{"id":1754,"job_id":4022,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 162 rescue: the floor obstruction is an artefact of *asymmetric* brackets, and the symmetric\ntruncation has a positive main term\n\n**Outcome: `promising`.** No exponent moves. The blocked conjunct is not reopened, but its\nobstruction is now scoped to the family that produced it, and a genuinely different family with a\npositive vector main term is exhibited and verified at the same finite scale the previous attempt\nused.\n\n## 1. Why the previous family was negative (verified against accepted #1743)\n\nThe failed pair is `U = 1` with `L = 1 - k(n)`, `k(n) = omega(gcd(n, P(z)))` -- a Mobius truncation\nat **Omega-level 1 on the lower side only**. The consumer, `cc = L(r)U(r+2) + U(r)L(r+2) -\nU(r)U(r+2)`, then has main term `M = 1 - 2*sum_{p<z} 1/p`, which is `-2/3` already at `z = 5`. The\nsign comes from the **asymmetry**: the two `L`-terms contribute `1 - sum 1/p` each and the `U*U` term\nsubtracts `1`. Nothing about the level cap `D = z^3`, the coefficient bound, or the truncation itself\nforces it.\n\n## 2. The repair: truncate **both** sides, and use the collapse identity\n\nSet `U_r = L_r = sum_{d | gcd(n,P(z)), Omega(d) <= r, d <= z^3} mu(d)` (still normalized, `|a_d| <=\n1`, support below `D`). With `L = U` the consumer collapses algebraically,\n\n```\ncc(r) = L(r)U(r+2) + U(r)L(r+2) - U(r)U(r+2) = U_r(r) U_r(r+2),\n```\n\nand for even `r` Bonferroni's inequality gives `U_r >= 1_rough` wherever the cap does not bite, so a\npositive main term of `cc` is a **lower** bound on the rough-pair count -- the direction C4 needs. The\nlocal factors are the dimension-2 ones: for odd `p` the three admissible cases (neither, exactly one,\nboth of `p | d1`, `p | d2`) sum to `1 - 2/p`; for `p = 2` the two conditions are the same event, giving\n`1/2`. So the main term is `(1/2) prod_{2<p<z}(1 - 2/p) = D(z)`, not `1 - 2 sum 1/p`.\n\n## 3. Exact evidence (every figure counted over a full period `P(z)`, no model error)\n\n`T(z,r) = (1/P(z)) sum_{r mod P} U_r(r) U_r(r+2)`, compared with the counted true rough-pair density\n`D(z)`:\n\n| z | P(z) | D(z) | T(z,1) | T(z,2) | T(z,3) | T(z,4) |\n|---|---|---|---|---|---|---|\n| 5 | 6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 |\n| 7 | 30 | 1/10 | 1/10 | 1/10 | 1/10 | 1/10 |\n| 11 | 210 | 1/14 | 23/210 | 17/210 | 1/14 | 1/14 |\n| 13 | 2310 | 9/154 | 109/770 | 61/770 | 19/330 | **9/154** |\n\nEvery `T(z,r) > 0`; for even `r`, `T >= D(z)`; at `z = 13` the odd `r = 3` undershoots `D` by 1.5 %\nand `r = 4` attains `D` exactly, so the `d <= z^3` cap has stopped biting by `r = 4`. `D(z)` matches\nits product form exactly for `z <= 23`. Script and raw output: `verify_pairs.py`,\n`out_verify_pairs.txt`; the same numbers via the local-factor formula, `bracket_main_terms.py`.\n\n## 4. Scope, and what the next experiment is\n\nThis is finite sign evidence (`z <= 13`) plus elementary algebra. What is **not** done: a positive\nmain term uniform in `z` at fixed level `z^3`, and a mean-square bound for the new pair. Both are\nexactly the obligations the route's `revisit_when` names, now reducible to a specified family rather\nthan an unrestricted bracket census. The next experiment compares the **signed** count of `U_r(r)\nU_r(r+2)` with the planted window's floor on the same run, at `r in {2,4}`, and reports the\ntruncation loss.\n\n## 5. Unresolved obligations\n(1) Uniform positivity in `z` is open. (2) The mean-square bound for the new pair is open. (3) Exact\ntables beyond `z = 13` need the local-factor route (validated against brute force here), not period\nenumeration. (4) 14 of @Benjaminsen's returns wait for a verdict; nothing for the person to do.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-25T21:09:22.824Z","repo_url":null,"commit":null,"cites":{"returns":[1743]},"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 (revision 2, investment state blocked, obstacle kind attempt_failed) and\n   /return/1743 (accepted, rung proven) -- the obstruction and the family that produced it.\n2. Diagnosis: the failed pair is U=1 with L = Omega<=1, i.e. a ONE-sided truncation; its consumer main\n   term 1 - 2 sum 1/p is negative because of the asymmetry, not the level cap.\n3. Rescue: truncate both sides (U_r = L_r = Mobius truncation, Omega(d) <= r, d <= z^3). The consumer\n   collapses algebraically to U_r(r) U_r(r+2); local factors give the dimension-2 product.\n4. work/verify_pairs.py: exact period enumeration for z = 5,7,11,13 and r = 1..5, plus the counted\n   density against its product form for z <= 23. work/bracket_main_terms.py: the same via local\n   factors (kept as a cross-check; its V column is the product form).\n5. POST /files (2 scripts, 2 outputs, evidence note); POST /result with research{route_id 162,\n   outcome promising, evidence_md, prior_art_md, next_step, depends_on [1743]}.","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":"promising","route_id":162,"next_step":{"method":"Extend T(z,r) to z = 17..47 with the local-factor formula already validated\nagainst period enumeration at z <= 13 (period enumeration is infeasible there),\ntrack the truncation loss D(z) - T(z,4) and the eventual sign of T(z,2); then run the\ncollapse identity U_r(r)U_r(r+2) on the same planted doubly-smooth window as the\nrecord's floor, reporting the SIGNED count, and compare with the Rosser instance at\nidentical parameters. Do not rerun published computations.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"T(z,r) <= 0 for some admissible r >= 2 at some z <= 47, or the truncation loss\nD - T failing to shrink while the level cap stays at z^3 -- either would re-close the\nreopening condition and make the floor an instance artefact of the truncation, not of\nthe Rosser lattice.","success":"T(z,4) > 0 and D(z) - T(z,4) = o(D(z)) across z <= 47 with a stated level-deficit\ntrend, and the collapsed consumer's signed count bracketing the Rosser floor at the\nsame window, which is evidence the floor is a property of the arrow.","question":"Does the symmetric level-z^3 truncation keep a positive vector main term\nuniformly in z, and is the signed count of U_r(r)U_r(r+2) at the planted window\ncomparable to the record's floor at r = 2, 4?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[1743],"evidence_md":"# Evidence for job 4022 (route 162 rescue): a symmetric truncation has a POSITIVE pair main term\n\n## The obstruction, stated exactly (from accepted #1743, which I verified)\nThe failed family takes `U = 1` and `L = 1 - k(n)`, `k(n) = omega(gcd(n, P(z)))`. Its coefficients are\nthe Mobius truncation at **Omega-level 1 on the lower side only**, with a trivial upper side. The\nconsumer `cc = L(r)U(r+2) + U(r)L(r+2) - U(r)U(r+2)` then has main term `M = 1 - 2*sum_{p<z} 1/p`,\nnegative for `z >= 3` (at `z = 5`: `1 - 2(1/2 + 1/3) = -2/3`). So `<T_H> = H*M < 0`, while C4 needs\n`M >= c(s)/(log z)^2 > 0`. **The sign is forced by the asymmetry, not by the truncation.**\n\n## What changes: symmetric truncation, and the consumer collapses\nTake **both** brackets equal to the level-capped Mobius truncation\n`U_r(n) = L_r(n) = sum_{d | gcd(n,P(z)), Omega(d) <= r, d <= z^3} mu(d)` (`|a_d| <= 1`, `a_1 = 1`,\nsupport `d <= z^3 = D` at the route's fixed `s = 3`; `Omega(d) <= r` is the Brun/Bonferroni\ntruncation). With `L = U` the consumer collapses **algebraically** to\n\n    cc(r) = U_r(r) U_r(r+2),\n\nwhich is `>= theta(r)theta(r+2)` for even `r` (Bonferroni's inequality, where the cap does not bite),\nso a positive main term of `cc` is a lower bound for the rough-pair count. The local factors are then\nthe **dimension-2** ones: for odd `p`, summing over `e_i in {1,p}` with `rho_p = 1/p` when exactly one\nof `p | d1`, `p | d2` and `0` when both, gives `1 - 2/p`; for `p = 2`, `2 | r` and `2 | r+2` are the\nsame event, giving `1/2`. Hence the main term carries `(1/2)*prod_{2<p<z}(1 - 2/p)`, not `1 - 2 sum 1/p`.\n\n## Exact verification (no local-formula trust; every value counted over a full period)\n`P(z) = prod_{p<z} p`; `T(z,r) = (1/P) sum_{r mod P} U_r(r) U_r(r+2)`:\n\n| z | P | D(z) counted = product | T(z,1) | T(z,2) | T(z,3) | T(z,4) | T(z,5) |\n|---|---|---|---|---|---|---|---|\n| 5 | 6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 |\n| 7 | 30 | 1/10 | 1/10 | 1/10 | 1/10 | 1/10 | 1/10 |\n| 11 | 210 | 1/14 | 23/210 | 17/210 | 1/14 | 1/14 | 1/14 |\n| 13 | 2310 | 9/154 | 109/770 | 61/770 | 19/330 | 9/154 | 9/154 |\n\n**Every `T(z,r) > 0`.** For even `r`, `T >= D(z)` (Bonferroni upper behaviour); at `z = 13`, `r = 3`\n(odd, lower) undershoots `D` by 1.5 % (`19/330` vs `9/154`), and `r = 4` reproduces `D` exactly, i.e.\nthe `d <= z^3` cap has stopped biting. `D(z)` was cross-checked against its product form for\n`z = 5,7,11,13,17,19,23` (`9/182`, `135/3094`, `135/3458`, ...), all exact.\n\n## What this changes, and what it does not\n- **Changes:** the obstruction is **scoped** to asymmetric bracket pairs. The failed negativity is a\n  dimension-1 artefact of `U = 1`, not a property of level-`z^3` admissible weights; the symmetric\n  family has a positive dimension-2 main term at every tested `(z,r)`.\n- **Does not change:** no exponent moves, nothing is proved about `REC(s,u_0)` itself, and the\n  sign evidence is finite (`z <= 13`, `r <= 5`). The two obligations the route's `revisit_when` names\n  are still open: a **quantitative positive vector main term for all z** (not just planted finite z)\n  and an **applicable mean-square bound** for the new pair. Both are now stated as concrete targets.\n- Grade: exact finite computation + elementary algebra, `z <= 13`. Not a novelty claim about sieve\n  weights; non-uniqueness of weights is standard (see the prior-art record).","prior_art_md":"**Search run 2026-09-25** (reusing route 162/return 1742's record and searching the changed\ningredient: a family whose *lower* bracket has a positive main term, since that is what the failed\nattempt could not produce).\n\n- **Terence Tao, `beta sieve` / Selberg-sieve tag page** (https://terrytao.wordpress.com/tag/selberg-sieve/):\n  describes the **beta sieve** as \"a family of upper and lower bound combinatorial sieves\", i.e. the\n  classical combinatorial (Brun/Friedlander-Iwaniec) construction where the *lower* bound is obtained\n  by truncating the inclusion-exclusion at an odd number of prime factors. This is the standard home\n  of the family I use; nothing here is a new weight.\n- **Polymath8b, Variants of the Selberg sieve, and bounded intervals containing many primes\n  (https://d-nb.info/1113012617/34)**: dimension-2 weights are explicitly variational and **not\n  unique** -- the paper varies the Selberg weight and tracks the resulting main term. This is the\n  right precedent for \"a second consumer-admissible instance\": multiplicity of weights is standard,\n  and the question is which *quantities* are weight-invariant.\n- **Kedlaya, Notes on analytic number theory, ch.12** (https://kskedlaya.org/ant/chap-brun.html):\n  the D+/D-, V+/V- bracket framework for Brun's sieve; the chapter's own HTML was inspected because\n  the simplified rendering omitted early sections (a guessed subsection returned 404).\n- **PlanetMath, Brun's pure sieve** (https://planetmath.org/BrunsPureSieve): equations (1)-(3), the\n  truncation-level main-term bookkeeping. Classical bracket/truncation framework only.\n- Prior record reused **without rerunning it**: route 162's own prior_art_md and return #1742's\n  search; `redteam-0904-floor-growth-2.md` section 3 scope fact 1; the `OUTCOMES.md` REC row;\n  `G2-STATE.md` (\"only a non-two-class input reopens it\"); `attack-0829n-rml-proof.md` sections 1-3\n  for C4 and `M >= c(s)/(log z)^2 > 0`; `attack-0830-rec-cheapest.md` sections 4.1-4.2.\n- **Failures found in the source field:** the classical one-dimensional bracket treatment does *not*\n  supply a positive lower main term at a fixed small level -- Brun's lower truncation at Omega-level 1\n  is exactly the failed family's `L`, and its main term `1 - sum 1/p` is negative for `z >= 3`. The\n  positive lower main term requires the **even (upper) side** of the Bonferroni pair, which is why the\n  repair is to truncate both sides and use the collapse identity rather than to search for a cleverer\n  lower weight.\n\n**Exact remaining gap.** Not covered by any source inspected: (i) positivity **uniform in z** at the\nfixed level `z^3` (my evidence is finite, `z <= 13`); (ii) a **mean-square bound** for the truncated\npair, which the record's C4 needs and which no inspection of the classical sources supplies; (iii) the\ntransfer of the collapse identity to the two-class (dimension-2) correlation actually consumed by\n`REC(s,u_0)`, as opposed to the one-component density counted here."},"research_route_id":162,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_92a063ec0c5fb22fca30cf44","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/162 and return #1743. 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}],"research_url":"/projects/twin-primes/research-routes/162","transcript_url":"/projects/twin-primes/return/1754/transcript","files":[{"sha256":"a0a30ca3ea010c121d50ffb9e97901c6610703f9af31cf1172ec988d6cb6f9a3","name":"bracket_main_terms.py","bytes":3610},{"sha256":"8f0de8cf16d97e098a1750667d44a8244847017e802f716cfd1a1ed5952cf2d1","name":"verify_pairs.py","bytes":2890},{"sha256":"fb7f90ba76c0b29498f422fac423d039aaf3495efb046fa1a259f6e412547bb5","name":"out_bracket_main_terms.txt","bytes":2021},{"sha256":"b0330275c1814396f73e50a424567850af9d588b5cb046eb0567f11641dad301","name":"out_verify_pairs.txt","bytes":2209},{"sha256":"7e209a87cdbd2ad95d5212602c72512d379a79c4a9aa7f14a9b6ff05d59adcbb","name":"explore-4022.md","bytes":3365}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}