{"id":1994,"job_id":4330,"problem_id":1,"lane_id":2,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4330 (explore, first look, route 170) — the pairing-multiplicity joint law is NOT the product of its marginals, but the extremal covering is near-tight\n\nRun `t5-2026-09-27` · launch `t5-dc53f901257bbb8a` · attempt `7a180d180e843328914091f9c7b80a20` · job 4330 ·\nsession `a112f6eb644e90cfe65ccf27` · type `explore`, stage `first_look`, lane adversarial,\nguidance `research-2026-09-27.2` (`framework-e6ecee8d58b8`). All arithmetic below ran locally\nunder this run's own Windows job object; files and hashes are attached.\n\n## 1. What was asked\n\nRoute 170's central uncertainty is one sentence: *the pairing multiplicities `mu_q` (how many run\npositions a single killer takes) are not already determined by their marginals — if they are, the\nroute collapses back into the closed density class.* Its recorded next experiment is exact:\nenumerate every killed run at the four smallest corridor rows, record `(mu_q)_{q in Q}`, and compare\nthe empirical joint distribution against the product of the marginals; extend to `s = 13` if the\nfirst four are inconclusive. That experiment is what this return does, at **five** rows.\n\n## 2. Definitions used (quoted, not invented)\n\n`T_P = { r in [0,P) : gcd(r,P) = gcd(r+2,P) = 1 }` (the slots); `q` kills `r` iff `q | r` or\n`q | r+2`; the period is `M = P * prod(Q)`; a **killed run** is a maximal stretch of consecutive\nslots of `T_P` (consecutive in the slot order of `[0,M)`) with every slot killed by some `q in Q`;\n`K*(P,Q)` is the longest such run. Corridor row `s`: `P = prod` of the primes `< s`,\n`Q = { prime q : s < q <= 2s }`. This is route-023 / return #1936's own object.\n\n## 3. The engine gate — rung: verified\n\nBoth the K* engine (`killed_runs` in `mu_joint.py`) and the mu census are the object the record\nuses, because the engine reproduces **all six** K* values on record:\n\n| P | Q | K* here | recorded | source |\n|---|---|---|---|---|\n| 210 | {11,13} | 3 | 3 | `lib/covering` test, #609 control |\n| 210 | {11} | 1 | 1 | `lib/covering` test |\n| 210 | {11,13,17} | 5 | 5 | route-026 table |\n| 210 | {11,13,17,19} | 8 | 8 | route-026 table (chain 1->3->5->8) |\n| 2310 | {13,17,19} | 6 | 6 | route-026 table |\n| 30 | {7,11,13} | 6 | 6 | multi-kill control (`6 > |Q| = 3`) |\n\n`python3 test_mu_joint.py` re-runs this and the other checks: **7 checks, 7 pass, exit 0.**\n\n## 4. The measured answer, part 1 — the joint law is not the product of the marginals\n\nOver every maximal killed run in one period, at five corridor rows:\n\n| s | P | Q | |Q| | period M | K* | maximal runs | TV(joint, product of marginals) |\n|---|---|---|---|---|---|---|---|\n| 7 | 30 | {11,13} | 2 | 4,290 | 3 | 108 | **0.2682** |\n| 9 | 210 | {11,13,17} | 3 | 510,510 | 5 | 9,990 | **0.2132** |\n| 10 | 210 | {11,13,17,19} | 4 | 9,699,690 | 8 | 194,400 | **0.1961** |\n| 11 | 210 | {13,17,19} | 3 | 881,790 | 5 | 15,750 | **0.2401** |\n| 13 | 2310 | {17,19,23} | 3 | 17,160,990 | 6 | 220,546 | **0.2795** |\n\nSo route 170's stated failure branch — *\"the joint distribution equals the product of marginals at\nevery tested row\"* — **does not fire**, at any of the five rows, and it does not fire within the\nextreme-length stratum either (`s=7` L=3: 0.500; `s=9` L=5: 0.372; `s=10` L=7: 0.236; `s=11` L=4:\n0.564). Rung: **measured** (finite exact enumeration; the object is gated in §3).\n\n## 5. The measured answer, part 2 — but the dependence is *small* once covering is removed, and the extremal covering is near-tight\n\nA difference from the product of marginals is not by itself news, because the covering condition\n`sum_q mu_q >= L` alone forces a deterministic dependence (the product law puts positive mass on\nvectors no run can have). Stratifying by the run length `L` and renormalising the product of the\n**conditional** marginals onto `{sum_q mu_q >= L}` is the honest null \"marginals + covering\":\n\n| s | L=1 | L=2 | L=3 | L=4 | L=5 | L=6 | L=7 | L=8 |\n|---|---|---|---|---|---|---|---|---|\n| 7 | .364 -> .253 | .000 -> .000 | **.500 -> .333** | | | | | |\n| 9 | .340 -> .235 | .337 -> .252 | .309 -> .209 | .360 -> .228 | **.372 -> .267** | | | |\n| 10 | .306 -> .211 | .281 -> .207 | .289 -> .199 | .354 -> .275 | .382 -> .321 | .312 -> .242 | .236 -> .194 | **.000 -> .000** |\n| 11 | .383 -> .262 | .380 -> .283 | .332 -> .216 | .564 -> .442 | **.260 -> .202** | | | |\n| 13 | .426 -> .289 | .426 -> .311 | .379 -> .255 | .416 -> .305 | .268 -> .200 | **.000 -> .000** | | |\n\n(`TV(joint, product of marginals) -> TV(joint, marginals conditioned on covering)`, bold = the\nextreme length `L = K*`.)\n\nTwo facts, both measured:\n\n1. **Covering accounts for only part of it.** Conditioning on `L` and on the covering floor removes\n   roughly 15–35 % of the distance; a residual of **0.19–0.44** survives in every stratum with a\n   large sample, and at the big strata (n = 10^4–10^5) that residual is far above sampling noise.\n2. **The extremal covering is near-tight.** At `L = K*` the incidence excess `sum_q mu_q - L`\n   (i.e. the number of doubly-killed positions in the run) is at most **1** at every row, and 0 at\n   `s = 7` and `s = 13`:\n\n   | s | K* | extreme-length mu vectors | max(sum mu - K*) |\n   |---|---|---|---|\n   | 7 | 3 | (1,2) x2, (2,1) x2 | **0** |\n   | 9 | 5 | (1,2,2) x6, (2,1,2) x14, (2,2,1) x4, (2,2,2) x4 | **1** |\n   | 10 | 8 | (2,2,2,2) x10, (3,2,2,2) x12 | **1** |\n   | 11 | 5 | (1,2,2) x4, (2,1,2) x6, (2,2,1) x6, (2,2,2) x8 | **1** |\n   | 13 | 6 | (2,2,2) x16 | **0** |\n\n   That is a named structural cause in the route's own sense (a family of vectors the marginals\n   cannot see): **a longest covering run partitions almost exactly among its killers** — the primes\n   share the run out in near-equal, almost disjoint shares (`mu_q ~ K*/|Q|`), and double-kills are\n   essentially absent at the extremal length.\n\n## 6. What this does and does not establish\n\n- It establishes (measured) that the joint law of `(mu_q)` is **not** the product of its marginals\n  on the corridor, at five rows and in every length stratum — so route 170's weakest assumption is\n  **not refuted at first look**, and the route does not collapse into the closed density class by its\n  own stated criterion.\n- It establishes (measured) the cause above and its limits.\n- It does **not** establish that the cause is *useable*: a bound on `K*` needs a ceiling on how much\n  the killers can jointly take, and `sum_q mu_q <= K* + 1` is a statement about the *covered* runs,\n  which is exactly the region a certificate must reason about. Whether the near-tightness can be\n  turned into an incidence ceiling below the density bound's `K* * G * sum 1/q` is **open and is the\n  proposed next step**.\n- The extremal strata are thin samples: `s=7` L=3 rests on **4** runs, `s=13` L=6 on **16**;\n  `s=9` L=5 on 28, `s=10` L=8 on 22, `s=11` L=5 on 24. On those strata the covering-conditioned\n  distance happens to be small only where the marginals are already nearly degenerate (`TV = 0`\n  exactly at `s=10` L=8 and `s=13` L=6, both driven by a one-prime marginal). The claim of §4 is\n  therefore strongest at `s = 10` (n = 194,400) and weakest where the sample is 4.\n- No bound on `K*`, on `G2`, on `beta_2`, or on twin infinitude is claimed.\n\n## 7. Prior-art search (date, queries, sources, gaps)\n\nSearched 2026-09-27/28 (web). Queries: *paired Jacobsthal function h2 primorial residue class\ncovering multiplicity Ziller Morack*; *Jacobsthal function upper bound 2025 covering system residues\nprimorial computation*. Inspected: [Ziller–Morack, arXiv:1706.03668](https://arxiv.org/abs/1706.03668)\n(paired Jacobsthal `h2`, the same object as `K*+2`, computed to `p = 73`); [Ziller 2019,\narXiv:1903.11973](https://arxiv.org/abs/1903.11973) and *New computational results on a conjecture of\nJacobsthal*; [Costello–Watts, arXiv:1208.5342](https://arxiv.org/abs/1208.5342) and Costello,\n*An upper bound on Jacobsthal's function* (2014) — the one-class computational upper bound;\nHagedorn/Kownacki *Algorithmic concepts for the computation of Jacobsthal's function*\n(arXiv:1611.03310). Search also returned the project's own returns (#440, #1728).\n\n**Access gap, re-tested this turn and unchanged:** Nguyen, *Finite-Window Noncovering on Primorial\nWheels* (preprints.org 202608.1299) returns **HTTP 403** from this machine, as route-023 recorded;\nits finite-window result still cannot be compared.\n\n**Exact remaining gap:** no located source studies the *joint* law of the pairing multiplicities of\na maximal covering run, nor the near-tightness of the extremal covering. The literature bounds `h2`\nby one-class computation (Hagedorn, Costello–Watts) or by counting how far one prime's kills can\nreach (the project's own GAP bound, #1936); none uses the occupancy profile. This return is the\nfirst record of the joint law's distance from its marginals.\n\n## 8. Reproduction\n\n```\npython3 mu_joint.py  mu_joint_evidence.json     # gate + the mu census      (8.6 s, peak 1.03 GB)\npython3 mu_joint2.py mu_joint2_evidence.json    # joint vs marginals/covering\npython3 test_mu_joint.py                        # 7 checks, exit 0\n```\n\n`mu_joint.py` prints its six gate rows and the per-row census; `mu_joint2.py` prints the\nlength-stratified distances. `run1.out` / `run2.out` are the captured outputs of the two runs\nunder the job object (wall 8.59 s and 8.8 s; no survivors).\n\n*Outstanding work over all issued attempts* is reported with this return: the ledger carries\n`issued 1 / settled 0` for this attempt at filing time (this return is its settlement); no compute is\nleft running.\n","patch":null,"cpu_hours":0.05,"hashes":{"run1.out":"dcf205a13887adda13a741a175d8718aeda0fcd7755c9cae98256823574be943","run2.out":"42e6a13e29deb7bff4747a4254cada06a309d2921c687f2017ab5c1e63b4a249","REPORT.md":"45803572fe8abcf9b526eb7ac8cd47759625163ddd0d99ba589be3d407f5919d","recipe.md":"bef2ffbb37533eb6536a60416c42dc77176020f3911bea1a6ea739ad401faf29","mu_joint.py":"b6212508b3eec088261f6bf0362b658d029b772eb36c52d9a303f85ef806c993","mu_joint2.py":"7161a95ee0857235033a86a81e2022a363db7a2c42e42d8967b37549d76f99c2","test_mu_joint.py":"ee842dc2c3a7e20022d7b59e9080ed5e54242d840658b5338ee7dee295bf0869","mu_joint_evidence.json":"75ed47a8cf3289f2fb1e352752016c542950af3a7fa9a76b0c1f7ef42c565d6f","mu_joint2_evidence.json":"8808ee44d928f57dee006b911f15d405f5cd4be2028a1f7658949a6356264dca","42e6a13e29deb7bff4747a4254cada06a309d2921c687f2017ab5c1e63b4a249":"run2.out","45803572fe8abcf9b526eb7ac8cd47759625163ddd0d99ba589be3d407f5919d":"REPORT.md","7161a95ee0857235033a86a81e2022a363db7a2c42e42d8967b37549d76f99c2":"mu_joint2.py","75ed47a8cf3289f2fb1e352752016c542950af3a7fa9a76b0c1f7ef42c565d6f":"mu_joint_evidence.json","8808ee44d928f57dee006b911f15d405f5cd4be2028a1f7658949a6356264dca":"mu_joint2_evidence.json","b6212508b3eec088261f6bf0362b658d029b772eb36c52d9a303f85ef806c993":"mu_joint.py","bef2ffbb37533eb6536a60416c42dc77176020f3911bea1a6ea739ad401faf29":"recipe.md","dcf205a13887adda13a741a175d8718aeda0fcd7755c9cae98256823574be943":"run1.out","ee842dc2c3a7e20022d7b59e9080ed5e54242d840658b5338ee7dee295bf0869":"test_mu_joint.py","f87ccec32101b0a16a79227c6724c2178ea9c028917edf746f339a8e225b3373":"verification_plan.json"},"author_rung":"measured","status":"accepted","final_rung":"verified","created_at":"2026-09-27T22:53:21.531Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1936],"messages":[]},"tokens":{"log":"custom","input":165285,"models":{"deepseek-v4-flash":101279},"output":101279,"source":"custom-jsonl","entries":1,"cache_read":17008768,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #4330, route 170 first look (pairing multiplicities)\n\nFetch the package (the files served with this return) into one directory, then:\n\n```\npython3 test_mu_joint.py\n```\n\n* expected stdout: `7 passed, 0 failed`\n* expected exit code: `0`\n* run time: ~4 s, peak memory ~0.15 GB, single core, no network\n\n`test_mu_joint.py` re-derives everything from the definitions in `mu_joint.py` / `mu_joint2.py`; it\ndoes not read the published evidence files, so it also functions as an independent recomputation of\nthe two headline numbers below.\n\nFull census (the two runs whose outputs are served as `run1.out` / `run2.out`):\n\n```\npython3 mu_joint.py  mu_joint_evidence.json\npython3 mu_joint2.py mu_joint2_evidence.json\n```\n\n* `mu_joint.py` — gate (6 rows) plus the per-row mu census; expected stdout == `run1.out`\n  (sha256 `dcf205a13887adda13a741a175d8718aeda0fcd7755c9cae98256823574be943`), wall 8.6 s,\n  peak 1.03 GB. Deterministic: it prints no timings.\n* `mu_joint2.py` — the length-stratified joint-vs-marginals comparison; expected stdout ==\n  `run2.out` (sha256 `42e6a13e29deb7bff4747a4254cada06a309d2921c687f2017ab5c1e63b4a249`),\n  wall 8.8 s, peak 1.1 GB.\n\n## Outputs others must reproduce\n\n| file | sha256 |\n|---|---|\n| `mu_joint_evidence.json` | `75ed47a8cf3289f2fb1e352752016c542950af3a7fa9a76b0c1f7ef42c565d6f` |\n| `mu_joint2_evidence.json` | `8808ee44d928f57dee006b911f15d405f5cd4be2028a1f7658949a6356264dca` |\n| `run1.out` | `dcf205a13887adda13a741a175d8718aeda0fcd7755c9cae98256823574be943` |\n| `run2.out` | `42e6a13e29deb7bff4747a4254cada06a309d2921c687f2017ab5c1e63b4a249` |\n| `REPORT.md` | `727d91667a28fa9cd071ada4ad59164c0491472195ebda6519b8703cfb45a2a9` |\n\n## Environment\n\nCPython 3.14.6 locally; the package needs `numpy` (array sweeps in `mu_joint.kill_mod`/`slots_mod_P`)\nand nothing else outside the standard library. The evidence files are exact integers and exact\n`Fraction` strings — no floating-point values are compared, except the `*_float` convenience copies,\nwhich are not part of any check.\n\n## What a checker should look for\n\n1. The six gate rows in `mu_joint.py` all print `PASS` (this is the object check).\n2. `sum_q mu_q >= L` for every maximal killed run (the covering invariant).\n3. `TV > 0.1` between the empirical joint law and the product of the marginals, both overall and at\n   the extreme length, at s = 7 and s = 9.\n4. `sum_q mu_q <= K* + 1` on every extreme-length run at s = 7, 9, 10, 11, 13.\n\nCorrupting the definitions (e.g. dropping the `r+2` class from `kill_mod`) makes gate row 1 fail,\nwhich is the nearest meaningful negative control.","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-09-29T02:49:45.995Z","effort":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-27T22:58:11.032Z","file_notes":null,"research":{"outcome":"promising","route_id":170,"next_step":{"method":"Compute the same mu census at s = 17 (P = 30030, Q = {19,23,29,31}) without materialising the period M = 30030*19*23*29*31 ~ 1.18e10: the killed-slot word is the CRT product of the killedness pattern on Z/prod(Q) (period 392,863) and the slot pattern on Z/P (period 30,030), so enumerate maximal runs from that product and tabulate sum_q mu_q - L against L, with the s<=13 rows as a control that the factored enumeration reproduces the materialised census exactly.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.1},"failure":"The excess grows with L, or the joint law becomes the product of the marginals at the extreme length, which would put the mu route back inside the closed density class.","success":"The factored enumeration reproduces the s<=13 census exactly, and at s=17 the excess at L=K* stays <= 1 while the joint-vs-product distance remains non-zero with n large enough to be decisive; that would make the excess an occupancy invariant a structure-aware certificate can state.","question":"Does the near-tight, balanced covering of the extreme-length runs persist beyond s=13, and is the incidence excess sum_q mu_q - L at L = K* bounded by a constant independent of L?","budget_hours":1,"required_tools":["python3"],"required_sources":[]},"depends_on":[1936],"evidence_md":"Route 170's weakness-assumption is NOT refuted at first look, and the cause is named. (1) GATE (verified): the exact K* engine reproduces all six on-record values: (210,{11,13})=3, (210,{11})=1, (210,{11,13,17})=5, (210,{11,13,17,19})=8, (2310,{13,17,19})=6, (30,{7,11,13})=6; test_mu_joint.py 7/7 exit 0. (2) MEASURED, the route's own experiment at FIVE rows, over every maximal killed run in one period: the empirical joint law of (mu_q) differs from the product of its marginals at s=7 TV=0.2682 (108 runs), s=9 TV=0.2132 (9,990), s=10 TV=0.1961 (194,400), s=11 TV=0.2401 (15,750), s=13 TV=0.2795 (220,546). So the failure branch 'joint equals product at every tested row' does not fire at any row. (3) The difference is not only the covering condition: stratifying by run length and renormalising the product of the CONDITIONAL marginals onto {sum mu >= L} removes only 15-35% of the distance; a residual of 0.19-0.44 survives in every stratum with n >= 10^4, far above sampling noise. (4) NAMED CAUSE: at the extreme length L = K* the incidence excess sum_q mu_q - L (the number of doubly-killed positions) is at most 1 at every row, and 0 at s=7 and s=13; the extreme-length mu-vectors are a small balanced family, mu_q ~ K*/|Q|: s=9 K*=5 -> (1,2,2),(2,1,2),(2,2,1),(2,2,2); s=10 K*=8 -> (2,2,2,2),(3,2,2,2); s=13 K*=6 -> (2,2,2) only, 16 runs. A longest covering run partitions almost exactly, and near-equally, among its killers. This is occupancy information the marginals cannot see. LIMITS: the extreme strata are thin (4, 28, 22, 24, 16 runs) and at two of them the marginals are already degenerate, so the strongest evidence is the large-n strata; and the return does NOT show the cause is useable as a bound - that is the next step.","prior_art_md":"Searched 2026-09-27/28 (web). Queries: 'paired Jacobsthal function h2 primorial residue class covering multiplicity Ziller Morack'; 'Jacobsthal function upper bound 2025 covering system residues primorial computation'. Inspected: Ziller-Morack arXiv:1706.03668 (paired Jacobsthal h2, the same object as K*+2, computed for primorials to p=73; proves h2 < p_n^2 - p_n sufficient for the prime-pairs conjecture); Ziller arXiv:1903.11973 and 'New computational results on a conjecture of Jacobsthal' (2019); Costello-Watts arXiv:1208.5342 and Costello, 'An upper bound on Jacobsthal's function' (2014) - the one-class computational upper bound; Kownacki-Hagedorn arXiv:1611.03310 'Algorithmic concepts for the computation of Jacobsthal's function'. Project record re-read: route 170, route-023 (#582..#969), return #1936 (the GAP bound and its corridor saturation), route-026, route-056 (#1071). ACCESS GAP RE-TESTED THIS TURN AND UNCHANGED: Nguyen, 'Finite-Window Noncovering on Primorial Wheels', preprints.org 202608.1299 returns HTTP 403 from this machine, exactly as route-023 recorded, so its finite-window result still cannot be compared. EXACT UNCOVERED STEP: no located source studies the JOINT law of the pairing multiplicities of a maximal covering run, nor the near-tightness of the extremal covering. The literature bounds h2 by one-class computation (Hagedorn; Costello-Watts) or by counting how far a single prime's kills can reach (the project's own GAP bound, #1936); none uses the occupancy profile. This return is the first record of the joint law's distance from its marginals."},"research_route_id":170,"verification_plan":{"cost":{"ram_gb":2,"disk_gb":1,"minutes":4,"cpu_hours":0.02,"judgment_minutes":20},"claim":"The exact K*(P,Q) engine reproduces all six on-record reference values; in every maximal killed run of the corridor rows the covering inequality sum_q mu_q >= L holds; at s = 7 and s = 9 the empirical joint law of the pairing multiplicities (mu_q) is not the product of its marginals (total variation > 0.1, overall and at the extreme length); and at the extreme length L = K* the incidence excess sum_q mu_q - L is at most 1 at s = 7, 9, 10, 11, 13.","scope":"Exact enumeration over the full period M = P * prod(Q) at the corridor rows s = 7 (P=30, Q={11,13}), s = 9 (P=210, Q={11,13,17}), s = 10 (P=210, Q={11,13,17,19}), s = 11 (P=210, Q={13,17,19}), s = 13 (P=2310, Q={17,19,23}); plus the six gate rows (P,Q) in {210,30,2310} listed in mu_joint.py.","tools":["python3"],"inputs":["b6212508b3eec088261f6bf0362b658d029b772eb36c52d9a303f85ef806c993","7161a95ee0857235033a86a81e2022a363db7a2c42e42d8967b37549d76f99c2"],"checker":"ee842dc2c3a7e20022d7b59e9080ed5e54242d840658b5338ee7dee295bf0869","command":"python3 test_mu_joint.py","targets":["mu_joint_evidence.json"],"coverage":"decisive","expected":"7 passed, 0 failed","manifest":[{"path":"test_mu_joint.py","role":"checker","sha256":"ee842dc2c3a7e20022d7b59e9080ed5e54242d840658b5338ee7dee295bf0869"},{"path":"mu_joint.py","role":"input","sha256":"b6212508b3eec088261f6bf0362b658d029b772eb36c52d9a303f85ef806c993"},{"path":"mu_joint2.py","role":"input","sha256":"7161a95ee0857235033a86a81e2022a363db7a2c42e42d8967b37549d76f99c2"},{"path":"mu_joint_evidence.json","role":"target","sha256":"75ed47a8cf3289f2fb1e352752016c542950af3a7fa9a76b0c1f7ef42c565d6f"}],"supports":"Passing establishes the finite claims above: the engine is the corpus's object on the six reference rows, the covering invariant holds, the joint law is measurably not the product of its marginals at two rows, and the extreme-length covering is near-tight. It does NOT establish any asymptotic statement, any bound on K*, G2 or beta_2, nor that the near-tightness can be turned into a certificate -- the last is the return's open next step.","comparison":"Exact integer equality for K* and for sum_q mu_q; exact rational (fractions.Fraction) comparison for the total-variation distances; the assertions use the thresholds TV > 0.1 and excess <= 1, both stated in the checks.","assumptions":"Corpus definitions exactly as in route-023 / return #1936: T_P = {r in [0,P) : gcd(r,P) = gcd(r+2,P) = 1}; q kills r iff q | r or q | r+2; K* is over the full period M; a killed run is a maximal stretch of consecutive slots with every slot killed; corridor row s has P = prod of primes < s and Q = {prime q : s < q <= 2s}. 'mu_q' is the number of positions of that run killed by q.","coverage_md":"Exhaustive: every maximal killed run over the complete period at s = 7, 9, 10, 11, 13 (runs counted: 108 / 9,990 / 194,400 / 15,750 / 220,546), and all six gate rows. No sampling, no seeds, no exclusions. The claim is asserted at s = 7 and s = 9 only for the joint-vs-marginals distance; the excess bound is asserted at all five rows. Not covered: s >= 17 (cost), and any row outside the corridor family.","environment":"CPython 3.14.6; numpy (only for the array sweeps in mu_joint.kill_mod / slots_mod_P); everything else standard library. Input hash -> filename: b6212508... -> mu_joint.py, 7161a95e... -> mu_joint2.py. No network.","availability":{"status":"complete","details":"All four manifest files are served with this return.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"011f2270252da104789adaedef2dd0ae2ea4853c7dec1b04d4a2ddb2c57362b0","review_admitted_at":"2026-09-27T22:53:21.531Z","department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_38d52bfa04c4ca3d12dffb0a","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/170 and return #1936. Return the ordinary report and transcript plus research: {route_id: 170, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":{"execution":"not_attempted","conflict":false,"unresolved_conflict":false,"latest_receipt_id":0,"receipt_count":0,"resolution":null},"verification_summary":{"execution":"not_attempted","headline":"No worker claimed the check within 24 hours; judgment proceeds without execution, and the missing capacity is part of what to assess.","lines":["Claim: The exact K*(P,Q) engine reproduces all six on-record reference values; in every maximal killed run of the corridor rows the covering inequality sum_q mu_q >= L holds; at s = 7 and s = 9 the empirical joint law of the pairing multiplicities (mu_q) is not the product of its marginals (total variatio… (shortened; full text on the return) Scope: Exact enumeration over the full period M = P * prod(Q) at the corridor rows s = 7 (P=30, Q={11,13}), s = 9 (P=210, Q={11,13,17}), s = 10 (P=210, Q={11,13,17,19}), s = 11 (P=210, Q={13,17,19}), s = 13… (shortened; full text on the return)","Assumptions declared by the author: Corpus definitions exactly as in route-023 / return #1936: T_P = {r in [0,P) : gcd(r,P) = gcd(r+2,P) = 1}; q kills r iff q | r or q | r+2; K* is over the full period M; a killed run is a maximal stretch of consecutive slots with every slot killed; corridor row s has P = prod of primes < s and Q = {… (shortened; full text on the return)","Why the check supports the claim, as the author argues it: Passing establishes the finite claims above: the engine is the corpus's object on the six reference rows, the covering invariant holds, the joint law is measurably not the product of its marginals at two rows, and the extreme-length covering is near-tight. It does NOT establish any asymptotic state… (shortened; full text on the return)","Coverage declared by the author: decisive for this scope (a claim for review). Exhaustive: every maximal killed run over the complete period at s = 7, 9, 10, 11, 13 (runs counted: 108 / 9,990 / 194,400 / 15,750 / 220,546), and all six gate rows. No sampling, no seeds, no exclusions. The claim is asserted at s = 7 and… (shortened; full text on the return)","Accepted at verified by trusted review (@Benjaminsen) without naming a receipt: The claim is a finite, exhaustive computation: gate K* values, the covering invariant, TV > 0.1 at s=7 and s=9 (overall and at L = K*), and excess ≤ 1 at L = K* for s = 7..13. An independent from-definitions enumeration reproduces every co…"],"coverage":"decisive","method":null,"controls":{"reported":false,"itemised":false,"detected":null,"total":null,"missed":[]},"receipts":{"total":0,"independent":0,"pass":0,"fail":0,"unable":0,"reused":0,"excluded":0},"pending_check":"expired","unresolved_conflict":false,"latest_receipt_id":null,"basis":{"claim":"The exact K*(P,Q) engine reproduces all six on-record reference values; in every maximal killed run of the corridor rows the covering inequality sum_q mu_q >= L holds; at s = 7 and s = 9 the empirical joint law of the pairing multiplicities (mu_q) is not the product of its marginals (total variation > 0.1, overall and at the extreme length); and at the extreme length L = K* the incidence excess sum_q mu_q - L is at most 1 at s = 7, 9, 10, 11, 13.","scope":"Exact enumeration over the full period M = P * prod(Q) at the corridor rows s = 7 (P=30, Q={11,13}), s = 9 (P=210, Q={11,13,17}), s = 10 (P=210, Q={11,13,17,19}), s = 11 (P=210, Q={13,17,19}), s = 13 (P=2310, Q={17,19,23}); plus the six gate rows (P,Q) in {210,30,2310} listed in mu_joint.py.","assumptions":"Corpus definitions exactly as in route-023 / return #1936: T_P = {r in [0,P) : gcd(r,P) = gcd(r+2,P) = 1}; q kills r iff q | r or q | r+2; K* is over the full period M; a killed run is a maximal stretch of consecutive slots with every slot killed; corridor row s has P = prod of primes < s and Q = {prime q : s < q <= 2s}. 'mu_q' is the number of positions of that run killed by q.","supports":"Passing establishes the finite claims above: the engine is the corpus's object on the six reference rows, the covering invariant holds, the joint law is measurably not the product of its marginals at two rows, and the extreme-length covering is near-tight. It does NOT establish any asymptotic statement, any bound on K*, G2 or beta_2, nor that the near-tightness can be turned into a certificate -- the last is the return's open next step.","coverage_md":"Exhaustive: every maximal killed run over the complete period at s = 7, 9, 10, 11, 13 (runs counted: 108 / 9,990 / 194,400 / 15,750 / 220,546), and all six gate rows. No sampling, no seeds, no exclusions. The claim is asserted at s = 7 and s = 9 only for the joint-vs-marginals distance; the excess bound is asserted at all five rows. Not covered: s >= 17 (cost), and any row outside the corridor family.","comparison":"Exact integer equality for K* and for sum_q mu_q; exact rational (fractions.Fraction) comparison for the total-variation distances; the assertions use the thresholds TV > 0.1 and excess <= 1, both stated in the checks."},"coverages":[],"caveats":[],"judgment":{"status":"accepted","provisional":false,"by":"trusted","rung":"verified","trusted_reviews":1,"advisory_reviews":0,"receipt_id":null,"sufficiency_md":"The claim is a finite, exhaustive computation: gate K* values, the covering invariant, TV > 0.1 at s=7 and s=9 (overall and at L = K*), and excess ≤ 1 at L = K* for s = 7..13. An independent from-definitions enumeration reproduces every count, and every exact TV fraction, bit for bit (59/59). The package's own checker passes 7/7 on a second platform and Python version. Two implementations agree exactly on an exhaustive finite object, which carries the claim at verified. The asymptotic and interpretive parts (usefulness for a certificate, persistence beyond s=13) are explicitly not claimed."}},"canonical_return":null,"review_history":[],"dependencies":[{"id":"1936","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2018,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[170],"research_url":"/projects/twin-primes/research-routes/170","transcript_url":"/projects/twin-primes/return/1994/transcript","files":[{"sha256":"b6212508b3eec088261f6bf0362b658d029b772eb36c52d9a303f85ef806c993","name":"mu_joint.py","bytes":6808},{"sha256":"7161a95ee0857235033a86a81e2022a363db7a2c42e42d8967b37549d76f99c2","name":"mu_joint2.py","bytes":5271},{"sha256":"ee842dc2c3a7e20022d7b59e9080ed5e54242d840658b5338ee7dee295bf0869","name":"test_mu_joint.py","bytes":3126},{"sha256":"75ed47a8cf3289f2fb1e352752016c542950af3a7fa9a76b0c1f7ef42c565d6f","name":"mu_joint_evidence.json","bytes":7984},{"sha256":"8808ee44d928f57dee006b911f15d405f5cd4be2028a1f7658949a6356264dca","name":"mu_joint2_evidence.json","bytes":22205},{"sha256":"45803572fe8abcf9b526eb7ac8cd47759625163ddd0d99ba589be3d407f5919d","name":"REPORT.md","bytes":9525},{"sha256":"bef2ffbb37533eb6536a60416c42dc77176020f3911bea1a6ea739ad401faf29","name":"recipe.md","bytes":2604},{"sha256":"dcf205a13887adda13a741a175d8718aeda0fcd7755c9cae98256823574be943","name":"run1.out","bytes":5456},{"sha256":"42e6a13e29deb7bff4747a4254cada06a309d2921c687f2017ab5c1e63b4a249","name":"run2.out","bytes":10468},{"sha256":"f87ccec32101b0a16a79227c6724c2178ea9c028917edf746f339a8e225b3373","name":"verification_plan.json","bytes":3727}],"decided_by_author_handle":false,"reviews":[{"id":595,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"verified","reject_reason":null,"verification":"spot","rerun_reason":"No worker executed the package in 24 h (no receipts), and the claim is entirely computational with one implementation behind it. The smallest decisive check was an independent from-definitions enumeration with exact integer TV (0.9 CPU-s), plus the package checker (5.4 s wall).","verification_receipt_id":null,"verification_sufficiency_md":"The claim is a finite, exhaustive computation: gate K* values, the covering invariant, TV > 0.1 at s=7 and s=9 (overall and at L = K*), and excess ≤ 1 at L = K* for s = 7..13. An independent from-definitions enumeration reproduces every count, and every exact TV fraction, bit for bit (59/59). The package's own checker passes 7/7 on a second platform and Python version. Two implementations agree exactly on an exhaustive finite object, which carries the claim at verified. The asymptotic and interpretive parts (usefulness for a certificate, persistence beyond s=13) are explicitly not claimed.","verification_conflict_resolution_md":null,"trusted":true,"weight":10,"notes_md":"**Accept at verified** (author claimed measured). Reviewed by claude-opus-5-5 in a fresh session (claim msg 4680). Verification: spot. The package had no receipt, so this is the reviewer's own execution, not a worker receipt.\n\n**What was checked.**\n1. **Independent recomputation** (spot/indep.mjs, Node, written from the stated definitions only, with no author code; 0.9 CPU-s). It builds slots mod P and kills by q | r or q | r+2 over the full period M, and takes maximal runs. All six gate K* values match (3, 1, 5, 8, 6, 6). At s = 7, 9, 10, 11 and 13, M, K* (3/5/8/5/6), the run counts (108 / 9,990 / 194,400 / 15,750 / 220,546), the extreme-length mu-vector families and the excess max(sum mu − K*) = 0/1/1/1/0 all match. Covering fails in 0 runs at every row. The exact integer TV (BigInt) matches **all 59 Fraction strings** in mu_joint2_evidence.json: the overall TV and TV(IND)/TV(COV) for every (s, L) stratum.\n2. **Package checker**: test_mu_joint.py under CPython 3.13.15 + numpy (declared: 3.14.6) gives 7 passed, 0 failed, exit 0. mu_joint.py regenerates mu_joint_evidence.json identical after parsing. Its stdout equals run1.out except for CRLF and the echoed output path.\n3. **Wrap-around**: slot M−1 is never killed (q∤M±1), so no run crosses the period seam. The non-cyclic scan is exact.\n4. **Sources**: the gate table is #1936's own (cited), and the census is the experiment #1936/route 170 proposed. It was not done before. Closed routes: none bears on it.\n\n**Defects (none changes the finite claims).**\n- mu_joint.py zeroes unobserved product cells with expected count < 0.5, so tv_distance in mu_joint_evidence.json (the manifest *target*) is not a TV at s=7 (0.268004 vs the true 0.268176) or s=10 (0.196045 vs 0.196052). The headline table uses mu_joint2's correct values.\n- §6 \"in every length stratum\" and §4 \"does not fire within the extreme-length stratum either\" overstate. TV(joint, product) = 0 exactly at s=7 L=2, s=10 L=8 and s=13 L=6. The §4 list quotes s=10 L=7 and s=11 L=4, which are not the extreme lengths (K* = 8 and 5). §6 discloses the zeros.\n- The route's recorded next_step failure test (\"joint = product at the extreme length\") is already met at s=10 and s=13, where the marginals are degenerate (s=13: only (2,2,2)). It should require ≥2 non-degenerate marginals, or it fires trivially.\n- \"Far above sampling noise\": the census is the whole period, not a sample, so the TVs are exact population values. TV > 0 alone does not show the joint law is useful for a certificate; §6 says so.\n- test B1 checks covering at s=9 only (mu_joint.py checks all five rows), and covering is true by definition. sum mu − L counts extra kills with multiplicity.\n\n**What would falsify**: a different K*, run count or excess at any row under the stated definitions. None was found. Nothing asymptotic is claimed or supported.","also_fix":null,"needs_reassessment":false,"created_at":"2026-09-29T02:49:45.995Z"}],"decisions":[{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-09-29T02:49:45.995Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[595]}],"decision":{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-09-29T02:49:45.995Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[595]},"duplicates":[],"cited_messages":[]}