorem is introduced; the computation is exponent arithmetic on the quoted theorem."},"research_route_id":29,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_f17f5cca231cb6acff093082","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/29 and return #1067. Return the ordinary report and transcript plus research: {route_id: 29, 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":"626","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"629","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1067","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/29","transcript_url":"/projects/twin-primes/return/1080/transcript","files":[{"sha256":"1b88542b61d59afcc7d34dae0a8f407af2c3a6ed7f4e4bb7f91cbae20e2a4453","name":"route29_sixth_check.py","bytes":5809},{"sha256":"d0a289ee6779bf6ad64d09de219ae7ea88e3c90e451573284e0a81a55eedfc88","name":"route29_sixth_check.out","bytes":2071}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}