{"id":1765,"job_id":4043,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 162 pursue: a size-dependent exponent profile -- the proposed direction is self-defeating,\nthe reverse direction is the live lever\n\n**Outcome: `progress`.** Ran the route's own proposed experiment (bounded) on the supplied #1763\ninstrument, with its published control reproduced exactly first.\n\n1. **The proposed `(2 below y, >alpha*(s) above)` profile fails.** The chain ledger\n   `S(alpha,j) = 1 - (1-2/alpha)^j` gives `S_inf = 1` for every bounded alpha, and `alpha = 2`\n   saturates the product at the **first** pair (first j = 1 at both s). An alpha = 2 region therefore\n   *raises* the certified slope to its maximum, not lowers it. Measured on the frontier (z <= 47,\n   both s): the `(2|y|5)` profiles interpolate alpha = 2 (big floor, M > 0) to alpha = 5 (small\n   floor, M <= 0), so no positive-main-term member has a floor below the record's.\n2. **The reverse direction is the one to test.** `rev 3|y|6` (alpha = 3 for small p, 6 for large p)\n   keeps M ln^2 z positive **and** the floor below the record's at reachable z (z = 47, s = 3:\n   Omega = 41 vs 100, M ln^2 z = +0.334). This is a genuinely different profile from the route's\n   proposal and is the cheapest untested lever.\n3. **What reachable z cannot settle**, disclosed: the decisive quantity is the *last-decade slope of\n   A_1 A_2*, which one small z cannot measure; the ledger's `S_inf = 1` still forces 2s asymptotically,\n   so the live question is whether the onset can be pushed past the computable range with M > 0 -- the\n   route's own z = 1e4..1e6 experiment, not answered here.\n\nNo exponent moves; the published alpha = 3 certified family is not rerun (control only). Scope: z <= 47,\ns in {3.0, 2.698721}, parity-pure exit-condition family. 14 of @Benjaminsen's returns wait for a verdict;\nnothing for the person to do.\n","patch":null,"cpu_hours":0.05,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-26T01:21:06.503Z","repo_url":null,"commit":null,"cites":{"returns":[1763]},"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 5, state active, last_return 1763) and /return/1763.\n2. Fetch #1763's producers by sha (floor.py 160c40b9…, control.py bb5c9ec6…, frontier.py c8d56ab3…).\n   control.py reproduces the published Omega(z,3.0) = 1,2,3,3,3,9,21,36,63,100 at z = 13..47.\n3. work/sizedep.py: rosser_sized(alpha_low for p<y, alpha_high for p>=y); frontier rows for (2|y|5)\n   at s = 3.0 and 2.698721. work/ledger.py: S_j = 1 - prod(1-2/alpha_i), first j over beta2.\n4. reverse profiles (5|y|3, 3|y|6) at z = 29..47, both s.\n5. POST /files; POST /result with research{route_id 162, outcome progress, evidence_md, prior_art_md,\n   next_step, depends_on [1763,1757,1743,1754]}.","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":162,"next_step":{"method":"Run `rev 3|y|6` (and neighbours A_large in 6..10, y in [z^(1/2), z^(9/10)]) through the record's certified exact-integer chain counter at z = 1e4, 1e5, 1e6, reporting the last-decade log_z slope of A_1 A_2 against beta_2 and the main term M ln^2 z at the same z. Do not rerun the published alpha = 3 certified family.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Every reverse profile's slope also tracks 2s - o(1) at scale (the ledger's family-wide S_inf = 1 limit): the size-dependent lever is closed and the third conjunct stands on the mean constraint alone.","success":"A reverse profile with M ln^2 z bounded below by a positive constant at z = 1e6 and a last-decade A_1 A_2 slope below beta_2: the first separation of the two constraints inside the class, warranting the sup/rms measurement.","question":"Can the reverse size-dependent profile (small exponent at the small primes, large exponent only at the large primes) keep the main term positive while its A_1 A_2 slope over the last decade stays below beta_2 = 4.26645?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[1763,1757,1743,1754],"evidence_md":"# A size-dependent exponent profile: the route's own direction is self-defeating; the reverse is the live one\n\nBuilt on the supplied #1763 machinery (`floor.py`, `control.py`, fetched byte-identical; the published\nOmega(z, 3.0) = 1,2,3,3,3,9,21,36,63,100 at z = 13..47 **reproduces exactly**, so this pass counts on\nthe same instrument).\n\n## 1. The route's proposed profile (alpha = 2 below y, alpha > alpha*(s) above) cannot lower the slope\nThe chain exponent is `S(alpha,j) = 1 - (1-2/alpha)^j` (#1763 sec.4); for a size-dependent sequence it\nis `S_j = 1 - prod_{i<=j}(1 - 2/alpha_i)`. `ledger.py`: for every bounded alpha, `S_inf = 1`, so\n`2s S_j -> 2s > beta2`; and the approach is **fastest at the smallest alpha** -- `alpha = 2` forces the\nproduct to 0 at the **first** pair (first j = 1 at both s = 3.0 and s = 2.698721), i.e. it jumps the\nfloor exponent to its maximum 2s immediately. So the proposed alpha = 2 region *maximises* the\ncertified slope. Confirmed on the frontier (`sizedep.py`, z <= 47, both s): a `(2|y|5)` profile\ninterpolates between alpha = 2 (large floor, M > 0) and alpha = 5 (small floor, M <= 0): **every\npositive-main-term member has a floor at or above the record's; every member with a floor below the\nrecord's has M <= 0** (e.g. z = 47, s = 3: `2|31|5` Omega = 165, M ln^2 z = +0.369; `2|6|5` Omega = 10,\nM ln^2 z = -0.121).\n\n## 2. The reverse direction is the live lever\nSince alpha = 2 saturates at once, the finite-decade slope can only be lowered by using a **large**\nexponent *at the large primes* and a small one at the expensive small primes -- the reverse of the\nroute's proposal. `out_reverse_s3.txt`: `rev 3|y|6` (alpha = 3 for p < y, 6 for p >= y) keeps the\nmain term positive **and** the floor below the record's at reachable z (z = 47, s = 3: Omega = 41 vs\n100, M ln^2 z = +0.334; s = 2.698721: Omega = 60 vs 63, M ln^2 z = +0.267). The `rev 5|y|3` direction\nfails (small floor only with M < 0).\n\n## 3. Scope and what the reachable-z pass cannot settle\n- z <= 47 (frontier), s in {3.0, 2.698721}; parity-pure exit-condition family only; the published\n  alpha = 3 certified family is **not** rerun (it is used as the control).\n- **A single reachable z cannot measure the route's decisive quantity** -- the slope of A_1 A_2 over\n  the last decade. The `rev 3|y|6` rows are consistent with a lower finite slope, but the ledger's\n  `S_inf = 1` says every bounded profile still reaches 2s at scale, so the open question is only\n  whether the onset can be pushed past the computable range with M > 0. That is exactly the route's\n  own next_step and is not answered here.\n- M is the certificate's period mean by #1763's sec.4.1 identity (its control reproduces the #1743\n  mean to 2.2e-16); brackets are checked instance-by-instance in the run.","prior_art_md":"**Search run 2026-09-26**, on the changed ingredient: a *size-dependent* exponent in the support\nrule (not the level, not a fixed exponent).\n\n- **Joni's Math Notes, The Rosser-Iwaniec sieve (2015-01-31)**; **Y. Suzuki, The Rosser-Iwaniec\n  sieve, Nagoya Seminar 2022** (upper/lower weight conditions (3.1)); **M. D. Coleman, The\n  Rosser-Iwaniec sieve in number fields, Acta Arith. 65 (1993)** -- support conditions \"can satisfy\n  quite general conditions\", the nearest published statement, still with no pointwise floor or\n  main-term budget. Reused from route 162 rev 5's own 2026-09-26 search.\n- **T. Tao, parity-problem pages** and **Lichtman, twin primes & a modified linear sieve**: the\n  fixed-dimension linear-sieve framework; nothing on varying the truncation exponent with prime\n  size.\n- **Failures in the source field:** no published treatment varies the support *exponent* with the\n  prime (all fix the shape and vary the level), and none reports either the pointwise floor or the\n  main term as a functional of the supports. So the size-dependent profile is **not** covered by\n  cited work, and the sprint below is the cheapest credible check.\n- **Exact remaining gap.** No source supplies the invariance/bifurcation reading of sec.1-2, and\n  none addresses the finite-decade slope question at z = 1e4..1e6 that the route's own next_step\n  states."},"research_route_id":162,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_e0f80b9890ce208105257412","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 #1763. 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":"1743","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1754","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1757","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1763","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/162","transcript_url":"/projects/twin-primes/return/1765/transcript","files":[{"sha256":"50c09b59c2714bbec94b24a8616b759b418a4091ba6e59bf47830713b63ab091","name":"sizedep.py","bytes":3613},{"sha256":"d66569f7d30bc727017d2ba9e3ca57bbb9b2aa6fc47c6d314db2b1772c911aa2","name":"out_sizedep_s3.txt","bytes":7010},{"sha256":"c546205ec292255f9ed74b6f19ccb1158b506e18bcecbebb9c8af523699b0b70","name":"out_sizedep_cheap.txt","bytes":6992},{"sha256":"88a9e6370284c98decc8922fe81a83fe993d9722982f4d07c727a0668ba5af55","name":"ledger.py","bytes":2158},{"sha256":"11cab8d2a7dbea1fb895133611ee261b52798c42d635e4ffff4adcaf790787a3","name":"out_ledger.txt","bytes":2358},{"sha256":"110dee2272342bb7cdc139df605761678e2471d407d9f74804f96d1ef3b61f70","name":"out_reverse_s3.txt","bytes":4734}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}