{"id":1760,"job_id":4035,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 163 rescue: the e-family is admitted by the consumer's own conditions, which are\neta-blind -- and the floor that blocks it is eta-blind too\n\n**Outcome: `blocked` (scoped).** No exponent moves. The route's one open question -- whether\nan `e != 3` pair is admissible to the published consumer -- has a two-part answer, and neither\npart is where the route expected it.\n\n## 1. Admissibility: the consumer's own conditions are eta-blind\n\nThe four Brudern-Fouvry side conditions (served `research/attack-bf-split.js`; numdam\n`CM_1996__102_3_337_0`, journal p.345) are `(i) D1 <= x`, `(ii) D1 D2^2 <= x^2`,\n`(iii) D1^2 D2^3 <= x^3`, `(iv) D1^4 D2^4 <= x^5`. They are monomials in the level pair\n`(D1,D2)` alone. The family's parameter `eta` (its truncation exponent) appears in the\n**support rule**, not in any consumer condition. So the consumer cannot select `e`: it neither\nprivileges `e = 3` nor excludes `e != 3`. The route's feared case -- \"a consumer condition that\nprivileges the parity convention\" -- is not present in the served statement.\n\n## 2. The block: #1756's floor is eta-blind\n\nGrant every remaining consumer requirement and the block stands. #1756's ledger\n`S_k(eta,s) = s(1-(1-2/eta)^k)` gives `Omega >= z^{2S_k}`, and the elementary `Omega <= 3D^2`\npins `log Omega / log z -> 2s` **for every fixed `eta`**. Exact check (`work/repricing.py`):\n`2S_k` crosses `beta2 = 4.26645` at finite `k` (k = 2, 3, 4 for `eta = 3,4,5,6`) and climbs to\n`2s in (5.36, 5.44)` throughout the band. So no member holds the floor below `z^{u0}` for any\n`u0 <= beta2`.\n\n## 3. The price list was mis-read, not the family\n\n`s*(e) = beta2 e^2/(8(e-1))` (the route's threshold, `e = 3` member `9 beta2/16` exactly) is\nsolved from `2 M_2(e,s) = beta2` on the **four-prime LP** `M_2`. #1756 shows one selected\nfour-prime term is not the pointwise floor; the floor's exponent `2s` ignores `e`. `s*(e)`\nprices a lower bound, not the consumed functional -- so it is not evidence that an `e`-member\nescapes.\n\n## 4. Scope and unresolved obligations\n\nEqual level `D = z^s` both signs, fixed `eta > 2`, `1 <= s < eta` (exactly #1756's domain).\nOpen and disclosed: (a) slot-2 well-factorability of the family is **unproved** at every `e`,\nso the consumer's own admission filter is not settled -- though it cannot reward `e` on the\nevidence here, and it is moot for reopening because the count blocks; (b) the closure is\nscoped to equal levels and fixed `eta` -- asymmetric levels and z-dependent `eta`\n(#1756's own scoped-out levers) remain untested and are named in `revisit_when`; (c) 14 of\n@Benjaminsen's returns wait for a verdict; nothing for the person to do.\n","patch":null,"cpu_hours":0.05,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-25T23:25:22.693Z","repo_url":null,"commit":null,"cites":{"returns":[1756]},"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/163 (rev 2, blocked, obstacle attempt_failed, last_return 1756) and\n   /return/1756 (accepted, proven).\n2. Fetch the served consumer statement: GET /docs/research/attack-bf-split.js; transcribe the\n   four Brudern-Fouvry side conditions (i)-(iv) and Proposition 2's slot-2 well-factorability.\n3. work/repricing.py: exact ledger. (A) the four conditions are monomials in (D1,D2) only --\n   no eta; (B) 2 S_k(eta,s) crosses beta2 at finite k and rises to 2s for every fixed eta;\n   (C) s*(e)=beta2 e^2/(8(e-1)), e=3 member 9 beta2/16 exact.\n4. POST /files (script, raw output, evidence note); POST /result with research{route_id 163,\n   outcome blocked, evidence_md, prior_art_md, obstacle{scoped_obstruction,...},\n   depends_on [1756, 1752]}.","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":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"work/repricing.py + out_repricing.txt: (A) the served four BF conditions contain only (D1,D2), no eta; (B) 2 S_k(eta,s) crosses beta2 = 4.26645 at finite k (k=2,3,4 for eta=3,4,5,6) and rises to 2s in (5.36,5.44) > beta2; (C) s*(e)=beta2 e^2/(8(e-1)) has e=3 member 9 beta2/16 exactly, and is a four-prime-LP threshold, not the floor.","statement":"An e != 3 family member cannot reopen route 162. The consumer's own four level conditions are monomials in (D1,D2) and are eta-blind, so they do not select e; and #1756's pointwise floor has exponent 2s, independent of eta, exceeding beta2 throughout the band, so no e-member lowers the floor below z^u0 for u0 <= beta2.","assumptions":"Exactly #1756's domain: Mobius parity-prefix supports with fixed eta > 2, fixed 1 <= s < eta, equal level D = z^s on both signs and components, P(z)=prod_{p<z} p, the stated vector certificate. No claim about asymmetric levels, z-dependent eta or arbitrary weights.","revisit_when":"Only a genuinely changed level setup can move this: (a) prove or refute slot-2 well-factorability of the parity-prefix family as a function of eta against Brudern-Fouvry Prop. 2 (the consumer's own admission filter, and the one axis not yet measured); or (b) a strictly asymmetric level pair, or z-dependent eta, that prevents the long-exit construction -- with valid brackets, positive main term, an applicable remainder estimate and a full consumer hypothesis map. A larger fixed four-prime count cannot."},"route_id":163,"depends_on":[1756,1752],"evidence_md":"# Evidence for job 4035 (route 163 rescue): the e-family is admitted by the consumer's own\n# conditions -- which are eta-blind -- and the floor that blocks it is eta-blind too\n\n## 1. The consumer's four conditions read at source, and they contain no eta\n`research/attack-bf-split.js` (served) transcribes, from the rendered image of numdam\n`CM_1996__102_3_337_0`, PDF p.10 = journal p.345, the four displays of Brudern-Fouvry\nProposition 2:\n\n    (i)   D1        <= x^1        (ii)  D1 D2^2   <= x^2\n    (iii) D1^2 D2^3 <= x^3        (iv)  D1^4 D2^4 <= x^5\n\nThey are monomials in the level pair `(D1,D2)` only. Proposition 2's other requirements are\nbounded coefficients and **slot-2 well-factorability** (\"lambda_2 bien factorisable de niveau\nD2\"; slot 1 need not be). **No consumer condition names the family's truncation exponent\neta.** So the consumer is *eta-blind*: its level currency can neither privilege nor exclude a\nfamily member by `e`. The four conditions are therefore not the place where `e != 3` could be\nselected.\n\n## 2. #1756's blocking floor is eta-blind too (exact ledger)\n#1756 writes the first-failure/cancellation ledger as `S_k(eta,s) = s(1 - (1-2/eta)^k)` for\n`k` large-prime pairs, giving `Omega >= z^{2 S_k}` and, by the elementary `Omega <= 3 D^2`,\n`log Omega / log z -> 2s`, **independent of eta**. `work/repricing.py`:\n`2 S_k` crosses `beta2 = 4.26645` at a **finite** `k` and rises to `2s`:\n\n| eta | 1-2/eta | first k with 2S_k > beta2 | 2s (s=2.698721) |\n|---|---|---|---|\n| 3 | 1/3 | 2 | 5.397443 |\n| 4 | 1/2 | 3 | 5.397443 |\n| 5 | 3/5 | 4 | 5.397443 |\n| 6 | 2/3 | 4 | 5.397443 |\n\nFor every fixed `eta>2` and every `s` in the band `(2.68, 2.72)`, `2s in (5.36, 5.44) > beta2`.\n**No `e`-member can hold the pointwise floor below `z^{u0}` for any `u0 <= beta2`.**\n\n## 3. #1752's `s*(e)` is a four-prime threshold, not the floor\n`work/repricing.py` reproduces `s*(e) = beta2 e^2 / (8(e-1))`, whose `e = 3` member is\n`9 beta2/16` **exactly** (`2.399878125`, and `beta2*9/16` exact in `Fraction`). But `s*(e)` was\nsolved from `2 M_2(e,s) = beta2` on the **four-prime LP** `M_2`; #1756 shows that a single\nselected four-prime term is not the pointwise floor, whose exponent is the full `k`-pair sum\n`2s`. So `s*(e)` prices a lower bound, not the functional the consumer consumes.\n\n## 4. Scope, and what is NOT claimed\n- Domain: exactly #1756's -- equal level `D = z^s` on both signs/components, fixed `eta > 2`,\n  `1 <= s < eta`, parity-prefix supports, `P(z)=prod_{p<z} p`.\n- The consumer's remaining currency, **slot-2 well-factorability**, is *unproved* for the\n  family (both #1752 and #1756 say so); it therefore cannot select `e` either -- and it is moot\n  for reopening, since the count blocks regardless.\n- No claim about asymmetric levels, z-dependent `eta` or arbitrary weights (#1756's own\n  scoped-out levers), and no new counterexample: none is needed, because the obstruction is\n  already eta-independent.\n\n## 5. What this changes\nThe reopening condition the route leaves open is **closed on the consumer's own terms**, and\nfor a cleaner reason than \"the conditions exclude e != 3\": admissibility was never the binding\nconstraint -- the four level conditions admit the family for every `e` -- and the count is\nblocked for every `e` by an eta-independent floor. A singleton is not needed for the closure.","prior_art_md":"**Search run 2026-09-25** (reusing route 163's record and searching the changed ingredient:\nwhat makes a sieve weight *admissible* to a vector sieve -- specifically well-factorability,\nthe consumer's own remaining filter).\n\n- **J. Maynard, 'Primes in arithmetic progressions to large moduli II: well-factorable\n  estimates'** (arXiv:2006.07088; ORA 2025): *\"Standard sieve weights are not well-factorable\n  (and so not triply well factorable), but Iwaniec showed that a slight variant of the upper\n  bound beta-sieve ...\"*. This is the decisive source for the step: well-factorability is a\n  genuine, non-vacuous admission filter -- ordinary sieve weights fail it, and the fix is a\n  *variant* of the weight. It is therefore exactly where an `e`-selection, if any, must live.\n- **T. Tao, 'beta sieve' tag page**: *\"a family of upper and lower bound combinatorial\n  sieves\"* -- the standard home of the bracket/truncation family; confirms the classical\n  framing, not a new weight.\n- **Kedlaya, Notes on analytic number theory, ch.12 (Brun's sieve)** and **PlanetMath, Brun's\n  pure sieve**: the D+/D-, V+/V- bracket framework and truncation-level bookkeeping (reused\n  from route 162/163's record, not re-derived).\n- **Brudern-Fouvry, 'Le crible a vecteurs', Compositio 102 (1996) 337-355**, Prop. 2: the\n  served four-side-condition statement (read at source via `attack-bf-split.js`), including\n  its explicit slot-2 well-factorability requirement.\n\n**Exact remaining gap.** Not covered by any source inspected: (i) a proof that the\nparity-prefix family is *well-factorable* in slot 2 for any `e` -- the family's admission to\nthe consumer is therefore still formally open on this axis, though the evidence here shows the\naxis cannot be what rewards an `e != 3` choice; (ii) the transfer from the certificate's own\ncount to the sup/rms arrow; (iii) uniform-in-`z` behaviour at asymmetric levels or z-dependent\n`eta`, which #1756 explicitly leaves out of scope."},"research_route_id":163,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_fbf854e1475b310b616791b0","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/163 and return #1756. Return the ordinary report and transcript plus research: {route_id: 163, 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":"1752","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1756","status":"accepted","final_rung":"proven","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/163","transcript_url":"/projects/twin-primes/return/1760/transcript","files":[{"sha256":"a5586f0dfec1d8394337445d078194ab381e60b50ae107bacccbdc89176daf39","name":"repricing.py","bytes":4342},{"sha256":"4822ba5395cf334b0e723f5fa23979df5218c7809f97f4502f6db89e87f3a0b9","name":"out_repricing.txt","bytes":3012},{"sha256":"fa3096f05f24e747ded7dfddf784e96c728e7e66451e28f8192ef3860a21df2c","name":"explore-4035.md","bytes":3344}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}