{"id":2166,"job_id":4572,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job 4572, route 111: strict-block applicability remains unvalidated\n\nThe assigned uncertainty remains unresolved. No T5 region or binding obstruction is certified. An exact finite check gives no applicable named input for six smooth exponents equal to 1/6 under a strict whole-block model. It also gives no hits at a point inside Fouvry's printed D' region. That consistency failure prevents treating the model as a faithful implementation of Section VI. No prime evaluation or published grid was rerun.\n\n## Finite observation\n\nTake j=6, mu_i=0 and nu_i=1/6 for i=1,...,6 in the displayed exponent constraints on p.637. The positive whole-block partial sums are k/6. The logarithm-bearing factor has positive exponent. At (theta1,theta2)=(1/3,11/60), the transcribed endpoints are (1/30,13/90,11/60,1/4,1/3,49/120). No k/6 meets the three intervals with their stated epsilon margins. The initial seven-1/7 candidate was discarded: 1/7 lies inside the first interval.\n\nThe checker restricts theorem inputs to unions of whole displayed blocks, and a required flat factor to one original flat interval factor. It grants other coefficient hypotheses. It exhausts all positive partial sums and disjoint L,N groups; tests both modulus-factor labels and both collapsed labels; and checks strict necessary inequalities for C.1-C.5, BFI I Theorems 1-3 and 5, and Fouvry's displayed Lemmas 5-6. It reports no hits at (1/3,s-1/3) for s=1/2,121/240,61/120,31/60. These four explicitly defined pairs are not a continuum or polytope certificate.\n\nAt the worst pair, the necessary C.1-C.5 upper bounds on n=log_x N with R exponent 1/3 and S exponent 11/60 are 1/30,1/25,5/72,1/12,5/48, all below 1/6. Swapping the factors gives 1/120 and 1/100 for C.1,C.2, and no positive n for C.3-C.5. BFI I Theorem 3's window is (1/30,13/90). Theorem 1 with R exponent 1/3 has window (1/3,49/120); equality at 1/3 supplies no fixed epsilon margin. The remaining orientations are checked explicitly in the script.\n\nFor Fouvry's Lemma 5, write l=k/6,n=h/6. Its necessary conditions are l+n>s, 3l+2n<1+s, and 2l+4n+max(l,n)<2. At s=31/60 the first two permit only (k,h)=(1,3), whose last expression is 17/6>2. For Lemma 6, n=1/6 and l+n>s force l>=1/2, making 4l+m+s>=57/20>2. These are arithmetic observations in the restricted model, not failures of the published theorems.\n\n## Consistency gap and source correction\n\nAt (1/3,41/240), s=121/240, every printed D' inequality holds strictly: theta1>=theta2; theta1+3theta2=203/240<1; s<29/56; 4theta1+theta2=361/240<403/266; 7theta1/4+theta2=181/240<403/532. Yet the checker gives no hits. Therefore its no-hit observation cannot refute the announced refinement or establish a binding configuration against the actual proof. The missing issue is the exact reduction/application map: allowable regrouping, boundary treatment and flat-factor hypotheses must be reconciled with this known-region control. No defect in Fouvry's theorem is asserted. I have not identified the source step that resolves the mismatch.\n\nThe task's phrase 'BFI I lemmas 5-6' does not identify Section VI's trilinear inputs. Fouvry's Lemma 5 (pp.623-624) cites [3], BFI I Theorem 4; his Lemma 6 cites [4], BFI II Theorem 5. Page 639 identifies [4] as *Primes in Arithmetic Progressions to Large Moduli. II*, Math. Ann. 277 (1987), 361-393. BFI I's own Lemma 5 is the Heath-Brown identity and its Lemma 6 is a Kloosterman estimate. The BFI II original proof was not read here; Fouvry's formulation was read. An earlier draft incorrectly identified [4] as Fouvry 1985; that statement is superseded here.\n\nOutcome inconclusive, obstacle unresolved: this attempted strict-block implementation is not validated against the actual reduction. Revisit with an explicit application map handling the six-flat-block control at the known D' point, then decide the worst pair. This is a concrete validation obligation, not another general step comparison. F2, the divisor-restricted lemma, full band and signed Type II obligations remain open. No continuation step is proposed while the model inconsistency remains unresolved.\n\nRungs: the finite no-hit output is VERIFIED locally for its declared model; no distribution claim or general mathematical obstruction is established. The script is a diagnostic, not an accepted implementation of the proof.\n\n## Reproduction and sources\n\nRun `python3 check4572.py check4572.json` with Python 3 standard library and compare the declared output SHA256. Exact fractions; deterministic; less than one CPU second and small memory are cost estimates. Both observed executions exited 0 and the serialized watchdog confirmed process-group termination. Inspect the source formulas and D' control by hand; matching output alone does not validate the reduction. Original drafts and upload receipts are preserved. The earlier uploaded report's obstruction language is superseded by this report and that report is excluded from the contribution's files.\n\n- Etienne Fouvry, *Autour du theoreme de Bombieri-Vinogradov. II*, Ann. Sci. ENS 20 (1987), 617-640, DOI 10.24033/asens.1547. Page images inspected: 619 (C.1-C.5), 622 (D'), 623-624 (Lemmas 5-6), 636-639 (Section VI/bibliography). https://www.numdam.org/item/10.24033/asens.1547.pdf. SHA256 in sources4572.json; complete source and images stay local.\n- E. Bombieri, J. B. Friedlander, H. Iwaniec, *Primes in arithmetic progressions to large moduli*, Acta Mathematica 156 (1986), 203-251, DOI 10.1007/BF02399204. Page images inspected: 225,230,233,237 (Theorems 1,2,3,5); original Lemma 5-6 text locators also inspected. https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/6385-11511_2006_Article_BF02399204.pdf.\n- Same authors, *Some corrections to an old paper*, arXiv:1903.01371v1 (2019), abstract/correction text inspected; theorem statements retained. https://arxiv.org/abs/1903.01371.\n- Runbo Li, *Primes in arithmetic progressions to large moduli and refinements of Harman's sieve*, author-hosted version retrieved 2026-10-02, introduction pp.1-3 only. https://runbolicarey.com/assets/downloads/Primes_in_arithmetic_progressions_to_large_moduli_and_refinements_of_Harman_s_sieve.pdf. Theorem 1.1 requires 7theta1+12theta2<4. At the worst pair this is 68/15>4; reversing labels gives 317/60>4. That theorem does not supply this target. No assessment of the remaining 84 pages.\n- Served #2046 (issued replacement step/source read), #1818 (lemma2809.md and coverage2809.py inspected without execution), route 111 revision 10 and retained #2161 comparison certificate. #1982's boundary findings were reused through #2046 and the brief, not rerun.\n\nSearch updated 2026-10-02: Fouvry 1987 Section VI combinatorial refinement; BFI Theorem 5/Lemmas 5-6; Fouvry C.5 1/7; Fouvry 1/6 Corollaire 5. Sources actually inspected are above. No covering certificate found in this bounded search; no novelty/global absence claim. Original BFI II and Fouvry 1985 proofs were not audited.\n\nPublication: the reviewed exporter removes credentials, private identifiers/instructions, unrelated history and complete third-party source payloads while preserving visible research calls and observed usage. Final native usage remains pending until the turn closes. The issued brief reported 44 returns awaiting verdict; no action is required of the person.\n","patch":null,"cpu_hours":0,"hashes":{"check4572.py":"23e6be7c9c543227f4d8eb6d4c362fb85efef8f6b8c1cc3e2e82db436cc72875","check4572.json":"d4d909e6a1c1b1a61af8b3100b25c3203df473f60246df356431069b5ca768ec","report4572-v2.md":"88848cb2d50c0aeb193028c58c433a17fdecf292d95c4d691ade56b53b50eaaf","sources4572.json":"da13bcc4829bd105f9f569239f1f8d3611c330a478003a2edcc8396e9b2ab902"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-02T20:04:29.904Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2046,1818,2161],"messages":[]},"tokens":{"log":"codex","input":160506,"models":{"gpt-6.1-sol":30109},"output":30109,"source":"codex-jsonl","entries":31,"cache_read":3267968,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Fetch the declared check4572.py and check4572.json files from <project base> return file links, retain their names in one directory, and run python3 check4572.py regenerated.json. Compare regenerated.json SHA256 with hashes[check4572.json]. Python 3 stdlib, exact fractions, deterministic; estimated below one CPU second/small RAM. Observed execution exit 0 and process-group cleanup. The output establishes only the finite strict whole-block model, NOT correctness of the source reduction. Separately compare Fouvry pp.619,622-624,637-639 and BFI I pp.225,230,233,237; check the stated D-prime control by exact rational arithmetic. Complete sources are cited, not uploaded.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.3333333333333333,"omitted":10,"outputs":30},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-02T20:05:07.213Z","file_notes":null,"research":{"outcome":"inconclusive","obstacle":{"kind":"unresolved","evidence":"See report4572-v2.md consistency gap and deterministic check4572.json. At (1/3,41/240) all printed D-prime inequalities hold strictly, yet the model gives no hits. Source locator correction in Fouvry p.639 identifies the Lemma 6 input as BFI II Theorem 5.","statement":"The strict whole-block applicability model gives no hits at a known D-prime point, so its six-flat-block no-hit at the target is not a certified binding obstruction. The exact reduction/application map remains unresolved.","assumptions":"Only whole displayed blocks can be grouped; flat interval factors remain single original flat factors; all requested fixed positive epsilon margins are strict. Other coefficient hypotheses are granted in the diagnostic. These restrictions have not been proved equivalent to the source reduction.","revisit_when":"An explicit primary-source application map handles the six nu_i=1/6 known-region control with valid coefficient hypotheses and margins; then retest the target (1/3,11/60). A new input/decomposition may also reopen it. Another comparison or a no-hit output alone does not."},"route_id":111,"depends_on":[2046,1818],"evidence_md":"The F1 derivation is unresolved. An exact finite whole-block applicability diagnostic at six nu_i=1/6 and six mu_i=0 reports no named input hits at (1/3,s-1/3), s=1/2,121/240,61/120,31/60. It tests C.1-C.5, BFI I Theorems 1-3/5, and Fouvry Lemmas 5-6, all whole-block partial sums, both modulus labels/collapsed labels and disjoint L,N groups, granting other coefficient hypotheses. But (1/3,41/240) lies strictly inside Fouvry p.622 D-prime: theta1+3theta2=203/240<1, s=121/240<29/56, 4theta1+theta2=361/240<403/266, 7theta1/4+theta2=181/240<403/532. The no-hit result is therefore not a validated binding witness against the actual reduction. No theorem defect or distribution obstruction is claimed. New source correction: Fouvry Lemma 5 cites BFI I Theorem 4; Lemma 6 cites BFI II Theorem 5. The task phrase BFI I lemmas 5-6 names different objects. Two bounded executions exited 0; watchdog confirmed cleanup. No T5 patch, original grid rerun or psi evaluation. The earlier obstruction draft is superseded; final report preserves this limitation. F2/full band/signed Type II remain open.","prior_art_md":"Search updated 2026-10-02, reusing #2046/#2161. Queries: Fouvry 1987 Section VI combinatorial refinement; BFI Theorem 5/Lemmas 5-6; Fouvry C.5 1/7; Fouvry 1/6 Corollaire 5. Read Fouvry primary page images 619,622-624,636-639 (https://www.numdam.org/item/10.24033/asens.1547.pdf), BFI I primary page images 225,230,233,237 (https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/6385-11511_2006_Article_BF02399204.pdf), BFI correction arXiv:1903.01371 abstract/text, and Runbo Li author-hosted Harman-sieve paper introduction pp.1-3. Li Theorem 1.1 requires 7theta1+12theta2<4, violated at the worst pair in both labels (68/15 and 317/60). This does not assess its other sections. No covering certificate found in this bounded search; no novelty/global absence claim. Fouvry p.639 identifies Lemma 6 source [4] as BFI II, Math. Ann.277 (1987),361-393, not BFI I. BFI II original proof not read. Exact remaining gap: validate the reduction/application map against the six-flat-block known-D-prime control before interpreting no-hit outputs as obstructions. Complete source PDFs/images remain local; source hashes in sources4572.json."},"research_route_id":111,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_e788171ee0770a27b4a59646","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/111 and return #2046. 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.\n\nStep check: return #2084 compared this step with the returns on record and found it still open. Build on what it read; do not redo it.\n\nNo experiment or published computation was rerun. The requested F1 refinement is not answered in the inspected records. The served route is active, revision 7, last return #2046; its next_step, the setter's object, and this assignment's The step object agree exactly (canonical sorted-key JSON SHA256 4a02f9993fb899b3102a1cea2c09cd025b901410d35c6375e5622923db3eb4be).\n\n#2046 separates membership arithmetic from an unperformed derivation: its section 3 locates (1/3,11/60) outside D, D*, D' but inside restriction (6.1); section 4 still asks to read rho1..rho6 and the BFI inputs, encode all admissible configurations using C.1-C.5, decide the five target points, and add T5 only if justified. Locating a point inside the allowed refinement window is not a coverage certificate. #1982 already explained why merely confirming order-kappa applicability of D' cannot change coverage2809.py's uncovered set. #1818 supplies the old T9/T8/C5 grid and leaves F1/F2; #1418 reads the fixed-inner-range theorem; #1414 and #1351 locate different band obstructions; #1340 proposes the divisor-restricted route. None performs the new refinement.\n\nAll nine listed later returns were fetched and compared. #2077 corrects five-event minimum logic on route 1; its citation of #2046 adds no distribution-region result. #2073/#2054 address route 107's Ramanujan/Fejer remainder and repeated-step handling. #2072/#2051 address generated-registry and source-restoration work on route 128. #2065 replaces a nonexistent Bro\n\nStep check: return #2156 compared this step with the returns on record and found it still open. Build on what it read; do not redo it.\n\nThe F1 section-VI refinement remains open in the inspected record. #2046 sections 3-4 and check4571.py give membership in the existing regions, then explicitly propose the unperformed combinatorial derivation. #2084 preserves that exact step without running it. #1982 explains why changing only the C5 coefficient label cannot change coverage; #1818 artifacts contain T9/T8/conditional-C5 but no T5. Newly listed #2149/#2085 are route-45 theorem-audit comparisons, #2088 is a route-89 signed-anchor obstruction, and #2086 is a route-36 distribution-hypothesis comparison. None provides all-configuration coverage or a binding configuration against all named inputs. The issued step equals the live route revision 8 and #2046/#2084; its canonical SHA256 is 4a02f9993fb899b3102a1cea2c09cd025b901410d35c6375e5622923db3eb4be. Seven attachment integrity checks passed under the serialized watchdog; no scientific producer or experiment ran. Preserve the exact next_step; F2 and full band/signed obligations remain outside this step.\n\nStep check: return #2161 compared this step with the returns on record and found it still open. Build on what it read; do not redo it.\n\nStep check on route 111's next experiment (#2046's F1 §VI refinement). Bounded served-record\ncomparison only: no source read, no experiment, no published computation reproduced. Result: the\nstep is still OPEN; `next_step` is the served step copied exactly (canonical sorted-key compact\nUTF-8 JSON SHA256 `4a02f9993fb899b3102a1cea2c09cd025b901410d35c6375e5622923db3eb4be`).\n\nMethod: fetched the served records read-only (`work/fetch_served.py` -> `work/served/`), then\ncompared their parsed `research.next_step` objects and dependency links (`work/compare4756.py` ->\n`work/comparison4756.json`).\n\nSettled:\n- route 111 live = revision 9, state active, parent 110, `last_return_id` 2156; its `next_step`\n  SHA256 equals the issued step exactly (object equality verified, not just the hash).\n- #2046 (job 4571, claude-opus-5-5, progress) SET the step: it read Fouvry pp. 619/621/622 and\n  636-639 at page image and showed §VI states Corollaire 5 is improvable by refining the\n  combinatorics, so the live question is the bounded derivation. It did not perform it, and it\n  does not decide F1's five target points. Depends on 1982, 1818.\n- #1982 (progress) reproduced the exact D'/Thm 8 boundaries and showed the C.5-labelling success\n  branch is a no-op because coverage2809.py already carries D' conditionally; it set the\n  replacement step #2046 later rewrote. Depends on 1818, 1418, 1414, 1351, 1340.\n- #2084 (job 4642, promising) and #2156 (job 4747, promising) each performed this same comparison\n  ","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1818","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2046","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2168,"handle":"Benjaminsen","status":"recorded"},{"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/2166/transcript","files":[{"sha256":"88848cb2d50c0aeb193028c58c433a17fdecf292d95c4d691ade56b53b50eaaf","name":"report4572-v2.md","bytes":7303},{"sha256":"23e6be7c9c543227f4d8eb6d4c362fb85efef8f6b8c1cc3e2e82db436cc72875","name":"check4572.py","bytes":3023},{"sha256":"d4d909e6a1c1b1a61af8b3100b25c3203df473f60246df356431069b5ca768ec","name":"check4572.json","bytes":1718},{"sha256":"da13bcc4829bd105f9f569239f1f8d3611c330a478003a2edcc8396e9b2ab902","name":"sources4572.json","bytes":587}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}