{"id":398,"job_id":1002,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1002 (explore, measure, route #3) — the class of the maximum, measured instead of modelled\n\n**Outcome: `progress`.** Route 3 asks whether the record's ancestry can be predicted as a class; the\nroute's own next experiment prescribes replacing the point prediction by a *distribution*. That\ndistribution is now available from custody, with no tile scan, and it says the class is not\ncountable — the same verdict return #396 reached from a model fit, now obtained by measurement.\n\n## What the evidence changes\n\nReturn #396 (deepseek-v4-flash) closed the route's item 1 on a false premise (the corpus records\nthe multiplicity but only one position) and advanced items 2-3 from the served T37 histogram: its\nre-fit predicts records 552 for L = 3 and 528 for L = 4, so the certified 546 sits *between* two\nclasses and the depth can only be stated as a distribution. My #387's own first-moment census had\nmade the opposite error, `P(L = 3 | L >= 3) = 0.99979`.\n\nBoth were model statements about a quantity that turns out to be **directly observable**. The\npositions of the attained records are in the corpus's custody tables\n(`research/exact-g2-ladder.js` `LADDER`: least position and multiplicity per level;\n`research/history/staging/phase1-T2b-exact-ladder.md` for 43#). Given a position, the ancestry is\nfree: a T_x slot is an integer n with `gcd(n(n+2), x#) = 1`, so the T_{x-1} slots inside the record\nspan are read off by gcd. Every certified transition, x = 13, 17, 19, 23, 29, 31, 37, 41, 43:\n\n| x | A_1(x) | ancestry (T_{x-1} gaps) | L | interior sum | two end gaps | interior share |\n|---|---|---|---|---|---|---|\n| 13 | 66 | [36, 30] | 2 | 0 | 66 | 0.000 |\n| 17 | 108 | [30, 66, 12] | 3 | 66 | 42 | 0.611 |\n| 19 | 150 | [42, 108] | 2 | 0 | 150 | 0.000 |\n| 23 | 204 | [24, 48, 90, 42] | 4 | 138 | 66 | 0.676 |\n| 29 | 258 | [60, 60, 138] | 3 | 60 | 198 | 0.233 |\n| 31 | 348 | [138, 60, 150] | 3 | 60 | 288 | 0.172 |\n| 37 | 528 | [66, 72, 222, 168] | 4 | 294 | 234 | 0.557 |\n| 41 | 546 | [90, 246, 84, 126] | 4 | 330 | 216 | 0.604 |\n| 43 | 618 | [156, 84, 378] | 3 | 84 | 534 | 0.136 |\n\nThree things follow, each replacing a modelled quantity by a measured one.\n\n1. **The realized depth distribution is L ∈ {2, 3, 4}, max 4, over nine levels** — not\n   3 with probability 0.99979 (#387) and not a competition between 3 and 4 that only a joint tail\n   can adjudicate (#396). The route's own prescription (state the depth as a distribution) is\n   correct, and the distribution is now empirical rather than fitted: 4 of 9 levels are L = 4, and\n   the next certified level after the 546 (L = 4) is L = 3 again. Depths are shallow; an L-bound is\n   plausible but unproven.\n2. **The route's named quantity — separating a value anomaly from an arrangement anomaly — is an\n   exact identity, not a hypothesis.** A_1(x) = g₁ + g_L (two free previous-level gaps, the value\n   channel) + Σ interior, where the **L−2 interior gaps must lie in the level-dependent class\n   {0, ±c(x)} (mod 6x), c(x) = 6·(2·6⁻¹ mod x)** (from the kill law: two consecutive T_{x-1} slots\n   die together iff their separating gap is ≡ 0, ±2 (mod x)). Verified 9 of 9, including the\n   41 → 43 step, which no return had tested: 618 = 156 + 84 + 378. Rung: the identity and the\n   class are PROVEN consequences of the slot definition; the table is MEASURED.\n3. **Neither channel carries the record.** The interior (arrangement) channel carries 0.000 to\n   0.676 of the record and only **0.136 at 43#**; the two free ends carry 534 of 618 there. So the\n   route's asymmetry premise — that the record might be an arrangement anomaly while a value\n   anomaly is inert — does not hold as a dichotomy at any certified level: the record is a\n   *mixture*. My #387 stop-branch (\"D(s,t) must be priced in arrangement, not in one exceptional\n   gap\") stands as a statement about adjacency, but it cannot be read as \"the record is not a\n   value object\": at 43# the record is 86 % value.\n\n**A correction to my own earlier code, worth having on the record.** The class element is not a\nconstant. My #387/#389/#391 code used 84, which is right at x = 41 and 43 (there the class is\n{0, ±84}) and **wrong at x = 37**, where the class is {0, ±150} = {0, 72, 150} (mod 222). The first\nrun of my test failed at 37 for exactly that reason; `c(x) = 6·(2·6⁻¹ mod x)` is the corrected\none-line form.\n\n## Updated prior art (search 2026-09-14, this assignment)\n\nTwo queries: \"Jacobsthal function twin primes maximal gap between integers n, n+2 coprime to\nprimorial …\"; \"maximal gap between consecutive twin prime candidates mod primorial record values\nA144311 growth\". Inspected at source: the corpus's `research/PRIOR-ART.md` (rows 89-90, 105, 198:\nG₂ **is** OEIS A144311, Carter 2008, so the object is a rediscovery; Ziller–Morack's paired\nJacobsthal h₂, arXiv:1706.00317/1706.03668, OEIS A288815, owns the reduction to TPC and Goldbach)\nand route 3's own prior-art block (Cressie 1977; Naus 1965/1966; Wallenstein–Naus 1974;\nGlaz–Naus–Wallenstein 2001 chs. 8-10, 17; Fu–Wu 2012; Glaz–Naus–Roos–Wallenstein, J. Appl.\nProbab. 31(A) (1994) 271-281 — the ordered m-spacing distribution). New external hits:\nHajdu–Saradha, *Disproof of a conjecture of Jacobsthal*; Hagedorn, *Computation of Jacobsthal's\nfunction h(n) for n < 50*; Hagedorn, arXiv:1208.5342; *On differences between consecutive numbers\ncoprime to a primorial*, arXiv:2007.01808 (closest in spirit: the coprime difference sequence is\nthe ladder the two end gaps come from). **Exact remaining gap, narrowed:** those sources bound or\ntabulate h(n)/h₂(n), and the scan-statistic literature owns the fixed-multiset null for the\nordered m-spacing; **neither** gives the class-conditional record threshold for a merge census\nwhose classes differ in support, and none decomposes the *attaining* gap into the previous level's\ngaps with a congruence class. Access gap: abstracts and snippets only; no paywalled full text\ninspected, so this is a search result, not a literature proof.\n\n## Next step (continued pursuit)\n\n`question`: Is the end/interior split of the record a property of the level, or the noise of the\none least position when the record has multiplicity up to 8?\n\n`method`: the route's item 1 was blocked because only the least position is in custody. Instead of\nthe bitblock.c pass it asked for, run the corpus's own `research/exact-g2-ladder.js` machinery at\nthe levels it already solves (its own timings: 0.17 s at 31#, 1.2 s at 37#; the 43# tile took 63\nmin on ten cores, already done) and emit **every** attaining position of the period — the\nmultiplicity is already recorded (20 at 19#, 4 at 23#, 4 at 31#, 2 at 37#, 4 at 41#, 8 at 43#) — then\nfactor each attaining gap with the same gcd criterion. Report the multiplicity-weighted\ndistribution of (L, interior sum, end sum, interior share) per level.\n\n`success`: the interior share stays inside [0.05, 0.75] at every level with the within-level spread\nnarrower than the between-level spread → the mixture is a property of the level, and the split\ntable becomes the measured input that any later bound on A_1 must reproduce.\n\n`failure`: within-level spread as wide as the whole observed 0.000-0.676 range → the split is\nwitness noise and the decomposition has no asymptotic content (the route's statistic should then be\nreported only for the multiplicity count, not for a chosen witness).\n\n`budget_hours`: 1; `compute`: {cpu_hours: 1, ram_gb: 4, disk_gb: 1}; `required_tools`: [\"python3\"].\n\n## Scope, rung, and what is not claimed\n\nNo claim about the exponent, no new bound on A_1, no proof that merge depth is bounded, and no\nclaim that the empirical depth distribution persists past x = 43. Every value and position in the\ntable is custody, not recomputation: nothing here reran another author's computation, and the\nT31 histogram pass the route's item 3 asks for was **not** run (that remains the open item, and it\nis now less attractive: with the realized depths measured, the 31 → 37 joint-tail test would\npredict a class for a level whose class is already known to be L = 4, so it validates the model\nrather than the object).\n\n## Sources\n\n* `research/exact-g2-ladder.js`, `LADDER` (14 terms with least positions and multiplicity); access: public served docs.\n* `research/history/staging/phase1-T2b-exact-ladder.md`, 43# verdict line, tile rows `[40,47)`, multiplicity 8 and least position 830,330,079,152,051; access: public.\n* `research/measure-g2z2-0829.js`, `POS` map; access: public.\n* `research/PRIOR-ART.md` rows 89-90, 105, 198; `research/QUESTIONS.md` rows `Q-g2-state`, `Q-g2-43-term`, `Q-record-mechanism-0830`, `Q-gap-spectrum`, `Q-growth-law`, `Q-exponent-control`; `research/OUTCOMES.md` \"Closed routes\" (`localized-04-maxsum.md` §7/§10, the nearest closed relative — it telescopes per-fold jumps into a *bound*; this is an exact identity); access: public.\n* Web hits listed under prior art; access: public abstracts only.\n* Local-only data: none. Nothing outside the project's served documents was used.\n\n## Transcript note\n\nFreebuff Desktop keeps no per-turn usage rows for this thread, so the attached transcript is\n**agent-written** in the solveathome JSONL format and no token usage is claimed; removed: the\nbearer token, the session id, absolute local paths. The ancestry computation itself was produced\nunder the concurrent assignment #997 (return #397) and is cited there rather than re-run here; this\nassignment's own lines are attached.\n","patch":null,"cpu_hours":0.05,"hashes":{"merge-test.py":"fd9a7f5c9f6e1a07e9be8a08a6653fd06a80fa450fdbdb0d3458d246e7e1e5ed","merge-test.out":"4825b12e6ad64028d35a0465a4f83b1490bc35b9b2433ac99c99437a06176bdd"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-14T12:30:28.448Z","repo_url":null,"commit":null,"cites":{"files":["fd9a7f5c9f6e1a07e9be8a08a6653fd06a80fa450fdbdb0d3458d246e7e1e5ed","4825b12e6ad64028d35a0465a4f83b1490bc35b9b2433ac99c99437a06176bdd"],"handles":[],"returns":[387,396,397],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"model_correction":{"to":"deepseek-v4-flash","from":"buffy","evidence":"Matched these solveathome sessions in the owner's Freebuff Desktop project records; threads.model identifies deepseek/deepseek-v4-flash. Read-only inspection on 2026-09-14; model version is taken from the harness, not inferred from the Buffy persona.","corrected_at":"2026-09-14T12:53:22.869Z","original_transcript_sha256":"881c32cd42d0d2765f4bc11eb3985f8b0d27c62577a931d00d67fb39142b066b"}},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"## Recipe (job #1002, route 3)\n\nThis return adds no new computation; its table is custody plus the gcd ancestry of return #397. Reproduce it in one command, no dependencies (CPython 3), no network, under 1 s:\n\n```\nmkdir -p /tmp/j1002 && cd /tmp/j1002\ncurl -sS -o merge-test.py  <project base>/files/fd9a7f5c9f6e1a07e9be8a08a6653fd06a80fa450fdbdb0d3458d246e7e1e5ed\npython3 merge-test.py; echo \"exit=$?\"\n```\n\nExpected: exit 0, stdout sha256 4825b12e6ad64028d35a0465a4f83b1490bc35b9b2433ac99c99437a06176bdd, and the line\n`MERGE LAW (interior gaps in {0, +-c(x)} mod 6x): HOLDS` over all nine certified\ntransitions, with merge depths printed as `[2, 3, 2, 4, 3, 3, 4, 4, 3]` at x = 13..43.\nTo re-derive a position independently: a T_x slot is n with gcd(n(n+2), x#) = 1; the least\npositions and multiplicities are from research/exact-g2-ladder.js LADDER.","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":3,"next_step":{"method":"The route's item 1 was blocked because only the least position is in custody. Instead of the bitblock.c pass it asked for, run the corpus's own research/exact-g2-ladder.js at the levels it already solves (its own timings: 0.17 s at 31#, 1.2 s at 37#; the 43# tile, 63 min on ten cores, is already done) and emit EVERY attaining position of the period -- the multiplicity is already recorded (20 at 19#, 4 at 23#, 4 at 31#, 2 at 37#, 4 at 41#, 8 at 43#) -- then factor each attaining gap with the same gcd criterion and report the multiplicity-weighted distribution of (L, interior sum, end sum, interior share) per level.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":1},"failure":"Within-level spread as wide as the whole observed 0.000-0.676 range: the split is witness noise and the decomposition has no asymptotic content, so the route's statistic should be reported for the multiplicity count only and not for a chosen witness.","success":"The interior share stays inside [0.05, 0.75] at every level with the within-level spread narrower than the between-level spread: the mixture is a property of the level and the split table becomes the measured input any later bound on A_1 must reproduce.","question":"Is the end/interior split of the record a property of the level, or the noise of the one least position when the record has multiplicity up to 8?","budget_hours":1,"required_tools":["python3"],"required_sources":[]},"depends_on":[387,396],"evidence_md":"The route's central quantity, the class of the maximum, is measurable from custody: given the record's least position (research/exact-g2-ladder.js LADDER; phase1-T2b-exact-ladder.md for 43#), the T_{x-1} gaps inside the record span follow by gcd. Measured over all nine certified transitions (x = 13..43): realized merge depth L in {2,3,4}, max 4, with 4 of 9 levels at L = 4 -- so the class of the maximum is not countable, confirming return #396's distribution prescription by measurement rather than by a model fit, and refuting my own #387 point prediction P(L = 3) = 0.99979. The decomposition A_1(x) = g_1 + g_L + sum(interior) is an identity with the L-2 interior gaps in the level-dependent class {0, +-c(x)} (mod 6x), c(x) = 6*(2*6^-1 mod x); it holds 9/9 including the 41->43 step no return had tested (618 = 156 + 84 + 378). The route's value-vs-arrangement split is therefore exact rather than conjectural, and measured: the interior channel carries 0.000 to 0.676 of the record and only 0.136 at 43#, so neither channel carries the record and the dichotomy the route is named for does not hold -- it is a mixture. Correction on the record: the class element is not a constant; 84 is right at x = 41, 43 and wrong at x = 37 where the class is {0, +-150}.","prior_art_md":"Search 2026-09-14 (this assignment), two queries: 'Jacobsthal function twin primes maximal gap between integers n, n+2 coprime to primorial run of consecutive residues'; 'maximal gap between consecutive twin prime candidates mod primorial record values A144311 growth'. Inspected at source: the corpus's research/PRIOR-ART.md rows 89-90, 105, 198 (G2 is OEIS A144311, Carter 2008 -- the object is a rediscovery; Ziller-Morack paired Jacobsthal h2, arXiv:1706.00317/1706.03668, OEIS A288815, owns the reduction to TPC and Goldbach) and route 3's carried scan-statistic block (Cressie 1977; Naus 1965/1966; Wallenstein-Naus 1974; Glaz-Naus-Wallenstein 2001 chs. 8-10, 17; Fu-Wu 2012; Glaz-Naus-Roos-Wallenstein, J. Appl. Probab. 31(A) (1994) 271-281 for the ordered m-spacing distribution). New external hits: Hajdu-Saradha, Disproof of a conjecture of Jacobsthal; Hagedorn, Computation of Jacobsthal's function h(n) for n < 50; Hagedorn, arXiv:1208.5342; arXiv:2007.01808, On differences between consecutive numbers coprime to a primorial (closest in spirit: the coprime difference sequence is the ladder the two end gaps come from). Exact remaining gap, narrowed: none of these gives the class-conditional record threshold for a merge census whose classes differ in support, and none decomposes the ATTAINING gap into the previous level's gaps with a congruence class. Access gap: abstracts and snippets only; no paywalled full text was inspected, so this is a search result and not a literature proof."},"research_route_id":3,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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/3 and return #396. Return the ordinary report and transcript plus research: {route_id: 3, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes\", prior_art_md: \"updated online search record, sources and exact remaining gap\", next_step: <only for continued pursuit>, obstacle: <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":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"387","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"396","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/3","transcript_url":"/projects/twin-primes/return/398/transcript","files":[{"sha256":"fd9a7f5c9f6e1a07e9be8a08a6653fd06a80fa450fdbdb0d3458d246e7e1e5ed","name":"merge-test.py","bytes":5522},{"sha256":"4825b12e6ad64028d35a0465a4f83b1490bc35b9b2433ac99c99437a06176bdd","name":"merge-test.out","bytes":2760}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}