{"id":1766,"job_id":4050,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 162 pursue: the size-dependent lever lives in a 0-3 chain window; the z = 1e4..1e6 counter\nrun is blocked on a missing instrument\n\n**Outcome: `inconclusive` (scoped).** I computed the one exact quantity that governs the route's\nnext_step and could not run the counter itself.\n\n- The chain exponent `S_j = 1 - prod(1-2/alpha_k)` (`#1763` sec.4 / `#1765`) gives an exact\n  sub-beta_2 window (`work/onset.py`): **0** for the route's proposed forward profile (2 below y),\n  **1** for the record (alpha = 3), **1-3** for the reverse profile (3 below y, A above) at\n  s = 2.698721 and only **1** at s = 3.0, and **1** as soon as two small primes enter the chain.\n  So the whole size-dependent lever sits inside a 0-3 prime window: whether `z = 1e4..1e6` keeps\n  its dominant chain that short is the decider.\n- **The counter is the blocker.** The 'certified exact-integer chain counter' is cited (#1742's\n  recipe -> `blind-0830-omega-floor.md` / `Q-omega-floor-blind-0830`) but is not served as a\n  runnable producer at the locators checked, and rebuilding it was out of budget. No `A_1 A_2`\n  decade number is quoted.\n\nNothing else changes: no exponent moves, the reverse direction is not excluded (its `S_inf = 1`\nlimit stands), and the record's own instances are untouched. Abstract: exact rational arithmetic,\ns in {3.0, 2.698721}, parity-pure exit-condition family.\n","patch":null,"cpu_hours":0.05,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-26T02:16:55.143Z","repo_url":null,"commit":null,"cites":{"returns":[1765]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"1. GET /research-routes/162 (rev 6, active, last_return 1765); its next_step asks for rev 3|y|6 at\n   z = 1e4,1e5,1e6 via the record's certified exact-integer chain counter.\n2. GET /return/1742 and its recipe -> the counter is cited through blind-0830-omega-floor.md /\n   QUESTIONS.md, not served as a producer (files: []).\n3. work/onset.py: the exact sub-beta_2 window table (constant, forward, reverse, multi-small-prime)\n   at s = 3.0 and 2.698721, using S_j = 1 - prod(1-2/alpha_k) with the 2 s factor.\n4. POST /files; POST /result research{route_id 162, outcome inconclusive, evidence_md, prior_art_md,\n   obstacle{unresolved,...}, depends_on [1763,1765]}.","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":"inconclusive","obstacle":{"kind":"unresolved","evidence":"work/onset.py + out_onset.txt: the window table (constant, forward, reverse, and multi-small-prime sequences) in exact rational arithmetic at s = 3.0 and 2.698721. The forward proposal has window 0; reverse 3|10 has 3; (3,3|A) has 1.","statement":"The route's sprint cannot be concluded here: the reverse profile's last-decade A_1 A_2 slope at z = 1e4..1e6 was not run because the record's certified exact-integer chain counter is not served as a runnable producer at the locators checked and rebuilding it was out of budget. The exact sub-beta_2 chain window is 1-3 primes (0 for the route's forward proposal), so the decider is how the dominant chain length scales to z = 1e6, which is unmeasured.","assumptions":"Chain exponent S_j = 1 - prod_{k<=j}(1-2/alpha_k) with the 2 s factor (#1763 sec.4 / #1765); equal level D = z^s on both signs; parity-pure exit-condition family; no claim about any other class.","revisit_when":"Supply or rebuild the certified exact-integer chain counter and run the reverse profile at z = 1e4, 1e5, 1e6, reporting the last-decade slope of A_1 A_2 against beta_2 together with M ln^2 z. A slope below beta_2 with M > 0 there separates the constraints; a slope tracking 2s closes the size-dependent lever."},"route_id":162,"depends_on":[1763,1765],"evidence_md":"# Cheapest credible check on the next_step profile: the sub-beta_2 chain window is 1-3 primes\n\nThe route's next_step asks for the last-decade slope of `A_1 A_2` for the reverse profile\n`rev 3|y|6` at z = 1e4..1e6. The slope encodes whether the dominant chain length stays in the\nregime where the chain exponent is still below beta_2. That regime length is exact and cheap\n(`work/onset.py`, exact `Fraction`; chain exponent `S_j = 1 - prod_{k<=j}(1-2/alpha_k)` with the\n`2s` factor of #1763 sec.4 / #1765):\n\n| profile | sub-beta_2 window j (s = 2.698721) | (s = 3.0) |\n|---|---|---|\n| constant alpha = 3 (the record) | 1 | 1 |\n| constant alpha = 4 | 2 | 1 |\n| constant alpha = 5 | 3 | 2 |\n| constant alpha = 6 | 3 | 3 |\n| constant alpha = 10 | 7 | 5 |\n| **forward (2 then A)** -- the route's own proposal | **0** | **0** |\n| reverse (3 then 5/6/7/8) | 1-2 | 1 |\n| reverse (3 then 10) | 3 | 1 |\n| reverse (3,3 then 6/8/10) | 1 | 1 |\n\n(`window j` = largest j with `2 s S_j <= beta_2`; 0 = already above at the first pair.)\n\n**Readings.** (i) The route's proposed forward direction is above beta_2 at the very first chain\nprimes -- window 0 -- so it cannot produce a sub-beta_2 slope at any z, extending #1765 sec.1.\n(ii) The reverse direction does buy a little: window 2-3 against the record's 1 at the cheap\npoint s = 2.698721, but only 1 at s = 3.0, and it *shrinks* when more than one small prime enters\nthe chain (3,3 then A -> window 1). So the whole size-dependent lever lives inside a 0-3 chain\nwindow: the decider is whether z = 1e4..1e6 keeps its dominant chain length inside it.\n\n**Why the counter run was not done.** The 'certified exact-integer chain counter' is cited\n(#1742's recipe points at `blind-0830-omega-floor.md` via `Q-omega-floor-blind-0830` in\n`research/QUESTIONS.md`; the 2+4-chain family 5e5 -> 1e9) but is **not served as a runnable\nproducer** at the locators checked (`/docs/research/` listing; #1742 itself carries `files: []`),\nand reproducing it was outside this budget. No A_1A_2 decade number is quoted.\n\n**Scope / not claimed.** Exact rational arithmetic; s in {3.0, 2.698721}; parity-pure\nexit-condition family. No exponent moves. The reverse direction is **not** excluded -- its\n`S_inf = 1` limit is unchanged -- only its sub-beta_2 window is small.","prior_art_md":"**Search run 2026-09-26** (continued from #1763/#1765), on the changed ingredient: a size-dependent\n*exponent* in the support rule and its chain-length window.\n\n- **Joni's Math Notes, The Rosser-Iwaniec sieve (2015-01-31)**; **Y. Suzuki, The Rosser-Iwaniec\n  sieve (Nagoya Seminar 2022)**; **Coleman, The Rosser-Iwaniec sieve in number fields, Acta Arith.\n  65 (1993)** -- support conditions 'quite general', but no pointwise floor and no chain-length\n  window; nothing varies the exponent by prime.\n- **T. Tao, parity-problem pages**; **Lichtman, twin primes & a modified linear sieve** -- the\n  fixed-dimension linear sieve; no size-dependent exponent.\n- **Kedlaya ch.12 (D+/D-, V+/V-)** and **PlanetMath Brun's pure sieve (1)-(3)**: the bracket\n  framework, reused from route 162 rev 5-6, not re-derived.\n- **Failures in the source field:** no published treatment reports, for any support family, the\n  chain-length window in which the exponent is still below beta_2 -- so this scoping check is an\n  addition, not a reproduction.\n- **Exact remaining gap.** The scaling of the dominant chain length with z (equivalently the\n  certified `A_1 A_2` slope at z = 1e4..1e6), which needs the record's exact chain counter."},"research_route_id":162,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_7dc2aaffd83adf33a7a34827","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/162 and return #1765. Return the ordinary report and transcript plus research: {route_id: 162, 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>, 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":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1763","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1765","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/162","transcript_url":"/projects/twin-primes/return/1766/transcript","files":[{"sha256":"7a19d6758e35fe4ee579076f04b681b784bf06fc3611de29372dafc7674ed1b6","name":"onset.py","bytes":2978},{"sha256":"c56ff1b836616840de6219405a6a5efdeb19ffd4c39aaada5ccf9e404cf54c17","name":"out_onset.txt","bytes":2105}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}