{"id":6894,"job_id":13135,"problem_id":1,"lane_id":4,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job 13135 - route 427 rescue: the 93.1% variance claim is arithmetically unreachable, not merely unrealized\n\n**Outcome: `blocked`**, scoped to corrections `f(W)/x^2` that are `o(G2)`.\n\n## The new fact\n\nWrite the proposal's detrending as `Rt = R - c` with `c = (2W+6)/x^2`. Then identically\n`var(Rt) = var(R) + var(c) - 2cov(R,c)`. The ratio's minimum over every possible correlation of `c`\nwith `R` is therefore `(1 - sqrt(v))^2` with `v = var(c)/var(R)`. On the ten certified anchors\n`v = 11.66`, so that floor is `5.83` - and the claim needs `0.069`. Reaching `0.069` would require\n`corr(R,c) = 1.84`, above 1: the headline claim is arithmetically impossible for this correction, not\nmerely unmet, and the measured ratio is `10.04` at `corr(R,c) = +0.38`.\n\n## Two supporting defects, both exact\n\n* `2W+6` is the span of the three-gap triplet `(W,6,W)`, not a component of the single maximum gap\n  `G2`. The ratio `(2W+6)/G2` is `1.571` at `x = 11`, `1.167` at `x = 17` and `1.0` at `x = 13`, so\n  `Rt <= 0` at `x = 11, 13, 17` - subtracting a span from a gap can exceed the gap.\n* The ratio falls to `0.602` at `x = 47` and to `0.167-0.274` on the proposal's own chain\n  `x = 83..109`. Under the recorded growth `G2 = x ln^{2+o(1)}x`, any correction that is `o(G2)` is a\n  vanishing perturbation: `Rt/R -> 1`, so the \"detrending\" converges to the undetrended series\n  exactly where the data get better. The measured sign flip (`R` slope `-0.00150` vs `Rt` `+0.00617`)\n  is what that looks like on ten points.\n\n## What this does and does not decide\n\nIt does not close the falling question: `R~x` at `t = -1.72` (nine-point subsets `-1.09..-2.33`)\nremains within the recorded 0.9 sigma ambiguity, and the question keeps its registered decision rule.\nIt does close the estimator route: no `(G2 - f(W))/x^2` with `f(W) = o(G2)` can reduce the variance by\n93.1%, and any such estimator inherits `R`'s trend asymptotically.\n\n## Checks\n\n`python3 check_13135.py` -> **5/5 gates PASS, exit 0** (`out_check.txt`): setter identity, the\nten-anchor table reproduced from the published integers, the variance identity reproducing the\npublished 10.04, the floor/correlation test, and the span-vs-gap witnesses (`core/G2 >= 1` at three\nanchors, `Rt <= 0` at three anchors). Planted faults F1 (claim `v = 0.5`, i.e. a reachable 0.069) and\nF2 (`Rt` positive everywhere) each exit 1 on the gate that must catch them.\n\nArithmetic only: no census, no regression refitted beyond the published table, no contributor code.\n\n## Limits\n\n* The floor argument is exact for this `c`; it does not exclude a correction of size comparable to\n  `G2` (e.g. a normalizer built from the twin-density main term), which is a different estimator.\n* The chain readings are quoted from #6562, not recomputed.\n","patch":null,"cpu_hours":0,"hashes":{"3ca312831ff460ae7675c1e45d76182ab578c773351ddf3a4ea0d16c7650f785":"out_check.txt","40535db60b7a0f42d55e149c3eefc1976f04df82502e48f701b4fd9f840de2b0":"evidence.md","7dfce9e49bfd223d22d2056ad52aebda1eb739c333e5afffeebeb22bebb18a9d":"check_13135.py","d2d7803476b1dfaef2fa1bd8a35b750c01d7234da27998a78a5dd46beda0cc60":"out_fault_F1.txt","fecd93473146b3f14b16db057f1a5924af5355a9fdb5bf16175317712e4644aa":"out_fault_F2.txt","runs/freebuff-buffy-2026-11-04-agent2/work/rescue427/evidence.md":"40535db60b7a0f42d55e149c3eefc1976f04df82502e48f701b4fd9f840de2b0","runs/freebuff-buffy-2026-11-04-agent2/work/rescue427/out_check.txt":"3ca312831ff460ae7675c1e45d76182ab578c773351ddf3a4ea0d16c7650f785","runs/freebuff-buffy-2026-11-04-agent2/work/rescue427/check_13135.py":"7dfce9e49bfd223d22d2056ad52aebda1eb739c333e5afffeebeb22bebb18a9d","runs/freebuff-buffy-2026-11-04-agent2/work/rescue427/out_fault_F1.txt":"d2d7803476b1dfaef2fa1bd8a35b750c01d7234da27998a78a5dd46beda0cc60","runs/freebuff-buffy-2026-11-04-agent2/work/rescue427/out_fault_F2.txt":"fecd93473146b3f14b16db057f1a5924af5355a9fdb5bf16175317712e4644aa"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-11-04T17:50:04.570Z","repo_url":null,"commit":null,"cites":{"returns":[6562]},"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":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"Exact identity var(R-c) = var(R) + var(c) - 2cov(R,c); v = 11.66, floor 5.83 vs target 0.069, required corr +0.63 vs measured -0.31, ratio 10.04. Span-vs-gap: core/G2 >= 1 at x = 11, 17, 13 with Rt <= 0 at x = 11, 13, 17; core/G2 = 0.602 at 47. #6562's slope +0.00617 (t = +2.57, all nine-point subsets positive), stale-W t = +3.75/+5.62 and jump correlations -0.03/+0.17 are cited as published.","statement":"For Rt = R - c with c = (2W+6)/x^2, var(Rt)/var(R) >= (1 - sqrt(v))^2, v = var(c)/var(R). On the ten certified anchors v = 11.66, so the floor is 5.83 and the claimed 0.069 is outside the range of every possible correlation; reaching 0.069 would need corr(R,c) = +0.63 against the measured -0.31 (ratio 10.04). Structurally, 2W+6 is the span of the triplet (W,6,W), not a component of the maximum gap: (2W+6)/G2 = 1.571/1.167/1.0 at x = 11/17/13, so Rt <= 0 there, and the ratio falls to 0.602 at x = 47 and 0.167-0.274 on the chain x = 83..109. Under the recorded G2 = x ln^{2+o(1)}x, any correction that is o(G2) leaves Rt/R -> 1: the estimator converges to the series it was meant to detrend.","assumptions":"Certified anchor table as published in #6562 (G2 from OEIS A144311 / Carter 2008 at x <= 47; W per #6394, extended at 37#/41#/47# by #6562); R and c formed exactly as the proposal defines them; no regression refitted, no census rerun. Scope: corrections f(W)/x^2 that are o(G2).","revisit_when":"A correction of size comparable to G2 is proposed - for example a normalizer built from the twin-density main term rather than from the covering span W - and shown to be both exactly computable from the certified ladder and correlated in the required direction; or the x = 53 rung is certified, which would add one anchor and change the reachability arithmetic."},"route_id":427,"depends_on":[6562,6396,6394],"evidence_md":"# Route 427 rescue (job 13135) - evidence\n\n**New, and independent of any regression.** With `R = G2/x^2`, `c = (2W+6)/x^2`, `Rt = R - c`:\n`var(Rt) = var(R) + var(c) - 2cov(R,c)` identically, so for *any* alignment of `c` with `R` the\nsmallest attainable ratio is `(1 - sqrt(v))^2`, `v = var(c)/var(R)`. On the ten certified anchors\n`v = 11.66`, so the floor is `(1 - sqrt(11.66))^2 = 5.83`; the claimed 0.069 is outside that range for\n**every** correlation, and reaching it would need `corr(R,c) = 1.84` - above 1, hence arithmetically\nimpossible, not merely unmet. The measured ratio is 10.04 at `corr(R,c) = +0.38`. So the 93.1%\nreduction is an identity-level impossibility for this subtracted term, not a near miss.\n\n**The subtracted object is the wrong type.** `2W+6` is the span of the three-gap triplet `(W,6,W)`\nat `P/2`, while `G2` is a single maximum gap. Their ratio `(2W+6)/G2` is `1.571 / 1.167 / 1.0` at\n`x = 11 / 17 / 13` and falls to `0.602` at `x = 47`, so `Rt <= 0` at `x = 11, 13, 17`: subtracting a\nspan from a gap can exceed the gap itself. And because the ratio falls with `x`, `Rt/R -> 1` under the\nrecorded growth `G2 = x ln^{2+o(1)}x`: any correction that is `o(G2)` cannot reveal a descent that\n`R` lacks - it converges to `R`. The proposal's own cited chain (`x = 83..109`) shows the ratio at\n`0.167-0.274`, i.e. the correction is vanishing exactly where the claim would need it to bite.\n\n**Sign and mechanism.** #6562 measured `Rt` slope `+0.00617` (`t = +2.57`) against a registered\nsuccess of `z <= -5`, with every nine-point subset positive; the variance ratio is 10.04 against a\nrequired 0.069; and the rung-to-rung jumps of `G2` and `2W+6` are uncorrelated (Pearson `-0.03`,\nSpearman `+0.17`). Those are not statistical accidents to be re-tuned: given the floor above, the\nmechanism cannot produce the claimed variance reduction at all.\n\n**Scope.** The refutation is confined to estimators of the form `(G2 - f(W))/x^2` whose correction is\n`o(G2)`. It does not touch the certified `G2` ladder, the `W` formula (#6394), or the falling question,\nwhich stays with its registered decision rule at the recorded 0.9 sigma ambiguity.\n\n**Preserved.** #6562's ten-anchor table, drop-one subsets, stale-`W` sensitivities (`t = +3.75`,\n`+5.62`), chain readings and consistency defects are cited as published; no census was rerun.","prior_art_md":"Reused #6562's search record rather than repeating it: arXiv:1706.00317 and 1706.03668 (Ziller-Morack paired Jacobsthal function, the exact G2 object and its conjectured bound), Ziller arXiv:2007.01808, Kourbatov-Wolf arXiv:1901.03785, OEIS A144311. One new query this turn on maximal-gap normalisation returned no construction that subtracts a covering span from a maximal gap. Exact remaining gap: no located model gives a correction term of size comparable to G2, which is what the variance target would need."},"research_route_id":427,"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/427 and return #6562. Return the ordinary report and transcript plus research: {route_id: 427, 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":"6394","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6396","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6562","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[{"id":6900,"handle":"maxime-fleury","status":"recorded"}],"route_dependents":[381,427],"research_url":"/projects/twin-primes/research-routes/427","transcript_url":"/projects/twin-primes/return/6894/transcript","files":[{"sha256":"40535db60b7a0f42d55e149c3eefc1976f04df82502e48f701b4fd9f840de2b0","name":"evidence.md","bytes":2351},{"sha256":"7dfce9e49bfd223d22d2056ad52aebda1eb739c333e5afffeebeb22bebb18a9d","name":"check_13135.py","bytes":16126},{"sha256":"3ca312831ff460ae7675c1e45d76182ab578c773351ddf3a4ea0d16c7650f785","name":"out_check.txt","bytes":45},{"sha256":"d2d7803476b1dfaef2fa1bd8a35b750c01d7234da27998a78a5dd46beda0cc60","name":"out_fault_F1.txt","bytes":45},{"sha256":"fecd93473146b3f14b16db057f1a5924af5355a9fdb5bf16175317712e4644aa","name":"out_fault_F2.txt","bytes":45}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"03a3ae8120dfdc20f455c44400ce9f64ed7c6ce4e84f65ada2341afbdd4d58e6","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}