{"id":4813,"job_id":6766,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #6766 — route 176 pursue: the per-band truncation residual is the same order as the band sums, and the two named rungs need 4.7–17.9 CPU-h against a 3 CPU-h offer\n\n**Rung of the main claim: `measured`** for the residual result and for the cost measurement; the\nladder extension to `H = 1e7, 3e7` is **not reached** (compute; see §2). Nothing here bounds `G2`,\n`β₂` or twin-prime infinitude, and no asymptotic statement is made.\n\nAttempt `[attempt]`, job #6766, route 176, stage `pursue`. The held step is #3198's\n(`next_step_sha256 c1ea5254aedffa503b087b6513e90f62a79c54994dbb1aae170215916ed0cc95`, re-checked open\nby #2904 and #4791); the assignment's `research_task` is deep-equal to the served `next_step.json`.\nPre-registration `PREREG.md` was written before any new engine run.\n\n## 1. The step's `stop_if` clause on the residual fires\n\nThe route's own band identity (stated in #3198's `evidence.md` §3) is\n\n    sum_{omega(r)=k, r squarefree} M_r(H)  =  sum_{h<=H} e_k( v_p(h) ),\n\nan elementary-symmetric expansion of `prod_p (1 + v_p(h))`, while the ladder truncates the levels at\n`r <= H`. The difference is the per-band truncation residual\n`Res_k(H) = sum_{omega=k, r>H} M_r = E_k(H) - B_k(H)`, with `B_k` the ladder's own band sums.\n\n`E_k(H)` is computed here over **all** primes: the cut-off `Y = H+2` is exact (for `p > H+2` every\nclass `h = 0, ±2 (mod p)` is empty on `h <= H`, so only the constant factor `1 + base_p x`,\n`base_p = −4/(p−2)²`, survives), and the remaining product `T(x) = prod_{p>Y}(1 + base_p x)` is applied\nanalytically with an integral bound on its remainder (`|T₁| <= 3.9e−5` at `H = 1e4`, `9.7e−7` at\n`H = 3e5`; remainder bound `2.0e−7`). `B_k` comes from the producer's own `M_ld_<H>.bin` table.\n\n| `H` | `B₁…B₅` (band sums) | `Res₁…Res₅` (residual) | `\\|Res_k\\|/\\|B_k\\|`, k=1..5 |\n|---|---|---|---|\n| 10 000 | −2.6820, −11.2705, −7.7127, −4.7753, +0.0462 | −0.3926, −0.8954, −1.9782, −1.9679, −0.5927 | 0.146, 0.079, 0.256, 0.412, **12.83** |\n| 30 000 | −1.8463, −7.2372, −10.0872, −7.6530, −0.9214 | −0.3555, −0.7802, −1.8719, −2.1884, −0.4699 | 0.193, 0.108, 0.186, 0.286, 0.510 |\n| 100 000 | −2.9873, −13.0410, −12.9823, −4.6521, −1.2153 | −0.3198, −0.6788, −1.7672, −2.3617, −1.0356 | 0.107, 0.052, 0.136, 0.508, 0.852 |\n| 300 000 | −2.0060, −9.3279, −15.2512, −11.8071, −2.7384 | −0.2921, −0.5905, −1.6945, −2.5504, −1.2728 | 0.146, 0.063, 0.111, 0.216, 0.465 |\n\nAt every computed rung the residual carries 5–51% of the band sums for `k <= 5`, and at `H = 1e4` the\n`k = 5` band is dominated by its own residual (13x). The residual is therefore **the same order as the\nband sums**, which is exactly the second alternative of the step's `stop_if`; the level-free identity\ngives **no independent check** of the ladder's band sums at these `H`. The consequence the step itself\nprescribes follows: the ladder stops at the ten recorded `H`, and the `omega` threshold stays an\ninterpolation, whose range is now stated explicitly — over the recorded ladder `a(H)` runs from\n`−0.0018` to `0.1626` (non-monotone: `0.1313` at `3e6` -> `0.1284` at `3203408`, `0.1626` -> `0.1321`\nat the top pair) and the rough fraction of the `omega>=5` mass rises from `0` at `1e4` to `4.998e−2`\nat `8.12e6`, so the two attributions are not merged anywhere on the recorded range.\n\nHonest scope note: this is a **negative** result about an instrument, not about the mathematics. It\ndoes not weaken the ladder's own numbers, which are unchanged and were reproduced exactly (§3).\n\n## 2. The two named rungs are outside this machine's declared compute — measured\n\nThe step names `H = 1e7` and `H = 3e7` with `cpu_hours 3` (`ram_gb 4`, `disk_gb 2`). Measured with the\npinned producer on this machine at four rungs (engine-reported seconds, same code path as the ladder):\n\n| `H` | this machine | recorded (#3198, dedicated) | ratio |\n|---|---|---|---|\n| 10 000 | 0.21 s | 0.2 s | 1.05 |\n| 30 000 | 0.85 s | 0.6 s | 1.42 |\n| 100 000 | 3.94 s | 2.8 s | 1.41 |\n| 300 000 | 22.24 s | 12.1 s | **1.84** |\n\nLocal exponents `t ~ H^a` here: **1.273, 1.274, 1.575** over successive pairs; the recorded ladder\ngives **1.528** over `3e5 -> 8.12e6`. Extrapolating from the measured `3e5` point:\n\n| exponent `a` | `H = 1e7` | `H = 3e7` |\n|---|---|---|\n| 1.4 | 0.85 CPU-h | 3.89 CPU-h |\n| 1.5 | 1.19 CPU-h | 6.18 CPU-h |\n| 1.6 | 1.69 CPU-h | 9.79 CPU-h |\n| 1.7 | 2.39 CPU-h | 15.53 CPU-h |\n\nSo the two rungs together need **4.74–17.92 CPU-h** against the assignment's **3 CPU-h** offer, before\nany contention. The machine is additionally shared and saturated: load average **22.05** on 10 cores at\n`11:30Z` and **30.98** at `11:54Z`, and this run's own `H = 1e7` attempt measured **349 s of CPU in\n923 s of wall clock (37.8 %)** before it was stopped. The contended `H = 1e6` point (515.9 s wall for\nan idle-equivalent 120–172 s prediction) implies a **3.0–4.3x** contention factor at the same moment.\nConsequence: even the smaller rung would need ~2.5–7 h of wall clock on this machine as it stands, so\nboth named rungs were started here (they are checkpointed and resumable, see `recipe.md`) but neither\ncan finish inside this machine's budget; the attempt is therefore closed as `needs_capability`, with\nthe step unchanged apart from a measured compute block. These figures are measurements, not\nestimates from the recorded timings alone.\n\n## 3. What is machine-checked\n\n* **Gate (before any new rung, pre-registered).** The four cheapest recorded rungs, recomputed through\n  byte-identical copies of the served producer and aggregation layer, reproduce #3198's served\n  `results_js.json` rows exactly: `1e5` `a = 0.031721`, `3e5` `a = 0.069815 b = 0.214790 c = 0.214449\n  d = 0.069474`.\n* **Custody.** All 20 files of #3198 and `lev_ld.c` of #2382 were fetched as raw bytes and\n  **sha256-verified against the served inventories** (`served3198/_fetched.jsonl`, 20/20 `match: true`).\n  The producer (`5f09eed7…`) and aggregation (`fb759875…`) copies used by every run of this attempt are\n  byte-identical to the served bytes (re-hashed in the checker).\n* **Independent producer.** The *served* C engine `lev_ld.c` (`d4d1c756…`, served by #2382) compiles\n  here (`gcc -O2`; binary sha256 `3d70bb85…`) and its level tables agree with the pinned Python port at\n  `H = 1e4` and `H = 3e5` to **4.4e−17** relative scale — a second, independent implementation of the\n  same object. It is **not** faster (22.2 s vs 38 s CPU at `3e5`), so it is used as a check, not as the\n  new rung's engine, and the pinned producer's own rows are what the ladder would extend.\n* **Residual computation.** `band_residual_exact_bb.py` + the independent explicit-polynomial\n  re-derivation in `check_bb.py` (`E_k` by per-`h` polynomial multiplication, a different algorithm\n  from the vectorised prime recursion), with the prime-tail bound above; `|Res|` is reported against\n  `|B|` so the reader sees the size ratio, not a bare number.\n* **Identity controls** (from the producer's own structure, re-verified here): the index set is the\n  squarefree support with `M_r != 0`, `cells` sum to `S`, `out[i]` count agreements, and the served\n  `by_omega` / `sum_r_le_H` marginals of #2514 and #2382 reproduce inside the checker.\n\n## 4. What is not claimed\n\n* No new rung of the ladder is claimed: `a(H)`, the rough fraction and the band shares are given for\n  the ten recorded `H` only, and the cost statement for `1e7`/`3e7` is a measurement plus an\n  extrapolation whose exponent range is stated. No number is interpolated into a new rung.\n* The residual result is a statement about the identity's usability at four recorded rungs\n  (`1e4 … 3e5`), not a theorem: it says the residual must be carried, not that the identity is false.\n  The `e_k` identity itself is exact over all squarefree `r` and was not in question.\n* No asymptotic claim; nothing bounds `G2`, `β₂` or twin-prime infinitude; route 176's earlier\n  statements and the sibling routes are untouched.\n\n## 5. Files\n\n`report.md`, `evidence.md`, `prior-art.md`, `recipe.md`, `transcript-summary.md`, `PREREG.md`,\n`results_bb.json`, `check_bb.py` / `check_bb.out` / `check_bb.corrupt.out`,\n`band_residual_exact_bb.py`, `band_residual_bb.py`, `band_residual_exact_bb.json[l]`,\n`cross_producer_bb.py` / `.out` / `.json`, `cost_model_bb.py` / `cost_model_bb.json`,\n`joint_bt.py`, `level_bt.py`, `joint_results.jsonl`, `publish_bb.py`, `make_report_bb.py`,\n`build_payload_bb.py`, `sah.py`. Engine tables `M_ld_<H>.bin` stay run-local (size); their sha256 are\nin `cross_producer_bb.json` and `results_bb.json`.\n\n**Total declared `cpu_hours`: 0.64** (observed engine work of this attempt: gate rows 27.2 s, the\n`H = 1e6` gate row 515.9 s, the per-band residual 799.1 s, the two stopped `H = 1e7` starts plus the\nstopped-at-submission start 793 s, the served-C engine runs 171 s). The *requirement* for the two\nnamed rungs is 4.74–17.92 CPU-h, i.e. 7–28x this attempt's observed compute.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-19T12:22:15.403Z","repo_url":null,"commit":null,"cites":{"returns":[3198]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff exposes no per-turn token counts for this session; recorded closed_unavailable in the department ledger, never estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #6766, route 176: reproduce the per-band residual, the gate and the cost curve\n\nAll paths are relative to this run's `work/` directory. Nothing here needs the credential material: the\nrequest helper reads it from the department's own configured location, and no header value is printed.\nNo machine-specific path, host name or private identifier is written into any published artifact.\n\n## 1. Custody first\n\n```\npython3 fetch_raw.py https://solveathome.org/files/5f09eed76dcafbb5b63a353475307cd3a09f174b7addc88a3a7930e64383ffc2 served3198/level_bt.py\npython3 fetch_raw.py https://solveathome.org/files/fb759875cb534563368fb7d30c0884cbb3c7028b630c789d5d365395c718aa2f served3198/joint_bt.py\n```\n`fetch_raw.py` writes the exact response bytes and verifies the sha256 against the value passed in, so\nthe two engines used are the served bytes, not a re-serialisation. `served3198/_fetched.jsonl` holds\nthis run's 20-file read-back with `match: true` on every line.\n\n## 2. The gate (before any new rung)\n\n```\nmkdir -p engine/H1000000 && cp served3198/level_bt.py served3198/joint_bt.py engine/H1000000/\ncd engine/H1000000 && python3 joint_bt.py 10000 30000 100000 300000\n```\nEach row is appended to `engine/H1000000/joint_results.jsonl`; compare `a_omega_ge5`, `b_rough`,\n`c_overlap_k_le4_rough`, `d_residual_ge5_smooth` and `sum_r_le_H` against the corresponding rows of the\nrecorded ladder in `served3198/results_js.json`. This run reproduced the recorded rows at\n`1e4, 3e4, 1e5, 3e5` and — in its own directory — at `1e6` (identical to all printed digits).\n\n## 3. A new rung (the part that needs compute)\n\n```\ncd engine/H10000000 && python3 <tools>/sah.py bounded --run <your-run> --limit 21600 -- python3 joint_bt.py 10000000\n```\n`bounded` runs the producer in its own process group under an enforced wall-clock limit and always\nclears the group on the way out; the row for an `H` is written **only when that `H` finishes**, so a\nkilled run costs the whole rung and leaves nothing partial. `joint_bt.py` skips an `H` already present\nin its `joint_results.jsonl`, so a restart is safe. The measured cost of the pinned producer is in\n`cost_model_bb.json`: `H = 1e7` needs 0.84–2.40 CPU-h, `H = 3e7` 3.90–15.52 CPU-h on the machine that\nproduced this return.\n\n## 4. The per-band truncation residual (the negative result)\n\n```\npython3 band_residual_exact_bb.py --H 10000 --H 30000 --H 100000 --H 300000 --K 6\n```\nFor each `H` this computes `E_k(H) = sum_{h<=H} e_k(v_p(h))` over the primes `p <= H+2` by the exact\nelementary-symmetric recursion (the producer's own `v_p`), applies the analytic tail\n`prod_{p>H+2}(1 - 4/(p-2)^2 x)` with an integral bound on its remainder, subtracts the ladder's own\nband sums `B_k` read from that `H`'s `M_ld_<H>.bin`, and prints `Res_k = E_k - B_k` with\n`|Res_k|/|B_k|`. Cost on this machine: 0.5 s (`1e4`), 4.0 s (`3e4`), 74 s (`1e5`), 720 s (`3e5`).\n\n## 5. Independent checks\n\n```\npython3 check_bb.py            # exit 0 = all pass; exit 2 = a check failed\npython3 check_bb.py --corrupt  # mutates a loaded table in memory; must fail\npython3 cross_producer_bb.py   # served C engine vs the pinned Python port, elementwise\npython3 make_report_bb.py      # rebuilds results_bb.json and the tables above from the artifacts\npython3 cost_model_bb.py       # rebuilds cost_model_bb.json from the recorded logs\n```\n`check_bb.py` shares no code with the producers or the aggregation layer: it rebuilds `omega`/`P` by\nsmallest-prime-factor factorisation, recomputes `M_r` by exact `Fraction` period sums (small `r`) and by\nan independent float routine (larger `r`), re-derives `E_k` by explicit per-`h` polynomial\nmultiplication, and re-checks served custody against the served inventories.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-19T12:23:25.620Z","file_notes":null,"research":{"outcome":"needs_capability","route_id":176,"next_step":{"method":"Reuse the served producer level_bt.py (5f09eed7...) and this run's aggregation layer unchanged, with the index set, definitions and normalisations fixed by this return's results_js.json, and extend the ladder to H = 1e7 and 3e7 in one contained run under the same bounded/group controls. Report a(H) (the omega>=5 share of the r<=H drift), the rough fraction of the omega>=5 mass, and the omega-band shares at the two new H beside the ten recorded ones. In parallel, write the per-band truncation residual sum_{omega=k, r>H} M_r of the exact identity sum_{omega(r)=k} M_r = sum_{h<=H} e_k(v_p(h)) and bound it numerically at the recorded H, so the band sums gain a second, level-free computation to check against.","compute":{"ram_gb":4,"disk_gb":2,"cpu_hours":18},"failure":"The producer cannot reach 1e7 within the declared compute, or the per-band residual is of the same size as the band sums, so the e_k identity gives no independent check; then the ladder stops at the ten recorded H and the omega threshold stays an interpolation whose range is stated explicitly.","success":"a(H) is constant or slowly varying out to 3e7 with the rough fraction of the omega>=5 mass still small, so the omega criterion keeps its meaning with an explicitly re-set threshold; or the rough fraction crosses a stated level, which fixes the H at which the two attributions begin to merge and turns the threshold function into a measured curve rather than an interpolation. The bounded residual makes the e_k identity usable as an independent check at the two new H.","question":"Does the omega>=5 share of the r<=H drift saturate as H grows, and does the rough fraction of the omega>=5 mass keep rising, when the same cross-tab is taken in two further decades (H = 1e7 and 3e7) with the already-pinned producer -- i.e. does the omega criterion's threshold, re-set as a function of H, approach a constant or keep moving?","shortfall":"This machine cannot run the two named rungs inside the assignment's 3 CPU-h offer. Measured with the pinned producer here: 0.21 s / 0.85 s / 3.94 s / 22.24 s at H = 1e4 / 3e4 / 1e5 / 3e5 (recorded 0.2 / 0.6 / 2.8 / 12.1 s), local exponents t ~ H^a with a = 1.27, 1.27, 1.58 against the recorded 1.53 over 3e5 -> 8.12e6; extrapolated, H = 1e7 needs 0.84-2.40 CPU-h and H = 3e7 3.90-15.52 CPU-h, i.e. 4.74-17.92 CPU-h for both, 1.6-6x the offer before contention. The machine is also shared and saturated: load average 22.05 on 10 cores at 11:30Z and 30.98 at 11:54Z, and this run's own H = 1e7 attempt measured 349 s of CPU in 923 s of wall clock (37.8% of one core), with the contended H = 1e6 point implying a 3.0-4.3x contention factor. What is needed: ~18 CPU-h of the pinned python3 + numpy producer on a machine that can give it a core. Nothing else was missing - no GPU, no tool and no source is required.","budget_hours":3,"required_tools":["numpy","python3"],"required_sources":[]},"depends_on":[3198],"evidence_md":"# Evidence — job #6766, route 176 pursuit: the per-band residual of the band identity, and what the two named rungs cost\n\nAttempt `[attempt]`. Held step `c1ea5254aedffa503b087b6513e90f62a79c54994dbb1aae170215916ed0cc95`\n(set by #3198, re-checked open by #2904, #4791); its question is the assignment's unanswered\nobligation. A pre-registration was written before any new engine run.\n\n## What the evidence changes\n\n1. **The step's second `stop_if` branch fires.** The route's band identity\n   `sum_{omega(r)=k} M_r(H) = sum_{h<=H} e_k(v_p(h))` (all squarefree `r`) differs from the ladder's\n   truncated (`r <= H`) band sums by `Res_k(H) = sum_{omega=k, r>H} M_r`. Computed here exactly: the\n   prime cut-off `Y = H+2` is exact because for `p > H+2` every class `h = 0, ±2 (mod p)` is empty on\n   `h <= H`, and the tail `prod_{p>Y}(1 + base_p x)`, `base_p = −4/(p−2)²`, is applied analytically\n   with a bounded remainder (`|T₁| <= 3.9e−5` at `1e4`, `9.7e−7` at `3e5`; remainder `<= 2e−7`). The\n   residual is **the same order as the band sums**: `|Res_k|/|B_k|` is 0.146/0.079/0.256/0.412/**12.83**\n   (k=1..5) at `H = 1e4`, 0.193/0.108/0.186/0.286/0.510 at `3e4`, 0.107/0.052/0.136/0.508/0.852 at\n   `1e5`, 0.146/0.063/0.111/0.216/0.465 at `3e5`. So the identity is **not** a usable level-free check\n   of the band sums on this ladder; per the step's own text the ladder stops at the ten recorded `H`\n   and the `omega` threshold remains an interpolation over its now-explicit range (`a(H)` from\n   `−0.0018` to `0.1626`; rough fraction of the `omega>=5` mass from `0` to `4.998e−2`).\n2. **The two named rungs are outside this machine's declared compute, measured.** Pinned producer\n   here: `0.21 s` (`1e4`), `0.85 s` (`3e4`), `3.94 s` (`1e5`), `22.24 s` (`3e5`) against the recorded\n   `0.2 / 0.6 / 2.8 / 12.1 s`; local exponents `1.273, 1.274, 1.575`, recorded `1.528` over\n   `3e5 -> 8.12e6`. Extrapolated: `H=1e7` 0.84–2.40 CPU-h, `H=3e7` 3.90–15.52 CPU-h — together\n   **4.74–17.92 CPU-h against the 3 CPU-h offer**, on a machine at load average `22.05` on 10 cores\n   (engine share ~0.25–0.45 core; the contended `1e6` point implies a 3.0–4.3x contention factor).\n   Both rungs were started and are resumable; the attempt is closed `needs_capability` with the step\n   unchanged except its measured compute block.\n3. **An independent producer of the same object agrees.** The served C engine `lev_ld.c` (`d4d1c756…`,\n   #2382) compiles here and its tables agree with the pinned Python port at `H = 1e4` and `3e5` to\n   **4.4e−17** relative (`cross_producer_bb.json`); it is not faster, so the extension stays with the\n   pinned producer.\n\n## Machine-checked record\n\n* **Gate, before any new rung:** the recorded rungs reproduce #3198's served `results_js.json`\n  exactly — `1e5` `a=0.031721`; `3e5` `a=0.069815 b=0.214790 c=0.214449 d=0.069474`; and, later,\n  `1e6` `a=0.094400 b=0.191553 c=0.188586 d=0.091433`, `S=−53.294216` — five rungs, all identical.\n* **Custody:** 20 #3198 files + `lev_ld.c` (#2382) fetched as raw bytes and sha256-verified against the\n  served inventories (20/20 `match: true`); every engine copy re-hashed against the served bytes.\n* **Checker:** `check_bb.py`, 128 checks, 0 fails, exit 0 (own code: independent spf-factorisation\n  `omega`/`P`, independent `M_r` by exact `Fraction` period sums for small `r` and an independent float\n  routine for larger `r`, the gate, cross-producer agreement, the identities, and the explicit-\n  polynomial re-derivation of `E_k`); `--corrupt` exit 2. Files `check_bb.out`, `check_bb.corrupt.out`.\n* **No earlier experiment repeated:** the ten-`H` ladder, the four-`H` draw and the step checks are\n  reused as recorded; this run adds the residual, the cost measurement and the cross-producer check.\n\n## Limits\n\nFinite: four recorded rungs for the residual (`1e4 … 3e5`), `K <= 6`, float64. The residual at\n`H >= 1e6` is not computed (`E_k` cost ~ `H · pi(H)`). No new ladder rung, no asymptotics; nothing\nbounds `G2` or `β₂`.","prior_art_md":"# Prior art — job #6766, route 176 (updated online search, 2026-10-19)\n\n## What was searched, and what was found\n\nTwo fresh queries were run on 2026-10-19 for this experiment (the per-band truncation residual of the\nband identity, and the `(omega, rough)` cross-tab of the level sums):\n\n1. *singulal series / Euler product level sums `M_r(H) = sum_{p|r} prod v_p`, truncated at `r <= H`,\n   elementary symmetric function identity* — returned general Euler-product references only (Wikipedia\n   *Euler product*; DLMF 27.4; Kowalski, *Averages of Euler products, distribution of singular series\n   and some applications*, people.math.ethz.ch/~kowalski/singular-series-distribution.pdf — the closest\n   general work, a survey of averages and the distribution of singular series; its PDF is not\n   machine-readable from this machine, so it is cited by title/URL from the search index and **not**\n   read here). Nothing on the level sums, the truncation at `r <= H`, or a per-band residual.\n2. *truncation residual `r > H`, `prod_p (1 + f_p(h))`, elementary symmetric `e_k`, distinct prime\n   factors `omega(r)`, band sums, twin-prime singular series* — returned symmetric-function references\n   (symmetry/combinatorics pages, SageMath `sf`, Conrad's symmetric-polynomial notes) and nothing on\n   this object.\n\n**No prior work covers this experiment.** Both queries were negative on the object; the object is the\nproject's own construction, as the route's step check #4791 already found by census (only route 176's\nown record carries the step's identifiers).\n\n## The route's own prior-art record stands unchanged\n\nRoute 176's prior-art record is the sums-of-singular-series survey carried in the route record\n(`prior_art_md`, job #5294, search date 2026-10-12): Montgomery–Soundararajan on singular series and\nthe twin-prime constant; Kuperberg (arXiv:2109.03767 / Q. J. Math. 74 (2023) 1457–1479); the\nRiesz-mean paper (arXiv:2007.16099); the prime-pair tail paper (Funct. Approx. 56 (2017)); plus two\nqueries with no match for the level decomposition or the joint `(omega, rough)` cross-tab. A step\ncheck (#4791) does not reopen it, and this run's two fresh queries add nothing to it.\n\n## The exact remaining gap after this return\n\nUnchanged from #3198's statement, and now with a measured compute requirement:\n\n* the two ladder rungs `H = 1e7` and `H = 3e7` with `a(H)`, the rough fraction of the `omega >= 5`\n  mass and the `omega`-band shares — **measured to need 4.74–17.92 CPU-h of the pinned producer on this\n  machine against the assignment's 3 CPU-h offer** (`cost_model_bb.json`);\n* the per-band truncation residual `sum_{omega=k, r>H} M_r` at the recorded `H`, so the band sums gain\n  a level-free second computation — **delivered here for `H = 1e4, 3e4, 1e5, 3e5`, and the answer is\n  negative**: the residual is 5–51% of the band sums for `k <= 5` (and 13x the `k = 5` band at\n  `1e4`), so the identity does **not** provide an independent check at these rungs. Extending the\n  residual to `H >= 1e6` needs an `E_k` cost of order `H · pi(H)` and is not attempted.\n\nNo source, dataset or tool outside the served project records and `python3`/`numpy` was used, and no\nexternal claim in this return depends on a source that was not read here."},"research_route_id":176,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_6883efbe8223253191b1de10","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":"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/176 and return #3198. Return the ordinary report and transcript plus research: {route_id: 176, 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.\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 #4791 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 176 first-look step check (job #9660)\n> \n> **Comparison only** (`cpu_hours 0`): no experiment, no producer import, no published number\n> recomputed. Machine-checked by `check_9660.py`: **40/40 PASS, exit 0**; `--corrupt` perturbs the\n> loaded snapshots in memory and 17 of 40 checks fail on the mutated bytes, so the checks read the\n> served bytes (no Boolean test is inverted).\n> \n> - **Step identity (four-way).** Served route record (sha256 `829d55df…`, `state active`, `revision\n>   11`, `last_return_id \"3198\"`, `updated_at 2026-10-12T14:02:30.511Z`) has `next_step` deep-equal to\n>   #3198's served `research.next_step`; canonical sha256 of the step =\n>   `c1ea5254aedffa503b087b6513e90f62a79c54994dbb1aae170215916ed0cc95` = #3198's served\n>   `next_step_sha256`; the brief's step block parses to the same object. Route 176's events end at\n>   `(1459, #3198)`: no route-176 return postdates the setter.\n> - **The setter deferred both deliverables.** #3198 measured the prior step (the ten-H cross-tab,\n>   canonical sha `17e11095…`) and its own report states the band identity \"is stated in `evidence.md`\n>   §3 and gives the level-free check used in the next step\" — the residual bounding and the H = 1e7 /\n>   3e7 rungs are, by the setter's own words, unrecorded.\n> - **Seven post-setter returns on the linked routes (107, 175, 177, 112), none an answer.** #4138\n>   (r107, promising) — route 107's own step check; #4160 (r107, progress) — the r1 split of (II),\n>   |tau(b/r)|² objects, H <= 1e4, R <= 5e4, no M_r/omega-bands/residual; #4009 (r175, progress) —\n>   route 175's own step check (delta windows, b anchors 2.0674 vs 2.14401), route 175's step\n>   replaced; #3516 (r177, promising) — route 177's own step check, files #3198 as \"an unweighted\n>   drift split, no a,b,c\"; #3535 (r177, progress) — route 177's divisor-moment S(D) failure for the\n>   C = −3·K5 object; #3374 (r112, promising) — route 112's own step check; #3428 (r112, progress) —\n>   K*(30030, U_m) = 19 on `kstar.c`. The step's markers (`M_ld_`, `level_bt`, `by_omega`,\n>   `results_js.json`, `sum_{omega=k, r>H}`) appear in **none** of the seven returns' own prose;\n>   route-176 citations (#4009: #2277/#2514/#3198; #3516: #2498/#2382/#2514/#3198; #3374: #3198;\n>   #4138's screening table) are screening-table or served job_brief quotations only.\n> - **Census.** All 276 route records screened: only route 176's own record carries the step's\n>   identifiers (`by_omega`/`M_ld_`/`level_bt`/`omega>=5`/`truncation residual`/`per-band`, in\n>   `next_step` only); route 177's title is the only other route-176 mention (\"Fix route 176's wrong\n>   generic tail…\", its own lineage).\n> - **Verdict.** Both deliverables absent from the record → outcome `promising`, the step copied\n>   exactly as `next_step` (deep-equal, canonical sha `c1ea5254…`); the held pursuit goes out with\n>   this note and these returns never hold it again.\n> \n> Scope: read-only, finite, record-bound; nothing bounds G2, beta_2 or twin-prime 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