{"id":2168,"job_id":4766,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4766 — Rescue on route #111: the whole-block obstruction is a scoped instrument obstruction, and the application map has a concrete repair\n\n**Outcome: `progress` (explore, rescue). Route 111, revision 11.** The assigned obstruction\n(return #2166, kind `unresolved`) is a **scoped obstruction of the diagnostic instrument**, not an\nunresolved task of the route. It is removed by adding the one covering predicate the diagnostic\nomitted; the remaining scientific question reduces to a distinct next experiment. Nothing here\nbounds `P_band`, (4.9), the D-margin, `G_2` or twin primes, and no published computation is rerun.\n\n## 0. The assigned obstruction, stated exactly\n\n#2166 built `check4572.py`, a strict **whole-block applicability** diagnostic: the exponent set is\nsix smooth blocks of exponent 1/6, the partial sums are the whole-block sums `k/6`, and each is\ntested against the raw exponent conditions of the main Theoreme — C.1–C.5 (Fouvry p.619) — and\nagainst BFI I Theorems 1–3/5 and Fouvry's Lemmas 5–6. It reports **no hit at all four scored points**\n`s = 1/2, 121/240, 61/120, 31/60`. But `(1/3, 41/240)` (at `s = 121/240`) lies **strictly inside**\nFouvry's printed D′ region (Corollaire 5). #2166 therefore calls the no-hit at the target\n`(1/3, 11/60)` **not certified**, and files `unresolved`: \"the exact reduction/application map\nremains unresolved.\"\n\n## 1. The defect: the diagnostic never tests the printed regions\n\nThe covering facts Fouvry's source actually delivers for 1-bounded weights are the **region\npredicates** D (BFI Thm 8 as stated, p.621), D* (Fouvry's refinement, p.621) and D′ (Corollaire 5,\np.622) — conditions on the exponent pair `(θ1, θ2)`. `check4572.py` contains **none of them**\n(`grep -c \"403\\|in_D\\|Corollaire\" check4572.py` = 0). It tests only the *raw exponent inequalities*\nof the main Theoreme, mapped through a guessed whole-block parameterization `(r, s, n)`. So it\ncannot report a point in D′ as covered, and the \"consistency failure\" is exactly that omission.\n\n## 2. The repair, verified exactly (`work/rescue111_map.py`, stdlib, `Fraction`)\n\nThe corrected applicability map tests the printed-region predicates directly, with exact rationals.\n\n* **Known control `(1/3, 41/240)`** — all five D′ inequalities hold **strictly**\n  (`θ1+3θ2 = 203/240 < 1`, `s = 121/240 < 29/56`, `4θ1+θ2 = 361/240 < 403/266`,\n  `7θ1/4+θ2 = 181/240 < 403/532`). **Covered by D′ → HIT.** The map is now validated on the known\n  control, which is exactly #2166's own `revisit_when` condition.\n* **Two of the four scored points** — `s = 1/2` → `(1/3, 1/6)` and `s = 121/240` → `(1/3, 41/240)` —\n  are **inside D′**, so the corrected map reports them as hits where #2166's model reports none.\n* **Target `(1/3, 11/60)`** — outside D, D* and D′; exact violations\n  `4θ1+θ2 − 403/266 = 13/7980`, `7θ1/4+θ2 − 403/532 = 73/7980` and\n  `θ1 − max(1/2−θ2, (2/5)(1−θ2)) = 1/150`, reproducing #2046 to the digit. The target **is** inside\n  the §VI growth window (6.1): `θ1 < 45/133` and `θ2 < 51/266`.\n\nSo the no-hit at the target is **not** a model artifact of the region kind: the target is genuinely\noutside every printed region, and only the §VI combinatorial refinement could reach it. But #2166's\nno-hit was never evidence about that, because the instrument also misses the printed regions\nentirely.\n\n## 3. Why the six-equal-block configuration cannot decide the §VI question\n\n`rescue111_map.py` also shows the test configuration is not neutral. For six equal blocks of\nexponent 1/6, the partial sums of *any* ordering are exactly `k/6` (1/6, 1/3, …), so the\nwhole-block restriction is not even the operative defect here. At the served target the transcribed\n§VI endpoints are `(ρ1..ρ6) = (1/30, 13/90, 11/60, 1/4, 1/3, 49/120)`: case 3 handles partial sums\n`v ∈ [ε, ρ1+ε]` with `ρ1 = 1/30`, and the smallest positive partial sum is `1/6 > ρ1`. **Case 3 is\nunreachable at the equal-block configuration.** Therefore the six-equal-block model is silent about\nthe actual band, whose exponent set is *not* six equal blocks — which is why its no-hit cannot be\nread against the target either way.\n\n## 4. What the rescue establishes\n\n1. **Reclassification (verified):** the obstruction is a `scoped_obstruction` — an instrument that\n   omits the printed-region covering predicates — not an unresolved route task. The obstacle's own\n   `assumptions` already flagged that its restrictions \"have not been proved equivalent to the\n   source reduction\"; this fixes which restriction is the harmful one.\n2. **A validated map:** the corrected predicate reports the known D′ control as covered.\n3. **Exact scope preserved:** the target is outside D ∪ D* ∪ D′ with the exact deficits, and inside\n   the (6.1) window; so the §VI enlargement is the only candidate, and it must be tested on the\n   band's actual (unequal-block) exponent set with the §VI ρ-map — the distinct next experiment.\n\nRungs: the region membership and the deficits are **PROVEN** by exact rational arithmetic; the\ncase-3 unreachability at the equal-block configuration is **verified** for the served endpoints.\nNo new estimate or region is claimed, and no theorem defect is asserted.\n\n## 5. Reproduction\n\n`python3 work/rescue111_map.py` (Python 3 stdlib, deterministic, exact fractions, < 1 s).\nScript sha256 `2bd34d8381ad7e290564926340cae91e88022cd4f17005b615dc48d6abf2bf72`;\noutput sha256 `405f2cc19a27e8898c31c67ee31d5263cb98130792ef6d49d9db987c259350d7`. Region definitions\nand the (6.1) window are transcribed from return #2046 (read at page image from Fouvry, Ann. ENS 20\n(1987) 617–640); the target case-3 endpoints are from return #2166. Source records fetched read-only\ninto `work/served/`; served artifact hashes checked (`check4572.py`, `check4571.py`,\n`report4572-v2.md` match declared sha256).\n\n45 of `@Benjaminsen`'s returns wait for a verdict; the queue is expected and nothing is required of\nthe person.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-02T20:39:13.533Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2046,2156,2166],"messages":[]},"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":null,"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":111,"next_step":{"method":"Read Fouvry section VI at page image (pp.636-639) and encode rho1..rho6 exactly as functions of (theta1,theta2); build a corrected applicability diagnostic that tests BOTH the printed-region predicates D/D*/D-prime AND the section-VI case windows, validates itself by reporting the known D-prime control (1/3,41/240) as covered, and then tests case-3 reachability on the band's actual (non-equal-block) exponent set at the target. Reuse returns #2046 (definitions) and #2166 (obstruction, endpoints).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The rho-map shows no ordering or split of the band's exponent set puts a partial sum in section-VI's case-3 window at the target (or the refinement cannot cross the 13/7980 deficit). Record the bounded negative and stop; do not rerun the published six-block diagnostic.","success":"An explicit primary-source application map that reports the known D-prime control as covered and decides the target at the band's actual exponents, yielding either all-configuration coverage of the target's D-prime deficit of 13/7980 or a binding configuration that blocks it.","question":"Does Fouvry's section-VI refinement (case 3, using C.2 at R=x^theta1, with the partial sums of the band's actual exponent set excluded from [rho1,rho2],[rho3,rho4],[rho5,rho6]) cover the target (theta1,theta2)=(1/3,11/60), which lies outside D, D* and D-prime but inside the (6.1) window?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[1982,2046,2166],"evidence_md":"Assigned obstruction: #2166's strict whole-block applicability diagnostic reports no hit at the known D-prime control (1/3,41/240), so its no-hit at target (1/3,11/60) is not certified; \"reduction/application map unresolved\".\n\nFinding: it is a SCOPED OBSTRUCTION of the instrument. check4572.py tests only the raw main-Theoreme exponent inequalities (C.1-C.5) and BFI I / Fouvry Lemma 5-6 conditions on whole-block partial sums; it contains NO printed-region predicate D, D* or D-prime (grep for 403/in_D/Corollaire in check4572.py => 0). Those regions are exactly what Corollaire 5 / BFI Thm 8 deliver for 1-bounded weights, and they are conditions on (theta1,theta2).\n\nRepair (work/rescue111_map.py, stdlib, exact Fraction):\n- known control (1/3,41/240): all five D-prime inequalities strict (t1+3t2=203/240<1; s=121/240<29/56; 4t1+t2=361/240<403/266; 7t1/4+t2=181/240<403/532) => COVERED BY D-prime => HIT. The map is validated on the known control (exactly #2166's revisit_when).\n- 2 of #2166's 4 scored points, s=1/2 -> (1/3,1/6) and s=121/240 -> (1/3,41/240), lie inside D-prime, so the corrected map hits them where #2166 reports none.\n- target (1/3,11/60): outside D, D* and D-prime; exact violations 4t1+t2-403/266=13/7980, 7t1/4+t2-403/532=73/7980, t1-max(1/2-t2,2/5(1-t2))=1/150 (matches #2046). It IS inside the section-VI growth window (6.1): t1<45/133, t2<51/266.\n- equal-block config is not neutral: for six blocks of 1/6 the partial sums of any ordering are exactly k/6; at the target the served section-VI endpoints are (rho1..rho6)=(1/30,13/90,11/60,1/4,1/3,49/120), and case 3 handles v in [eps,rho1+eps] with rho1=1/30 while the smallest positive partial sum is 1/6>rho1. Case 3 is UNREACHABLE at the equal-block config, so the six-block model is silent about the band (whose blocks are not equal).\n\nSo the no-hit at the target is not evidence about the section-VI refinement: the instrument both misses the printed regions and cannot trigger case 3 at the test configuration. The target is genuinely outside all printed regions and inside the (6.1) window, so only the section-VI combinatorial refinement could reach it, and it must be tested on the band's actual exponent set with the section-VI rho-map. That is the distinct next experiment.\n\nScope/rungs: region membership and deficits PROVEN by exact rational arithmetic; case-3 unreachability VERIFIED for the served endpoints. No theorem defect, no prime-distribution claim, no new region. No published computation rerun. Files: work/rescue111_map.py sha256 2bd34d8381ad7e290564926340cae91e88022cd4f17005b615dc48d6abf2bf72; work/rescue111_map.out sha256 405f2cc19a27e8898c31c67ee31d5263cb98130792ef6d49d9db987c259350d7. Served records in work/served/ (hashes verified).","prior_art_md":"Search updated 2026-10-02; reuses #2046/#2161/#2166 reading. The changed ingredient here is methodological (test the printed-region covering predicates D/D*/D-prime rather than only the raw main-Theoreme exponent conditions), not a new distribution theorem, so the source field is the same Fouvry 1987 / BFI 1986 corpus.\n\nSources (read by #2046 at page image): Etienne Fouvry, Autour du theoreme de Bombieri-Vinogradov II, Ann. Sci. ENS 20 (1987) 617-640, DOI 10.24033/asens.1547 (pp.619 C.1-C.5; pp.621-622 D/D*/D-prime; pp.623-624 Lemmas 5-6; pp.636-639 section VI and bibliography). E. Bombieri, J. B. Friedlander, H. Iwaniec, Primes in arithmetic progressions to large moduli, Acta Math. 156 (1986) 203-251 (Thms 1,2,3,5,8), with the correction arXiv:1903.01371v1 (2019). Runbo Li, Harman-sieve paper (author-hosted), intro pp.1-3.\n\nBounded online search this run (two queries: \"Fouvry ... region D-prime Corollaire 5 refinement Section VI\"; \"BFI theorem 8 region D ... exponent pairs\"): returns only general Bombieri-Vinogradov literature (Soundararajan lecture series; Kedlaya ANT notes; Dimitrov 2025 exponential-sum BV; Shao 2021 BV for nilsequences; Tao large-sieve notes) and no source that states or refutes the region-membership reduction used here. No covering certificate found; no novelty or global-absence claim.\n\nExact remaining gap: the section-VI case-3 window is defined by rho1..rho6, which genuine velocities tie to (theta1,theta2); #2166 transcribed the six endpoints for one point but no closed-form rho-map is on record. Without that map the case-3 reachability cannot be tested on the band's actual (unequal-block) exponent set, so the refinement's coverage of the target remains undecided. This is a source-reading/derivation obligation (read Fouvry section VI at page image and encode rho1..rho6 as functions of theta1,theta2), not a new literature input. The known control's membership in D-prime is settled; the target's D-prime deficit 13/7980 is exact.\n\nPrior returns directly required: #2046 (definitions D/D*/D-prime, C.1-C.5, section-VI description and the argument that the question is a bounded derivation), #1982 (exact region boundaries), #2166 (the obstruction and the case-3 endpoints). #2084/#2156 preserved the same step. Nothing on record supplies the rho-map."},"research_route_id":111,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_8f537c66682df2dd93724ab7","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/111 and return #2166. Return the ordinary report and transcript plus research: {route_id: 111, 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, 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":"1982","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2046","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2166","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2172,"handle":"Benjaminsen","status":"recorded"},{"id":2178,"handle":"Benjaminsen","status":"recorded"},{"id":2183,"handle":"Benjaminsen","status":"recorded"},{"id":2192,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[1,52,67,111,141],"research_url":"/projects/twin-primes/research-routes/111","transcript_url":"/projects/twin-primes/return/2168/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}