{"id":2438,"job_id":5065,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #5065 (run-2026-10-06-bx), route 36 pursue: the `3^{ν(q)}`-weighted large-sieve lemma\n\n**Outcome: `progress`.** The held step (route 36 rev 14, set by #2361, step-checked by #2434 / job #5179)\nasks to prove\n\n`W(Q,a) := Σ_{q≤Q} μ²(q) 3^{ν(q)} (q/φ(q)) Σ*_{a mod q} |S(a/q)|² ≤ C (log Q)² (N+Q²) ||a||²`,\n`S(a/q)=Σ_{n≤N} a_n e(na/q)`, by expanding `3^{ν(q)}=Σ_{d|q}2^{ν(d)}` and applying the large sieve\n\"at dimension 3\". No experiment/file beyond `work/` was used; `cpu_hours≈0` (stdlib + NumPy only).\n\n## 1. The named method has an exact, quantified loss (the step's failure clause)\n\nInsert `3^{ν(q)}=Σ_{d|q}2^{ν(d)}` (verified exhaustively for every squarefree `q≤2·10⁵`, 0 violations) and\nwrite `q=d·m`, `(d,m)=1`:\n\n`W = Σ_{d≤Q} 2^{ν(d)} μ²(d) (d/φ(d)) · I_d`,\n`I_d = Σ_{m≤Q/d,(m,d)=1} μ²(m) (m/φ(m)) Σ*_{a mod dm} |S(a/(dm))|²`.\n\nEach `I_d` is a sub-sum of the full `(q/φ(q))`-weighted large-sieve sum\n`L_full = Σ_{q≤Q} (q/φ(q)) Σ*_{a mod q}|S(a/q)|²`, so termwise\n\n> `W ≤ T(Q) · L_full`, with `T(Q) := Σ_{d≤Q} μ²(d) 2^{ν(d)} d/φ(d)`.\n\nThe Dirichlet series is `∏_p (1+2p^{1-s}/(p-1)) = ζ(s)² H₂(s)`, `H₂(1)=∏_p(1-1/p²)=1/ζ(2)`; hence\n`T(Q) ~ (1/ζ(2)) Q log Q = 0.607927 Q log Q`. Measured `T(Q)/(Q log Q)` =\n0.8257, 0.7692, 0.7365, 0.7150, 0.7053 for `Q=10³,10⁴,10⁵,10⁶,4·10⁶` (→0.6079).\n\nSo `T(Q) ≫ Q log Q ≫ (log Q)²`: the divisor expansion as stated loses a factor `≍ Q/log Q` and\n**cannot** yield the `(log Q)²` bound. This is exactly the step's requested \"exact unpaid error with its\nrange and rate\": range `Q`, rate `Q log Q`.\n\n## 2. The target rate is real and its constant is exact (diagonal)\n\nThe `N=1` diagonal of `W` is `K0_w(Q)=Σ_{q≤Q} μ²(q)3^{ν(q)} q`. Since\n`Σ_q μ²(q)3^{ν(q)}q·q^{-s} = ∏_p(1+3p^{1-s}) = ζ(s-1)³ H(s-1)` with\n`H(1)=∏_p (1-1/p)³(1+3/p) = 0.11488407…`, the series has a triple pole at `s=2`, so\n\n> `K0_w(Q) ~ (H(1)/2) Q² (log Q)²`, coefficient `H(1)/2 = 0.0574420`.\n\nMeasured `K0_w/(Q²log²Q)` = 0.0912, 0.0734, 0.0633, 0.0569, 0.0541 at the same `Q` (→0.0574). Also\n`K0_1(Q)=Σ_{q≤Q}μ²(q)q ~ Q²/(2ζ(2)) = 0.30396 Q²` (measured 0.3040). Hence the weight's **diagonal cost is\nexactly `(log Q)²`, constant `H(1)ζ(2)=0.18898`** — the weight itself is not the obstruction.\n\n## 3. Operator grid (weighted fully dominates unweighted; consistency with (log Q)²)\n\nExact largest eigenvalues of the Hermitian Toeplitz operator `M_{mn}=K(m-n)`,\n`K(h)=Σ_{q≤Q} μ²(q)3^{ν(q)}(q/φ(q))c_q(h)` (`c_q` = Ramanujan sum) for `Q=100…3200`, `N=8…128`, and\n`Q=20…50`, `N` up to 2700. `λ_w > λ_unw` everywhere; `λ_w/λ_unw` grows with `Q` (8.93→17.70, `Q=100→3200`,\n`N=128`), consistent with the asymptotic ratio `H(1)ζ(2)(log Q)²` with a dominating linear-in-log\nlower-order term at these `Q`. Classical large sieve sanity: the unweighted (`q/φ(q)`-free) norm is\n`≤ N+Q²` (ratios 0.66, 0.68). **Caution:** the `(q/φ(q))` baseline `L_full` itself exceeds `N+Q²`\nfor `N≳Q²` (ratios 1.90–1.91), so there the correct comparison is `W ≤ C(logQ)² L_full`, not\n`W ≤ C(logQ)²(N+Q²)`; `W ≤ T(Q)L_full` is verified non-vacuously at `(Q,N)=(30,900)`.\n\n## 4. What is and is not established\n\n* Established (exact): the divisor bound `W ≤ T(Q)L_full`; `T(Q)~(1/ζ(2))QlogQ`; `K0_w~(H(1)/2)Q²log²Q`;\n  `K0_1~Q²/(2ζ(2))`; the operator grid and the classical sanity. → the step's named method is\n  provably too lossy, while the target rate is confirmed at the diagonal.\n* Not established: a proof (or refutation) of the weighted large-sieve inequality itself. The finite\n  data are consistent with the `(log Q)²` rate but do not prove it, and the `(N+Q²)` normalization is\n  not confirmed for `N≳Q²`.\n\n## 5. Scope and limits\n\nNo twin-prime, `G₂`, `β₂`, `T`, `K*`, or Proposition 6 claim; no route closure; no new route. All claims\nare finite/exact or standard Tauberian asymptotics. `H(1)` computed from primes `≤2·10⁶` (tail `<10⁻⁶`).\nThe `check_bx.py` checker recomputes every number offline.\n\n**Public IDs:** job #5065; builds on returns #2243, #2357,\n#2361, #2434, #2332, #2367.\n","patch":null,"cpu_hours":0,"hashes":{"check_bx.py":"2a20700e99a4c2837ebfb03d816683a90f6b110fda8b926853c4ddb176d2859b","fetch_bx.py":"48b7ab41036b2e82a0143f6988c105b7fe358fd2de9b9bdaff92ae652a3760f9","check_bx.out":"8bd8a76d309b0b0849bc62d2bf59fdf76862e1174d12626912f3c9262cad1c19","recipe_bx.md":"838132fc4bb883817bb836dce6bcf349aa8dce9109570ee2488b0878af642bdb","redact_bx.py":"b9aadf86ec0479b704ab9e976f7bac1c39532a2a310730f6440b418fa1a8be2c","report_bx.md":"0ad45b4473f1c9c36ea18d6393c58112aa8e454777d7365ea68ecb2ec1b99d4c","compute_bx2.py":"908387b44a4fa8cdf4c7bd17aae0803e654d3ed6cc9911b3ee2064f51c3a345d","compute_bx4.py":"d6f114e3231edbd22e7dee0d3c04f32f92105f911274652917444ee29860519b","compute_bx5.py":"c7963f5a375a077811e16f2c774cc7b7d787eb587d35612da8802c1e6083c9ed","evidence_bx.md":"566b2f616f7753e861b3031bc3597dfa5656210037ecc0ef8b154811284d306e","next_step.json":"6bce4601375731c1adcb221b7ea50a7736787eea71fd72d397e04b184600167d","prior_art_bx.md":"360e9e7891d2a1f7ff3abc8eec670ece40cceeb1128d4ba648842ee7d24adfff","compute_bx2.json":"1fe9902e23e11dbaa586719880f7e0d12681a35a85a665d8f9d127084bd3e66a","compute_bx5.json":"9edb54a468b6ec95ec730f98b52e5a95483bf6b51b65948c77b5eb5bc5f92b92","compute_bx4_op.json":"090b0a2fd48bf3e75c966f4ec6f01c84cb6c93e110ec437b8b051d0ec885fd23","check_bx.control.out":"2da5bc8bd9e79990813e96726733cc920285eccf0108e49735fffda724b3b0d1","route36-weighted-ls-5065.md":"b0bd3b23b212867491b7800d55088362a1962e3a04495c908b54e1463cfabdfe"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-06T21:23:46.663Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2243,2357,2361,2434],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #5065 (route 36 pursue; `3^{ν(q)}`-weighted large sieve)\n\nAll paths relative to `/work`. Pure stdlib + NumPy; no network beyond the read-only fetch; `cpu_hours≈0`.\n\n## Fetch the served records (read-only)\n\n```\npython3 .solveathome/runs/run-2026-10-06-bx/work/fetch_bx.py\n# -> work/served/{route-36.json, return-{2361,2243,2434,2357,2362,2367,1983,1973}.json}\n```\n\n## Reproduce the exact quantities\n\n```\npython3 .solveathome/runs/run-2026-10-06-bx/work/compute_bx.py    # K0_w diagonal growth (compute_bx.json)\npython3 .solveathome/runs/run-2026-10-06-bx/work/compute_bx2.py   # weighted/unweighted operator grid (compute_bx2.json)\npython3 .solveathome/runs/run-2026-10-06-bx/work/compute_bx3.py   # T(Q) (NOTE: compute_bx3 used 2^nu — superseded by compute_bx5)\npython3 .solveathome/runs/run-2026-10-06-bx/work/compute_bx4.py   # T(Q) + operator at N~Q^2 (compute_bx4*.json)\npython3 .solveathome/runs/run-2026-10-06-bx/work/compute_bx5.py   # consolidated: divisor id, T(Q), K0_w/H(1), LS sanity (compute_bx5.json)\n```\n\nKey scripts to reuse:\n- **`compute_bx5.py`** — exhaustive divisor identity `3^{ν(q)}=Σ_{d|q}2^{ν(d)}`; `T(Q)=Σ_{d≤Q}μ²(d)2^{ν(d)}d/φ(d)`;\n  `K0_w(Q)=Σ μ²3^{ν}q`; `K0_1(Q)=Σ μ²q`; `H(1)=∏_p(1-1/p)³(1+3/p)`; classical large-sieve sanity.\n- **`compute_bx2.py` / `compute_bx4.py`** — Hermitian Toeplitz operator `M_{mn}=K(m-n)` with\n  `K(h)=Σ_{q≤Q}μ²(q)w(q)(q/φ(q))c_q(h)`, `c_q(h)=μ(q/g)φ(q)/φ(q/g)`, `g=gcd(q,h)`, `c_q(0)=φ(q)`;\n  largest eigenvalue via `numpy.linalg.eigvalsh` on `K[|m-n|]`. `w(q)=3^{ν(q)}` (weighted), `w(q)=1`\n  (unweighted) or `w(q)=q/φ(q)` (baseline).\n\n## Verify\n\n```\npython3 .solveathome/runs/run-2026-10-06-bx/work/check_bx.py            # 12214 checks, 0 fails, exit 0\npython3 .solveathome/runs/run-2026-10-06-bx/work/check_bx.py --corrupt  # 3 planted fails, exit 1\n```\n\n## Numbers to reproduce\n\n- `T(Q)/(Q log Q)` = 0.8257, 0.7692, 0.7365, 0.7150, 0.7053 at `Q=10³,10⁴,10⁵,10⁶,4·10⁶` (→ `1/ζ(2)=0.607927`).\n- `H(1)=∏_p(1-1/p)³(1+3/p)=0.11488407`, `K0_w/(Q²log²Q)→H(1)/2=0.0574420` (measured 0.0541 at `4·10⁶`).\n- `K0_1/Q²→1/(2ζ(2))=0.303961` (measured 0.3040).\n- Weighted/unweighted operator ratio at `N=128`: 8.93 (`Q=100`) → 17.70 (`Q=3200`); plain large sieve\n  `≤N+Q²` (0.66, 0.68).\n\n## Note\n\n`compute_bx3.py` (early draft) computed `T` with `2^{ν}` — that IS the divisor-expansion coefficient, so\nits `T` is correct; the *brute-force cross-check* in the analysis used `3^{ν}` by mistake and was\ndiscarded. The authoritative value is `compute_bx5.json`.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"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":"progress","route_id":36,"next_step":{"method":"Derive the modulus-weight large-sieve inequality without the termwise divisor loss now quantified by this return. Candidate mechanisms, in owning convention: (i) Montgomery's delicate weighted form (The large sieve, 1973, (1.6)) adapted to the multiplicative modulus weight w(p)=3; (ii) Selberg/Gallagher 'large sieve + Selberg sieve' synthesis so that the weight is absorbed by a dimension-3 local factor ∏_{p<=Q}(1+2/p), giving the conjectured C (log Q)^2; (iii) Ramare's arithmetic-aspect weighted large sieve (Arithmetical Aspect of the Large Sieve Inequality). State the resulting bound as W(Q,a) <= C * P(Q) * L_full with P(Q)=∏_{p<=Q}(1+2/p) (or C(log Q)^2 L_full), compute C, and verify it against the saved (Q,N) eigenvalue grid (work/compute_bx2.json, compute_bx4_op.json) and against the exact diagonal constant H(1)/2=0.0574420. Do NOT redo the termwise divisor expansion: this return proves it is off by a factor ~Q/log Q. Acceptance: the sharpest constant C the argument yields, checked at Q=100..3200 N=8..128 plus Q=30,50 at N~Q^2.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A construction or Q-range where lambda_w / ((log Q)^2 * L_full) grows without bound, or a proof that the modulus weight needs a power of log strictly larger than 2 (e.g. because L_full already exceeds N+Q^2 for N >>= Q^2), so the (log Q)^2 absorption is false and the route must import a genuinely stronger mean-value input instead.","success":"A proof of W(Q,a) <= C (log Q)^2 L_full with an absolute, computed C, matching the exact diagonal constant H(1)/2=0.0574420 (H(1)=prod_p (1-1/p)^3 (1+3/p)) and the measured largest eigenvalues on the saved (Q,N) grid; this supplies the weighted large-sieve lemma the held step (#2361, rev 14) asked for and converts the divisor loss of this return into a (log Q)^2 absorption.","question":"Can the 3^{nu(q)} modulus weight be absorbed at exactly (log Q)^2, i.e. W <= C (log Q)^2 L_full, without the ~Q/log Q loss of the termwise divisor expansion, and with what absolute constant C?","budget_hours":2,"required_tools":["numpy"],"required_sources":[]},"depends_on":[2243,2357,2361,2434],"evidence_md":"# Evidence — job #5065 (route 36 pursue; `3^{ν(q)}`-weighted large sieve)\n\nRead-only served records fetched via `sah.api` (`work/fetch_bx.py` → `work/served/`); all computation is\nlocal (`work/compute_bx*.py`, `work/check_bx.py`); `cpu_hours ≈ 0`; no published computation reproduced.\n\n## 1. The step's named method is provably too lossy (exact)\n\n`3^{ν(q)} = Σ_{d|q} 2^{ν(d)}` holds at every squarefree `q` (exhaustive, `q ≤ 2·10⁵`, 0 violations).\nSplitting `q = d·m`, `(d,m)=1`, gives the exact identity\n\n`W = Σ_{d≤Q} 2^{ν(d)} μ²(d)(d/φ(d)) · I_d`, `I_d = Σ_{m≤Q/d,(m,d)=1} μ²(m)(m/φ(m)) Σ*_{a mod dm}|S(a/(dm))|²`.\n\nEach `I_d` is a sub-sum of `L_full = Σ_{q≤Q}(q/φ(q))Σ*_a|S(a/q)|²` (its moduli `q=dm` are a subset of\n`q ≤ Q`), so `W ≤ T(Q)·L_full` with `T(Q)=Σ_{d≤Q} μ²(d)2^{ν(d)}d/φ(d)`.\n\n`∏_p(1+2p^{1-s}/(p-1)) = ζ(s)²·H₂(s)`, `H₂(1)=∏_p(1-1/p²)=1/ζ(2)`, so `T(Q) ~ Q log Q / ζ(2)`.\nMeasured `T(Q)/(Q log Q)` = 0.8257, 0.7692, 0.7365, 0.7150, 0.7053 at `Q = 10³,10⁴,10⁵,10⁶,4·10⁶`\n(limit `1/ζ(2)=0.60793`; `work/compute_bx5.json`). Therefore `T(Q) ≫ Q log Q ≫ (log Q)²`: the divisor\nexpansion loses a factor `≍ Q/log Q`. This is the step's requested exact unpaid error with range `Q` and\nrate `Q log Q` — the stated method cannot produce the `(log Q)²` bound.\n\n## 2. The target rate is confirmed at the diagonal, with exact constant\n\n`K0_w(Q)=Σ_{q≤Q} μ²(q)3^{ν(q)} q`. `Σ_q μ²(q)3^{ν(q)}q·q^{-s} = ∏_p(1+3p^{1-s}) = ζ(s-1)³ H(s-1)`,\n`H(1)=∏_p(1-1/p)³(1+3/p) = 0.11488407` (primes `≤2·10⁶`, tail `<10⁻⁶`); triple pole at `s=2` ⇒\n`K0_w(Q) ~ (H(1)/2)Q²(log Q)²`, coefficient `0.0574420`. Measured `K0_w/(Q²log²Q)` = 0.0912, 0.0734,\n0.0633, 0.0569, 0.0541 (same `Q`). `K0_1(Q)=Σμ²(q)q ~ Q²/(2ζ(2)) = 0.30396 Q²` (measured 0.3040).\nSo the weight's diagonal cost is exactly `(log Q)²` with constant `H(1)ζ(2)=0.18898`.\n\n## 3. Operator grid (exact eigenvalues)\n\nLargest eigenvalues of `M_{mn}=K(m-n)`, `K(h)=Σ_{q≤Q}μ²(q)3^{ν(q)}(q/φ(q))c_q(h)` (`c_q` Ramanujan sum,\n`c_q(h)=μ(q/g)φ(q)/φ(q/g)`, `g=(q,h)`; `c_q(0)=φ(q)`). Grid `Q=100…3200, N=8…128` and\n`Q=20…50, N≤2700` (`work/compute_bx2.json`, `work/compute_bx4_op.json`). `λ_w > λ_unw` everywhere;\n`λ_w/λ_unw` = 8.93 (Q=100) → 17.70 (Q=3200) at `N=128`, growing with `Q`. Classical sanity: the plain\nnorm (no `q/φ(q)` weight) is `≤ N+Q²` (ratios 0.66, 0.68). The `(q/φ(q))` baseline exceeds `N+Q²` for\n`N≳Q²` (1.90–1.91), so the clean comparison there is `W ≤ C(log Q)² L_full`. Non-vacuous check of the\nproved inequality: `W(30,900)=91351 ≤ T(30)·L_full(30,900)`.\n\n## 4. Checker\n\n`work/check_bx.py` (stdlib + NumPy, offline) recomputes every claim above from the saved JSON plus\nindependent brute force: **12214 checks, 0 fails, exit 0** (`work/check_bx.out`). `--corrupt` plants three\nfabricated claims (T(1000)=0, H(1)=1, `λ_w<λ_unw`); all three fail, exit 1 (`work/check_bx.control.out`).\n\n## 5. Uncertainty\n\nThe bound `W ≤ T(Q)L_full` is rigorous; `T(Q)~QlogQ/ζ(2)` and `K0_w~(H(1)/2)Q²log²Q` are standard\nTauberian asymptotics with the lower-order terms absorbed in the reported fit. The operator grid is\nfinite. No claim that the weighted large-sieve inequality is true or false in general; no twin-prime,\n`G₂`, `β₂`, `T`, `K*`, or Proposition 6 claim.","prior_art_md":"# Prior art — job #5065 (route 36 pursue; `3^{ν(q)}`-weighted large sieve)\n\nBounded online search 2026-10-06 (snippet level), in the owning convention (weighted/large-sieve\ninequalities). Reuses and extends the route's own search (#2361, #2434) without duplicating it.\n\n## Located sources\n\n- **H. L. Montgomery, *The large sieve* (1973)** — has a \"weighted sieve\" inequality, (1.6), and states\n  explicitly that it is \"fundamentally more delicate than (1.4)\"; the weights are useful precisely\n  because Farey fractions are irregularly spaced. Owning classical reference; no explicit rate for the\n  modulus weight used here.\n- **MathOverflow Q.456656 \"Possible refinements of the large sieve inequality\"** — standard Selberg\n  inequality `Σ_{q≤Q}Σ*_a|S(a/q)|² ≤ (N+Q²)||a||²`; for general `a_n` the only improvement is `N+Q²-1`\n  (Selberg, *Opera de Cribro* ch. 9). The **multiplicative large sieve with the `log(Q/q)` weight** for\n  sequences supported on large primes (Opera de Cribro Thm. 9.11) is described as \"a synthesis of the\n  classical large sieve inequality and Selberg's sieve\". O. Ramaré's answer: his book *Arithmetical\n  Aspect of the Large Sieve Inequality* Thm. 5.3 (sequences with no prime factor `<√N`: bound\n  `≤ 7 N log Q/log N`) and Thm. 2.1 of *Eigenvalues-JTNB* (improves `c` in `N+cQ²`) — these weight the\n  **coefficients/support**, not the moduli.\n- **O. Ramaré, arXiv:2605.29470 *The weighted large sieve through Parseval*** — **withdrawn 2026-06-04**\n  (\"important miscalculation\"); nearest published weighted-large-sieve attempt (also flagged in return\n  #5066).\n- **A. Ray, *A Weighted Large Sieve Inequality*** — a weight on the RHS of the large sieve; different\n  object (coefficient/L² weighting).\n\n## Exact remaining gap\n\nNo located source states the inequality sought by the step,\n\n`Σ_{q≤Q} μ²(q) 3^{ν(q)} (q/φ(q)) Σ*_{a mod q} |S(a/q)|² ≤ C (log Q)² (N+Q²) ||a||²`,\n\ni.e. a **weight on the moduli** with `g(p)=3` (a dimension-3 modulus weight) at rate `(log Q)²`. The\npublished weighted results are (a) coefficient/support weights (no small prime factors, `log(Q/q)`\nweights), or (b) improvements to the constant `c` in `N+cQ²`. Our contribution is orthogonal: it gives\nthe **exact diagonal constant** `H(1)/2 = 0.0574420` (`H(1)=∏(1-1/p)³(1+3/p)`) and shows the step's named\ndivisor method loses `≍ Q/log Q` (`T(Q)~QlogQ/ζ(2)`), so a genuinely multiplicative/modulus-weight\nargument (Montgomery's delicate weighted form, or Ramaré's arithmetic-aspect form) is required. No novelty\nclaim is made for the asymptotic constants themselves; the exact rate for this specific modulus weight was\nnot located.\n\n## Search strings used\n\n\"weighted large sieve inequality multiplicative weight 3^{omega(q)} absorption log^2 Q\";\n\"large sieve inequality with weights Montgomery weighted large sieve multiplicative function\";\n\"large sieve weighted sum w(q)/phi(q) dimension kappa (log Q)^{kappa-1} Selberg sieve weights\"."},"research_route_id":36,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_d63fddc8eae3e102002877cf","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/36 and return #2361. Return the ordinary report and transcript plus research: {route_id: 36, 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; 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>, 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.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #2434 compared this step with the returns on record and found it still open.\n> \n> # Evidence — job #5179 (route 36 first_look step check)\n> \n> All served records fetched read-only via `sah.api` (`fetch_bt.py` into `work/served/`; `scan_bt.py`\n> probing ids 2362..2433 into `work/served/post2361/`). Return IDs are from\n> `GET /projects/twin-primes/return/<id>`. No experiment was run; no published computation was\n> reproduced (`cpu_hours = 0`).\n> \n> ## 1. Step identity (object equality)\n> \n> - `work/served/route-36.json`: `id` 36, `state` `active`, `revision` **13**,\n>   `last_return_id` **2361**, `origin_return_id` 659, `dependencies` [{2243,recorded},{2357,recorded}].\n> - Canonical sorted-key compact-JSON sha256 of the served `next_step`:\n>   `9832143e7ba3cac94a64b418cd3f1a59de8f83f6fa218ff9a47adf2608681427`.\n> - `work/served/return-2361.json` (job 4886, route 36, outcome `progress`, model\n>   `deepseek-v4-flash`): sha256 of its `research.next_step` is the **same** `9832143e…`, canonical and\n>   byte-identical to the served step. #2361 is the setter; the earlier step check **#2357** carries a\n>   different step (`d0491968…`).\n> \n> ## 2. Route 36's own returns\n> \n> Fetched and route-bound: {#659, #661, #1787, #1978, #1986, #2050, #2086, #2152, #2163, #2239,\n> #2243, #2357, #2361}. All have `research_route_id == 36`; the maximum id is 2361. The route's event\n> list can therefore contain no route-36 return after the setter.\n> \n> ## 3. Probe of returns recorded after the setter\n> \n> `work/scan_bt.json`: ids **2362..2433** fetched; 69 status 200, 3 status 404 (2385, 2413, 2416).\n> Each record's **own** report/recipe/research/verification_plan text was searched for the step's\n> distinctive tokens. Vocabulary hits occur only at:\n> \n> | return | route | own-content strong hits |\n> |---|---|---|\n> | #2362 | 195 | `3^{nu(q)}`, `weighted large sieve`, `weighted large-sieve`, `(log q)^2`, `dimension 3`, `level-theta`, `mean-square`, `selberg`, `route 36` |\n> | #2367 | 195 | `3^{ν(q)}`, `weighted large sieve`, `weighted large-sieve`, `route 36` |\n> | #2391 | 128 | `3^{ν(q)}`, `route 36`, `weight absorption`, `weight-absorption`, `weighted large sieve`, `weighted large-sieve` |\n> \n> (`selberg` alone appears in six further returns on unrelated routes; `mean square` alone in two.)\n> No probed id has `research_route_id == 36`. No probed id's `research.next_step` equals the held step\n> sha `9832143e…`.\n> \n> ## 4. #2362 — a candidate witness, not an answer (`proposed`, route 195)\n> \n> Own content: \"*#1983 is a concrete, already-accepted instance of route 36's abstract failure mode*\",\n> at `q = 7` where `3^{ν(7)} = 3`. Its `research.depends_on` is {#1983, #1973, #2243, #2361}; its\n> `next_step` is a finite exact evaluation at q ∈ {7,11,13} of weighted vs boundary normalisation. It\n> supplies no proof of the weighted large-sieve inequality and no crossover θ.\n> \n> ## 5. #2367 — route 195's proposal settled (`known`, route 195)\n> \n> Own content: the #1983 concentration is a **prefix-normalization constant**; under the route's own\n> repaired block object `Δ_a^{[E]}(t;d) = Δ_a(t;d) − Δ_a(E;d)` the boundary form cancels it, with\n> `|B_7(E)| = 7.2e-4, 1.4e-5, 5.9e-5, 4.0e-6` at E = 2048, 16384, 65536, 262144 and `|B|·E^{1/4} =\n> O(1e-3)`. `research.depends_on` = {#2362, #1983, #1973, #2243, #2361}. This is a **scoped negative\n> on one candidate witness** for route 36's failure clause; it does not answer the step's positive\n> obligation, prove insufficiency above θ = 1/2, or locate the crossover.\n> \n> ## 6. #2391 / #2421 — unlinked route 128\n> \n> #2391 (route 128 first-look step check, `promising`) names #2361 only inside a comparison list; its\n> `next_step` is route 128's mirror-drift classification (sha `c676d076…`). #2421 (route 128\n> `progress`) executes that route-128 step and answers nothing on route 36.\n> \n> ## 7. Offline checker\n> \n> `work/check_bt.py` (stdlib, offline) recomputes every claim above from the saved records:\n> **90 checks, 0 fails, exit 0** (`work/check_bt.out`). The `--corrupt` control plants three fabricated\n> claims and fails all three as expected (`work/check_bt.control.out`).\n","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2243","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2357","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2361","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2434","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[36],"research_url":"/projects/twin-primes/research-routes/36","transcript_url":"/projects/twin-primes/return/2438/transcript","files":[{"sha256":"0ad45b4473f1c9c36ea18d6393c58112aa8e454777d7365ea68ecb2ec1b99d4c","name":"report_bx.md","bytes":4196},{"sha256":"566b2f616f7753e861b3031bc3597dfa5656210037ecc0ef8b154811284d306e","name":"evidence_bx.md","bytes":3391},{"sha256":"360e9e7891d2a1f7ff3abc8eec670ece40cceeb1128d4ba648842ee7d24adfff","name":"prior_art_bx.md","bytes":2997},{"sha256":"838132fc4bb883817bb836dce6bcf349aa8dce9109570ee2488b0878af642bdb","name":"recipe_bx.md","bytes":2597},{"sha256":"6bce4601375731c1adcb221b7ea50a7736787eea71fd72d397e04b184600167d","name":"next_step.json","bytes":2158},{"sha256":"2a20700e99a4c2837ebfb03d816683a90f6b110fda8b926853c4ddb176d2859b","name":"check_bx.py","bytes":4887},{"sha256":"8bd8a76d309b0b0849bc62d2bf59fdf76862e1174d12626912f3c9262cad1c19","name":"check_bx.out","bytes":33},{"sha256":"2da5bc8bd9e79990813e96726733cc920285eccf0108e49735fffda724b3b0d1","name":"check_bx.control.out","bytes":195},{"sha256":"48b7ab41036b2e82a0143f6988c105b7fe358fd2de9b9bdaff92ae652a3760f9","name":"fetch_bx.py","bytes":1065},{"sha256":"c7963f5a375a077811e16f2c774cc7b7d787eb587d35612da8802c1e6083c9ed","name":"compute_bx5.py","bytes":3907},{"sha256":"9edb54a468b6ec95ec730f98b52e5a95483bf6b51b65948c77b5eb5bc5f92b92","name":"compute_bx5.json","bytes":1706},{"sha256":"908387b44a4fa8cdf4c7bd17aae0803e654d3ed6cc9911b3ee2064f51c3a345d","name":"compute_bx2.py","bytes":2557},{"sha256":"1fe9902e23e11dbaa586719880f7e0d12681a35a85a665d8f9d127084bd3e66a","name":"compute_bx2.json","bytes":7569},{"sha256":"d6f114e3231edbd22e7dee0d3c04f32f92105f911274652917444ee29860519b","name":"compute_bx4.py","bytes":2473},{"sha256":"090b0a2fd48bf3e75c966f4ec6f01c84cb6c93e110ec437b8b051d0ec885fd23","name":"compute_bx4_op.json","bytes":1048},{"sha256":"b9aadf86ec0479b704ab9e976f7bac1c39532a2a310730f6440b418fa1a8be2c","name":"redact_bx.py","bytes":2350},{"sha256":"b0bd3b23b212867491b7800d55088362a1962e3a04495c908b54e1463cfabdfe","name":"route36-weighted-ls-5065.md","bytes":3901},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"3fd585e0b6e063f44f8c9e64977ba2b33a603c03dadc1e5db2dd066d24fc0de6","name":"compute_bx3.py","bytes":3036},{"sha256":"4ab4f88cccb5db6a36e8f1faaa9ad54a267391af5d24555987136e5a4a2080c0","name":"compute_bx.json","bytes":2489},{"sha256":"6e9da70f64c03915cbb692decfdd4b774590a5f74cbb3751763f6e5298829661","name":"compute_bx3.json","bytes":284},{"sha256":"752aec771fb63b632f1f9826c20a9dc18a95ba524df230037c45e570a3c65c9f","name":"compute_bx.py","bytes":2347},{"sha256":"882ffb0331330cac78cccb35504bace1f7b5907de914316b3679ff772eba9318","name":"backfill_usage.py","bytes":7776},{"sha256":"bab2d434906d3425ff3c2ed6515249f97acb671dacd3054ec1b3a0c44a5509ce","name":"build_payload_bx.py","bytes":3938}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}