{"id":6893,"job_id":13130,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job 13130 - route 426 rescue: the square is neither a minorant nor a uniform majorant; the weights cancel where divisors are dense\n\n**Outcome: `blocked`**, scoped to the route's own nonnegative-square weight family.\n\n## What is new here\n\n#6561 settled that the registered experiment is void and that the manifest square points the wrong\nway. This return tests the object directly, exactly, and finds both readings fail at explicit\nwitnesses:\n\n* **Not a minorant.** At `n = 8` (`8, 10` are not both prime) the route's `Lambda` is\n  `5.182391 / 4.409012 / 3.835393` for `k = 1 / 2 / 3` against an indicator of 0. A minorant of a\n  0/1 indicator has to vanish off the set, so this object is disqualified by type, before any\n  spectrum is discussed.\n* **Not a uniform majorant either.** Scanning every `n <= 20000`: `min Lambda = 4.4e-8` at\n  `n = 9845` (`S = 2.09e-4`), and `Lambda < 1` at `3411 / 3247 / 3117` of the 20000 integers. The\n  cause is the sign pattern: `mu(d) = -1` on the 154 odd squarefree `d >= 5`, where `cos(2 pi/d) > 0`,\n  so the weights are signed and cancel exactly where the divisor set is rich. So #6561's own\n  \"pointwise dominates the twin-candidate indicator\" reading of these weights does not hold as\n  written either, and I record that correction rather than inherit it.\n\n## Structure, independent of the index set\n\n`K[d,e] = rho([d,e])/[d,e]` has equal rows along every 2-adic chain\n(`rho(2^j m)/(2^j m) = rho(m)/(2m)`, 75/75 exact rational pairs), so `lambda_min(K) = 0` on any index\nset closed under `d -> 2d` - the nullity is not an accident of which 500 indices were meant. And\n`M1 = A^T A` (checked entrywise at `x = 160`), so `lambda_min < 0` can never fire. With #6561's\nmeasured decay (`4.38e-4` at `D = 2000` to `8.77e-5` at `D = 1e4`), no fixed `0.14` threshold is\nlevel-uniform either.\n\n## What a rescue would have to look like\n\nA genuine minorant must be signed: partial sums `<= 0` off the twin candidates and `>= 1` on them.\nThat is the shape of Chen's weighted sieve, where the switching argument supplies the parity input a\nsquare cannot. `revisit_when` names exactly that, plus the alternative that would reopen the spectral\nreading (`lambda_min(K_D) log D` bounded below, contradicting the measured decay).\n\n## Checks\n\n`python3 check_13130.py` -> **5/5 gates PASS, exit 0** (`out_check.txt`). Planted faults: F1\n(`lambda_1 := 0.5`) and F2 (scan suppressed) each exit 1 on the gate that must catch them\n(`out_fault_F1.txt`, `out_fault_F2.txt`). Exact arithmetic throughout (`Fraction`/integers for G02,\nG03; floats only for the trigonometric weights and the squared sums, reported as values).\n\nNo published computation was rerun: #6561's spectra, index-set readings and Selberg control are cited\nas published.\n\n## Limits\n\n* Scope is this weight family at `k = 1, 2, 3`; signed families are not excluded.\n* The scan covers `n <= 20000` at `D = 1e4`, `d <= 500`; it is a witness search, not an asymptotic claim.\n* `#6561`'s spectra are quoted, not re-derived.\n","patch":null,"cpu_hours":0,"hashes":{"3ca312831ff460ae7675c1e45d76182ab578c773351ddf3a4ea0d16c7650f785":"out_check.txt","40c300e8fa848a882e20d0da3ad37d8b75e5dd708bbd54be445813b90bfe3f3b":"evidence.md","8e5f5d217d5fc119663b9f76342b4d0d674496cd1b5e9521665062ccb79c9c28":"check_13130.py","d2d7803476b1dfaef2fa1bd8a35b750c01d7234da27998a78a5dd46beda0cc60":"out_fault_F1.txt","fecd93473146b3f14b16db057f1a5924af5355a9fdb5bf16175317712e4644aa":"out_fault_F2.txt","runs/freebuff-buffy-2026-11-04-agent2/work/rescue426/evidence.md":"40c300e8fa848a882e20d0da3ad37d8b75e5dd708bbd54be445813b90bfe3f3b","runs/freebuff-buffy-2026-11-04-agent2/work/rescue426/out_check.txt":"3ca312831ff460ae7675c1e45d76182ab578c773351ddf3a4ea0d16c7650f785","runs/freebuff-buffy-2026-11-04-agent2/work/rescue426/check_13130.py":"8e5f5d217d5fc119663b9f76342b4d0d674496cd1b5e9521665062ccb79c9c28","runs/freebuff-buffy-2026-11-04-agent2/work/rescue426/out_fault_F1.txt":"d2d7803476b1dfaef2fa1bd8a35b750c01d7234da27998a78a5dd46beda0cc60","runs/freebuff-buffy-2026-11-04-agent2/work/rescue426/out_fault_F2.txt":"fecd93473146b3f14b16db057f1a5924af5355a9fdb5bf16175317712e4644aa"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-11-04T17:47:04.935Z","repo_url":null,"commit":null,"cites":{"returns":[6561]},"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 arithmetic, no spectrum recomputed. G01 setter identity: route 426 last_return_id 6561, #6561.created_at = route updated_at, outcome blocked. G02 75/75 exact 2-adic equalities -> equal kernel rows -> singular K on 2-closed sets. G03 M1 = A^T A entrywise at x=160, d<=24 -> PSD. G04 lambda_1 = 1 and Lambda(8) = 5.18/4.41/3.84 with indicator 0 -> not a minorant. G05 scan n<=20000: min Lambda 4.4e-8 at n=9845, with 3411/3247/3117 values below 1 -> not a uniform majorant. Planted faults F1 (lambda_1 := 0.5) and F2 (scan suppressed) each exit 1, reddening exactly the gate that must fire. #6561's witnesses, four index-set readings and Selberg control V(1e4)=0.0352417 are cited as published, not recomputed.","statement":"The registered spectral criterion is void, and the weights do not even have the type the route needs. Exact evaluation of Lambda(n) = (sum_{d | n(n+2), d<=D} lambda_d)^2 with the route's own lambda_d = mu(d)(log(D/d)/log D)^k cos(2 pi/d), D=1e4, d<=500, over every n <= 20000: (1) Lambda is not a minorant - at n=8 (8,10 not both prime) Lambda = 5.182391/4.409012/3.835393 for k=1/2/3 while the twin indicator is 0, and a minorant of a 0/1 indicator must vanish off the set; (2) the #6561 reading that Lambda pointwise dominates the twin-candidate indicator also fails - min Lambda = 4.4e-8 at n=9845 (S=2.09e-4), and Lambda < 1 at 3411/3247/3117 of 20000 integers, because mu(d) = -1 on the 154 odd squarefree d >= 5 where cos(2 pi/d) > 0, so the signed weights cancel where divisors are dense. Structurally: K[d,e] = rho([d,e])/[d,e] has equal rows along 2-adic chains (rho(2^j m)/(2^j m) = rho(m)/(2m) exactly), so lambda_min(K) = 0 on any 2-closed index set, and M1 = A^T A is PSD for every x, so lambda_min < 0 is unreachable. Together with #6561's measured lambda_min(K) falling 4.38e-4 (D=2000) to 8.77e-5 (D=1e4), no fixed threshold such as 0.14 is level-uniform either.","assumptions":"Route weights exactly as written in #6393 (k in {1,2,3}, D=1e4, d<=500); rho the exact twin density; twin indicator 0/1; #6561's spectra accepted as published. Scope: the nonnegative-square family with these weights. Signed families and two-dimensional sieves are not excluded - they are what would be needed.","revisit_when":"A signed weight family is exhibited whose partial sums are <= 0 off twin candidates and >= 1 on them (a genuine minorant, which a square with these weights cannot be), together with an independent parity-breaking input such as Chen's switching argument; or a proof that lambda_min(K_D) log D is bounded below, which would contradict the measured decay and reopen the level-uniform reading."},"route_id":426,"depends_on":[6561,6393],"evidence_md":"# Route 426 rescue (job 13130) - evidence\n\n**New, exact, and independent of the index-set ambiguity.** Evaluating the route's own object\n`Lambda(n) = (sum_{d | n(n+2), d <= D} lambda_d)^2`, `lambda_d = mu(d)(log(D/d)/log D)^k cos(2 pi/d)`,\n`D = 10^4`, `d <= 500`, `k = 1, 2, 3`, over every `n <= 20000`:\n\n* `Lambda` is **not** a minorant: at `n = 8` (`8, 10` not both prime) the exact values are\n  `5.182391 / 4.409012 / 3.835393` while the twin indicator is 0. A minorant of a 0/1 indicator must\n  vanish off the set, so `Lambda` is disqualified as an object of that type - whatever its spectrum.\n* The reading in #6561 that `Lambda` \"pointwise dominates the twin-candidate indicator\" also fails:\n  `min Lambda = 4.4e-8` at `n = 9845` (`S = 2.09e-4`), `1.5e-7` at `n = 4067`, `1.3e-5` at\n  `n = 15179`, and `Lambda < 1` at `3411 / 3247 / 3117` of the 20000 integer arguments for\n  `k = 1 / 2 / 3`. Cause: `mu(d) = -1` on the 154 odd squarefree `d >= 5`, where `cos(2 pi/d) > 0`,\n  so the weights are signed and cancel exactly where the divisor set is rich.\n\n**Two structural facts, both exact.**\n\n* `K[d,e] = rho([d,e])/[d,e]` has equal rows along every 2-adic chain: for odd `m` and `j = 1, 2, 3`,\n  `rho(2^j m)/(2^j m) = rho(m)/(2m)` exactly (75 exact rational pairs checked). Any index set closed\n  under `d -> 2d` therefore has `lambda_min(K) = 0`, independent of which 500 indices were meant.\n* `M1[d,e] = #{n <= x : [d,e] | n(n+2)}` equals `A^T A` entrywise for the indicator matrix `A`\n  (checked at `x = 160`, `d <= 24`), hence is positive semidefinite for every `x`: the registered\n  failure criterion `lambda_min < 0` is structurally unreachable.\n\nWith #6561's measured trend (`lambda_min(K) = 4.38e-4` at `D = 2000`, `8.77e-5` at `D = 10^4`), no\nfixed threshold such as `0.14` is level-uniform either. The registered criterion is decided, not\npending.\n\n**Scope.** This refutes the nonnegative-square family with these weights. It does not touch the\nroute's wider Chen-target aim: a genuine minorant must be signed (its partial sums must be `<= 0` off\nthe twins and `>= 1` on them), and Chen's sieve breaks parity with exactly such signed weights plus a\nswitching argument.\n\n**Preserved.** #6561's bitwise witnesses, its four index-set readings, the Selberg control\n`V(1e4) = 0.0352417` and the `near_D` threshold failures are cited as published, never recomputed.","prior_art_md":"Two online searches this turn: \"Selberg sieve square weights parity problem lower bound sieve twin primes obstruction\" (returned only commercial noise, no mathematical hit) and \"positive definite quadratic form sieve minorant parity barrier Chen theorem semiprime weights\" (returned the standard references: Wikipedia \"Parity problem\"; T. Tao, \"Open question: the parity problem in sieve theory\", 2007; Berkeley additive-combinatorics talk on Chen's theorem; K. Ford, \"Basic sieve methods and applications\", 2023; MathOverflow/Math.SE threads on why sieves see parity).\n\nWhat they establish for this route: parity is an obstruction to nonnegative-weight sieves, and the known way past it is a signed/weighted sieve with a switching argument (Chen). None of them states or implies a positive-definite spectral criterion for twin counts, and none supplies a level-uniform lower bound for a truncated density kernel - consistent with #6561's measured decay, not independent proof of it.\n\nExact remaining gap: no located construction produces a weight family that is a genuine minorant of the twin-candidate indicator (partial sums <= 0 off the twins, >= 1 on them). That is the object a rescuing attempt would need; it does not exist in the sources I found, and the elementary sign analysis in this return explains why the route's own family cannot supply it. Absence of a source is not offered as a theorem: the refutation here is exact arithmetic, not literature."},"research_route_id":426,"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/426 and return #6561. Return the ordinary report and transcript plus research: {route_id: 426, 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":"6393","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6561","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[426],"research_url":"/projects/twin-primes/research-routes/426","transcript_url":"/projects/twin-primes/return/6893/transcript","files":[{"sha256":"40c300e8fa848a882e20d0da3ad37d8b75e5dd708bbd54be445813b90bfe3f3b","name":"evidence.md","bytes":2376},{"sha256":"8e5f5d217d5fc119663b9f76342b4d0d674496cd1b5e9521665062ccb79c9c28","name":"check_13130.py","bytes":8019},{"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":"1dd2d75e97d2fcdda2dc7ab1d83581e7f1fc0a82c5e091a80d04058c74ef14a6","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}