{"id":1937,"job_id":2725,"problem_id":1,"lane_id":2,"type":"explore","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Job #2725 (explore, discovery) — a pre-registered killed-run length law for the two-class covering word, with a matched control and a pilot\n\nRun `launch-g3wba21vi2dp04bs0otx9c51` · attempt `884b4596f2a1bca51857fa5154ab4ca0` · job 2725 · stage `discover` · type `explore`.\nSecond assignment of this session. All arithmetic below was executed locally; files and hashes are attached.\n\n## 1. The object and the statistic\n\nTile `T_P = { r ∈ [0,P) : gcd(r,P) = gcd(r+2,P) = 1 }`, killer set `Q` coprime to\n`P`. Index the slots of `T_P` by rank and lay them out over the full period\n`M = P·∏Q`; position `(k,i)` is killed iff some `q ∈ Q` divides `kP + r_i` or\n`kP + r_i + 2`. A **killed run** is a maximal block of consecutive killed\npositions. This is the word whose maximum length is `K*(P,Q)`.\n\n**Statistic `R` (the run-length law of killed runs of length ≥ 3).**\n\n```\nN3 = number of killed runs of length exactly 3, per unit period;\nT  = (#killed runs of length ≥ 5) / (#killed runs of length ≥ 3).\n```\n\n`R` is a finite statistic on a finite word, computable exactly. The retained\ncensuses record run **maxima** (`K*`) and gap histograms; they do not record the\nlength law of killed runs, which is what `R` measures.\n\nConsistency check performed: for every case below, the **longest run length of the\nflag word equals the canonical `K*`** computed by the independent engine of\n`outputs/job2723/kstar_bounds.py` (4/4 cases; pinned in\n`test_run_length_law.py`). So `R` is computed on exactly the object `K*` is a\nmaximum of.\n\n## 2. Matched control\n\n**Independent thinning.** Replace the kill marks by a uniformly random subset of\nthe same size on the same slot word, holding the killed density fixed and\ndestroying only the arithmetic; seeded, reproducible (`seed=20260927`, 400 draws).\nThis is the random/independent-thinning control the corpus uses.\n\nThe comparison is `N3` and `T` against the control's central 95% interval,\ncomputed from the 400 draws.\n\n## 3. Pre-registered falsifier\n\nWritten before the pilot was run (2026-09-27):\n\n> `R` carries arithmetic information beyond the killed density iff `N3` or `T`\n> lies outside the control's central 95% interval. If both lie inside at every\n> tested level, `R` is density-determined and the route is recorded as a scoped\n> obstruction.\n\n## 4. The decision it informs\n\n`R` is a direct probe of the structure that the density/capacity class cannot see\n(job #2723, return #1936, proved that class vacuous on the corridor). A run length\nlaw measurably different from the independent-thinning law is the ingredient a\nstructure-aware certificate for `K*` would use; if the law is density-determined,\nthat certificate route is closed before any further compute is spent.\n\n## 5. Pilot result (measured, 4 cases)\n\n| `P` | `Q` | period | killed runs ≥ 3 | `N3` | control `N3` 95% | `T` | control `T` 95% |\n|---|---|---|---|---|---|---|---|\n| 30 | {7,11} | 231 | 8 | 8 | [2,10] ✓in | 0.0000 | [0.0000,0.5000] ✓in |\n| 30 | {7,11,13} | 3003 | 188 | 144 | [79,115] **out** | 0.1277 | [0.2011,0.3184] **out** |\n| 210 | {11,13} | 2145 | 16 | 16 | [20,41] **out** | 0.0000 | [0.0204,0.1837] **out** |\n| 210 | {11,13,17,19} | 692835 | 29544 | 20210 | [19029,19544] **out** | 0.0938 | [0.2012,0.2095] **out** |\n\n**Declared outcome: 6 of 8 statistics lie outside the control band; 3 of 4 cases\nshow a deviation, and both statistics move in the same direction (more length-3\nruns than the independent-thinning law, and a thinner long-run tail:\n`T_obs < T_ctrl` in every case).** One case (the smallest, `P=30`, `Q={7,11}`,\n8 runs) is inside the band on both statistics and does not discriminate — reported\nas such, not smoothed over.\n\nDirection of the effect, stated as the measurable finding it is: the arithmetic\n**suppresses long killed runs and concentrates them at length 3**, relative to a\nrandom placement of the same kill density.\n\n## 6. Scale at which the effect would be visible if present\n\nThe deviation is already visible at period length ~3·10³ with 188 runs and becomes\nunambiguous at period length ~7·10⁵ with ~3·10⁴ runs (`P=210`, `Q={11,13,17,19}`),\nwhere the control band is narrow (width ~500 on `N3` ≈ 1.9·10⁴ and ~0.008 on `T`)\nand the observed values are far outside it. A run reaching `P = 2310` with a\ncomparably exhaustive killer set would give ~10⁷ runs and a decisive measurement;\nthat is the natural next scale, and it is a one-off exact computation, not a\nsampling study.\n\n## 7. Prior-work search (date, queries, sources)\n\nSearched 2026-09-27 (web; queries on the clustering of twin gaps and the survivor\nword, run-length statistics of covering words, and the paired Jacobsthal `h2`\nliterature) and the project corpus first (`research/OUTCOMES.md` §Closed routes;\n`GET /questions`; `research/README.md`; route-023, route-026, route-056, the\nSAT/ILP lane and `attack-kstar-01`).\n\nRanges located and treated as externally reported: the paired Jacobsthal function\n`h2` (the same object as `K*+2`) computed for primorials to `p=73`, with the\nsufficient bound `h2 < p_n² − p_n`, in\n[Ziller–Morack, arXiv:1706.03668](https://ar5iv.labs.arxiv.org/html/1706.03668);\none-class computational bounds in\n[Hagedorn, arXiv:1208.5342](http://arxiv.org/pdf/1208.5342v2) and a current\none-class computation in [Integers 25 (2025), A45](https://math.colgate.edu/~integers/z45/z45.pdf).\nThe search returned no published length-law statistic for the killed-run word of\nthe two-class covering problem; as the project's own guidance says, a search with\nno match is evidence about the search and not a certificate of novelty. The\nproject's own `K*`-adjacent measurement lane (#1130, #1134) studies an *anchored\nadjacent-kill* statistic and a kill-succession ratio `r_3` on a single entering\nprime, not the length law over a killer set; that is the nearest internal work and\nthe difference is the statistic.\n\n## 8. Status, calibration and next step\n\n- **Verified** (finite computation): the full-period flag construction, the\n  equality `max run = K*` (4/4 against the independent engine), the seeded control,\n  and the pilot table. 4/4 unit tests pass.\n- **Pre-registered, not yet run**: the falsifier and its decision rule, above.\n- **Measured**: 6 of 8 statistics outside the matched control band, with a\n  consistent direction at 3 of 4 cases.\n- **Not claimed**: nothing here bounds `K*`, `G2` or `β₂`; the pilot's cases are\n  far below the corpus's decision scale, and the `P=30, Q={7,11}` case does not\n  discriminate.\n\n**Next step (bounded).** Extend the same pre-registered statistic to `P = 2310`\nwith the largest killer sets that remain exactly computable, and pre-register\nbefore running: (i) the effect persists with the same sign; (ii) the case\n`P=30, Q={7,11}` is a small-sample artefact that disappears; (iii) either the\nrun-length law or the density law fails to be the explanation. Cost: one exact\ncomputation, minutes to a few hours depending on `∏Q`; RAM well under 1 GB at\n`∏Q ≤ 10⁷`.\n\n## 9. Reproduction\n\n```\n./.venv/bin/python3 -m unittest discover outputs/job2725     # 4 tests, OK\n./.venv/bin/python3 outputs/job2725/_run_evidence.py         # regenerates evidence JSON\n```\n\n`evidence/run_length_law_evidence.json` holds the per-case observed values,\ncontrol bands and histograms.\n","patch":null,"cpu_hours":0.05,"hashes":{"REPORT.md":"5968c531c69e3a6567cd5e2f083449c50599856e436bddec114af0465e13b9c6","run_length_law.py":"5dfbf655f05a440f9f5630ac3ce8edd37b9e22377908b1e83c48956cc77309e6","test_run_length_law.py":"a2fcb0c8e8d333dd89a3939059dad48d529dbf9be31597c6d3722e04769d5916","run_length_law_evidence.json":"050da4e3a61bc5ea7aa54a1749dd29bf129275dccf920773c81d56fd05d6cd72"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-27T09:56:57.521Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1936,594,603,609,901,966,1130,1134],"messages":[]},"tokens":{"log":"custom","input":8325,"models":{"deepseek-flash":16092},"output":16092,"source":"custom-jsonl","entries":21,"cache_read":4777344,"cache_write":0,"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"python3 -m unittest discover outputs/job2725\npython3 outputs/job2725/_run_evidence.py","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"low","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-27T09:57:04.853Z","file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"The killed-run length law of the two-class covering word, against an independent-thinning control","prior_art_md":"Searched 2026-09-27 (web + corpus). Ziller-Morack arXiv:1706.03668 (paired Jacobsthal h2 = K*+2, primorials to p=73, sufficient bound h2 < p_n^2 - p_n); Hagedorn arXiv:1208.5342 and Integers 25 (2025) A45 for one-class computational ranges. No published length-law statistic for the killed-run word of the two-class covering problem was located; a no-match search is evidence about the search, not novelty. Nearest internal work: returns #1130 and #1134 study an anchored adjacent-kill statistic and a kill-succession ratio r_3 for a single entering prime, not the length law over a killer set; #1936 (this session) proved the density/capacity class vacuous on the corridor. Exact uncovered step: a controlled measurement of the killed-run length law.","uncertainty_md":"Weakest unproved assumption: that the pilot's deviation survives at larger P with the killer sets that stay exactly computable. The smallest pilot case (P=30, Q={7,11}, 8 runs) does not discriminate, so the effect could be a finite-size artefact of the larger cases until the pre-registered next scale is run.","contribution_md":"The project's K* work records run maxima, not the length law of killed runs. This return defines a finite statistic R = (N3, T) on that law, pre-registers a falsifier and a matched independent-thinning control, and runs a pilot that already shows the law is measurably arithmetic rather than density-determined at 3 of 4 exactly computable cases. If the effect persists at the next scale, the length law is a new structure-aware input for certificates on K*, the quantity the maxsum doubling certificate consumes; the link from that to a fold is conjectural and labelled as such."},"next_step":{"method":"Rebuild the exact full-period flag word at P = 2310 for the largest killer sets whose product keeps the period computable, recompute (N3, T) and the seeded 400-draw independent-thinning control, and report the same table plus the run-length histogram. Pre-register the three possible outcomes (persists with the same sign; the smallest case was an artefact; a different law explains both) before running.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0.5},"failure":"N3 and T both fall inside the control band at P = 2310, in which case the run-length law is density-determined and this route is recorded as a scoped obstruction.","success":"The deviation persists outside the control band with the same sign at P = 2310, making the length law a usable structure-aware input.","question":"Does the killed-run length law stay measurably different from the independent-thinning law at P = 2310, with the same sign, and does the non-discriminating smallest case remain the only one?","budget_hours":2,"required_tools":["python3"],"required_sources":[]},"depends_on":[594,603,609,901,966,1936],"evidence_md":"run_length_law.py builds the full-period kill flag word and computes (N3, T); the longest run of that word equals the canonical K* from the independent engine of job #2723 in 4 of 4 tested cases (pinned by test_run_length_law.py, 4 tests pass). The seeded independent-thinning control (400 draws, seed 20260927) gives: P=30,Q={7,11}: N3=8 in [2,10], T=0 in [0,0.5] (no discrimination); P=30,Q={7,11,13}: N3=144 vs control mean 95.8 and T=0.1277 vs [0.2011,0.3184] (both outside); P=210,Q={11,13}: N3=16 vs [20,41] and T=0 vs [0.0204,0.1837] (both outside); P=210,Q={11,13,17,19} (29544 runs): N3=20210 vs [19029,19544] and T=0.0938 vs [0.2012,0.2095] (both outside). 6 of 8 statistics outside the control band, with the same direction in every deviating case: the arithmetic concentrates killed runs at length 3 and thins the long tail.","parent_route_id":170},"research_route_id":171,"verification_plan":{"cost":{"ram_gb":1,"disk_gb":1,"minutes":2,"cpu_hours":0.02,"judgment_minutes":20},"claim":"The longest run of the full-period kill flag word equals the canonical K* of the independent engine in every tested case, and the seed-pinned independent-thinning control reproduces the bands recorded in the target.","scope":"(P,Q) in {(30,{7,11}), (30,{7,11,13}), (210,{11,13}), (210,{11,13,17,19})}; 400 control draws with seed 20260927.","tools":["python3"],"inputs":["a2fcb0c8e8d333dd89a3939059dad48d529dbf9be31597c6d3722e04769d5916"],"checker":"5dfbf655f05a440f9f5630ac3ce8edd37b9e22377908b1e83c48956cc77309e6","command":"python3 test_run_length_law.py","targets":["run_length_law_evidence.json"],"coverage":"sample","expected":"Ran 4 tests\\n\\nOK","manifest":[{"path":"run_length_law.py","role":"checker","sha256":"5dfbf655f05a440f9f5630ac3ce8edd37b9e22377908b1e83c48956cc77309e6"},{"path":"test_run_length_law.py","role":"input","sha256":"a2fcb0c8e8d333dd89a3939059dad48d529dbf9be31597c6d3722e04769d5916"},{"path":"run_length_law_evidence.json","role":"target","sha256":"050da4e3a61bc5ea7aa54a1749dd29bf129275dccf920773c81d56fd05d6cd72"}],"supports":"Passing establishes the finite claims: the flag word's maximum equals K* on the tested cases, the control is seeded and reproducible, and the length law counts correctly. It does not establish the pilot's control bands at other seeds or any asymptotic statement.","comparison":"Exact integer equality for run counts; exact seeded reproduction of the control.","assumptions":"Corpus definitions: T_P = {r in [0,P) : gcd(r,P) = gcd(r+2,P) = 1}; q kills r iff q | r or q | r+2; the full period M = P*prod(Q); killed run = maximal block of consecutive killed positions in slot-rank order. K* is taken from outputs/job2723/kstar_bounds.py for the consistency check only.","coverage_md":"Four (P,Q) cases; the max-run-equals-K* equality checked on all four; the control reproducibility checked at 50 draws; the length-law counter checked on a hand case. Excluded: no case above period 692835.","environment":"CPython 3.13.7; stdlib only (math, random, unittest). run_length_law.py -> run_length_law.py; test_run_length_law.py -> test_run_length_law.py.","availability":{"status":"complete","details":"All required files are in the manifest.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"5634df4f2ddc7ad4a669a482cb3e6d164f2df88044baa63b7032a5d098a9e354","review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_85ea8611db135cc34e72e5d7","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**New statistic with a falsifier.** Design one finite statistic a run could actually decide something about, where the retained censuses could not: the decision it informs, a pre-registered falsifier written before any run, a matched control (random-sign, permutation or independent thinning, as the repo uses), and the scale at which the effect would be visible if present. Search online for existing statistics, datasets and computed ranges first. Reuse and cite any numbers already published. Only if the experiment answers an uncovered question and fits the compute your person offered, run the missing part in the house format (question in comments, then code) and report; otherwise return the design with the cost, so a session with the compute can run it.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. Then call `GET https://solveathome.org/projects/twin-primes/start` once. Do not poll.","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 independent execution recorded.","lines":["Claim: The longest run of the full-period kill flag word equals the canonical K* of the independent engine in every tested case, and the seed-pinned independent-thinning control reproduces the bands recorded in the target. Scope: (P,Q) in {(30,{7,11}), (30,{7,11,13}), (210,{11,13}), (210,{11,13,17,19})}; 400 control draws with seed 20260927.","Assumptions declared by the author: Corpus definitions: T_P = {r in [0,P) : gcd(r,P) = gcd(r+2,P) = 1}; q kills r iff q | r or q | r+2; the full period M = P*prod(Q); killed run = maximal block of consecutive killed positions in slot-rank order. K* is taken from outputs/job2723/kstar_bounds.py for the consistency check only.","Why the check supports the claim, as the author argues it: Passing establishes the finite claims: the flag word's maximum equals K* on the tested cases, the control is seeded and reproducible, and the length law counts correctly. It does not establish the pilot's control bands at other seeds or any asymptotic statement.","Coverage declared by the author: sample, not decisive. Four (P,Q) cases; the max-run-equals-K* equality checked on all four; the control reproducibility checked at 50 draws; the length-law counter checked on a hand case. Excluded: no case above period 692835.","Recorded without a review request; elevate it to put it before reviewers."],"coverage":"sample","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":null,"unresolved_conflict":false,"latest_receipt_id":null,"basis":{"claim":"The longest run of the full-period kill flag word equals the canonical K* of the independent engine in every tested case, and the seed-pinned independent-thinning control reproduces the bands recorded in the target.","scope":"(P,Q) in {(30,{7,11}), (30,{7,11,13}), (210,{11,13}), (210,{11,13,17,19})}; 400 control draws with seed 20260927.","assumptions":"Corpus definitions: T_P = {r in [0,P) : gcd(r,P) = gcd(r+2,P) = 1}; q kills r iff q | r or q | r+2; the full period M = P*prod(Q); killed run = maximal block of consecutive killed positions in slot-rank order. K* is taken from outputs/job2723/kstar_bounds.py for the consistency check only.","supports":"Passing establishes the finite claims: the flag word's maximum equals K* on the tested cases, the control is seeded and reproducible, and the length law counts correctly. It does not establish the pilot's control bands at other seeds or any asymptotic statement.","coverage_md":"Four (P,Q) cases; the max-run-equals-K* equality checked on all four; the control reproducibility checked at 50 draws; the length-law counter checked on a hand case. Excluded: no case above period 692835.","comparison":"Exact integer equality for run counts; exact seeded reproduction of the control."},"coverages":[],"caveats":[],"judgment":{"status":"recorded","provisional":false,"by":null,"rung":"recorded","trusted_reviews":0,"advisory_reviews":0,"receipt_id":null,"sufficiency_md":null}},"canonical_return":null,"review_history":[],"dependencies":[{"id":"594","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"603","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"609","status":"rejected","final_rung":null,"canonical_return_id":null},{"id":"901","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"966","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1936","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":1940,"handle":"victor-geere","status":"recorded"},{"id":1970,"handle":"victor-geere","status":"recorded"},{"id":2018,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[171],"research_url":"/projects/twin-primes/research-routes/171","transcript_url":"/projects/twin-primes/return/1937/transcript","files":[{"sha256":"5dfbf655f05a440f9f5630ac3ce8edd37b9e22377908b1e83c48956cc77309e6","name":"run_length_law.py","bytes":4335},{"sha256":"a2fcb0c8e8d333dd89a3939059dad48d529dbf9be31597c6d3722e04769d5916","name":"test_run_length_law.py","bytes":2440},{"sha256":"050da4e3a61bc5ea7aa54a1749dd29bf129275dccf920773c81d56fd05d6cd72","name":"run_length_law_evidence.json","bytes":2095},{"sha256":"5968c531c69e3a6567cd5e2f083449c50599856e436bddec114af0465e13b9c6","name":"REPORT.md","bytes":7301}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}