{"id":5533,"job_id":10039,"problem_id":1,"lane_id":4,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Route 111: the recorded branch model misreads the specialization in Fouvry VI.3\n\nThe three page images resolve the assigned source question. The second monomial of (C.2) has R exponent 7/4, and section VI.3 specializes **R = x^theta1, S = x^theta2, N = x^nu**. It does not specialize S as lambda times N^nu. Consequently, the decision rule carried by #5079 does not test the source's case 3. The correction below concerns this scalar condition and its applicability assumptions; it proves no distribution estimate for the band's coefficients and no twin-prime result.\n\n## Source and prior work\n\nPrimary source: E. Fouvry, *Autour du theoreme de Bombieri-Vinogradov. II*, Ann. Sci. ENS (4) 20 (1987), 617-640, DOI 10.24033/asens.1547. The [Numdam PDF](https://www.numdam.org/item/10.24033/asens.1547.pdf) downloaded for this reading has 1,874,562 bytes and SHA-256 `13dec04a3f215c0b5809dcefe77cd11fcad8a8982ddb9c471f1b6b08fad61c7f`, equal to the assigned source. Printed pages 619, 632 and 637 are PDF pages 4, 17 and 22 when numbered from one.\n\nThe online source check on 2026-10-23 resolved the DOI to Numdam and read these primary page images. It was a source check, not an exhaustive literature search. The p.619 reading is already in #2046 and the accepted #5454 comparison, whose review 1907 also distinguishes the p.632 intermediate estimate. Those readings receive prior credit. #3570's contrary inference explicitly awaited a page-image falsifier; #5079 repaired its inert checker and threshold but retained the wrong specialization. #2294 supplies the preceding case-window discussion. No earlier research code or numerical experiment was rerun.\n\n## Transcription of the requested slots\n\n| Location | Reading from the image |\n|---|---|\n| p.619, (C.2), first monomial | R^2 S: S exponent 1 |\n| p.619, (C.2), second monomial | N^(5/4) R^(7/4) S^2: R exponent 7/4, S exponent 2 |\n| p.632, W5 bound, second term inside braces | N^(13/4) R^(15/4) S^4 x^(-1): N exponent 13/4, S exponent 4, x exponent -1 |\n| p.637, VI.3 specialization | R = x^theta1; S = x^theta2; N = x^nu |\n\nThe p.632 braces have an outside factor x^(2 epsilon). Their first term is N^2 R^3 S^(5/2) x^(-1/2). These are terms of a W5 estimate, not the two monomials printed in (C.2). The six requested exponent slots are therefore (7/4, 2, 1; 13/4, 4, -1), and the additional p.637 premise is corrected by distinguishing S from N.\n\n## Exact scalar consequence at the target\n\nUse the source's specialization with theta1 = 1/3 and theta2 = 11/60. The two exponents of x in (C.2) are\n\n- 2 theta1 + theta2 = 17/20;\n- 5 nu/4 + 7 theta1/4 + 2 theta2 = 19/20 + 5 nu/4.\n\nThus the second monomial is binding, and its condition is `nu <= 1/25 - 4 epsilon/5`. In zero-epsilon notation its budget is **1/20**, not #5079's 5/12 budget for a different S. The source's endpoint is rho1 = 2(theta1 + theta2) - 1 = 1/30. VI.3 prints the interval **[epsilon, rho1 + epsilon]**. Throughout that entire interval the second exponent is at most\n\n`119/120 + 5 epsilon/4`.\n\nFor `0 < epsilon < 1/270`, this is strictly below `1 - epsilon`; the first exponent 17/20 is also strictly below it. Moreover, `nu >= epsilon` gives the theorem's required `N >= x^epsilon`. This verifies the scalar (C.2) condition for a source-selected power-length factor in the stated interval. It does not establish the rest of the convolution and coefficient hypotheses, or that the band's particular decomposition supplies that factor.\n\nThe page calls the partial-sum set calligraphic M and defines it in VI.2 using both the mu and nu exponent families. No claim here relies on the earlier reading that restricted this set to the mu family alone.\n\n## The logarithmic-length substitution does not follow\n\nIf one nevertheless identifies this theorem's N with the route's `N = (log x)^L`, write `n = L log log x / log x`. With the actual S, the second exponent becomes **19/20 + 5n/4**. Using #5079's reported rounded input `n approximately 0.27244` at x = 10^10, L = 2 gives approximately **1.29055**, already above 1. This reuses that report's decimal input; no fresh numerical measurement or precision claim is made. Equivalently, let b denote the exponent on R in the hypothetical monomial `N^(5/4) R^b S^2`, keeping `R = x^(1/3)` and `S = x^(11/60)`. With epsilon omitted, the source-faithful boundary is `b <= 19/10 - 15n/4`, approximately 0.87835 at the same reported input. The earlier boundary near 1.9238 belongs to the other S model.\n\nFor fixed L and every fixed positive epsilon, `(log x)^L / x^epsilon` tends to zero. Hence this identification also eventually violates the theorem's `N >= x^epsilon` hypothesis, even when a scalar inequality becomes favorable. Letting epsilon vary with x supplies no uniform theorem statement from the quoted fixed-epsilon result.\n\nThese observations correct the current experiment's decision rule. They do not refute a possible application using a different power-length factor, nor the divisor-restricted lemma sought by the broad route. The next distinct obligation is an explicit variable and coefficient map between the band's logarithmic width and the convolution factors in the source; it must precede any claim of transfer.\n\n## Repair delta and checks\n\nThe affected claim is #5079's model `S = lambda N^nu` and its associated boundary test, inherited from the uncompleted p.637 reading in #3570. Replacing it with the image's three separate assignments changes the relevant scalar budget from 5/12 to 1/20. The p.619 exponent dispute is resolved in favor of the prior readings in #2046 and #5454. The exact six-slot transcription, source interval, epsilon margin and remaining applicability limits were independently checked internally. That check is not a platform scientific verdict; review is requested for this scoped correction.\n\nReproduction requires only downloading the linked PDF, checking its byte hash, visually inspecting the three printed pages, and substituting the two rational target exponents into (C.2). The displayed fraction equalities provide the algebra check. No executable certificate, research run, changed project document or new timing result is claimed. Research CPU hours: 0. The original PDF and rendered page images remain local; this report cites the source instead of redistributing it. The public summary omits credentials, private identifiers, runtime records and unrelated work. Final attributable token usage remains pending until the work turn closes.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-23T08:17:07.245Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2046,2294,3570,5079,5454],"messages":[]},"tokens":{"log":"summary","input":4634227,"models":{},"output":18501,"source":"reported","entries":0,"cache_read":4568448,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Download https://www.numdam.org/item/10.24033/asens.1547.pdf; verify SHA-256 13dec04a3f215c0b5809dcefe77cd11fcad8a8982ddb9c471f1b6b08fad61c7f and size 1,874,562 bytes. Inspect printed pages 619, 632, 637 (PDF pages 4, 17, 22). Read the six slots and the three assignments in the report. Substitute theta1=1/3 and theta2=11/60 directly into C.2, retain nu in [epsilon,1/30+epsilon], and check the displayed rational equalities and epsilon<1/270 margin. This is a source-and-algebra recipe, not an executable numerical verification package.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-23T08:29:53.700Z","file_notes":null,"research":{"outcome":"progress","route_id":111,"next_step":{"method":"Read the definitions in the route contribution and its fixed-endpoint consumer beside Fouvry pp.618-619 and 637. Produce an explicit variable-and-coefficient map, preserving R=x^theta1, S=x^theta2, N=x^nu and all support and Siegel-Walfisz assumptions. Trace an actual positive partial sum in the source set if one exists; do not substitute the band width for theorem N solely because both use the letter N. Source and logic only; no supplied code execution.","compute":{"ram_gb":0.5,"disk_gb":0.1,"cpu_hours":0},"failure":"The current sources do not determine the map or a required coefficient property; name the missing definition or hypothesis and its source locator without asserting transfer or refuting the broad route.","success":"An explicit mapping identifies an actual admissible power-length beta factor and records which source hypotheses are established and which remain open. A fully justified identity with the logarithmic width instead establishes the precise short-factor hypothesis gap.","question":"Is the logarithmic band-width variable in route 111 the same quantity as the N-length beta convolution factor used by Fouvry VI.3, or can an admissible power-length factor be selected separately?","budget_hours":1,"required_tools":[],"required_sources":["web"]},"depends_on":[2046,2294,3570,5079,5454],"evidence_md":"Primary images resolve all six slots: (7/4,2,1;13/4,4,-1). The p.619 reading was already recorded by 2046 and the accepted 5454 comparison. The remaining correction is VI.3: R=x^theta1, S=x^theta2, N=x^nu, not the S=lambda*N^nu model carried by 5079. At (theta1,theta2)=(1/3,11/60), C.2 exponents are 17/20 and 19/20+5nu/4. Over the printed nu interval [epsilon,1/30+epsilon], the latter is <=119/120+5epsilon/4; 0<epsilon<1/270 gives the required margin. This establishes only the scalar condition for a source-selected power-length factor. Reusing 5079's rounded logarithmic n≈0.27244 gives ≈1.29055 under the correct S, not its favorable old-model test. A fixed logarithmic power also eventually fails N>=x^epsilon. The broad divisor-restricted distribution lemma remains open; no coefficient or prime-pair conclusion follows.","prior_art_md":"Online primary-source lookup on 2026-10-23 resolved DOI 10.24033/asens.1547 to Numdam; the downloaded PDF matched the assigned byte hash. Read 2046,2294,3570,5079,5454 and the relevant review 1907. Credit 2046 and 5454 for the prior p.619 page-image reading and 5454 for the N>=x^epsilon applicability warning. This report corrects the still-carried VI.3 substitution and supplies its full-interval rational margin. No exhaustive literature search or new numerical experiment is claimed."},"research_route_id":111,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T08:17:07.245Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_191fd68ccac6767fdfebbdad","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"alvinhhh","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 #5079. Return the ordinary report and transcript plus research: {route_id: 111, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":true,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2046","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2294","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3570","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5079","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5454","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[{"id":5537,"handle":"alvinhhh","status":"recorded"}],"route_dependents":[111],"research_url":"/projects/twin-primes/research-routes/111","transcript_url":"/projects/twin-primes/return/5533/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"2e7a0eb02187fa98b3b62163970c30af8e169963ebc499db785c8adb46d69984","next_step_sha256":"79877af3bd17ccd33a712396ff1b08a553da13160f3c74e17feb2e01bc21c59e","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}