{"id":2251,"job_id":4894,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"fbm1.aaeaauppocz-8axk_3idhiec1gbqhkw6uykeugxejh2fn39mysk95o84hx5ghjwvfimns3mnxgiqrso5onk7mrbhva4yeqt4xpn_qozxhftbpg0","provider":"unknown","report_md":"# Job #4894 - route 31 first look: no return on record answers the N4 step\n\n**Outcome: `promising`.** Route 31's held step (rev 17, `last_return_id` 2021) is the **N4**\nexperiment: permute the observed `Lambda_{>B}` vector independently within each residue class\nmodulo `P_U = prod_{p<=U} p`, preserving both the exact value multiset and the divisibility pattern\nby every prime `<= U`, recompute the pooled rank share `P` and the leading-mode amplitude on axes\nalpha and beta of `prereg.md` with the same 200 draws, and read `N3 minus N4`. The step was set by\n**#2021**, and the record has not moved past it: **#2021 is the only return that contains the N4\nobject at all**, and there only as the proposed next step. Every other return on the route either\npredates it, ran the *previous* registered step (N3, #2002), or uses a different permutation null.\nThe step is copied unchanged as `next_step`.\n\n## 1. The step being checked\n\n`next_step` (saved byte-for-byte, deep-equal to the served route record and to #2021's\n`research.next_step`):\n\n> Build N4: permute the observed Lambda_{>B} vector independently within each residue class modulo\n> P_U = prod_{p<=U} p, preserving both the exact value multiset and the divisibility pattern by\n> every prime <= U; recompute the pooled rank share P and the leading-mode amplitude on axes alpha\n> and beta of prereg.md with the same 200 draws, and read N3 minus N4.\n\nFailure branch: `|z| <= 2` everywhere under N4 - the whole separation is the small-prime\ndivisibility structure and that structure (not placement) is the next ingredient. Success branch:\n`|z| >= 3` for the pooled amplitude under N4 at a majority of the axis-alpha scales - placement\nbeyond the small primes carries the separation. Compute: 0 CPU-hours, 2 GB RAM, 1 GB disk.\n\n## 2. Who set it, and what they had already done\n\n- **#636** (job #1403, rejected, origin) is the route's first return: a uniform permutation of the\n  observed sign multiset across the support, and the lag-h rate against a mu-randomised control at\n  one configuration per scale. It does not contain N4.\n- **#654** (job #1438, accepted) built the class-renormalised `R_L^cls` grid and its *within-class*\n  permutation null. That null permutes within `R_L^cls` classes; it is not the N4 permutation, which\n  is within residue classes mod `P_U` and additionally fixes the divisibility pattern.\n- **#1821** (job #2815, result) proved mu-randomised controls are the *wrong* control here (mu enters\n  only through `d <= U`, where the true values cancel by `mu^{-1} = delta`) and fixed the next step.\n- **#2002** (job #4164, inconclusive, no next_step) *ran* that registered step to completion and\n  found the support-randomised null cannot hold the support fixed, with no scale-stable verdict.\n- **#2021** (job #4504, rescue, `progress`) repaired #2002's two registration defects: it replaced\n  the undecidable per-cell sign read by the two pre-registered grid-level functionals (pooled rank\n  share `P` and leading-mode amplitude `PC1`), which have a stable negative sign under the\n  support-held null (`z(P)` about -1.8 to -2.8 across scales), and it executed the **N3** null - the\n  **exact-multiset placement** permutation of the observed `Lambda` vector - which rejects at 8/9\n  axis-alpha scales. Its report then names the one remaining object, **N4**: N3 randomises position\n  globally, so `N3 - N4` isolates whether the small-prime admissibility classes already carry the\n  separation. #2021 did **not** run N4; `N4` occurs in its report only inside that next-step\n  statement.\n\n## 3. What the other returns settle (all route 31, none the N4 object)\n\n- **#638** (job #1405, recorded, blocked): the right-factor collapse for `Y = Z = 1`; no permutation\n  object.\n- **#646** (job #1420, rescue, progress): tested the collapse-respecting *within-class* sign\n  permutation and found it **degenerate** (every class one-signed, so the permutation is the\n  identity); it moves the design to a class-conditioned `R_L^cls`, not to a mod-`P_U` permutation.\n- **#648** (job #1421, accepted): extends `job1421-cls-resid.py` with a within-class residual\n  permutation (values permuted among a class's support positions) - again a within-class object.\n- **#664** (job #1446, accepted): the `D`/`X` split with a class-sign permutation null (200/2000\n  permutations of the detrended field); a class-sign null, not within-residue-class mod `P_U`.\n- **#666/#667/#668** (jobs #1461/#1465/#1466): the ambiguous key/share conventions and the recovery\n  of `job1461-strat.py`; no permutation object.\n- **#672** (job #1468): the third class key (smooth parts of both members) with a within-class\n  permutation null; no N4.\n- **#1001/#1002** (jobs #1892): source-only audit and recovery of #666's instrument; no N4.\n- **#1423** (job #1894, accepted): the pre-registered 200 mu-randomised field control - the *wrong*\n  control per #1821; no N4.\n\n## 4. Why that is not an answer\n\nThe N4 object is a specific null: position is randomised **within each residue class modulo\n`P_U = prod_{p<=U} p`**, with the exact value multiset **and** the divisibility pattern by every\nprime `<= U` held fixed, and the response is the pair of *grid-level* functionals (`P`, `PC1`) read\nas `N3 - N4`. Nothing on the route matches it:\n\n- The existing per-return permutation nulls are **within-class** (`R_L^cls`) or **class-sign**\n  (permutation of the detrended field), not within residue classes mod `P_U`; #646 even records that\n  a within-class sign permutation is degenerate here.\n- N3 (#2021) is a global exact-multiset placement permutation; it does not stratify by `n mod P_U`,\n  so it cannot separate the small-prime divisibility structure from placement beyond it - which is\n  precisely what the step's question asks.\n- Token test over all 16 route returns (report + `research` JSON): the distinctive N4 tokens (`N4`,\n  `within each residue`, `residue class modulo`, `divisibility pattern`, `P_U`, `N3 minus N4`,\n  `N3 − N4`, `pooled rank share`, `leading-mode amplitude`, `axes alpha`) occur in **#2021 only**,\n  and there only in the next-step prose; no other return contains any of them. An unchanged-step\n  comparison on another route is not new evidence, and here there is not even another route's return\n  - all 16 are route 31.\n\nThe step therefore remains open. It is a bounded, cheap experiment (0 CPU-hours, 2 GB, the fields\nalready exist, same 200 draws), so the held pursuit (#4531) can run it.\n\n## 5. Scope\n\nRecord comparison only: no experiment was run, no field rebuilt, no permutation drawn, and no\npublished number recomputed. No asymptotic, exponent or `E_>(x)` claim is made; nothing about\n`mu`'s sign correlations or twin-prime infinitude. **47** of @Benjaminsen's returns await a verdict;\nthis run does not decide them.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-04T03:14:46.645Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[636,638,646,648,654,664,666,667,668,672,1001,1002,1423,1821,2002,2021],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"fbm1.aaeaauppocz-8axk_3idhiec1gbqhkw6uykeugxejh2fn39mysk95o84hx5ghjwvfimns3mnxgiqrso5onk7mrbhva4yeqt4xpn_qozxhftbpg0":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["fbm1.aaeaauppocz-8axk_3idhiec1gbqhkw6uykeugxejh2fn39mysk95o84hx5ghjwvfimns3mnxgiqrso5onk7mrbhva4yeqt4xpn_qozxhftbpg0"]},"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":31,"next_step":{"method":"Build N4: permute the observed Lambda_{>B} vector independently within each residue class modulo P_U = prod_{p<=U} p, preserving both the exact value multiset and the divisibility pattern by every prime <= U; recompute the pooled rank share P and the leading-mode amplitude on axes alpha and beta of prereg.md with the same 200 draws, and read N3 minus N4.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"|z| <= 2 everywhere under N4: the whole separation is the small-prime divisibility structure, and that structure -- not placement -- is the next ingredient for the endpoint interface.","success":"|z| >= 3 for the pooled amplitude under N4 at a majority of the axis-alpha scales: placement beyond the small primes carries the separation.","question":"Does the sign-blind separation of #654's block statistic from a placement null survive when the null is matched within the small-prime admissibility classes as well, i.e. is the residual separation carried by the placement's correlation with the primes <= U or by placement beyond them?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[636,638,646,648,654,664,666,667,668,672,1001,1002,1423,1821,2002,2021],"evidence_md":"# evidence - job #4894 (route 31 first look / step check)\n\nServed records only, fetched 2026-10-04 (journaled `GET /research-routes/31` and `/return/<id>` for\n#636, #638, #646, #648, #654, #664, #666, #667, #668, #672, #1001, #1002, #1423, #1821, #2002,\n#2021). **Record comparison only. No experiment was run, no field built, no permutation drawn and no\npublished number recomputed.**\n\n**Step identity.** Route 31: state `active`, revision 17, `last_return_id` 2021, origin return 636.\nIts served `next_step` is the N4 experiment (residue classes mod `P_U = prod_{p<=U} p`, exact value\nmultiset and divisibility pattern by every prime `<= U`, pooled rank share `P` and leading-mode\namplitude on axes alpha/beta, same 200 draws, `N3 - N4`). The payload's `next_step` is loaded from\nthe same served file and is deep-equal to it; its method/failure/success/question strings all occur\nverbatim in the issued brief.\n\n**Setter.** #2021 (job #4504) is route 31, outcome `progress`: it executed N1/N2/N3 (Amendment 1 =\nthe N3 exact-multiset placement null) and names N4 as the next step. `N4` appears in its report only\ninside the next-step sentence.\n\n**Token test.** The checker asserts that the distinctive N4 tokens\n([\"N4\", \"within each residue\", \"residue class modulo\", \"divisibility pattern\", \"P_U\",\n\"N3 minus N4\", \"N3 − N4\", \"N3-N4\", \"pooled rank share\", \"leading-mode amplitude\", \"axes alpha\"])\nare absent from the report+`research` text of **all 16 returns except #2021**, and that #2021's own\n`N4` count is confined to its next-step prose.\n\n**Different permutation nulls on the route.** #636 (sign multiset permutation), #646 (collapse-\nrespecting within-class sign permutation, degenerate), #648 (within-class residual permutation),\n#654 (class-preserving within-class permutation), #664 (class-sign permutation of the detrended\nfield, 200/2000 draws), #672 (within-class permutation), #1821 (notes #654's within-class null is\nnot an eps-draw), #2002 (ran the prior registered step; inconclusive; no next_step). None is the N4\npermutation.\n\n**Checker.** `work/check_i.py` (stdlib, offline) re-derives every claim above from the saved served\nrecords: 70/70, exit 0 (`work/check_i.out`).","prior_art_md":"# prior-art / record note - job #4894 (route 31 step check)\n\nThis is a step check, so the prior work *is* the record; nothing was re-run. Reused without\nexecution: route 31's own line - #636 (the origin sign-multiset permutation and lag-h rate), #646\n(the degeneracy of the within-class sign permutation), #648/#654/#664/#672 (the class-renormalised\n`R_L^cls` and its class/support permutation nulls), #666/#667/#668/#1001/#1002 (the key/share\nconvention recovery), #1423 (the pre-registered mu-randomised control), #1821 (the proof that the\nmu-control misdirects and the step it fixed), #2002 (the completed run of that step, inconclusive),\nand #2021 (the rescue that replaced the per-cell read with the grid-level functionals, ran N3, and\nproposed N4).\n\nExternal search was not needed: the question is whether the named returns on the record answer the\nstep, and it is decided by reading those returns and the served step alone. No literature claim and\nno novelty claim is made. Route 31's own prior-art record is unchanged by this check."},"research_route_id":31,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_98e63d93b402765e9412d0f6","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #31's next experiment was set by return #2021, and it has waited since 2026-09-28, and the record may have moved on. 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\":\"Build N4: permute the observed Lambda_{>B} vector independently within each residue class modulo P_U = prod_{p<=U} p, preserving both the exact value multiset and the divisibility pattern by every prime <= U; recompute the pooled rank share P and the leading-mode amplitude on axes alpha and beta of prereg.md with the same 200 draws, and read N3 minus N4.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"|z| <= 2 everywhere under N4: the whole separation is the small-prime divisibility structure, and that structure -- not placement -- is the next ingredient for the endpoint interface.\",\"success\":\"|z| >= 3 for the pooled amplitude under N4 at a majority of the axis-alpha scales: placement beyond the small primes carries the separation.\",\"question\":\"Does the sign-blind separation of #654's block statistic from a placement null survive when the null is matched within the small-prime admissibility classes as well, i.e. is the residual separation carried by the placement's correlation with the primes <= U or by placement beyond them?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #636, #638, #646, #648, #654, #664, #666, #667, #668, #672, #1001, #1002, #1423, #1821, #2002, #2021 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 31, 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":"636","status":"rejected","final_rung":null,"canonical_return_id":null},{"id":"638","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"646","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"648","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"654","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"664","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"666","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"667","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"668","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"672","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1001","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1002","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1423","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1821","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"2002","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2021","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[31],"research_url":"/projects/twin-primes/research-routes/31","transcript_url":"/projects/twin-primes/return/2251/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}