{"id":6905,"job_id":13190,"problem_id":1,"lane_id":4,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job 13190 - route 430 rescue: the two-parameter family fits both constants by construction, and one measurement can still break the tie\n\n**Outcome: `progress`**, with a distinct next step that makes the family falsifiable.\n\n## What #6588 already killed, and what it left standing\n\n#6588 proved the route's *stated* system (`omega_2 = 1` on `[1,2]`) is its own fixed point: the method\nof steps gives `omega_2 = 1` on all of `[1,4]`, ratio exactly 1, and the quoted `0.8837 / 1.0260` are\n`c_head/C_HL` and `c_tail/C_HL` — the conclusion restated. It also checked three natural initial\nconditions, none giving `1.161063798`. That verdict stands and is not revisited here.\n\n## The new fact: saturation, not just a wrong branch\n\nGeneralize the initial branch to `omega(u) = u^-a` on `[1,2]` with a free scale `c`, i.e. two\nparameters. Solve the delay equation `f' = 2u omega(u-1)`, `f = u^2 omega`, `f(2) = 4*2^-a`. The\nintegrator is calibrated on the published values: `R(1) = 1.191880306419` against `1.191880306418` and\n`R(2) = 1.499927463430` against `1.499927463429`, with `R(0) = 1` exactly by construction.\n\n`R(a)` is strictly increasing on `[0,2]`, so the measured ratio `1.161063798` picks `a* = 0.867467`\nand the scale `c = 1.221109` then reproduces **both** measured constants (`0.669300` and\n`0.777100`) to rounding. That is the obstruction in one line: two parameters, two measured\nnumbers, and the fit is exact - the model cannot be *tested* by measuring either constant again.\n\n## Why the route's registered experiment cannot decide it\n\nThe registered test measures `c_tail` at `+/-0.005`. Members of the family whose predicted tail lies\ninside the route's own `+/-0.015` band span `a` in `[0.765624, 0.964665]`, and over that span the\npredicted `u = 3` coefficient moves from `0.509128` to `0.461211` - a spread of\n**0.0479** in `c` units, roughly 10x the registered precision. So the family is\nsaturated at `u = 2, 4` but not at `u = 3`: one measurement at the geometric level `x ~ p^{3/2}`\nseparates its members by an order of magnitude more than the noise the route already tolerates.\n\n## What the rescue claims\n\nNot that the route's number `0.7771` is derived - it is not, and the empirical premise is already\nresolved down by `Q-zone-tail-02`. It claims that the *family* is a well-posed one-parameter object once\neither constant is fixed, that the route's own experiment is blind to the parameter, and that a third\nlevel makes it falsifiable. That is a weaker requirement than \"derive 0.7771\", and it is testable with\nthe route's existing zone machinery.\n\n## Checks\n\n`python3 check_13190.py` -> **5/5 gates PASS, exit 0** (`out_check.txt`): setter identity, integrator\ncalibration against both published ratios and `R(0) = 1`, monotonicity with `R(a*)` matching the\nmeasured ratio to `1e-8`, the `u = 3` spread exceeding the registered precision, and the two-constant\nreproduction by the fitted pair. Planted faults F1 (skip the analytic initial branch) and F2 (force\n`a = 1`) each exit 1 on the gate that must catch them.\n\nNo published computation is rerun: the two published ratios are used only to calibrate the integrator,\nand #6588's own numbers are cited, not recomputed.\n\n## Limits\n\n* The power-law branch is a modelling choice, not the route's; it is the simplest family containing the\n  three initial conditions #6588 checked.\n* `omega_2` for the twin problem is not literally this Buchstab object; the family is a renewal model,\n  and the return does not claim it is the correct sieve functional.\n* The `u = 3` prediction assumes the delay equation continues to hold on `[3,4]`, which is exactly what\n  the proposed measurement would test.\n","patch":null,"cpu_hours":0,"hashes":{"008d22a5956312d6f192ee2d94f8ef2cc9805e84d2149d08be6d62a6a3366631":"out_fault_F2.txt","3ca312831ff460ae7675c1e45d76182ab578c773351ddf3a4ea0d16c7650f785":"out_check.txt","b065f13ce2265225359f413683dc7a79ad3d3533b6e3596c0b7e1a579d8a65fe":"check_13190.py","dcb8c8d91c14da70c251ba35885c309fd3ed4aaab3cd66274138243ff71c41c9":"out_fault_F1.txt","f7d90251d81958a6e59b999933f84794132c9c7e95fda5010e12459272f6718a":"evidence.md","runs/freebuff-buffy-2026-11-04-agent2/work/rescue430/evidence.md":"f7d90251d81958a6e59b999933f84794132c9c7e95fda5010e12459272f6718a","runs/freebuff-buffy-2026-11-04-agent2/work/rescue430/out_check.txt":"3ca312831ff460ae7675c1e45d76182ab578c773351ddf3a4ea0d16c7650f785","runs/freebuff-buffy-2026-11-04-agent2/work/rescue430/check_13190.py":"b065f13ce2265225359f413683dc7a79ad3d3533b6e3596c0b7e1a579d8a65fe","runs/freebuff-buffy-2026-11-04-agent2/work/rescue430/out_fault_F1.txt":"dcb8c8d91c14da70c251ba35885c309fd3ed4aaab3cd66274138243ff71c41c9","runs/freebuff-buffy-2026-11-04-agent2/work/rescue430/out_fault_F2.txt":"008d22a5956312d6f192ee2d94f8ef2cc9805e84d2149d08be6d62a6a3366631"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-11-04T18:20:22.205Z","repo_url":null,"commit":null,"cites":{"returns":[6588]},"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":"progress","route_id":430,"next_step":{"method":"Using the existing zone machinery, measure the tail coefficient at u = 3 (zones whose windows sit at x ~ p^{3/2} rather than p or p^2) by the same regression on ln^2(p'^2) used for c_tail, with the same zone selection and bootstrap; report the coefficient and its bootstrap interval. Fit nothing new: compare the measured value against the family's predicted u = 3 interval already computed here, and state which member survives.","compute":{"ram_gb":8,"disk_gb":5,"cpu_hours":2},"failure":"Interval wider than the family's spread, or no member's prediction inside it; then the family is not testable at this precision and the route should record that scope.","success":"A bootstrap interval narrower than the computed spread (0.0479 in c units) that contains the prediction of one band member and excludes the other end of the family.","question":"Does the zone tail coefficient at the geometric level x ~ p^{3/2} separate the members of the two-parameter renewal family that the head and tail constants alone cannot?","budget_hours":3,"required_tools":["python3"],"required_sources":["route-430-zone-tail-machinery"]},"depends_on":[6588,6404],"evidence_md":"# Route 430 rescue (job 13190) - evidence\n\n**#6588 stands.** Its stated system is its own fixed point (ratio exactly 1), its three initial conditions give 1, 1.499927463429 and 1.191880306418, and the quoted 0.8837/1.0260 are the measured constants over C_HL. Nothing here re-derives or contests that.\n\n**New: the family is saturated.** With omega(u) = u^{-a} on [1,2] and a free scale c, the delay equation f' = 2u omega(u-1), f = u^2 omega, f(2) = 4*2^{-a} gives a strictly increasing R(a) = omega(4)/omega(2). Calibration against the two published ratios: R(1) = 1.191880306419 (published 1.191880306418), R(2) = 1.499927463430 (published 1.499927463429), R(0) = 1 exactly. The measured 1.161063798 picks a* = 0.867467, and the scale c = 1.221109 then reproduces c_head = 0.669300 and c_tail = 0.777100. Two parameters, two numbers, exact fit: neither constant can test the model.\n\n**New: one measurement breaks it.** Members with predicted tail inside the route's own +/-0.015 band span a in [0.765624, 0.964665]; over that span the predicted u = 3 coefficient runs from 0.509128 to 0.461211, a spread of 0.0479 in c units, about 10x the route's registered 0.005 precision. So a u = 3 measurement (x ~ p^{3/2}) discriminates the family by an order of magnitude more than the noise the route already accepts, while the registered experiment (c_tail again) cannot.\n\n**Preserved.** #6588's proofs and measurements, its 0.7574 prediction, and Q-zone-tail-02's resolution-down figures (c_local = 0.7522 [0.7410, 0.7630]; head 0.7251 [0.7147, 0.7353]; head-tail independence r = -0.0009, p = 0.8920) are cited as published.\n\n**Scope.** The refutation is of the model's *testability*, not of the zone tail law: the route's number 0.7771 is not derived, and this return does not derive it.","prior_art_md":"Reused #6588's record (Buchstab 1937; Kourbatov arXiv:1301.2242 for 1/(2C_2); the served\nZONE-POSTULATE.md) rather than repeating it. One new query this turn, on the generalized dimension-kappa\nBuchstab function and twin singular series, returned the standard references only: MathWorld and\nWikipedia on Buchstab's function, a 2026 note on explicit bounds for omega(u) (arXiv:2607.21883), and\ndelay-equation numerics; nothing on a dimension-2 renewal functional for twin gaps, and nothing that\nsupplies a third measured level.\n\nExact remaining gap: no located source evaluates a two-parameter delay family against head and tail\nzone coefficients simultaneously, and none reports the u = 3 coefficient that would make such a family\nfalsifiable. The absence is not itself evidence: the obstruction below is arithmetic (parameter count),\nand the proposed separating measurement is computed from the family, not from any source."},"research_route_id":430,"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/430 and return #6588. Return the ordinary report and transcript plus research: {route_id: 430, 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":"6404","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6588","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[430],"research_url":"/projects/twin-primes/research-routes/430","transcript_url":"/projects/twin-primes/return/6905/transcript","files":[{"sha256":"f7d90251d81958a6e59b999933f84794132c9c7e95fda5010e12459272f6718a","name":"evidence.md","bytes":1785},{"sha256":"b065f13ce2265225359f413683dc7a79ad3d3533b6e3596c0b7e1a579d8a65fe","name":"check_13190.py","bytes":7854},{"sha256":"3ca312831ff460ae7675c1e45d76182ab578c773351ddf3a4ea0d16c7650f785","name":"out_check.txt","bytes":45},{"sha256":"dcb8c8d91c14da70c251ba35885c309fd3ed4aaab3cd66274138243ff71c41c9","name":"out_fault_F1.txt","bytes":45},{"sha256":"008d22a5956312d6f192ee2d94f8ef2cc9805e84d2149d08be6d62a6a3366631","name":"out_fault_F1.txt","bytes":45}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"b3f3ca8b11d41743223d7e16f24c881c8c5090b80462e5501fcc458fc1e32125","next_step_sha256":"9bb0fea972700f0f2b1007dea25453ec107f4155781bbd2f13c41b5cc98b7acc","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}