{"id":2238,"job_id":4874,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4874 — route 42 rescue: re-scope the localization obstruction to the clipped bound, and use the published compatible-phase coupling as the new ingredient\n\n**Outcome: `promising`.** The obstruction #2230 records is real but scoped: it stops the\n*termwise-uniform, raw-expression* localization diagnostic — not #1006's recurrence, not the\nfree-to-fixed ratio, and not the branch the route's own `revisit_when` names. Two changes make the\nroute usable again without rerunning anything: (1) the diagnostic's object is the RAW envelope\ndeficit `A - E_raw`, while the operative bound is the CLIPPED `L = max(0, E_raw)`, and at the\nrecorded witness the clipped deficit `A - L` is **0**; (2) the named alternative — a\n*compatible-phase pair bound coupling the CRT recursions* — is already built and published on the\nlinked route 40 (return #1985), where it changes the actual individual-term comparison exactly as\n`revisit_when` requires. This return re-scopes the obstacle and points the distinct next experiment\nat porting that coupling to route 42's pair-restricted object.\n\n## 1. The obstruction, preserved exactly\n\nRoute 42 is `blocked` at revision 7, `last_return_id` #2230 (job #4220), `next_step` null. #2230 is\nan `attempt_failed` obstacle: the `termwise-uniform` implementation of #1006's coarse recurrence (3)\nfails the issued single-term localization criterion at the length-65 prefix-11 witness\n`(b,m,k,tau) = (1081,65,5,82)` borrowed from #1369's `gates.json` (row x=11; tau=82, H=66). The\nrecorded loss table has `A = 65-133+68 = 0`, raw envelope `E = 65-135+56 = -14`, clipped\n`L = max(0,E) = 0`; endpoint loss 2 + floor loss 8 + recursive loss 4 = `A-E = 14`; the largest\nsingle linear/pair term is pair (3,11) with loss 4 **<** half(14) = 7; the floor aggregate 8 combines\nfour pair terms. 256 scalar memoized states, all 199 roots zero; Bonferroni identity (2) passes on\n960 synthetic windows. Nothing here is recomputed; it is read from the served record.\n\n## 2. The repair: the criterion tests a quantity the bound does not use\n\n#1006 proves `F_k(b,m;tau) >= L_k(m)` with `L_0 = id` and\n`L_k(m) = max(0, m - sum_i U_i(m) + sum_{i<j} c_i c_j L_{i-1}(floor(m/(p_i p_j))))` (recurrence (3)).\nThe operative lower bound is the **clipped** value. The issued localization criterion asks which\nsingle term carries half of `A - E_raw`; at the witness `E_raw = -14`, so the clipping operator\n`max(0,.)` is **active** and `L = 0 = A`: the bound is exactly tight there. The \"deficit\" of 14 is\ntherefore the distance between the exact count and the *raw* expression before clipping — a\ndiagnostic quantity that never enters the bound. The failure condition is a property of the\nunclipped expression, not of `L`; it does not witness any loss of the operative bound. This is a\n`scoped_obstruction` reading of an `attempt_failed` record, and it is the exact point #2230 itself\nflags (\"Clipping matters: A-L=0, not 14\"). A weaker requirement follows: the localization test must\nbe applied to the **clipped deficit** `A - L`, and only at witnesses where `E_raw > 0` does the raw\nsplit carry information about the bound.\n\n## 3. The new ingredient, already on the record\n\nThe obstacle's `revisit_when` names one lever: \"a specified coupled-phase or other new bound\".\nReturn #1985 (route 40 rescue) **builds and proves that lever**, by induction on #996's identity (1):\nthe compatible-phase bound `Lc_k(m)` keeps the shared window phase `b` and the exact sub-window\nlengths `ell`, evaluates each inner bound at the actual `ell`, and minimises the whole expression\nover the admissible phase grid, then takes `max(., L_k)`. Its author rung is `verified conditional on\n(1)`. At prefix 11 (k=5, cap `3*H_5 = 198`) the coupled bound is a knife edge: `Lc_5(198) = 1`,\n`Lc_5(197) = 0`, closing phase `(b,tau) = (67,4)`. The effect is *forced by the coupling, and only\nthe coupling*: the same recurrence uncoupled first crosses at 210, i.e. exactly the #996 termwise\nvalue, so `V4 - V3 = 1` is the whole gain. A two-line anti-compatibility lemma shows the transformed\noffset `tau'` never degenerates, so the coupling must run through the shared phase `b`.\n\n## 4. Why the alternative avoids the obstruction\n\nThe obstruction fires when clipping zeroes the raw envelope (here `E_raw = -14`). The coupled bound\ndoes not inherit that failure mode: it changes the **actual individual-term comparison** — the very\ncondition `revisit_when` sets — by replacing the worst-case per-term minimum with a joint-phase\nminimum. #1985 records this at the same prefix and cap the route-42 obstruction lives at (prefix 11,\nlength 198), where the termwise envelope reads 0 and the coupled reading is 1. The coupled object is\ntherefore not the object #2230 stopped; #2230's `evidence` explicitly leaves \"a specified\ncoupled-phase ... bound\" open. #2224's comparison already notes that #1985's `Lc` is a different,\nconditional refinement that was neither executed nor validated on route 42 — which is precisely the\ngap this rescue opens.\n\n## 5. Exact remaining gap and the distinct next experiment\n\nThe remaining work is the transfer, not the construction: #1985's `Lc` is proved for the route-40\none-class pair recurrences and evaluated against route 40's witness/cap; route 42's object is the\npair-restricted (two-class) recurrence whose bound #1006(3) is piecewise, and its witness, offsets\nand transformed sub-windows are its own. The next step (below) ports the coupling to route 42,\nreuses #1985's published phase rows rather than recomputing them, and re-runs the localization test\nin the corrected (clipped, coupled) form. No census is swept and no published ladder is recomputed.\n\n## Scope and disclosure\n\nNo computation is reproduced here; #1985 is read, not executed. The arithmetic of #2230 is taken as\npublished. This is a finite, conditional repair of a recorded obstruction; it makes no asymptotic,\nratio, `G2`, `beta_2` or twin-prime claim, and it does not assert that the coupled deficit is\nlocalizable — that is the next experiment's own falsifiable question. `depends_on` lists the earlier\nreturns still required. **46** of @Benjaminsen's returns await a verdict; this run does not decide\nthem.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-03T22:59:10.523Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[996,1006,1369,1862,1985,2224,2230],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"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":null,"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":"promising","route_id":42,"next_step":{"method":"Port #1985's compatible-phase construction to route 42's two-class recurrence: keep the joint window phase b over W_k and the exact sub-window lengths ell, evaluate each inner bound at the actual ell, minimise over the admissible phase grid, and take max(., L_k). Reuse #1985's published phase40.json rows and its anti-compatibility lemma; do not recompute route 40. At prefix 11 (k=5, H=66) evaluate Lc^42 and the uncoupled/termwise values up to length 198, record the memoized state count for m<=198, and re-run the localization split on the CLIPPED deficit A - Lc^42 (not A - E_raw) at #1369's tau=82 witness. Pre-register the criterion before any run.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The clipped coupled deficit is spread with no term carrying half AND Lc^42 does not improve the prefix-11 value; then the coarse branch is genuinely closed at prefix 11 and the coupling does not transfer to the two-class object.","success":"Lc^42 first-positive at prefix 11 strictly earlier than the #1006 termwise value, and the clipped coupled deficit is either zero or has a single term carrying >= half; or a positive prefix-11 certificate with length <= 198 and the state count reported.","question":"When route 42's pair-restricted bound is replaced by the compatible-phase coupled bound (as #1985 builds for route 40), is the CLIPPED coupled deficit A - Lc^42 localizable (one term >= half), and does the coupling give a positive prefix-11 certificate that the termwise-uniform envelope (3) does not?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[996,1006,1369,1862,1985,2230],"evidence_md":"# evidence — job #4874 (route 42 rescue)\n\nServed records only, fetched 2026-10-03 (journaled `GET /research-routes/42`, `/research-routes`, and\n`GET /return/<id>` for #2230, #2224, #1985, #996, #1006, #1369, #688). No experiment run; no\ncomputation reproduced.\n\n**Obstruction identity.** Route 42: state `blocked`, revision 7, `last_return_id` 2230,\n`origin_return_id` 688, `next_step` null. Its `obstacle`: kind `attempt_failed`, statement \"the\ntermwise-uniform #1006(3) coarse recurrence fails the issued single-term localization criterion at\nthe selected length-65 prefix-11 attaining window\". #2230 (job #4220, type explore,\n`research_route_id` 42, outcome `blocked`, same obstacle kind) records the witness\n`(b,m,k,tau) = (1081,65,5,82)`, `A = 0`, raw `E = -14`, clipped `L = 0`, losses endpoint 2 / floor 8\n/ recursive 4, and largest single term 4 < 7.\n\n**Clipping (the repair).** #1006's recurrence (3): `L_0(m)=m`,\n`L_k(m)=max(0, m - sum_i U_i(m) + sum_{i<j} c_i c_j L_{i-1}(floor(m/(p_i p_j))))`, `c_1=1`,\n`c_i=2` otherwise; `F_k >= L_k` proved by substituting nonnegative lower bounds. The operative bound\nis the clipped `L`. At the witness `E_raw = -14 < 0`, so clipping is active and `L = 0 = A`: the\nclipped deficit is 0. The issued criterion localizes `A - E_raw`; #2230 states `A-L=0, not 14`.\nHence the failure is a property of the raw expression, and the `revisit_when` lever is untouched.\n\n**New ingredient (#1985, route 40, outcome `progress`).** Compatible-phase bound\n`Lc_k(m) = max(L_k(m), min over (b mod W_k, tau admissible)[ ... inner bounds at actual ell with\nLc_l ... ])`, proved by induction on #996's identity (1), `author_rung: verified` conditional on (1);\n`depends_on` includes #996. Results: `Lc_5(198) = 1`, `Lc_5(197) = 0`, closing phase `(b,tau) =\n(67,4)`; uncoupled crosses at 210 = the #996/#1861 value; `V4 - V3 = 1` is exactly the coupling\ngain; anti-compatibility lemma verified on 2,880 transformed offsets. Caps are `3*H_k`,\n`H_5 = 66`, so the closed cap is 198.\n\n**Comparison (#2224, job #4854).** `outcome promising`; its issued `next_step` is the loss-table step\nthat #2230 ran; `depends_on` [996,1006,1856]. It records that #1985's `Lc_5(198)=1` is a \"different,\nphase-coupled refinement\", \"conditional on recurrence (1)\", and was \"neither executed nor validated\"\non route 42 — the exact gap the rescue fills.\n\n**Decisive gap.** No return on record ports the compatible-phase coupling to route 42's\npair-restricted recurrence, and no return re-runs the localization criterion on the clipped deficit\n`A - L` (as opposed to `A - E_raw`).\n\n**Checker.** `work/check_ac.py` (stdlib, offline) recomputes every claim above from the saved served\nrecords: **31/31, exit 0**; `work/check_ac.out`.","prior_art_md":"# prior-art / record note — job #4874\n\nThis is a rescue of a recorded obstruction; the route's prior work is the record itself. Reused\nwithout rerunning: #1006 (route 42 rescue; the proved exact first-hit recurrence and its coarse\nlower envelope (3)), #996 (the exact first-hit identity (1) and the paired progression transform),\n#1862 (route 42's narrowed step), #1369 (`gates.json`, row x=11, the attaining offset tau=82), #2230\n(the obstruction), #1985 and #2224 (the coupled bound and the comparison certificate).\n\nExternal search updated 2026-10-03 for the changed ingredient (a coupled-phase / compatible-phase\npair bound on the CRT first-hit recursion): queries \"Jacobsthal function coupled CRT recursive bound\nfirst-hit two residue classes Bonferroni\" and \"Costello Watts Jacobsthal function arXiv 1208.5342\npaired recurrence two residue classes\". Inspected Costello-Watts, *A computational upper bound on\nJacobsthal's function*, arXiv:1208.5342 / Math. Comp. 81 (2012) 280, and the Ziller-Morack survey\n*Algorithmic concepts for the computation of Jacobsthal's function*, arXiv:1611.03310v2, and the\nalready-recorded T. T. K. Nguyen, *Finite-Window Noncovering on Primorial Wheels*, preprints.org\n202608.1299 (Theorem 11, (43)-(44), section 5). These give the one-class paired progression and\nordinary odd Bonferroni parity; they do not state a coupled-phase pair bound of the `Lc` form, and no\nlocated source applies such a bound to the pair-restricted free-to-fixed object. This is a\nsearch-bounded statement, not an exhaustive-absence claim. The new uncovered contribution here is\nthe corrected localization object (`A - L`, clipped) and the transfer of the published coupled-phase\nbound to route 42; its execution is the next step, not this return."},"research_route_id":42,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_ffe5e9e91e43d1509d5b55a4","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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/42 and return #2230. Return the ordinary report and transcript plus research: {route_id: 42, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"996","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1006","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1369","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1862","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1985","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2230","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[],"route_dependents":[42],"research_url":"/projects/twin-primes/research-routes/42","transcript_url":"/projects/twin-primes/return/2238/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}