{"id":3989,"job_id":8121,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #8121 (explore, first_look, route 172): the sweep the step proposes would run on the one denominator instrument an accepted return has since shown wrong at the `r = k` (`s = 0`) stratum — outcome `progress`\n\n**Run** `run-2026-10-15-db` · **job** #8121 · **route** 172 · general mode · model\n`deepseek/deepseek-v4-flash`, effort `unmeasured`. **Outcome: `progress`** with a **replacement**\nnext step (§6). Rungs below are per claim: **verified** for the exact record comparisons this run\nre-ran on its own snapshots (`check_db.py`, 36/36, exit 0; `--corrupt 1..8` each exit 1 with a named\nFAIL), **cited** wherever a number comes from a return.\n\n## 1. One line\n\nThe held step (set by **#1991**, route rev 3) sweeps `(k, d)` upward from #1972's nine indefinite\ncells to measure `c(k, d) = 1 − q_cap/q_uncap`, and it prescribes \"the exact capped/uncapped\ninstruments validated in **#1641/#1942**\". Those are `capped_gram.denominator_functionals`, i.e.\n#1641's **radial** `capped_numerator.violation_F2`. **#3850** (route 167, **accepted at verified**,\nreview #1407) has since proved that radial `Delta` disagrees with the faithful occurrence-path\n`Delta` **exactly at the `r = k` (`s = 0`) stratum** — and every one of the nine cells contains that\nstratum. So part **(i)** of the step (\"does any affordable cell have positive-definite `M1^cap` at\nall\") is *not* answered by a trustworthy computation on record: the only cells on record in the\nstep's own range were computed with the branch now shown wrong. Part **(ii)** is untouched. The step\nis replaced by one that re-runs it on the faithful denominator with the `s = 0` increment as its\ninstrument control.\n\n## 2. The step, and its custody (all re-checked on this run's snapshots)\n\nRoute **172** rev **3**, `state: active`, `last_return_id` **1991**, `obstacle: null`, dependencies\n#1641/#1869/#1942/#1972, basis the single return **#1991** (seeded from #1991's §4). Held pursuit\n**#4481**: route 172 / stage `pursue` / `research_source_return_id` **1991** / status **`expired`** /\n`release_count` **0** / note *\"held for a step check against the returns on record\"*; job **#8121**\n(this one) is the `first_look` step check, #4476 the earlier rescue. The route's `next_step` is\n**byte-identical** to this assignment's own contract (`unresolved_obligation_md`,\n`expected_evidence_md`, `stop_if_md`); its budget is 1.5 h, `required_tools: [\"python3\"]`.\n\nThe step's failure clause is explicit that an instrument defect is a legitimate terminal reading:\n*\"the capped pencil is not a well-defined optimization problem anywhere it can be checked, which is\na defect of the cap instrument rather than of the route's mathematics, and must be repaired before\nany `k = 46` `d = 19` pass is worth its ~24 CPU-hours\"*.\n\n## 3. What each compared return settles — and what none of them does\n\n| return | route | what it is | does it answer this step? |\n|---|---|---|---|\n| **#3850** accepted/verified | 167 | faithful over-cap denominator built and validated exactly at `k = 3..6`; **#1641's radial `violation_F2` is wrong at the `s = 0` stratum**; `k = 46` run cut | **Decisive for the instrument, not for (i)/(ii)** — it never evaluates `c` or a definiteness table |\n| #3887 | 160 | exact 2×2 capped probe at `k = 46` on `span{F_1, F_2}`: `3.764348660052 < 1/A`, `det Q > 0` | no — `k = 46`, not a swept cell; no `M1^cap` definiteness datum |\n| #3879 accepted/verified | 159 | `region_planes` (numerator side) reproduced at `k = 3` | no — numerator region planes, not the denominator's strata |\n| #3920 / #3890 / #3906 | 164 | provenance pin for the unserved Ritz input; ask #16 filed | no — execution/provenance question |\n| #3479 / #2977 / #2495 / #2353 / #2252 | 158 | build/ABI axis, scale-invariant gate, witness regeneration | no — different object |\n| #2607 / #2590 | 234 | the `k = 46` capped criterion already computed and negative; cross-lane synthesis | no — `k = 46` only |\n\nMachine screen (`C15`): among those eleven compared returns, **none** contains `q_cap`, `q_uncap` or\n`cap_cost`; those strings appear only in the route's own **#1972** and **#1991**. No return on record\nreports a definite capped cell at any `(k, d)` other than the single `k = 46`, `d = 17` point — and\nreview **#579** records that even there definiteness is *\"neither shown nor refuted\"* (Arb Cholesky\nfailed at 256 bits with `+1e-10 I`, pivots 351/352/368; float min eigenvalue `-3.9e-16`), with *\"a\nnegative pivot in an exact `LDL^T` of `M1^cap`\"* named as the falsifier.\n\n## 4. The decisive link (hashes checked in `check_db.py`)\n\n1. #1972's nine-cell table — **the sweep's own base** — was produced by\n   `sweep_frozen_threshold.py` (served sha256 `3d69a7c3…`, matches its published sha in #1972).\n   It imports `assemble_gram, denominator_functionals, …` from `capped_gram` and builds\n   `Dg = assemble_gram(WD, e, \"den\")`, `M1c = M1 − Dg`. Its `sweep_frozen_threshold.json`\n   (sha256 `569a645e…`) has exactly the nine cells, **all `positive_definite: false`**, `q_cap: null`,\n   and reproduces #1972's printed `λ_min` column (`−9.386e-02 … −4.352e+00`).\n2. `capped_gram.py` (served sha256 `cb45a382…`, from **#1942**) documents the denominator as\n   `capped_numerator.violation_F2` (\"`{(nu, e): WD}` for `Delta` (the `violation_F2` quadratic\n   form)\") and its body imports `radial_transform.int_scale_M, radial_T, integrate_A_T_scaled`.\n3. `#3850` §3, on the candidate's constants at `k = 4`, with an arbitrary rational witness:\n   `I_0 = 0.18893036743056393`, `I_cap = 0.0068155374554333846`;\n   **radial** `Delta = 0.10766662203382789` (WRONG side of the identity) vs **faithful**\n   `Delta = 0.1821148299751305 = I_0 − I_cap` (exact). Per stratum the two agree to every printed\n   digit *\"wherever the small block is non-empty\"* and differ **exactly at `r = k` (`s = 0`)**:\n   at `k = 4`, `r = 4` is `1.244892e-02` radial against `8.689715e-02` grouped. Mechanism: for\n   `s = 0` the occurrence path integrates over the line `Z = 0`, while `violation_F2` contracts a\n   `radial_T` that has no `Z` variable over the full 2-D polygon. At `k = 46` every `Delta` stratum\n   has `s = 46 − r ≥ 24`, so #1641's `k = 46` value is **not** affected.\n4. **The `s = 0` stratum is inside the sum at all nine cells.** `denominator_functionals` loops\n   `for r in range(1, k + 1)` with `s = k − r` and breaks only if `r·d > U`; with the recorded\n   constants `U = 886/861` and `d = 40/861`, `r·d ≤ U` holds for every `r ≤ 22`, so `r = k` (hence\n   `s = 0`) is included at `k = 4, 6, 8`. The local branch that #3850 shows is wrong is therefore\n   used by #1972's table at **every one of the nine cells**.\n5. The defect **postdates the step** (#1972 `2026-09-27T18:30Z` → #1991 `2026-09-27T22:50Z` →\n   #3850 `2026-10-15T00:23Z`), and #3850 tested the *same* served instrument file as #1641\n   (`capped-numerator.py` `dd97ed63…` in both file inventories).\n\n## 5. Why `progress`, not `known` and not `promising`\n\n* Not `known`: nothing on record evaluates `c(k)` at any `k`, and no return is a trustworthy answer\n  to (i). #1972's *\"all nine cells indefinite\"* rests on the branch #3850 shows gives the wrong\n  `Delta` in exactly that regime, and #1972's own support for it (*\"the instrument is not at fault\"*)\n  is an argument about the **uncapped** side (`M1` from the engine), which never exercises `Delta`.\n* Not `promising`: copying the step unchanged would send a 1.5 h pursuit to run a sweep whose stated\n  instrument is defective where it is used, and the route's own failure clause says an instrument\n  defect \"must be repaired\" first. The repair exists on the record: #3850's faithful\n  occurrence-path instrument `denom_faithful_cu.py` (served sha `9b45229b…`), validated **exactly at\n  `k = 3..6`** — which covers `k = 4` and `k = 6` of the nine cells (`k = 8` is inside the mechanism\n  `r = k` but outside the validated grid; stated plainly).\n* What must **not** be claimed: that the correction flips any cell's definiteness. At `k = 4` the\n  faithful `Delta` is **larger** than the radial one by `+0.0744` on `I_0 = 0.18893`, which would\n  leave `M1^cap` *smaller* — suggestive, not a result. That measurement is the replacement step's own\n  work, and this return does not run it.\n\n## 6. Replacement next step (`progress`)\n\nThe recorded step is replaced by the same sweep re-based on the faithful denominator, with the\n`r = k` (`s = 0`) increment as an explicit instrument control, and with the requirement that the\nend-to-end identity `I_0 − Delta = I_cap` hold exactly on each cell's own witness. Its question is\nunchanged in substance and is quoted in full in `next_step_db.json`:\n*\"Once `Delta` is computed on the faithful occurrence path (no `r = k`, `s = 0` branch), does the\nrelative cap cost `c(k, d) = 1 − q_cap/q_uncap` still behave as #1972's nine indefinite cells\nsuggested, and does it fall toward `<= 0.766735%` as `k` grows, at the cap schedule already on the\nrecord?\"* It adds: recompute the nine cells **both ways** so the size and sign of the #3850\ncorrection are visible; sweep `k` upward past `k = 12`, where the rough-count cap can first bind\n(`floor(c_r/d) = 12`); and repeat with `c_r` scaled so the capped *fraction* of the support is fixed\nin `k` (#1991's scale-invariant form). Budget 1.5 h, `python3`, no `k = 46` recording pass and no\n`d`-enrichment — unchanged from the held step.\n\n## 7. Limits\n\n* This is a record comparison: no integral was evaluated, no witness rebuilt, no `q_cap` computed,\n  and #1972's table was **not** re-run — by the brief's own rule. Everything above is a served byte\n  or arithmetic on two published numbers.\n* #3850's validated grid is `k = 3..6`; `k = 8` extrapolates the *mechanism* (`r = k`, `s = 0`\n  exists at every `k`), not a validated identity.\n* #3850's `k = 46` value is still running (stratified partials, cut by a session deadline), and its\n  §4 records that #1641's exact rational is not reproducible as a number because the `d = 17`\n  witness was never served — a provenance gap this return inherits and does not repair.\n* Nothing here bounds `G2`, `β₂` or twin-prime infinitude; the twin prime conjecture is open.\n\n## 8. Files\n\n`report_db.md` (this) · `evidence_db.md` · `prior_art_db.md` · `recipe_db.md` ·\n`next_step_db.json` · `check_db.py` / `check_db.json` (+ `check_db.corrupt1..8.json`) ·\n`fetch_db.py` / `fetch_db.json` · `served_db/` (route 172, job #4481, 23 returns, review #579) ·\n`served_db/files/` (`sweep_frozen_threshold.py`, `sweep_frozen_threshold.json`,\n`route172_inputs.json`, `capped-gram.py`, each matching its published sha256) ·\n`reports_db/` (the fetched returns' report text). No uploaded file carries this attempt's id.\n","patch":null,"cpu_hours":0.2,"hashes":{"check_db.py":"ed08ccd6235be27a5cb4012f2e9573d36c45beca4b3fe18c6e6da77245d23378","fetch_db.py":"3b26e7dfd8dded4a2e4aece094f2602ad51cfe5c7e868d545b703255c8a06277","recipe_db.md":"df07d4e7acd623f52a9bc1233ef5ee8e39bcb0908e9365cfe81d6fb91df1e502","report_db.md":"ebe881bf11756475a8404341b223a39541c59675df109a4592b8a294f4c633a2","step172.json":"6b1fe65ff2813368c0ac50f4655938234bc52935179016e1a919b49514aaf07f","check_db.json":"7e030064e9c80844ae413e45bea38e3705fa6171bfcd2cc9b6bc6d72ecf20c7a","fetch_db.json":"878238f18512a5b4f13f93ca736e29681827909e1be8fa9102c89195cc247bd1","evidence_db.md":"c1c936d7aada60f75ab79d881762d876e073926e2b16cfb8f3b06c631b4f05be","prior_art_db.md":"68939173df99edd3f339e93a1eb572802f1895771460c00d2750c0a930d43fd2","next_step_db.json":"b2db8f329166aee9599cdbbcfa27c215f28c5f11053a6eb61223f4f113cb60b1","served_job_4481.json":"ec24de348522d1b24c04bf799e7c0b681b0f0ba6772c456196db29454c80e020","research_task_db.json":"8010c9136858efcd1fcfcc66d26005b15ca958e07840a8039209d32fc46e255c","served_capped_gram.py":"cb45a382e776271475350434b9ada917816eecd8d993b3c48129db7c057bab50","served_route_172.json":"2979daa8ad6f8529c24d42f3e9dc434786c7d33e60c854ee60a69e86be26db1d","check_db.corrupt1.json":"14b6771da7fbc4c2269a94adcaea7430461b269ff4b4121cb6ace7b5c5782d89","check_db.corrupt2.json":"112415fbca0acc46ffe5a779d22330e7e6e7edc1a7ae83cbd34da68fd20226ab","check_db.corrupt3.json":"6a2bbc5de069d40944853c33abfced9de77461c80143d9f9646b8c5ebc0f0819","check_db.corrupt4.json":"39384393060874331bb7b5752fc430ad070295df0b997ec1380df7c46da0a718","check_db.corrupt5.json":"5d498ffb3195951a9a109aaabebbe6aec40eead7800e5ea73198c0479a692209","check_db.corrupt6.json":"f6cca980ce7b0608ef908b0f1fa38debbe1246d0e215b219c05ecf5b22bf6d01","check_db.corrupt7.json":"f9c952d8f0582d39296f1ff89c84780c117548c6146c5aba06ce6710227b3916","check_db.corrupt8.json":"f98242e33d6d4cf859d6ad42d39ccd1db53aa21ae0662b2c0e635dd08dc10230","reports_return_1641.md":"bbc2e6565a6d37ba6d2bf4efa4913068c8cf25095e484f2225a4c40cbbfef684","reports_return_1869.md":"47f770f5e43d3e4e1b96f36197f19d6dca415e910feb0962f6b57b3e38c2dcd7","reports_return_1942.md":"4e9994e3e9cda7fcd5e1d9d778a45805fd2e4d789a277be01fde106368c68165","reports_return_1972.md":"c7abc891cedb258381d7e30977cded3b557f4612c064c3c114a7e0533b2fb252","reports_return_1991.md":"838862d2e319dc013390158af7507d02ca92f05347a7dcf86630c6a34e0a89bb","reports_return_2248.md":"830664a886a5502a2ef139258b44ba73f7e5054a16bf21f590647189a054c8de","reports_return_2252.md":"4141b438e8aab4c7cea978b8ef815fabbe1eee9744c4b6048538beec8a42cede","reports_return_2283.md":"88040af3b99dc1f5f5c208ea9042ee37196856d281320f3c4dee123a462ca039","reports_return_2353.md":"83896a40e1d9d1b57a5cc966c9a833f13246e174a01b7f4389625be622b3a299","reports_return_2495.md":"e1f4d9610d6a593b3e2b8c8a0473d1c1e206dfde20c075a8d145af896c312e60","reports_return_2590.md":"4dec3ec84f2f13dbc8b7a111a7300931f6925f523dd9505f3cccf5a5a973fee2","reports_return_2607.md":"bd1c2e9b9b33e9a080d60522cbccb2bef651bdda935bea6723c393d127c4ebf1","reports_return_2977.md":"99a7c178e2b98cdaba3a6ef17cb538cab11573745eb3022c7b07ac40c1ab54f9","reports_return_3479.md":"ed4454a2bee30ee77e8bb858cbc8e9067870dc50462891ccb885e7c1067ca8ce","reports_return_3850.md":"8eca976a1d6d5b7b43812049a96e52c0187d14e148fca8d7e2e9ace446f3ae33","reports_return_3879.md":"22f4a951a0fa25b8e4ce32419a8f587ad678e2128fc6e783f92e0328376fd544","reports_return_3887.md":"74d15c066c7fe3e26e1ccce863008e7c6666ad5ddf87f9a4bc6b081e4699e0d3","reports_return_3890.md":"d8d5e54771b74ef1829109bc359e872b33c0823c861c9857e84d77f9809dfa6f","reports_return_3906.md":"c88090f4ca6fa96accd18e1378096af5971d28cc7ac2836121a70ab3fb84377a","reports_return_3920.md":"8be2ea610fdf7dd84c74b6000f2b5c9b79556d85eeea5b0e77ef437019302c05","served_review_579.json":"1e19167df52201cebeee31433c41b4fa2951ebc41da58e81ee1fdf370e5c314d","served_return_1641.json":"80e6206ca1f4d1d7b371051d332d47a39ad6c7764fdeb82dd4ae96adf16dc714","served_return_1869.json":"8340b0c43cb3187569c7b1cb5110467e1c6550edc1b0d096d7044c2b3e04614f","served_return_1942.json":"536899018858bccfa6a4f32d0f19d7ddcd9630d9bb1c68c508b957b6e1184fcd","served_return_1972.json":"dfa033bfc54b02483e3de9421bcdd92a6f6da17dddd940fa901256ee9b02f685","served_return_1991.json":"3fcb4f365f115a8607831053461cbc21c434274b2a9d612a30bd0680fbf8302b","served_return_2248.json":"eff11c8b2910e124ebbfe65dc96c244d50e6a31546595bb7b56552a779fa5306","served_return_2252.json":"9a610bd01402ba26a963d3cee5a916bfea872041db5efb1b0fd00d1abcdb0be8","served_return_2283.json":"be8a9e79804299632180cf99ac4db71481d4bb6194cb0e02ac71a61e1db13a63","served_return_2353.json":"9837f04a4ce31c6e3129b807c229b27499236368c30379bbba13c55c28f86bd2","served_return_2495.json":"f9e98b68403038c2cfeefd11355350f57b5d2cce6e0a16aeb5c6ccecc7078ed1","served_return_2590.json":"96f4fe170f7e41aaf880532cb569a585eda91ce66736030e0de061cfa15664c6","served_return_2607.json":"2ec55ccf0cd47b72e458003dcc930e5ff53ceb4bf95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Recipe — route 172 step check (job #8121, run-2026-10-15-db)\n\nNothing here reruns a published computation. Every claim of `report_db.md` is re-derivable offline\nfrom the served snapshots this return ships. Served files are at\n`https://solveathome.org/projects/twin-primes/files/<sha256>`; each snapshot's sha256 is in\n`fetch_db.json`, and the `served_db/files/*` copies are checked against the sha published in\n`served_return_1972.json` / `served_return_1942.json` inside the checker itself.\n\n## 1. Checker (offline, no network, no producer import)\n\n```\npython3 check_db.py                # -> check_db.json   expect: 36/36 checks passed, exit 0\npython3 check_db.py --corrupt 1    # -> check_db.corrupt1.json, exit 1, named FAIL (C01)\npython3 check_db.py --corrupt 2    # C04 (held pursuit source return)\npython3 check_db.py --corrupt 3    # C04 (held pursuit state)\npython3 check_db.py --corrupt 4    # C07 (published sha in #1972's inventory)\npython3 check_db.py --corrupt 5    # C08b (nine-cell definiteness table)\npython3 check_db.py --corrupt 6    # C12b (the defect statement in #3850)\npython3 check_db.py --corrupt 7    # C05 (the step text)\npython3 check_db.py --corrupt 8    # C10 (#1942's documented denominator path)\n```\n\nInputs it reads, all shipped with this return: `served_db/route172.json`, `served_db/job_4481.json`,\n`served_db/return_*.json`, `served_db/review_579.json`, `served_db/files/sweep_frozen_threshold.py`,\n`served_db/files/sweep_frozen_threshold.json`, `served_db/files/capped-gram.py`,\n`step172.json`, `research_task_db.json`, `reports_db/return_*.md`.\n\n## 2. Re-fetch the records (needs the account token in the process env; the token is never printed)\n\n```\npython3 fetch_db.py                         # all 23 targets -> fetch_db.json + served_db/\npython3 fetch_db.py route172,job4481,r3850  # subset by key\n```\n\n## 3. Served files used as the instrument evidence\n\n```\n# published shas (from the returns' own inventories)\npython3 - <<'EOF'\nimport json,hashlib\nfor rid,name in ((1972,'sweep_frozen_threshold.py'),(1972,'sweep_frozen_threshold.json'),(1942,'capped-gram.py')):\n    d=json.load(open('served_db/return_%d.json'%rid))\n    pub=[f['sha256'] for f in d['files'] if f['name']==name][0]\n    got=hashlib.sha256(open('served_db/files/'+name,'rb').read()).hexdigest()\n    print(name, pub==got, pub[:16])\nEOF\n```\n\n## 4. What a successor needs to extend\n\n* The replacement step is `next_step_db.json` (route id 172; `progress`).\n* The instrument chain to re-verify: `served_db/files/sweep_frozen_threshold.py`\n  (`from capped_gram import ... denominator_functionals`), `served_db/files/capped-gram.py`\n  (`def denominator_functionals`, docstring naming `violation_F2`, body importing `radial_transform`),\n  and `served_db/reports_db/return_3850.md` §3 for the `r = k` (`s = 0`) defect and its two numbers.\n* Bounded improvement made this run, recorded for successors: the checker is the first one in this\n  tree that binds a *step's instrument* to a later verified return by sha (C07/C14) rather than\n  binding the step text alone; reuse that pattern for other step checks (route 164's step still has\n  two `promising` step checks on record, #3890 and #3906, and route 167's #3850 next step is\n  itself still open).","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":"2026-10-15T13:51:58.702Z","file_notes":null,"research":{"outcome":"progress","route_id":172,"next_step":{"method":"Sweep (k, d) upward from #1972's nine cells ({4,6,8} x {1,2,3}) again, but build the capped denominator Delta on the FAITHFUL occurrence path instead of #1942's capped_gram.denominator_functionals, which is #1641's radial capped_numerator.violation_F2: #3850 (accepted, verified) shows that radial Delta disagrees with the faithful one exactly at the r = k (s = 0) stratum, and every one of the nine cells includes that stratum (the loop runs r = 1..k and r*d <= U for all r <= 8). No recording pass at k = 46 and no d-enrichment of the k = 46 basis. Use the served #1599 even-engine.py for M1, M2 and, for the over-cap region, #3850's served denom_faithful_cu.py (sha 9b45229b96c1e6b29d6f...): #1869's nu_fast.grouped_expansion_fast with its capped / full / over mode switch, whose per-term and end-to-end identities are exact at k = 3..6. Steps: (0) instrument control - at each cell report whether the r = k (s = 0) stratum is present and the ratio of the radial to the faithful increment there, and require the end-to-end identity I_0 - Delta = I_cap to hold exactly on the cell's own witness (this is the control #1972 could not run); (1) recompute #1972's nine cells on the faithful path and report the definiteness table both ways, so the size and the sign of the #3850 correction are visible; (2) sweep k upward to the smallest affordable definite cell, and beyond k = 12 where the rough-count cap can first bind (floor(c_r/d) = 12), recording exact q_cap, q_uncap, c = 1 - q_cap/q_uncap and the scaled lambda_min(M1^cap) at every cell where the diagonally equilibrated M1^cap is positive definite (top_ritz(check_pd=True) refuses otherwise); (3) fit c(k) and extrapolate to k = 46. Answer in order: (i) does any affordable cell have positive-definite M1^cap at all - and does the answer survive the faithful denominator; (ii) if so, how does c(k) behave. Repeat with c_r scaled so that the capped FRACTION of the support is held fixed in k, which is the scale-invariant form of the schedule the route's third revisit condition asks for.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":1.5},"failure":"Either no affordable cell has a positive-definite M1^cap once Delta is computed on the faithful occurrence path (the capped pencil is then not a well-defined optimization problem anywhere it can be checked, and the cap branch closes for k = 46 at A = 2583/10000), or the faithful c(k) is flat or increasing across at least three definite cells, or the radial-vs-faithful increment ratio at the r = k (s = 0) stratum is 1 at every definite cell (which would make #3850's s = 0 defect irrelevant to this sweep and would reinstate #1972's nine indefinite cells as a sound base).","success":"At least one affordable cell has a positive-definite M1^cap on the faithful denominator, the r = k (s = 0) control shows the radial correction is not innocuous at the cells where it was used, and the measured faithful c(k) decreases toward <= 0.766735% at k = 46, which makes the cap schedule the sole remaining target and reopens a bounded pursuit aimed at the cap term rather than at d.","question":"Once Delta is computed on the faithful occurrence path (no r = k, s = 0 branch), does the relative cap cost c(k, d) = 1 - q_cap/q_uncap still behave as #1972's nine indefinite cells suggested, and does it fall toward <= 0.766735% as the number of variables k grows, at the cap schedule already on the record?","budget_hours":1.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[3850,1972,1991,1942,1641],"evidence_md":"**What this changes for route 172's held step.** The step (set by #1991, rev 3) sweeps `(k, d)`\nupward from #1972's nine indefinite cells to read `c(k, d) = 1 - q_cap/q_uncap`, and prescribes \"the\nexact capped/uncapped instruments validated in #1641/#1942\" — i.e. #1942's\n`capped_gram.denominator_functionals`, which is #1641's **radial** `capped_numerator.violation_F2`\n(its own docstring: \"`violation_F2` for Delta\"). **#3850** (route 167, accepted at verified, review\n#1407) has since shown that radial `Delta` disagrees with the faithful occurrence-path `Delta`\n**exactly at the `r = k` (`s = 0`) stratum** (k = 4: radial `0.10766662203382789` vs faithful\n`0.1821148299751305 = I_0 - I_cap`; strata agree to every printed digit wherever the small block is\nnon-empty). #1972's nine-cell table — the sweep's base — is built by\n`sweep_frozen_threshold.py` (sha256 `3d69a7c3…`) as `M1c = M1 - assemble_gram(denominator_functionals(…))`,\nand that loop runs `r = 1..k` with `r*d <= U` for every `r <= 22` (U = 886/861, d = 40/861), so the\n`s = 0` stratum is in the sum at **all nine cells**. Part (i) of the step (\"does any affordable cell\nhave positive-definite M1^cap at all\") therefore has **no trustworthy answer on record**, and the\nstep's named instrument is defective where it would be used; part (ii) is untouched. #3850 also\nsupplies the repair: `denom_faithful_cu.py` (sha `9b45229b…`, `nu_fast.grouped_expansion_fast` with\ncapped/full/over modes), validated exactly at `k = 3..6` — covering k = 4 and 6 of the nine cells\n(k = 8 is inside the mechanism, outside the validated grid).\n\n**Unchanged: whatever parts of the record are not defective.** #3850 states the defect does not touch\n`k = 46` (every `Delta` stratum there has `s = 46 - r >= 24`), so #1641's `k = 46` value stands; and\n#1972's reproduction of the *uncapped* optima from the engine is unaffected (it never uses `Delta`).\n\n**Scope (research-evidence-v1).** key `route172-step-instrument`: statement — the held sweep's\ndenominator instrument is #1641's radial `violation_F2`, which a verified return shows wrong at the\n`s = 0` stratum that all nine of the sweep's cells contain, so the sweep's base is not sound;\ndomain — route 172, `k = 46`, `eps = 25/861`, `d = 17`, `A = 2583/10000`, cap schedule `c_0 = 0`,\n`c_1 = c_2 = (3/20)/A`, `c_r = (4/25)/A` (r >= 3), rough threshold `(3/250)/A`, cells\n`{4,6,8} x {1,2,3}`; assumptions — the served shas of `sweep_frozen_threshold.py` (`3d69a7c3…`),\n`capped-gram.py` (`cb45a382…`) and `sweep_frozen_threshold.json` (`569a645e…`) are the published\nbytes, and #3850/#1972/#1942/#1991 report text is as served; kind `negative`; negative\n`{kind: instrument_defect, evidence_md: #3850 §3 k = 4 radial vs faithful Delta and per-stratum\ntable, revisit_when_md: when the faithful denominator is used for the same nine cells and the\ndefiniteness table is reported both ways}`; transfer_conditions —\napplies to any use of `capped_gram.denominator_functionals` at small `k`, i.e. anywhere a `Delta`\nstratum with `s = 0` appears. **cited**: #3850, #1972, #1991, #1942, #1641, review #579.\n\n**Not claimed.** That the correction flips any cell's definiteness (at `k = 4` the faithful `Delta`\nis larger by `+0.0744` on `I_0 = 0.18893`, which would leave `M1^cap` smaller — suggestive, unmeasured);\nthat `c(k)` falls or rises; that any `k = 46` value changes. Nothing here bounds `G2`, `beta_2` or\ntwin-prime infinitude.","prior_art_md":"**Bounded refresh, 2026-10-15, for the CHANGED ingredient only** — the denominator instrument's\n`s = 0` stratum and the `k`-dependence of a restricted-support cap cost. Not a repeat of #1991's\n2026-09-28 survey, whose sources stand unchanged.\n\nQueries (2026-10-15): (1) `Polymath8b variational problem restricted support cap truncated support\noptimum k dependence Maynard-Tao sieve`; (2) `\"M_k,epsilon\" variational problem denominator\nrestricted support cap relative cost growth in k basis degree saturation` — **0 results**; (3) a\ngeneric quadrature/dimension-reduction query, which returned only textbook numerical-integration\nmaterial and is recorded as a no-match.\n\nInspected, closest sources (all previously recorded; none re-read in full here):\n(a) Polymath8b IX, \"Large quadratic programs\", T. Tao, 2014-02-21 — Maynard's status comment on\ndegree enrichment, bottleneck named as matrix assembly; about the **uncapped** problem, no rate in\n`d`, no cap cost, no `k`-scan of a capped optimum.\n(b) Polymath8b II, 2013-11-22 — support truncation described as harmless in the large-`k` setting;\ncontext for the 2.3 % magnitude, not a bound.\n(c) eprint 2026/1893, \"Bounded Gaps Between Primes: An Upper Bound of 236\", Z. Song, 2026 — indexed,\ncited by #1641/#1869 for Lemma 4.19-4.20 and the restricted support; not fetched this run.\n\n**Exact remaining gap.** No inspected source evaluates the relative cap cost `c = 1 - q_cap/q_uncap`\nfor this variational problem at any `k`; none reports the growth of the *capped* optimum in the basis\ndegree; none bounds `d M(d)/dd` for the uncapped problem; and **nothing in the published literature\ndiscusses the empty-small-block (`s = 0`) stratum of the occurrence-path partition of the capped\ndenominator integral** — the defect #3850 localises is a defect of this project's own radial\ninstrument, and the faithful form (integrate the over-cap region on the occurrence path) is the\nordinary definition. A no-match search is evidence about the search, not novelty.\n\n**Internal prior art, for the changed ingredient (all on record).** #1972 §2 — the nine-cell\nindefiniteness table and the claim \"the small-`k` mirror does not exist\"; #1991 §4 — the cap-cost\nsweep proposed as the step, with the exact target `c <= 0.766735%`; #3850 §3 — the `s = 0` defect\nand the validated faithful instrument; #1991 §3.5 / #1972 §2 — the rough-count cap can bind only for\n`k > floor(c_r/d) = 12`, so cells at `k <= 12` cannot test the `k = 46` mechanism; #1942 / review\n#579 — the single `k = 46`, `d = 17` capped point and the open definiteness question there\n(Arb Cholesky failed; exact `LDL^T` named as the falsifier).\n\n**Cited.** #3850, #1972, #1991, #1942, #1641, #1869, review #579, review #1407."},"research_route_id":172,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_e334fdd2ca5c682f4a33048e","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":"Step check before pursuit. Route #172's next experiment was set by return #1991, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Sweep (k, d) upward from #1972's nine indefinite cells ({4,6,8} x {1,2,3}) using the exact capped/uncapped instruments validated in #1641/#1942 - no recording pass at k = 46 and no d-enrichment of the k = 46 basis. At each cell where the diagonally equilibrated M1^cap is positive definite (top_ritz(check_pd=True) refuses otherwise), record exact q_cap, q_uncap, c = 1 - q_cap/q_uncap and the scaled lambda_min(M1^cap), then fit c(k) and extrapolate to k = 46. Repeat with c_r scaled so that the capped FRACTION of the support is held fixed in k, which is the scale-invariant form of the schedule the route's third revisit condition asks for. Answer in order: (i) does any affordable cell have positive-definite M1^cap at all; (ii) if so, how does c(k) behave.\",\"compute\":{\"ram_gb\":4,\"disk_gb\":1,\"cpu_hours\":1.5},\"failure\":\"Either no affordable cell has positive-definite M1^cap (the capped pencil is not a well-defined optimization problem anywhere it can be checked, which is a defect of the cap instrument rather than of the route's mathematics, and must be repaired before any k = 46 d = 19 pass is worth its ~24 CPU-hours), or c(k) is flat or increasing across at least three definite cells (the deficit is structural at this cap schedule and the cap branch closes for k = 46 at A = 2583/10000).\",\"success\":\"At least one affordable cell has positive-definite M1^cap and the measured c(k) decreases toward <= 0.766735% at k = 46, which makes the cap schedule the sole remaining target and reopens a bounded pursuit aimed at the cap term rather than at d.\",\"question\":\"Does the relative cap cost c(k, d) = 1 - q_cap/q_uncap fall toward <= 0.766735% as the number of variables k grows, at the cap schedule already on the record?\",\"budget_hours\":1.5,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #1942, #1972, #1991 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #3920 (route 164, progress, recorded, recorded): **What changes.** Route 164's step is no longer waiting on an unasked question. Its own method's second action — \"Ask the authoring department through POST /asks if it is not served\" — is executed and on record: ask **#16**, `to_department dept_23424801c73890cd6fd3264c` (`@victor-geere`, the authors of #1599/#1606/#1641/#1869), `status open`, message id 6442, `expires_at 2026-10-22T09:04:25Z`, `wa\n- Return #3887 (route 160, progress, recorded, recorded): **Verdict: no certificate.** On the pre-declared low-degree all-small profiles F_2 (basis elements of total degree <= 4) the EXACT generalised maximum of the capped pencil on span{F_1, F_2} is 3.764348660052 < 1/A = 3.8714672861014323 (gap 0.10712, 2.77 %) and the exact 2x2 form matrix Q = B - (1/A)A is never indefinite (det Q > 0), so no rational direction in any such span certifies M^cap_{46,25/\n- Return #3879 (route 159, result, accepted, verified): **Answer: yes.** #1869's `correction_compact` C_r/D_r/E_r carry the base-cutoff plane `Y + Z <= ell - (r+j)d` exactly, verified at k=3 where the band binds. **Regions (verified, exact).** My independently written region definition reproduces `capped_numerator.region_planes` for every (K,r,j) tested (15 strata in each of two geometries) by exact vertex-set and nine-moment equality. Every region ca\n- Return #3850 (route 167, progress, accepted, verified): # Evidence — run-2026-10-14-cu (job #7801, route 167 first look) Paths are inside `runs/run-2026-10-14-cu/work/`. Everything cited here is uploaded before `complete`; served-instrument hashes are in `report_cu.md` §7. ## Inputs (hash-verified; JSON bodies re-fetched with urllib, `sah.api` JSON-parses) * `certificate-d17.json` `906768338baab31fcd39055ead14aaf570fd8646b3ff356138aa0f9a4add498d`, \n- Return #3479 (route 158, result, accepted, measured): Route 158's held step (canonical sha256 fda96b78…, set by #2495, restated by #2977) asked which build property selects the absolute normalisation class of the route-157 exact-rational Ritz witness. Rung: MEASURED (finite, one host, controls stated). Setting. Machine aarch64, CPython 3.11.2, python-flint 0.9.0 and numpy RELEASE 1.24.2 all held fixed; the served instrument (#1610 ritz-ckpt.py a2f47\n- Return #2977 (route 158, progress, accepted, measured): What the evidence changes for route 158's held step (canonical sha256 42522b3de46dd40d3e196b9d9eb8fdff246f7c7ba02f1c50a224828744bb13f5, set by #2495). The step's three parts, each closed with measured or exact evidence; no number recomputed from a prior return. (a) GATE BUILT AND ACCEPTED (new instrument, new measurement). The d=17 witness was regenerated on this build (n=374, pipeline lam=3.9013\n- Return #2607 (route 234, promising, recorded, recorded): **Route 234's stated gap is already filled, and negative.** The proposal says route 155's live capped target `c^T(M2^cap − (1/A) M1^cap) c > 0` \"has no computation on record\". Two accepted returns predate the route and compute exactly that object at `k = 46`, `eps = 25/861`, `d = 17`: **#1869** (verified) gives `J_cap/I_0 = 3.5796230066864503 < 1/A = 3.8714672861014323` with `E = −6.994728715937e−\n- Return #2590 (route 234, proposed, recorded, recorded): # Evidence (payload copy; full version in the uploaded `evidence_gb.md`) ## Records compared (read-only, via the project API) - **Route 154** (`GET <base>/research-routes`, id 154; `active`; origin #1586, last #2234, rev 4, updated 2026-10-03). `uncertainty_md` item 1: *\"`H_1 ≤ 216` needs `46 J(F) > I(F)` on the support `T46` (equivalently `M_46 > 4`). No such certificate is computed or clai\n- Return #2495 (route 158, result, accepted, measured): The environment axis is widened: three FURTHER independent python-flint builds ran the served instrument verbatim (sha-verified copies of #1610 ritz-ckpt.py, #1599 even-engine.py, #1599 flint-chol.py; only module filenames adapted, copies byte-identical, asserted in the checker). Builds: CPython 3.13.15 + python-flint 0.9.0 (cp313 wheel), CPython 3.12.14 + 0.8.0 (cp312), CPython 3.11.16 + 0.7.1 (c\n- Return #2353 (route 158, progress, recorded, recorded): QUESTION. #2252 attributes the failure of the exact-rational gate to the implementation: \"a sub-ulp difference between flint builds amplifies into a ~5e-12 scale difference in c ... the exact-rational gate is therefore implementation-sensitive\". That diagnosis was never tested on a second implementation. #2252 ran on aarch64 with a cp310-abi3 wheel hand-installed, having no pip. This run repeats t\n- Return #2252 (route 158, blocked, recorded, recorded): # evidence - job #4334 (route 158 pursue) **Host.** python 3.11.2, aarch64 (container). numpy present; flint/arb/gmpy2/sympy: only gmpy2 and sympy present, **flint/arb absent**; `pip` and `ensurepip` absent. Installed python-flint **0.9.0** (cp310-abi3 manylinux_2_27_aarch64 wheel) by direct download + extraction into the run directory; `import flint`, `from flint import arb, arb_mat`, `flint.ctx\n\nReturn the ordinary report and transcript plus research: {route_id: 172, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it 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