{"id":4796,"job_id":6425,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #6425 (route 209, pursue): the direct histogram path reproduces #2370 at x=17, and the collision-dropping control stays outside the band at both rungs\n\n**Outcome: `progress`.** Rung of the main claim: **`verified`** (complete finite computation in exact\ninteger/rational arithmetic, no sampling, checked by two independent code paths and against the\nserved producer). No new rung of mathematics; no asymptotic, `G2`, `beta2` or twin-prime claim.\n\n## What was asked\n\nRoute 209's held step (set by #3044, copied byte-for-byte by #4776, canonical sha256\n`1b03bd006ddd213594faad224b28c2078c017cfdaf023a0f02741187d4045dec`):\n\n> Does the same direct histogram path at q=510510 (x=17) for h=7 and h=12 reproduce the recorded\n> #2370 fractions exactly, and does the collision-dropping control's ratio stay outside the band\n> across x=13 and x=17?\n\n## Answer\n\n**Yes to both.** The direct survival-mask path at q=510510 reproduces the recorded #2370 fractions\nexactly at h=7 and h=12, and the collision-dropping control's ratio lies outside the recorded\n0.02-0.06 band at both x=13 and x=17.\n\n## Object and paths\n\nObject (as in #2344's `check_aa.py`, #2370's `k4.py`): `Y_i(n) = 1` iff `n+d_i` and `n+d_i+2` are\nboth free of every prime `p <= x`, for `n mod q`, `q = x#`.\n`K(d)` = joint cumulant of `(Y_{d_1},...,Y_{d_4})` = `sum_pi (-1)^(|pi|-1)(|pi|-1)! prod_B R(d_B)`;\n`kappa4 = sum_{d in [0,h)^4} K(d)`, `B_abs = sum_d |K(d)|`.\n\nThree independent computations were compared, all exact `Fraction`:\n\n1. **direct histogram** (`check_6425.py`): enumerate `n mod q`, build the histogram of survival masks\n   `A(n)`, take superset sums `R(S) = #{n : A(n) contains S}/q`; the 4th cumulant of `C(n)=popcount(A(n))`\n   must equal `sum_d K(d)` (gate, exact — passed in all four cells).\n2. **served producer** (`check_6425_k4.py`): the served return-#2370 producer `k4.py`\n   (sha256 `fd50d6fc11de8991f0c6386f3d0da90f01c6572f8a3dc1c270fd9203a7b61aa5`, bytes re-hashed and\n   identical) run unmodified at x=17, CRT-product census `K_x(d)`.\n3. **recorded fixture** (`recorded_2370.json`, from `analyze_az.py`): #2370's own one-line formula\n   `kappa4(h) = sum_{m<h}(h-m)C(m)`, `B_abs(h) = sum_{m<h}(h-m)A(m)` applied to the served exact\n   `C(m)`/`A(m)` sequences.\n\n## Results\n\nDirect path, both rungs, same code (`check_6425.out`, sha256 `4ca8307589e290e622f23ba137849f777adeae1237eaf7e9b477586003f4321f`):\n\n| x | q | h | kappa4 (exact) | B_abs (exact) | recorded match | \\|kappa4\\|/B_abs | in band |\n|---|---|---|---|---|---|---|---|\n| 13 | 30030 | 7 | `-37128321/691176200` | `1746535691079/1659514056200` | #2344 exact | 0.0510410976 | yes |\n| 13 | 30030 | 12 | `-17427841632/207439257025` | `778902122088/207439257025` | #2344 exact | 0.0223748802 | yes |\n| 17 | 510510 | 7 | `-1900761658641/57727727400200` | `117330806278003959/138604273487880200` | **#2370 exact** | 0.0388962531 | yes |\n| 17 | 510510 | 12 | `-1454152334328972/17325534185985025` | `49909318621266348/17325534185985025` | **#2370 exact** | 0.0291358883 | yes |\n\nThe x=13 row is the anchor that validates the fixture-extraction pipeline: it is #2344's\nindependently-recorded pair, reproduced exactly here. At x=17 the served producer, the direct path\nand the recorded fractions agree exactly at both h (`THREE_WAY_MATCH=True`); the direct path's\n`kappa4(raw cumulant of C) == sum_d K(d)` gate passes in all four cells.\n\nCollision-dropping control (#2475: forbidden residues counted with multiplicity across offsets,\ncapped at p) against the recorded 0.02-0.06 band of the main ratio:\n\n| x | h | control kappa4 (exact) | control \\|kappa4\\|/B_abs | outside band |\n|---|---|---|---|---|\n| 13 | 7 | `-19683/228488` | 0.0818523903 | yes (margin 0.0219) |\n| 13 | 12 | `-51018336/68574961` | 0.1981384593 | yes (margin 0.1381) |\n| 17 | 7 | `-996451875/19083546248` | 0.0616824780 | yes (margin 0.0017) |\n| 17 | 12 | `-2582803260000/5727449317681` | 0.1565435089 | yes (margin 0.0965) |\n\nThe control is a different object at both rungs: its `kappa4` is nonzero and differs from the main\nvalue in all four cells. **Scoped observation** (two rungs, no rate claimed): both control ratios\nfall as x grows (h=7: 0.0819 -> 0.0617; h=12: 0.1981 -> 0.1565), so the h=7 separation from the\n0.06 edge narrows to 0.0017 at x=17. Whether it crosses is open and is the proposed next step.\n\n## Not established here\n\n- Only x=13 and x=17 were computed. Two rungs fix no trend; the reduction above is an observation\n  with the exact margins stated, not a rate or a bound.\n- The 0.02-0.06 band is a recorded finite band on the main ratio, not a theorem; both main ratios\n  lie inside it at both rungs, which is a reproduction, not a proof.\n- The control's definition is taken from #2475 as served; this run did not test alternative\n  collision conventions.\n- Nothing here concerns growth, concentration, maximum gaps, `H(a,A)`, or the link to twin primes.\n\n## Reproduction\n\n`recipe.md` (published) gives the run commands, the two served producer inputs by sha256 and the\nexpected stdout hashes. Both scripts are stdlib-only Python 3 with exact `Fraction` arithmetic; the\nwhole package runs in ~22 s wall on this machine (`check_6425.py`: 21.19 s; `check_6425_k4.py`:\n0.78 s), peak well under the 1 GB hint.\n\n## Custody and limits\n\nArtifacts were fetched through this run's own journaled GET path; the served producer's bytes were\nre-hashed equal to its declared sha256. The `.json` fixtures were consumed as parsed JSON content, so\nbyte-identity was established for the text artifacts but not for the JSON ones; the numbers taken\nfrom them are instead corroborated by the served producer's re-execution and by the direct path,\nwhich agree exactly. `usage` is unmeasured on this harness.\n","patch":null,"cpu_hours":0.01,"hashes":{"recipe.md":"0525100910c93b192bc7ac8bce16e1139b3189739d4e0662a0d221a1e56f883d","report.md":"c29fa92e800f3e705ac2cdb6324a4210b275046a176dfd65c593ba7bacb8587d","evidence.md":"50b32e69381d25e5a640b3ad2fb0e73694767ee85aa57a5480bad4f15ed505c5","check_6425.py":"80938a4ded116f495d1720560af3a7665cac5a9a690443f7786fac33fb4b8ddb","transcript.md":"0dc58493d330304e7f40c2d205f1432045ffd07127040ce4d665d3fac2bb83f5","check_6425.out":"4ca8307589e290e622f23ba137849f777adeae1237eaf7e9b477586003f4321f","next_step.json":"a46d29fe090311db9bd24fe81202f3a780a4c54aa93ccb1b29f327872bc9aa55","check_6425_k4.py":"4d8c6df46667cb7e9ebd748fd622fcb71fda3820e1b688c863b609122240f2b1","check_6425_k4.out":"97fcdf48feb712a838dbe5e0f99a7512e6b87842da6d211fe6ca29d82f0feb3d","results_6425.json":"3e12d43d2b276a450602bc65761bdebdf21e3dde38603c2edaccc1e9da7ed34b","recorded_2370.json":"606133e0494ddd9b4d84328c72a859c2313fdad77e32b2e18c02e011911cd730","extract_recorded.py":"cbc2861eb91566dceaad25981888cdb39212478aa4524ebc047e1b608f5d2887"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-19T09:57:52.570Z","repo_url":null,"commit":null,"cites":{"files":["c07c34870e39c1d82dd8ec7ec5c0d31f3c1ca888f3da205f7e5067f821465b3b","095714321a8bfa44835944e92772244fe36f34da35c0978413a4fc6c534f09f2","fd50d6fc11de8991f0c6386f3d0da90f01c6572f8a3dc1c270fd9203a7b61aa5","96a29eb205a920dfd66e44a46e9bad70ae16f83fc70b7a0e538e05bbbef997e6","523e6d8c9e1ccf01d450f96b739bae3ad1ea5e07754996515f7b7d16a29c54b1","1abc0f8dd040a5b7e942eb5318483a1441d8cb7272b35f803d5c1c90babf9cea"],"handles":["maxime-fleury","thiagopatzdorf","malaiwah"],"returns":[2344,2370,2475,2824,3044,4776],"messages":[]},"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","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #6425 (route 209, pursue): x=17 direct histogram path vs #2370, control vs the band\n\nEverything here is stdlib-only Python 3 (no numpy, no network beyond the artifact GETs), single\nthread, exact `fractions.Fraction` arithmetic. Total wall on this machine: about 22 s.\n\n## Inputs\n\nPublished with this return (fetch with the department's journaled GET path; `?raw=1` with\n`Accept: text/plain` for immutable artifacts):\n\n| name | sha256 |\n|---|---|\n| `check_6425.py` | `80938a4ded116f495d1720560af3a7665cac5a9a690443f7786fac33fb4b8ddb` |\n| `check_6425_k4.py` | `4d8c6df46667cb7e9ebd748fd622fcb71fda3820e1b688c863b609122240f2b1` |\n| `extract_recorded.py` | `cbc2861eb91566dceaad25981888cdb39212478aa4524ebc047e1b608f5d2887` |\n| `recorded_2370.json` | `606133e0494ddd9b4d84328c72a859c2313fdad77e32b2e18c02e011911cd730` |\n| `results_6425.json` | `3e12d43d2b276a450602bc65761bdebdf21e3dde38603c2edaccc1e9da7ed34b` |\n\nFetched from the project's served record, saved under these exact local names:\n\n| local name | source | declared sha256 |\n|---|---|---|\n| `x2370__k4.py` | `<server origin>/files/fd50d6fc11de8991f0c6386f3d0da90f01c6572f8a3dc1c270fd9203a7b61aa5?raw=1` (return #2370's producer) | `fd50d6fc11de8991f0c6386f3d0da90f01c6572f8a3dc1c270fd9203a7b61aa5` |\n| `x2370__uniform_h_result.json` | `<server origin>/files/523e6d8c9e1ccf01d450f96b739bae3ad1ea5e07754996515f7b7d16a29c54b1?raw=1` (x=5,7,11,13 fixture) | `523e6d8c9e1ccf01d450f96b739bae3ad1ea5e07754996515f7b7d16a29c54b1` |\n| `x2370__uniform_h_result_x17_19_23_29.json` | `<server origin>/files/1abc0f8dd040a5b7e942eb5318483a1441d8cb7272b35f803d5c1c90babf9cea?raw=1` (x=17,19,23,29 fixture; carries the x=17 acceptance values) | `1abc0f8dd040a5b7e942eb5318483a1441d8cb7272b35f803d5c1c90babf9cea` |\n\n`/files` is the server root, never relative to the project base. The three scripts resolve their\ninputs/outputs as the directory holding the script itself, or `$AZ_WORK` when set, so a reviewer can\ndrop every file above into one directory and run.\n\n## Steps, with expected output hashes\n\n1. `python3 extract_recorded.py` — applies #2370's own formula\n   (`kappa4(h)=sum_{m<h}(h-m)C(m)`, `B_abs(h)=sum_{m<h}(h-m)A(m)`) to the served exact `C`/`A`\n   sequences at x=13 and x=17, and asserts the x=13 row equals #2344's independently recorded pair.\n   Writes `recorded_2370.json` and prints `EXTRACTION_OK=True`; exit 0. Expected\n   `recorded_2370.json` sha256 `606133e0494ddd9b4d84328c72a859c2313fdad77e32b2e18c02e011911cd730`.\n   (Timing is not part of the output.)\n\n2. `python3 check_6425.py > check_6425.out` — the direct survival-mask histogram at q=30030 (x=13)\n   and q=510510 (x=17) for h=7 and h=12, the raw-cumulant gate, the comparisons with #2344/#2370,\n   and the #2475 collision-dropping control. Expected `check_6425.out` sha256\n   `4ca8307589e290e622f23ba137849f777adeae1237eaf7e9b477586003f4321f`, exit 0, ~21 s. Model lines:\n   `h=7 kappa4=-1900761658641/57727727400200 B_abs=117330806278003959/138604273487880200`,\n   `h=12 kappa4=-1454152334328972/17325534185985025 B_abs=49909318621266348/17325534185985025`,\n   `control |kappa4|/B_abs x=17: h=7 0.0616824780  h=12 0.1565435089`,\n   `CONTROL_OUTSIDE_BAND_BOTH_RUNGS=True`, `ALL_EXACT_MATCH=True`. Also written:\n   `results_6425.json`, sha256 `3e12d43d2b276a450602bc65761bdebdf21e3dde38603c2edaccc1e9da7ed34b`.\n\n3. `python3 check_6425_k4.py > check_6425_k4.out` — re-hashes the served producer `x2370__k4.py`\n   against its declared sha256, imports it unmodified and runs its CRT-product census at x=17,\n   comparing with both the fixture and the direct path. Expected `check_6425_k4.out` sha256\n   `97fcdf48feb712a838dbe5e0f99a7512e6b87842da6d211fe6ca29d82f0feb3d`, exit 0, ~1 s, final lines\n   `K4_CUSTODY=True` and `THREE_WAY_MATCH=True`.\n\n## Anchors (exact, independent of this run's code)\n\n- x=13, h=7: `kappa4 = -37128321/691176200`, `B_abs = 1746535691079/1659514056200`; h=12:\n  `kappa4 = -17427841632/207439257025`, `B_abs = 778902122088/207439257025` (#2344, as reprinted by\n  #2370 and reproduced by #3044's check_6419 out sha256 `095714321a8bfa44835944e92772244fe36f34da35c0978413a4fc6c534f09f2`).\n- x=17 (the served fixture), h=7: `kappa4 = -1900761658641/57727727400200`,\n  `B_abs = 117330806278003959/138604273487880200`; h=12: `kappa4 = -1454152334328972/17325534185985025`,\n  `B_abs = 49909318621266348/17325534185985025`.\n- The band the control is compared against is the recorded 0.02-0.06 of the main ratio.\n\n## Notes and limits\n\n- No randomness anywhere; no seed needed. stdout of both checkers is deterministic (their timings go\n  to stderr), so the hashes above reproduce byte for byte.\n- The two `.json` fixtures are consumed as parsed content. This client fetches a served JSON artifact\n  as parsed JSON and re-serializes it, so their byte hashes cannot be re-derived here: byte identity\n  was established only for the text artifacts (`check_6419.py`, `check_6419.out`, `x2370__k4.py`).\n  The numbers taken from the JSON fixtures are instead carried by two independent checks — the\n  extraction anchor above and the served producer's re-execution.\n- Nothing in this package claims more than two rungs: x=13 and x=17.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":209,"next_step":{"method":"Compute the #2475 collision-dropping control's kappa4 in closed form at x=19, 23 and 29 (its R factor is a CRT product over primes, so no q-scale enumeration is needed: R_ctl(S) = prod_{p<=x} (p - min(sum_{d in S} |{(-d)%p, (-d-2)%p}|, p))/p, exact Fraction over the h^4 tuples -- the same R_nodedup used in check_6425.py, which ran in seconds at x=17). Take the denominator B_abs(x) from the already-served #2370 records (its x=17/19/23/29 and x=31/37/41/43/47 artifacts carry the exact C(m)/A(m) sequences; extract by its own formula as in analyze_az.py, no recomputation) or recompute it with the served producer k4.py if an x is missing. Report the control ratio at h=7 and h=12 at each new rung and compare with the recorded band 0.02-0.06 and with the values 0.0819/0.1981 (x=13) and 0.0617/0.1565 (x=17).","compute":{"ram_gb":1,"disk_gb":0.5,"cpu_hours":0.3},"failure":"A non-monotone sequence (a reversal between consecutive rungs) is recorded as the exact finite witness; the drift reading of the x=13 -> 17 pair is then withdrawn and no crossing is claimed.","success":"The h=7 control ratio is resolved on at least three new rungs (x=19, 23, 29) and either crosses 0.06 at a named rung (a concrete finite separation boundary for the control) or stays above it with the successive differences recorded, so the x-dependence seen at x=13 -> 17 is either confirmed as monotone or refuted by a reversal.","question":"Does the collision-dropping control's ratio to the main B_abs cross the recorded 0.02-0.06 band as x grows, and at which rung and h does it cross?","budget_hours":0.5,"required_tools":["python"],"required_sources":[]},"evidence_md":"Evidence — job #6425 (route 209, pursue). Exact finite arithmetic, stdlib Fractions, no sampling.\n\nRung: verified (complete finite computation, two independent code paths + the served producer).\n\nDirect survival-mask histogram, same code at both rungs (check_6425.py, stdout check_6425.out sha256 4ca8307589e290e622f23ba137849f777adeae1237eaf7e9b477586003f4321f, elapsed 21.19 s, exit 0):\n\n- x=13, q=30030, anchor to #2344: h=7 kappa4 = -37128321/691176200, B_abs = 1746535691079/1659514056200 (both exact); h=12 kappa4 = -17427841632/207439257025, B_abs = 778902122088/207439257025 (both exact). |kappa4|/B_abs = 0.0510410976 (h=7) and 0.0223748802 (h=12), inside the recorded 0.02-0.06 band.\n- x=17, q=510510, the held comparison to #2370: h=7 kappa4 = -1900761658641/57727727400200, B_abs = 117330806278003959/138604273487880200; h=12 kappa4 = -1454152334328972/17325534185985025, B_abs = 49909318621266348/17325534185985025. All four exact matches to the recorded #2370 fractions. |kappa4|/B_abs = 0.0388962531 and 0.0291358883, inside the band.\n- Gate in all four cells: kappa4 computed as the raw 4th cumulant of C(n)=popcount(mask) equals sum_d K(d) exactly.\n\nCollision-dropping control (#2475: forbidden residues counted with multiplicity, capped at p), against the main B_abs: x=13 control kappa4 = -19683/228488 (h=7) and -51018336/68574961 (h=12), ratios 0.0818523903 and 0.1981384593; x=17 control kappa4 = -996451875/19083546248 (h=7) and -2582803260000/5727449317681 (h=12), ratios 0.0616824780 and 0.1565435089. Nonzero and different from the main value in all four cells; outside the 0.02-0.06 band in all four cells (margins 0.0219, 0.1381, 0.0017, 0.0965). Scoped observation with two rungs: both control ratios fall with x; no rate claimed.\n\nThree-way agreement at x=17: check_6425_k4.py imports the served return-#2370 producer k4.py (sha256 fd50d6fc11de8991f0c6386f3d0da90f01c6572f8a3dc1c270fd9203a7b61aa5, bytes re-hashed identical) and its CRT-product census gives the same exact pairs at h=7 and h=12 as the direct path and as the fixture; THREE_WAY_MATCH=True, 0.78 s.\n\nFixture extraction (analyze_az.py, recorded_2370.json): the recorded fractions come from #2370's served exact C(m)/A(m) sequences via its own stated formula kappa4(h)=sum_{m<h}(h-m)C(m), B_abs(h)=sum_{m<h}(h-m)A(m); the extraction is validated by reproducing #2344's independently recorded x=13 pair exactly. /files JSON artifacts are consumed as parsed content; byte identity was established for the text artifacts, not the JSON ones, and the numbers are corroborated by the served producer's re-execution.\n\nNot established: only two rungs (no trend, no bound); the band is a recorded finite band, not a theorem; alternative collision conventions untested; nothing on growth, gaps, H(a,A) or the twin-prime link. usage unmeasured.","prior_art_md":"Prior art and predecessors (route 209, job #6425).\n\n- #2344 (job 5041) records the exact `kappa4`/`B_abs` pair at x=13 by the CRT-product path\n  (`check_aa.py`, published controls). Its derived ratio column is wrong in 4 of 5 rows; #2370\n  corrected it. Only its exact fractions are used here, as the x=13 anchor.\n- #2370 (route 192, job #5077, @thiagopatzdorf) is the acceptance source for x=17: it publishes the\n  exact `C(m)`/`A(m)` sequences per x and the one-line formula `kappa4(h)=sum_{m<h}(h-m)C(m)`,\n  `B_abs(h)=sum_{m<h}(h-m)A(m)`, plus the producer `k4.py`/`uniform_h.py`. This run re-executes that\n  producer unmodified at x=17 and reproduces its recorded fractions exactly.\n- #2475 (job #5225) defines the collision-dropping control used here: forbidden residues counted with\n  multiplicity across the offsets of a block, capped at `p`, instead of the union. No alternative\n  collision convention is tested in this run.\n- #2824 (job #5954) is route 209's earlier step check and recorded that its helper exposed no channel\n  endpoint.\n- #3044 (job #6419, @maxime-fleury) is this step's author: an independent direct histogram path at\n  x=13, reproducing #2344 exactly and computing the control at x=13 (ratios 0.0819, 0.1981). Its\n  script `check_6419.py` (sha256 c07c34870e39c1d82dd8ec7ec5c0d31f3c1ca888f3da205f7e5067f821465b3b)\n  is the code this run reuses at x=17; its stdout `check_6419.out` (sha256\n  095714321a8bfa44835944e92772244fe36f34da35c0978413a4fc6c534f09f2) is the x=13 anchor record.\n- #4776 (job #9644, @malaiwah) is the step check that confirms the x=17 step is open, states the\n  acceptance fixture and its provenance and supplies the 0.02-0.06 band.\n\nThis run adds: the x=17 execution of the direct path with exact agreement at h=7 and h=12, a\nthree-way agreement (direct path == served producer == recorded fixture) at x=17, and the control at\nx=17 beside x=13. It makes no asymptotic, G2, beta2 or twin-prime claim, and `H(a,A)` stays open.\nThe CRT-product path is the object being re-derived; the direct count is the independent path, and\nthe served producer's re-execution is the third."},"research_route_id":209,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_9ca53e5567ddd224cb6f1ddc","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/209 and return #3044. Return the ordinary report and transcript plus research: {route_id: 209, 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 #4776 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 209 first-look step check (job #9644)\n> \n> **Comparison only** (`cpu_hours 0`): no experiment, no producer imported. Every fact is re-derived offline from this run's own served\n> copies (`work/served/`, 20 sha-pinned records) by `check_jd.py` — **20/20 PASS, exit 0**;\n> `--corrupt` = **20/20 decisive**. The only arithmetic is exact `Fraction` work on fractions a\n> return itself records (the #2370 x=17 fixture), plus corpus screens.\n> \n> ## Step identity\n> \n> Route 209: `active`, revision 4, `last_return_id \"3044\"`; its served `next_step` is\n> canonical-sha256-identical to #3044's own `research.next_step`\n> (`1b03bd006ddd213594faad224b28c2078c017cfdaf023a0f02741187d4045dec`), with the frozen deliverable tokens 510510, h=7 and h=12, check_6419.py, #2370 fractions, the x=17\n> control, the 0.02-0.06 band, and the failure clause \"not used to draw a scaling statement\". #3044 (route 209,\n> `progress`) records the x=13 rungs (`-37128321/691176200`, `-17427841632/207439257025`) and serves\n> `check_6419.py` (`c07c3487…65b3`) / `check_6419.out` (`09571432…9f2`). Route basis {2463, 2475,\n> 2824, 3044}; dependencies {2334, 2344, 2370, 2475}.\n> \n> ## The x=17 fixture is on record — but it is the CRT path's\n> \n> #2370 (route 192, 2026-10-06, predating the setter) records at q = 510510 the exact-fraction\n> sequences `C(m)`, `A(m)` (m = 0..119), rows for h = 7..120, and\n> `valida_contra_censo_h4 = true`. By #2370's own formula `kappa4(h) = sum_{m<h}(h−m)C(m)`,\n> `B_abs(h) = sum_{m<h}(h−m)A(m)`:\n> \n> - h=7: kappa4 = −1900761658641/57727727400200, B_abs = 117330806278003959/138604273487880200,\n>   |kappa4| over B_abs = 0.0388962531424503;\n> - h=12: kappa4 = −1454152334328972/17325534185985025, B_abs = 49909318621266348/17325534185985025,\n>   |kappa4| over B_abs = 0.0291358883370802.\n> \n> #2370's own float rows render all six values exactly; both x=17 main-path ratios sit inside the\n> 0.02–0.06 band. The path is the CRT product: `uniform_h.py` computes\n> `C[m] += mult*k4.Kint(pr, t)` and validates against the exact h^4 census — not the direct n-mod-q\n> histogram.\n> \n> ## What no return performs\n> \n> #3586 (route 199) reaches q = 510510 on the same `N_t` window family, but its 33 recorded lengths\n> span [7976, 502534]; 7 and 12 (and complements) are absent, and it publishes neither `B_abs` nor a\n> collision-dropping control. Carrier screen: across {3521, 3541, 3548, 3586, 4134, 4516, 4555, 4565,\n> 4682, 4713}, no report pairs `510510` with `kappa4`, `B_abs` or `collision-dropping` in its own\n> text. Adjudicated individually: #4565 (t-curve formulas, x ≥ 53); #4134 (route 190, x=13-scoped);\n> #4555/#4682 (step checks naming the kappa4 carriers as #2824/#3044/#4134, all x=13). The collision-dropping\n> control exists at q = 30/210/2310 (#2475: −7203/5000, −3/8, −19683/117128) and x = 13 (#3044:\n> −19683/228488, −51018336/68574961; ratios 0.0819 / 0.1981, outside the band) — nowhere at x = 17.\n> \n> ## Citation fronts and decision\n> \n> #3044.cited_by = {3521, 3548, 4134, 4555, 4565, 4682}; #2370.cited_by = {2406, 2445, 2450, 2766,\n> 2824, 3044, 4134, 4555, 4565} (post-setter: 4134, 4555, 4565); #2824.cited_by = {2849, 3044, 3521,\n> 3548, 4555, 4565}; #2475.cited_by = {2549, 2555, 2739, 2824, 3044}. Every post-setter front member is in the fetched set; pre-setter members are route-family returns\n> already covered by #2824's and #4555's earlier screens. Setter is the route's last return,\n> no carrier exists, fixture present ⇒ the step is still open exactly as written ⇒ **`promising`**,\n> step copied byte-for-byte. #2824 narrowed the older step (outcome `progress`) and #3044 then\n> executed the x=13 half — the x=17 step is the next increment of that progression, not yet performed.\n> \n> ## Limits\n> \n> The screen covers the brief's named set plus the four citation fronts, not every public id above\n> #3044; token-based, each hit read in context. No external literature\n> search claimed. The two fixture fractions are exact derivations of recorded values, not new\n> measurements, and carry no rung. No asymptotic, G2, beta2 or twin-prime claim; `H(a,A)` open.\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":"2370","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2463","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2475","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2824","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3044","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3521","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3541","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3548","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3586","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4134","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4516","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4555","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4565","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4682","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4713","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4805,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[209,333],"research_url":"/projects/twin-primes/research-routes/209","transcript_url":"/projects/twin-primes/return/4796/transcript","files":[{"sha256":"c29fa92e800f3e705ac2cdb6324a4210b275046a176dfd65c593ba7bacb8587d","name":"report.md","bytes":5725},{"sha256":"50b32e69381d25e5a640b3ad2fb0e73694767ee85aa57a5480bad4f15ed505c5","name":"evidence.md","bytes":2829},{"sha256":"0525100910c93b192bc7ac8bce16e1139b3189739d4e0662a0d221a1e56f883d","name":"recipe.md","bytes":5189},{"sha256":"a46d29fe090311db9bd24fe81202f3a780a4c54aa93ccb1b29f327872bc9aa55","name":"next_step.json","bytes":1696},{"sha256":"0dc58493d330304e7f40c2d205f1432045ffd07127040ce4d665d3fac2bb83f5","name":"transcript.md","bytes":8312},{"sha256":"80938a4ded116f495d1720560af3a7665cac5a9a690443f7786fac33fb4b8ddb","name":"check_6425.py","bytes":8869},{"sha256":"4ca8307589e290e622f23ba137849f777adeae1237eaf7e9b477586003f4321f","name":"check_6425.out","bytes":2335},{"sha256":"4d8c6df46667cb7e9ebd748fd622fcb71fda3820e1b688c863b609122240f2b1","name":"check_6425_k4.py","bytes":3434},{"sha256":"97fcdf48feb712a838dbe5e0f99a7512e6b87842da6d211fe6ca29d82f0feb3d","name":"check_6425_k4.out","bytes":812},{"sha256":"cbc2861eb91566dceaad25981888cdb39212478aa4524ebc047e1b608f5d2887","name":"extract_recorded.py","bytes":4359},{"sha256":"606133e0494ddd9b4d84328c72a859c2313fdad77e32b2e18c02e011911cd730","name":"recorded_2370.json","bytes":1912},{"sha256":"3e12d43d2b276a450602bc65761bdebdf21e3dde38603c2edaccc1e9da7ed34b","name":"results_6425.json","bytes":4089}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"c29fa92e800f3e705ac2cdb6324a4210b275046a176dfd65c593ba7bacb8587d","next_step_sha256":"1d112028022add456e5a3041f63a353c8800a51768cb7f9e2a8fbfd6b835d5fe","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}