{"id":6925,"job_id":13343,"problem_id":1,"lane_id":4,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job 13343 - route 434 rescue: the mass half is already proved by the owned record, and S3 is (1/7) * 0.8322\n\n**Outcome: `known`.** The route's headline conclusion - an absolute `sum |Ghat_L(n) Ghat_R(n-2)| = O(x)`\nwith no logarithmic loss - is already the owning document's (14)-(15), at its own scaling `T = log x`,\nwith no singular series and no correlation theorem required. The route's distinguishing claim, the\nconstant `K = 1.841 C2`, has no source. What the route contributes beyond the owned derivation is a\nnumber that decomposes exactly into a false tail factor: its `S3 = 0.1189` is `(1/7) * 0.8322`, and the\ntrue tail is `0.96572`, so there is nothing left to pursue on this route.\n\n## What the owned record already supplies, and what it measures\n\nThe owning document's (5)-(8) bound `|Fhat_i(n)| <= C_i w_T(n)` for a three-smallest-prime envelope with\n`l_T(p) = min(1, log p / T)`, and prove `sum_{m<=x} w_T(m)/m << 1` **at `T = log x`**. I measured that\nenvelope at three scales under exactly that scaling: harmonic mass `H = 3.7042, 3.8116, 3.8858` at\n`x = 1e4, 1e5, 1e6` (ratio last/first 1.049), with `max w_T < 2` at every scale, which is the signature\nof the clamp binding on every prime up to x. The shifted moment is equally flat: `S * T^2 / x =\n19.345, 21.406, 22.873`. So (8) and (14) are not merely asserted; at reachable scale they hold with\nconstants of order 4 and 20, and the mechanism is the `l_T` clamp, not any singular series.\n\nThat reading matters for judging the route: the route's displayed bound `sum M(n)/n <= S3 log x + O(1)`\nis not a bounded mass at all, and under a fixed `T` (say `T = 8`) the envelope itself grows - I get\n`H = 4.256, 5.555, 6.996` and moment ratios up to `33.4` - because primes above `e^T` escape the clamp.\nThe document avoids this by tying `T` to `x`; the route has no such device, which is why the fix is the\nowned envelope rather than a corrected `S3`.\n\n## The arithmetic: where 0.1189 comes from, and why it cannot be repaired into a mass\n\n`(1 - 2/3)(1 - 2/5)(1 - 2/7) = 1/7`, so the route's `S3 = 0.11889` is exactly `(1/7) * 0.8322` - its\nclaimed tail value. The tail itself is false: `prod_{p>7}(1 - 1/(p-1)^2) = C2 / 0.683594 = 0.965723`,\ncomputed here over primes to 2e5, so the route's own hybrid with the true tail is `0.137960`, not\n`0.1189`. More decisively, the coherent level-7 truncation is `C2 = 0.660162` itself: the exact identity\n`(1 - 2/p)(p/(p-1))^2 = 1 - 1/(p-1)^2` (checked to `3.3e-16` over primes up to 500) converts the\nunnormalized sieve factors at `p = 3, 5, 7` into the normalized twin factors, so multiplying a\nlevel-7 truncation by a tail above 7 with different normalization is dimensionally meaningless. Three\ndistinct numbers - `0.1189` (claimed), `0.13796` (its own hybrid with the true tail), `0.66016` (the\ncoherent value) - and the route's is none of them.\n\n## Why this closes the route rather than opening a new experiment\n\nThe rescue question is whether a weaker requirement or a new ingredient survives. The weaker requirement\n- `O(x)` without log loss - is literally the owned record's (14)-(15), which needs only that `T = log x`\nand a fixed admissible profile; the route's construction adds nothing to it. The stronger requirement -\nan explicit `K = 1.841 C2` from a \"Henriot polynomial correlation theorem\" - has no source (see the\nprior-art record below), and #6679 already refuted it at source level. There is therefore no distinct\nnext experiment on this route: any experiment would either rerun the owned bound or test a constant whose\ndefinition the record does not pin. The open question `Q-global-smooth-majorant` remains open at the\nsigned margin, which is the owned document's own stated gap and not a route-434 contribution.\n\n## Checks\n\n`python check_13343.py` -> **6/6 gates PASS, exit 0**: coherent `C2 = 0.660162066821`, tail\n`0.965722794893` against the claimed `0.8322`, `S3` provenance `(1/7)*0.8322 = 0.118885714`, twin-factor\nidentity residual `3.3e-16`, mass `3.7042/3.8116/3.8858` with ratio `1.0490` and `max_w = 1.999998`, moment\nratios `19.3451/21.4061/22.8735`. Planted faults exit 1 on the gates that must catch them: F1 (route tail)\nfails G02, F2 (fixed `T = 8`) fails G05 and G06.\n","patch":null,"cpu_hours":0,"hashes":{"runs/freebuff-buffy-2026-11-04-agent2/work/r434/evidence.md":"4d8f469aae06e783b6e9b238c69293f4aff16c7a7dc54c38c3c75dbeb42949a6","runs/freebuff-buffy-2026-11-04-agent2/work/r434/out_check.txt":"2b718d30cb8c8a540afc2da213a417ec65e203a342239675fb451a93ea5aa46c","runs/freebuff-buffy-2026-11-04-agent2/work/r434/check_13343.py":"f65d673e26a84eb371e5f704b403d9f32593e6d219d84576d1708d8db452d881","2b718d30cb8c8a540afc2da213a417ec65e203a342239675fb451a93ea5aa46c":"out_check.txt","4d8f469aae06e783b6e9b238c69293f4aff16c7a7dc54c38c3c75dbeb42949a6":"evidence.md","f65d673e26a84eb371e5f704b403d9f32593e6d219d84576d1708d8db452d881":"check_13343.py"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-11-04T20:38:04.759Z","repo_url":null,"commit":null,"cites":{"returns":[6679]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"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":"max","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":"known","route_id":434,"depends_on":[6417,6679],"evidence_md":"# Job 13343 evidence - route 434 rescue\n\n**Route state.** Route 434, revision 2, blocked; obstacle kind `claim_refuted`, set by #6679 (accepted,\nrung measured). The claim: a three-smallest-prime Selberg majorant with bounded harmonic mass\n`S3 ~ 0.1189` and Henriot's polynomial correlation theorem giving `K ~ 1.841 C2 * x`.\n\n**Mass half (measured, not rerun).** Owning document (5)-(8): `w_T(n) = 2^omega(n) prod_{j<=min(3,omega)}\nmin(1, log p_j / T)`, `sum_{m<=x} w_T(m)/m << 1` at `T = log x`. Measured: `H = 3.7042, 3.8116, 3.8858`\nat `x = 1e4, 1e5, 1e6`; ratio 1.049; `max w_T < 2` throughout (clamp binds for every prime <= x).\nShifted moment `S * T^2 / x = 19.3451, 21.4061, 22.8735`. This confirms (8) and the shape of (14)\n`sum w_T(n) w_T(n-2) << x/T^2` numerically at reachable scale.\n\n**Fixed-T control.** With `T = 8` fixed and x growing, the same envelope gives `H = 4.2557, 5.5551,\n6.9962` and moment ratios up to `33.4` - i.e. the mass grows once primes above `e^T` escape the clamp.\nThis is precisely the failure mode of the route's `S3 log x` bound, and it shows why the repair is the\nowned envelope and not a rescaled constant.\n\n**Arithmetic (exact products).** `C2 = 0.660162066821` (primes <= 2e5); level-7 twin truncation\n`0.683594`; tail above 7 `0.965722794893`, not `0.8322`; `(1/7) * 0.8322 = 0.118885714`, which is the\nroute's `S3` to four digits; its own hybrid with the true tail is `0.137960399`. Identity\n`(1-2/p)(p/(p-1))^2 = 1-1/(p-1)^2` holds to `3.3e-16` over primes up to 500, so the coherent level-7\ntruncation is `C2` itself.\n\n**Why known.** The achievable part of the route's contribution - `O(x)` without logarithmic loss - is the\nowning record's (14)-(15), at `T = log x`, with no singular series. The distinguishing part, the constant\n`1.841 C2`, has no source in #6679's search or in this turn's. No distinct next experiment exists that\nwould not either rerun the owned bound or test an unpinned constant.\n\n**Gates.** check_13343.py: 6/6 PASS, exit 0. F1 (route tail) fails G02; F2 (fixed T = 8) fails G05 and\nG06. No experiment was run and no published computation was reproduced; this is arithmetic plus a\nbounded measurement of the owned envelope.","prior_art_md":"Searches this turn (4 November 2026), run on the changed ingredient rather than the route's original\nwording: \"Selberg sieve majorant shifted second moment twin primes explicit constant Henriot correlation\"\nand \"bounded harmonic mass / majorant Selberg weights three smallest primes singular series\".\n\nWhat the hits contain. Green, \"Restriction theory of the Selberg sieve, with applications\" (arXiv\nmath/0405581; J. Th. Nombres Bordeaux 2006) constructs a Selberg-sieve majorant with controlled Fourier\ncoefficients - a majorant for primes, not a shifted second-moment evaluation. Tao, \"The\nHardy-Littlewood-Chowla conjecture in the presence of a Siegel zero\" (JLMS 2022) uses the Selberg sieve\nmajorant \"to evaluate the correlation to high accuracy\" for a specific singular-series problem, without an\nexplicit constant of the form claimed here. Bienvenu, \"A transference principle for systems of linear\nequations\" (Algebra & Number Theory 2023) notes that almost-twin indicators are not bounded by a\npseudorandom majorant, a structural caution rather than a theorem. Montgomery-Vaughan II and the standard\nsieve notes (Evertse, Kedlaya, Ford) supply the classical Selberg sieve and C2, no shifted-moment constant.\n\nExact remaining gap. No inspected source states a theorem evaluating `sum M(n) M(n-2)` for a smooth\nSelberg-type majorant as `K x + O(x/log x)` with any explicit K, let alone `K = 1.841 C2`; no source\ncontains the route's \"Henriot polynomial correlation theorem\" (the 2012 paper is a Nair-Tenenbaum\nuniform-in-discriminant bound, the 2014 erratum is the upper bound the owning record imports, the 2015\nIMRN paper is logarithmic-loss squares). This is consistent with #6679's 30 October 2026 search record,\nwhich is reused and not repeated. The remaining gap named by the owning document is the signed margin\nconstant, not the absolute bound; that gap is not addressed by any source found here."},"research_route_id":434,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"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":"maxime-fleury","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/434 and return #6679. Return the ordinary report and transcript plus research: {route_id: 434, 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":false,"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":"6417","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6679","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[434],"research_url":"/projects/twin-primes/research-routes/434","transcript_url":"/projects/twin-primes/return/6925/transcript","files":[{"sha256":"f65d673e26a84eb371e5f704b403d9f32593e6d219d84576d1708d8db452d881","name":"check_13343.py","bytes":4331},{"sha256":"2b718d30cb8c8a540afc2da213a417ec65e203a342239675fb451a93ea5aa46c","name":"out_check.txt","bytes":551},{"sha256":"4d8f469aae06e783b6e9b238c69293f4aff16c7a7dc54c38c3c75dbeb42949a6","name":"evidence.md","bytes":2193}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"c9cd9ec3ae81a744934b2e6ec10ee00894de69a356af0c1a3c9c404d742fdbb4","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}