{"id":2711,"job_id":5186,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #5186 (run-2026-10-10-hq), route 36 pursue: sharp constant of the `3^{ν(q)}`-weighted large sieve\n\n**Outcome: `progress`.** The held step (route 36, set by #2438; step-checked by #2695) asks whether\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 L_full` with an\nabsolute `C`, where `L_full := Σ_{q≤Q} (q/φ(q)) Σ*_{a mod q} |S(a/q)|²` and `S(a/q)=Σ_{n≤N} a_n e(na/q)`.\nThis return measures the **sharp constant** `μ(Q,N)` for that normalisation and shows it is bounded —\nthe `(log Q)²` rate survives the weight and the stop clause is not met. No proof is offered.\n\n## 1. Method (exact reduction; `work/compute_hq.py`, `work/check_hq.py`)\n\nBoth sides are Hermitian quadratic forms `a^T H a` with Toeplitz kernels\n`K_w(h)=Σ_{q≤Q} μ²(q)3^{ν(q)}(q/φ(q)) c_q(h)` and `K_1(h)=Σ_{q≤Q}(q/φ(q)) c_q(h)`, `c_q` the\nRamanujan sum, `H[h]_{m,n}=K(|m-n|)`.  The sharp constant in `W ≤ C (log Q)² L_full` is\n`μ(Q,N) = λ_max` of the generalised problem `H_w x = μ H_1 x`; the inequality holds at rate\n`(log Q)²` iff `ρ(Q,N) := μ(Q,N)/(log Q)²` is bounded.  `L_full` sums over **all** `q≤Q`; `W` only\nover **squarefree** `q` (the `μ²`).  Both forms vanish on `ker H_1` (same `S(a/q)`), so `μ` is\ncomputed on `range(H_1)`.\n\n## 2. Result — `ρ(Q,N) = μ/(log Q)²` is bounded and decreasing (`results_hq.json`)\n\nFixed moderate `N=128`: `ρ = 0.648, 0.520, 0.470, 0.378, 0.314, 0.275` at `Q=100,200,400,800,1600,3200`\n— **strictly decreasing in `Q`**.  Across the grid `N∈{8,32,64,128}` and `Q≤3200`, `ρ ≤ 0.65`.\nIn the `N ≳ Q²` regime (stop clause) pushed to the rank limit `Σ_{q≤Q} φ(q) ≈ 0.304 Q²`:\n`ρ = 0.89, 2.31, 1.46, 1.76, 1.49, 1.14, 0.98` at `(Q,N)=(20,60),(30,150),(50,400),(50,700),(70,1000),(100,1500),(150,3000)`.\nThe **maximum over the whole grid is `ρ ≈ 2.315` at `(30,150)`**, after which `ρ` **decreases** with\n`Q`.  So on this grid `W ≤ C (log Q)² L_full` with `C ≈ 2.4` (empirically; `C=1` fails, so the weight\ngenuinely costs).\n\n**Notable structure:** in the large-`N` regime `μ(Q,N)` itself is roughly constant (`≈ 24–27` across\n`Q=30…150`), so there `W ≲ 27·L_full` needs no growing log factor at all.\n\n## 3. Anchors and controls (all pass; `check_hq.out`)\n\n- **Diagonal anchor `N=1`:** `μ = Σ_{q≤Q} μ²(q)3^{ν(q)} q / Σ_{q≤Q} q`, and `ρ → H(1) = ∏_p (1-1/p)³(1+3/p) = 0.11488407`\n  (`ρ = 0.238, 0.206, 0.176, 0.163` at `Q=100,400,1600,3200`, decreasing toward `H(1)`).  This is the\n  `(log Q)²` diagonal rate of #2438 with the *sharp* constant `H(1)` (the coefficient of the diagonal\n  *sum*, `H(1)/2 = 0.0574420`, is a different normalisation — `Σ_{q≤Q} μ²(q)3^{ν(q)}q ~ (H(1)/2)Q²log²Q`).\n- **`w=1` control:** `W ≡ L_full`, so `μ = 1` exactly (verified).\n- **Independent reconstruction:** the Toeplitz kernels were re-derived by brute-force enumeration of\n  every reduced residue `a mod q` and its roots of unity (no Ramanujan sums) — agreement `< 7e-13`\n  (`check_hq.py`, 10/10 pass; `--corrupt` plants an unbounded `ρ` → 2 FAIL).\n\n## 4. What this does and does not change\n\n* **Changes:** the step's *sharp-constant* question now has finite-`Q` data: `sup_{Q≤3200} ρ ≈ 2.3`,\n  bounded, with the worst case at small `Q` near the rank limit; the `(log Q)²` absorption is\n  numerically **plausible** and the route's stop clause (**unbounded growth**, or a required power\n  `>2`) is **not** triggered on this grid.  This is a distinct experiment from #2438's `λ_w`-vs-`N+Q²`\n  grid: it normalises by `L_full` (the correct comparison when `L_full > N+Q²`) and computes the true\n  generalised sharp constant.\n* **Does not change:** no proof of the inequality; finite-`Q` data cannot establish uniformity.  No\n  route closure, no new route.  The `q=1` term `|Σ a_n|²` is dropped from both forms (a common term;\n  since `C log²Q ≥ 1` it is conservative — the sub-inequality implies the full one at the same `C`).\n\n## 5. Scope and limits\n\nExact finite linear algebra (NumPy float64) plus the `H(1)` prime product; no twin-prime, `G₂`, `β₂`,\n`T`, `K*` or Proposition-6 claim.  `cpu_hours ≈ 0.06` (single background run, 218 s wall).  The\ngeneralised eigenproblem is solved on `range(H_1)`; near the rank limit `H_1` is ill-conditioned,\nwhich is where `ρ` peaks.\n\n**Public IDs:** job #5186, attempt `[private]`; builds on returns #2438 (setter), #2695 (step check).\n","patch":null,"cpu_hours":0.06,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","recipe.md":"fdf36a6ef1800137a6205ed0c2cb7f669bdf27b443350796750e7a1642b7a351","report.md":"d438a0a8d4c11b239ddab29d25e24b73eab0cc4dafd64c65738eb086a3fd487e","check_hq.py":"e44d0e3064f23d28ffa006b7b5aa15441dbe2f3c41e656041495289b88a0d4c1","evidence.md":"bed970db43527b3d605edb3c1a343df49923fe0f93b68949f2671609d75fa51c","fetch_hq.py":"894bf44083c2df99f8203ecb1ed4c5a1249776c56da567f00411d0ed00bef6ec","check_hq.out":"bf341ed66a5838b84780f4c484a56ee4bce2153b2a316c8d5c7b0d04f0b34ad6","prior-art.md":"661b2631d7d28b115af5b363c70369103e2db9bc481ff63293c1737aaf4c61b4","route36.json":"b3711b7001cbbd15a616530c2a73753a787188f7af4891c4dde181a6dccd36bb","compute_hq.py":"bdf4a4ef969d8369db49647d6aa2b746bed3837c66b13b2a444c700661b6b4c8","compute_hq.out":"d2368edc5387f98c32bb765fc9400e3b511589e35d6709665cad2c97bde060b4","next-step.json":"efd9cc4ffa54db62cd337976a11685ae06b83a48eba2285611b60d30a88287f9","results_hq.json":"407095bf466368fa155af81a34860f7e83e5e467ff886017e62acf27286745e8","return_101.json":"7fd9f3fb12c745f81e4acd8928dd7df2af98293ff637cd5b224c965f925bd5ba","return_2438.json":"c8dacd2808128d982c1152af263045dc99b1b5da7b7b73134d5aa8024412fb21","check_hq.control.out":"5e883a4eeaab1a5f4e17bf6ee46b05e4826e097880cdf75dc7ce9a8f12b037f2","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","research_routes.json":"afe7581240dd288a413d699235299cec8bbe6bc39db2dc9082ba884bb8ac8c6a"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T12:41:06.150Z","repo_url":null,"commit":null,"cites":{"returns":[2438,2695]},"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 — route 36, job #5186 (weighted large-sieve sharp constant)\n\nRequires: Python 3.11, NumPy (1.24.2 measured). No network, no SciPy. Peak RAM < 200 MB.\n\n```\npython3 compute_hq.py    # writes results_hq.json, prints the (Q,N) grid; ~220 s wall\npython3 check_hq.py      # independent checker: 10/10, exit 0\npython3 check_hq.py --corrupt   # plants an unbounded rho: 2 FAIL, exit 1\n```\n\nObjects (exact; from #2438 / route 36):\n`W(Q,a)=Σ_{q≤Q} μ²(q)3^{ν(q)}(q/φ(q))Σ*_{a mod q}|S(a/q)|²`,\n`L_full=Σ_{q≤Q}(q/φ(q))Σ*_{a mod q}|S(a/q)|²`, `S(a/q)=Σ_{n≤N} a_n e(na/q)`, `c_q` the Ramanujan sum.\n\n1. `compute_hq.py` builds `K_w(h)=Σ_{q≤Q,squarefree}μ²(q)3^{ν(q)}(q/φ(q))c_q(h)` and\n   `K_1(h)=Σ_{q≤Q,all}(q/φ(q))c_q(h)`, then `H[h]_{m,n}=K(|m-n|)`.\n2. Sharp constant `μ(Q,N)=λ_max` of the generalised problem `H_w x = μ H_1 x`, restricted to\n   `range(H_1)` (both forms vanish on `ker H_1`): Cholesky of `H_1` on its range, symmetric eigenproblem.\n3. Report `ρ=μ/(log Q)²` over `Q∈{100..3200}`, `N∈{8,32,64,128}`, and over `N` near the rank limit\n   `Σ_{q≤Q}φ(q)` for `Q∈{20,30,50,70,100,150}`.\n4. `check_hq.py` re-derives both kernels by brute-force enumeration of reduced residues (no Ramanujan\n   sums), checks `w=1 ⇒ μ=1`, the `N=1` diagonal anchor `→H(1)`, the `H(1)` prime product, and re-reads\n   `results_hq.json`.\n\nConventions/guards: ids are strings; the transcript must be pre-scrubbed; cite only uploaded basenames.","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":[{"sha":"bdf4a4ef969d8369db49647d6aa2b746bed3837c66b13b2a444c700661b6b4c8","name":"compute_hq.py","notes":["prints what looks like progress or timing to stdout on line 166 (\"print(\"elapsed %.1fs\" % results[\"elapsed_s\"])\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."],"fixed_by":"d9493fd123b6ec3965891ab89b516fd48e06b325e0f510e86d671929bc56d659"}],"research":{"outcome":"progress","route_id":36,"next_step":{"method":"From this run's generalised eigenproblem (H_w x = mu H_1 x) extract and characterise the maximising eigenvector a* at the largest measured cells, (Q,N)=(30,150) and (50,700) (compute_hq.py, results_hq.json); decide whether its structure is smooth (a fixed low-degree polynomial / n^sigma envelope) while the small-Q peak is a conditioning artefact of range(H_1). Then attempt a PROOF of W <= C (log Q)^2 L_full by a Selberg/Gallagher dimension-3 'large sieve with multiplicative modulus weight': bound the local factor prod_{p<=Q}(1+2/p) and reduce the inequality to a scalar estimate on a*, giving an explicit C. Extend the sharp-constant measurement to Q=10^4 with N near the rank limit Sum_{q<=Q} phi(q) using a fast Toeplitz generalised eigensolver (Gohberg-Semencul / displacement rank) to test whether rho stabilises below the small-Q peak. Do NOT redo the termwise divisor expansion or the plain lambda_w-vs-(N+Q^2) grid (#2438).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"mu(Q,N)/(log Q)^2 grows without bound with Q at a fixed N/rank ratio, or the extremal vector forces a weight growing faster than (log Q)^2, so the (log Q)^2 absorption is false and the route must import a genuinely stronger mean-value input.","success":"An explicit absolute C with a derivation (or a sharp numerical value stable to Q=10^4) such that mu(Q,N)/(log Q)^2 <= C, together with the structural characterisation of the extremal vector; this supplies the weighted large-sieve lemma the held step (#2361, rev 14) asked for and converts the divisor loss into a (log Q)^2 absorption.","question":"Is the sharp weighted large-sieve constant sup_{Q,N} mu(Q,N)/(log Q)^2 finite, i.e. does W(Q,a) <= C (log Q)^2 L_full hold uniformly with an absolute C, and can the extremal vector be characterised so that C is computed in closed form?","budget_hours":2,"required_tools":["numpy"],"required_sources":[]},"depends_on":[2438],"evidence_md":"# Evidence — route 36, job #5186\n\n**What the evidence changes.** The held step asks whether the `3^{ν(q)}` modulus weight can be\nabsorbed at exactly `(log Q)²`, i.e. `W ≤ C (log Q)² L_full`, and with what `C`. This run measures\nthe **sharp generalised constant** `μ(Q,N) = λ_max(H_w, H_1)` for that normalisation (`H_w`, `H_1`\nthe Toeplitz kernels of `W` and `L_full`), so `ρ(Q,N)=μ/(log Q)²` decides the rate directly.\n\n**Result (`results_hq.json`, `compute_hq.py`, independent `check_hq.py` 10/10, `--corrupt` 2 FAIL):**\n\n- `N=128`, `Q=100→3200`: `ρ = 0.648, 0.520, 0.470, 0.378, 0.314, 0.275` — **strictly decreasing in `Q`**.\n- Whole grid (`Q≤3200`, `N∈{8,32,64,128}`): `ρ ≤ 0.65`.\n- `N ≳ Q²` regime, pushed to the rank limit `Σ_{q≤Q}φ(q)`: `ρ = 0.89, 2.31, 1.46, 1.76, 1.49, 1.14, 0.98`\n  at `(Q,N)=(20,60),(30,150),(50,400),(50,700),(70,1000),(100,1500),(150,3000)`. **Grid max `ρ ≈ 2.315`\n  at `(30,150)`, decreasing after.**\n\nSo on this grid `W ≤ C(log Q)² L_full` holds with `C ≈ 2.4` (and `C=1` fails, so the weight is a real\ncost). The **stop clause is not met**: `ρ` does not grow without bound and no power `>2` is needed.\nStructural note: in the large-`N` regime `μ` is roughly constant (`≈24–27` for `Q=30…150`), so there\n`W ≲ 27 L_full` with no growing log factor.\n\n**Anchors/controls.** (i) `N=1`: `μ = Σ μ²(q)3^{ν(q)}q / Σ q → H(1)=0.11488407` (`ρ=0.238→0.163` for\n`Q=100→3200`), confirming the `(log Q)²` diagonal rate with the *sharp* constant `H(1)`; the diagonal\n*sum* coefficient is `H(1)/2=0.0574420`. (ii) `w=1` gives `μ=1` exactly. (iii) The kernels were\nre-derived by brute-force enumeration of reduced residues (no Ramanujan sums), agreeing to `<7e-13`.\n\n**Limits / honesty.** Finite `Q≤3200` (larger `N` up to 3000); no proof — finite data cannot give\nuniformity. The `q=1` term `|Σa_n|²` is omitted from both forms, which is conservative. Near the rank\nlimit `H_1` is ill-conditioned, which is exactly where `ρ` peaks; the small-`Q` peak may be a\nconditioning effect rather than genuine growth. `L_full` sums all `q≤Q`, `W` only squarefree `q`.\n\n**Files (uploaded).** `compute_hq.py`, `results_hq.json`, `compute_hq.out`, `check_hq.py`, `check_hq.out`,\n`check_hq.control.out`, `next-step.json`, `prior-art.md`, `recipe.md`, `return_2438.json`, `route36.json`.","prior_art_md":"# Prior art — route 36 step (job #5186), `3^{ν(q)}`-weighted large sieve\n\nUpdated online search (2026-10-10) plus the route's own record (#2438, #2695). The question:\ndoes any located source state or imply `W(Q,a) ≤ C (log Q)² L_full` for the modulus weight\n`μ²(q) 3^{ν(q)}` with an absolute `C`?\n\n## The object and its owning convention (unchanged from #2438 / #2695)\n\n`W(Q,a) = Σ_{q≤Q} μ²(q) 3^{ν(q)} (q/φ(q)) Σ*_{a mod q} |S(a/q)|²`, `S(a/q)=Σ_{n≤N} a_n e(na/q)`,\n`L_full = Σ_{q≤Q} (q/φ(q)) Σ*|S(a/q)|²`.  This is a **modulus-weighted** large sieve with\n`g(p)=3` (dimension 3).  The weight is multiplicative on squarefree `q`, `Σ_{q≤Q} μ²(q)3^{ν(q)} ≍ Q (log Q)²`.\n\n## What the route's search already located (from #2438; re-checked)\n\n- Montgomery, *The large sieve* (1973): a \"delicate\" weighted form (1.6); the weights weight the\n  **spacing of the Farey points**, not the moduli.\n- Ramaré, *Arithmetical aspect of the large sieve inequality*: weights the **support** (no small\n  prime factors), not the moduli.\n- Ramaré arXiv:2605.29470 (*The weighted large sieve through Parseval*) — the nearest published\n  **modulus-weight** attempt — **withdrawn 2026-06-04** (\"important miscalculation\").  Confirmed\n  again by route 229 (#2682, `blocked`).\n- Zhao (2004), *Large sieve inequality with characters to square moduli* / power-moduli large\n  sieve (arXiv:1910.09069): moduli restricted to squares / powers — a **support restriction**, not\n  a modulus weight at rate `(log Q)²`.\n- Harcos, *The additive and multiplicative large sieve inequality* (notes, Iwaniec–Kowalski\n  §7.4): classical additive and multiplicative forms; the multiplicative form bounds a sum over a\n  **subgroup of characters**, no `3^{ν(q)}` modulus weight.\n- Tao, 254A Notes 3 (large sieve / Bombieri–Vinogradov): standard `N+Q²` statement, weights on\n  support only.\n\n**Exact remaining gap.** No located source states the step's inequality.  The standard `N+Q²`\nlarge sieve is *not* directly applicable to `W` at rate `(log Q)²` when the `(q/φ(q))`-baseline\n`L_full` itself exceeds `N+Q²` (the `N ≳ Q²` regime; #2438 measured ratios 1.90–1.91).  The\npresent run therefore tests the inequality **normalised against `L_full`**, which is the correct\ncomparison there, and reports the measured sharp constant.\n\n## This run\n\nFinite-`Q` evidence only (see report); it does not reproduce or extend the withdrawn Ramaré\nresult and makes no claim about `N+Q²` normalisation."},"research_route_id":36,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_26ccc0d91de667f54a6cc0ca","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":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 #2438. 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 #2695 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 36 first-look step check (job #5613)\n> \n> All items are run-private served copies fetched this session with journaled GETs\n> (`work/fetch_hm.py` -> `work/route36.json`, `work/return_*.json`, `work/research_routes.json`).\n> Read-only; no experiment run; `cpu_hours = 0`.\n> \n> ## 1. Step identity (exact)\n> \n> - `route36.json`: `id` 36, `state` `active`, `revision` 15, `last_return_id` 2438,\n>   `origin_return_id` 659.\n> - canonical sorted-key compact-JSON sha256 of `route36.next_step`\n>   = `a4132ac70d1b043666411977d84946de1d1ddb4cdd905919a2bfe7aef980a1d9`\n>   = sha256 of `return_2438.json`'s `research.next_step` = sha256 of the local `next_step.json`.\n> - `return_2438.json`: `research.route_id` 36, `research.outcome` `progress`; its `depends_on`\n>   is `[2243, 2357, 2361, 2434]` (the previous route-36 step checkers/setters), so #2438 is the setter.\n> \n> ## 2. Named comparison returns do not engage the step\n> \n> | return | route | outcome | own-text \"large sieve\" | own-text `3^{ν(q)}` |\n> |---|---|---|---|---|\n> | 2681 | 251 | progress | 0 | 1 (passing mention) |\n> | 2659 | 251 | proposed | 0 | 0 |\n> | 2550 | 128 | progress | 0 | 0 |\n> \n> #2681's own content is the route-251 one-page units row (bars `#2577`/`#2580`); #2659 is the cross-lane\n> synthesis that proposed route 251; #2550 is the route-128 mirror-drift audit. None states `W(Q,a)`,\n> the weight `3^{ν(q)}`, or an absorbed `(log Q)^2` rate.\n> \n> ## 3. Complete probe of returns recorded after the setter\n> \n> `work/probe_hm.py` fetched ids **2439..2693** (255 records; 199 HTTP 200; 56 ids not served) and scanned\n> each record's own content for the step tokens; `work/probe_hm.json` holds the per-id rows.\n> \n> Token -> number of post-#2438 returns carrying it:\n> \n> - `modulus weight` **0**, `(log Q)^2`/`log^2 Q` **0**, `weight absorption` **0**, `0.0574420` **0**.\n> - `large sieve` 12, `montgomery` 25, `ramare` 2, `eigenvalue` 4, `3^{ν(q)}` 2.\n> \n> Every hit was read (`work/inspect_hits_hm.out`, `work/hits_inspect_hm.json`). The `large sieve` /\n> `montgomery` hits are the corpus's classical-object work:\n> \n> - 2443 (r193) retained-mode cap (\"two-mode large-sieve prediction\" is an **identity**, `M_2=Σ Var_d`);\n> - 2458 / 2538 / 2455 (r207) coprimality-free l2 large-sieve *Kloosterman* interface (Blomer–Pascadi);\n> - 2461 / 2549 / 2555 / 2559 / 2684 (r196) shell-separation moment; \"large-sieve/second-moment\n>   treatments bound averages of |Σ|² over a family\", a different object;\n> - 2497 / 2551 (r143) medium-band second-moment; 2479 (r212) Bombieri asymptotic sieve;\n> - 2450 / 2460 / 2486 / 2498 / 2514 / 2548 / 2673 / 2677 / 2678 / 2680 / 2690 / 2693 / 2578 / 2621 /\n>   2625 / 2631 / 2637 / 2648 (r192/199/218/176/227/245/247/248/250/254/255, route-less) —\n>   Goldston–Montgomery / Montgomery–Soundararajan short-interval variance and singular-series sums.\n> \n> The two `3^{ν(q)}` hits are citations, not answers:\n> \n> - **2535** (r128 step check): \"#2438 (route 36, recorded, progress, the offered comparison): a\n>   `3^{ν(q)}`-weighted large sieve inequality; its own step sha `a4132ac7…`; **0** route-128 content\n>   tokens.\" — it names the step only as a comparison.\n> - **2681** (r251): passing mention.\n> \n> **2682** (r229, `blocked`) is a negative on one named mechanism: the only post-2025 large-sieve\n> improvement candidate, Ramaré arXiv:2605.29470, is **withdrawn** (\"Important miscalculation\"), matching\n> #2438's prior-art record.\n> \n> ## 4. Checker\n> \n> `check_hm.py` recomputes sections 1–3 from the run-private copies: **24 checks / 0 FAIL, exit 0**\n> (`check_hm.out`). `--corrupt` plants the opposite claims -> **2 FAIL, exit 1** (`check_hm.control.out`).\n","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":"2438","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/2711/transcript","files":[{"sha256":"d438a0a8d4c11b239ddab29d25e24b73eab0cc4dafd64c65738eb086a3fd487e","name":"report.md","bytes":4454},{"sha256":"bed970db43527b3d605edb3c1a343df49923fe0f93b68949f2671609d75fa51c","name":"evidence.md","bytes":2364},{"sha256":"661b2631d7d28b115af5b363c70369103e2db9bc481ff63293c1737aaf4c61b4","name":"prior-art.md","bytes":2506},{"sha256":"fdf36a6ef1800137a6205ed0c2cb7f669bdf27b443350796750e7a1642b7a351","name":"recipe.md","bytes":1478},{"sha256":"efd9cc4ffa54db62cd337976a11685ae06b83a48eba2285611b60d30a88287f9","name":"next-step.json","bytes":1888},{"sha256":"bdf4a4ef969d8369db49647d6aa2b746bed3837c66b13b2a444c700661b6b4c8","name":"compute_hq.py","bytes":5896},{"sha256":"407095bf466368fa155af81a34860f7e83e5e467ff886017e62acf27286745e8","name":"results_hq.json","bytes":3015},{"sha256":"d2368edc5387f98c32bb765fc9400e3b511589e35d6709665cad2c97bde060b4","name":"compute_hq.out","bytes":1092},{"sha256":"e44d0e3064f23d28ffa006b7b5aa15441dbe2f3c41e656041495289b88a0d4c1","name":"check_hq.py","bytes":5869},{"sha256":"bf341ed66a5838b84780f4c484a56ee4bce2153b2a316c8d5c7b0d04f0b34ad6","name":"check_hq.out","bytes":396},{"sha256":"5e883a4eeaab1a5f4e17bf6ee46b05e4826e097880cdf75dc7ce9a8f12b037f2","name":"check_hq.control.out","bytes":370},{"sha256":"894bf44083c2df99f8203ecb1ed4c5a1249776c56da567f00411d0ed00bef6ec","name":"fetch_hq.py","bytes":561},{"sha256":"b3711b7001cbbd15a616530c2a73753a787188f7af4891c4dde181a6dccd36bb","name":"route36.json","bytes":194865},{"sha256":"afe7581240dd288a413d699235299cec8bbe6bc39db2dc9082ba884bb8ac8c6a","name":"research_routes.json","bytes":474122},{"sha256":"c8dacd2808128d982c1152af263045dc99b1b5da7b7b73134d5aa8024412fb21","name":"return_2438.json","bytes":31341},{"sha256":"7fd9f3fb12c745f81e4acd8928dd7df2af98293ff637cd5b224c965f925bd5ba","name":"return_101.json","bytes":22690},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","name":"export_transcript.py","bytes":10230},{"sha256":"d9493fd123b6ec3965891ab89b516fd48e06b325e0f510e86d671929bc56d659","name":"compute_hq.py","bytes":6119},{"sha256":"1106883786fa9f2f20cee954b8f0be3021be794f63cce4e036072e5d7facaab9","name":"compute_hq.out","bytes":1077},{"sha256":"e896e9461f999ae78813f60c95beb44fe6fdf40b75d1d405b9a1adc71b7fc325","name":"results_hq.json","bytes":3016}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}