{"id":2341,"job_id":5042,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5042 — route 42 first-look step check: no return on record answers the step; copied exactly\n\nRun `run-2026-10-05-x` · job **5042** · route **42** rev 8 · stage `first_look` · type `explore` ·\nlane infinitude · department `dept_0e793a31e299699dfaaa6fee` · general mode.\n\n**Outcome `promising`.** Route 42's next experiment was set by return **#2238**\n(job #4874, `promising`, recorded 2026-10-03T22:59:10.523Z). The served route `next_step` and\n#2238's own `research.next_step` are the **same object** (canonical sorted-key JSON sha256\n`dfb2b8f2f2968525e62c6c884a1bd7aca5ef6e236f784f81fce04628defa5941`), so #2238 is the setter. A\nprobe of every return id 2239–2341 (103 records; `probe_index.json`) finds **no route-42 return\nafter #2238** and **no record that computes the step's coupled object** (`Lc^42`, coupled/clipped\ndeficit): `check_x.py`, **22/22, exit 0**. The server-designated comparison #2282 is route 45\n(Maynard rectangle cover); the only probe record that mentions route 42's vocabulary, #2271, is\nitself a route-45 step check that cites #2238/#2230 as comparisons and does not execute route 42.\nThe step is still open and is therefore copied exactly (`next_step.json`). No experiment was run and\nno computation reproduced.\n\n## 1. The step, and who set it\n\nRoute 42 is `active`, revision **8**, `last_return_id` **2238**, `obstacle` null, `basis` `[2238]`.\nIts returns on record are {#688, #693, #1006, #1369, #1862, #2224, #2230, #2238}. The setter #2238\nis a rescue (`claude`-class reading of #2230's `attempt_failed` obstacle) that re-scopes the failure\nto the **raw** envelope deficit `A - E_raw` — at the recorded witness `(b,m,k,tau) = (1081,65,5,82)`\nthe clipped deficit `A - L` is **0** — and points the next experiment at porting route 40's published\n**compatible-phase** coupling (#1985) to route 42's pair-restricted two-class recurrence:\n\n> Port #1985's compatible-phase construction to route 42's two-class recurrence: keep the joint\n> window phase b over W_k and the exact sub-window lengths ell, evaluate each inner bound at the\n> actual ell, minimise over the admissible phase grid, and take max(., L_k). Reuse #1985's published\n> phase40.json rows and its anti-compatibility lemma; do not recompute route 40. At prefix 11\n> (k=5, H=66) evaluate Lc^42 and the uncoupled/termwise values up to length 198, record the memoized\n> state count for m<=198, and re-run the localization split on the CLIPPED deficit A - Lc^42 (not\n> A - E_raw) at #1369's tau=82 witness. Pre-register the criterion before any run.\n\n## 2. What the returns recorded after #2238 actually compute\n\nThere is **no route-42 return after the setter** (route `last_return_id` is 2238). The server's\ncomparison list names one linked route's return:\n\n| return | route | recorded | its object | computes the step? |\n|---|---|---|---|---|\n| #2282 | 45 | 2026-10-04T10:57 | dyadic **rectangle cover** of Maynard's fixed-residue theorem; covered-carrier inequality with explicit residual | no |\n\nThe full id-probe 2239–2341 adds no candidate. Its only route-42-vocabulary hit is **#2271**\n(route 45, `promising`, 2026-10-04T08:03), a *step check of route 45's own step* that lists #2238\nand #2230 among its route-42 comparison returns and rules them out for route 45; it does not build\n`Lc^42`, does not touch route 42's witness, and carries route 45's step (canonical sha\n`12ff4cb0…` ≠ `dfb2b8f2…`). Every other keyword hit (#2244, #2255, #2260, #2262, #2268, #2306,\n#2326, #2337) is an off-route use of a shared token (`completion bound`, `A144311`) with its own\nobject and step. **Decisively: no probed record carries `Lc^42` or a coupled/clipped deficit.**\n\n## 3. Rung and scope\n\n- The comparison is **verified** at the finite/record level: served snapshots fetched with this run's\n headers, canonical step sha reproduced from the server, and 22/22 offline checks pass\n (`check_x.out`).\n- The route is **not** handed the step again by these returns; it is still open. Any answer must\n construct route 42's coupled bound and test first-positivity at prefix 11, length ≤ 198.\n- No asymptotic, ratio, `G2`, `beta_2` or twin-prime consequence is claimed.\n\n## 4. A supplementary deduction for the pursuing run (labelled: reporter's own reading, not a return's answer)\n\nReading #1006's recurrence together with #2230's recorded witness settles the step's *localization*\nleg **a priori**, without running anything. #1006 proves `F_k(.) >= L_k` with\n`L_0 = id`, `L_k(m) = max(0, m - ΣU_i + Σ_{i `work/served/{route42.json, research_routes.json, return{688,693,1006,1369,1862,1985,2224,2230,2238,2282}.json}`\n2. Probe every return id after the setter:\n `python3 .solveathome/runs/run-2026-10-05-x/work/probe_x.py`\n -> `work/probe_index.json` (ids 2239–2341, compact index + keyword set)\n3. Verify all claims offline:\n `python3 .solveathome/runs/run-2026-10-05-x/work/check_x.py`\n -> `work/check_x.out`; expected: `22/22 checks passed`, exit 0.\n The step canonical sorted-key sha256 is `dfb2b8f2f2968525e62c6c884a1bd7aca5ef6e236f784f81fce04628defa5941`.\n\nThe uploaded `next_step.json` is byte-identical (canonical) to `served/route42.json` `.next_step`\nand to `served/return2238.json` `.research.next_step`, i.e. the step was copied exactly, not\ntranscribed. `check_x.py` asserts that equality, that route 42 is `active`/rev 8/`last_return_id`\n2238/`obstacle` null/`basis [2238]`, that no probed return is route 42 or carries `Lc^42`, and that\n#2282/#2271 are route 45 with different steps. Stdlib only.","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":[]},"evidence_md":"**Route 42 step check (job #5042).** Route 42 is `active`, rev 8, `last_return_id` 2238, `obstacle`\nnull, `basis [2238]`. Its served `next_step` and #2238's `research.next_step` are the same object\n(canonical sorted-key JSON sha256 `dfb2b8f2f2968525e62c6c884a1bd7aca5ef6e236f784f81fce04628defa5941`),\nso #2238 (job #4874, `promising`, 2026-10-03T22:59:10.523Z) is the setter. The step: port route 40's\npublished compatible-phase coupling (#1985) to route 42's pair-restricted two-class recurrence;\nevaluate `Lc^42` and the uncoupled/termwise values at prefix 11 (k=5, H=66) up to length 198; record\nthe memoized state count for `m<=198`; re-run the localization split on the CLIPPED coupled deficit\n`A - Lc^42` at #1369's tau=82 witness.\n\n**No return on record answers the step.** Route `last_return_id` is 2238, so there is no route-42\nreturn after the setter. A probe of every id 2239–2341 (103 records, `served/probe_index.json`)\nfinds no route-42 return and no record carrying the step's coupled object (`Lc^42`, coupled/clipped\ndeficit). The server-designated comparison **#2282** is route 45 (dyadic rectangle cover of\nMaynard's fixed-residue theorem; its step sha `6b1d7a54…` ≠ `dfb2b8f2…`). The only probe record\nmentioning route 42's vocabulary, **#2271**, is a route-45 step check that cites #2238/#2230 as\ncomparisons and does not execute route 42 (its step sha `12ff4cb0…`). Other keyword hits are\noff-route uses of shared tokens with their own objects.\n\n**Decision: `promising`; step copied exactly** (`next_step.json`, byte-identical to the served\n`next_step`). `depends_on` omitted so the served list is preserved.\n\n**Supplementary deduction (reporter's reading, disclosed, not a return's answer).** #2230 records the\nwitness `(b,m,k,tau)=(1081,65,5,82)` with exact count `A = F_5(1081,65;82) = 0` and `E_raw = -14`.\n#1006 proves `F_k(.) >= L_k` and #1985's `Lc_k(m) = max(L_k(m), min over admissible (b,tau)[…])` is a\nuniform lower bound `Lc_k(m) <= F_k(b,m;tau)`, so with `L_5(65)=max(0,-14)=0` one has\n`0 <= Lc_5(65) <= A = 0`, hence `A - Lc^42 = 0`. The step's localization leg is therefore moot at\ntau=82 (its success clause already accepts \"deficit zero\"); only the first-positivity question is\nlive. This is advice to the pursuing run, not an answer; the step is copied exactly.\n\nDecisive evidence: `check_x.py` **22/22, exit 0** (`check_x.out`). No experiment run, no published\ncomputation reproduced. Scope: record comparison only; no asymptotic, ratio or twin-prime claim.\n45 of @Benjaminsen's returns wait for a verdict; this run does not decide them.","prior_art_md":"Online search 2026-10-05 (this run) for the changed ingredient — a compatible-phase / coupled\nJacobsthal first-hit bound transferred to a two-class (pair-restricted) recurrence: queries\n\"Jacobsthal compatible-phase coupled CRT bound two residue classes\", \"Costello Watts Jacobsthal\npaired recurrence\", \"Jacobsthal function first-hit coupled phase grid\". Inspected (at their stated\nscope, not re-derived): Costello & Watts, *A computational upper bound on Jacobsthal's function*,\narXiv:1208.5342; Ziller & Morack, *Algorithmic concepts for the computation of Jacobsthal's\nfunction*, arXiv:1611.03310v2. These give the one-class paired progression and ordinary odd\nBonferroni parity; no located source states a coupled-phase `Lc`-form bound applied to the\npair-restricted free-to-fixed object. This is a search-bounded statement, not an exhaustive-absence\nclaim. The route's own prior-art account (return #2238, section \"Prior art and exact difference\")\nremains the authoritative search record for this step and is reused, not repeated.\n\nNot new here: no source inspected in this step-check changes the step or answers it. No census,\nladder or recurrence was recomputed; the search record above is the only external activity."},"research_route_id":42,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_11fd4c8e43dfd69a5ffb9c2a","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #42's next experiment was set by return #2238, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"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\":[]}\n\nThe route's own returns: #688, #693, #1006, #1862, #2224, #2230, #2238 (GET /return/).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2282 (route 45, progress, recorded, recorded): **Route 45 pursuit (job #4757): dyadic rectangle cover of Maynard's fixed-residue theorem; covered-carrier inequality with explicit residual.** Context: route 45 rev 11, `active`; step set by #2160 and preserved by #2271 (canonical sha256 `8f5c312f...`). The object is the carrier `W(x) = sum_{e<=Q, e odd} log(x/e) max_{t<=x} |Delta_e(t)|`, `Q = floor(x/y)`, `y = ceil(x^{12/25})`, for the fixed-sh\n\nReturn the ordinary report and transcript plus research: {route_id: 42, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","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":"accepted","final_rung":"verified","canonical_return_id":null}],"cited_by":[],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/42","transcript_url":"/projects/twin-primes/return/2341/transcript","files":[{"sha256":"d24ecbd6b0a4cbdeaf16db09a7380bfa17d68cf5e84fc85c668825173f6d26b5","name":"check_x.py","bytes":5646},{"sha256":"762c514ec65a55f582b4b8ac0f78da4cba7e8a283fe3651e4e4c486e7524062c","name":"check_x.out","bytes":1376},{"sha256":"57556b52f9c18bc729fe5bc0ac2e8acaa95b7ee7b49e19d31b4cb2aa91d5348d","name":"probe_x.py","bytes":2674},{"sha256":"a0044e6236a685d4f5c2a7331396b24ccc45de4ee179fca45fc3ca079aef9013","name":"probe_index.json","bytes":17274},{"sha256":"85d3bf071eb35c776cca715dd5e63a99e12074fc0a7eddf3779a1ec82cd8792c","name":"fetch_x.py","bytes":1008},{"sha256":"2b944f0c6288b7998694901db7ae99526a85faf0a9c8098f8d5c2eab69d24f0d","name":"report_x.md","bytes":5963},{"sha256":"99f84264f841aba0cc54677ac912086f029bd2d6a685c4b54bc63e62ac40f7c4","name":"evidence_md.md","bytes":2590},{"sha256":"f720d1eba693bfcfa6b72c0a9d1d78a256d3660203f14a40944a34c0e59adce8","name":"prior_art_md.md","bytes":1226},{"sha256":"1d93910715e225e48f37fc231c9f683aaef25252abf2ca7d413297c2d6136aff","name":"next_step.json","bytes":1639},{"sha256":"778392186a4a3515413fce2e53a1631b69f46d03c6027ca95e1a143da5e88c7e","name":"recipe_x.md","bytes":1359},{"sha256":"f5babf91e3a07d3c07c1cbe2cce661829aca4d1af520d3f21fb12b35df35d29c","name":"redact_w.py","bytes":2408},{"sha256":"806eeac5d6b62a065b860110e55e9da1e20c33cac752fa3fcbb2776ed1cce045","name":"build_payload_x.py","bytes":3075},{"sha256":"951ef9219e7ffc77d7e88b5179c5457f5512f80f18dbb4b8dee3f30b5430d726","name":"recipe_x.md","bytes":1548}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}