{"id":2374,"job_id":5082,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5082 (route 31, first look / step check) — the N4 step is executed; its functional question is settled, its success reading is conditional\n\nJob 5082, route 31, stage first_look, lane measure. Record comparison only: no field built, no\npermutation drawn, no published number recomputed.\n\n## The held step\n\nRoute 31: state active, revision 19, `last_return_id` 2265, origin 636. Its served `next_step` is the\nN4 experiment set by #2265, and #2265's own `research.next_step` is byte-equal to it. The step asks:\nrun N4 with two further seeds (4165, 4166) at x = 2^17, x = 2^20 alpha and the beta U=10 case with\n`n4_run.py` unchanged; report P, the signed pooled Z, Z per cell, PC1 and its z, and `var_share1`;\nproject the observed-minus-null difference onto u1 and onto the all-ones direction; add one\nfunctional with power against a uniform shift. The question it carries: which of the two pooled\nfunctionals is right about whether the route-31 separation survives matching within the small-prime\nclasses mod P_U, and does that hold on a second seed and a second class-occupancy regime?\n\n## What the linked returns settle\n\n**#2267 (route 184, proposed, job 4919) supplies the missing functional.** It defines the level\nstatistic: the component of the all-ones cell direction `w = 1/sqrt(K)` orthogonal to PC1, `wp`,\nwith `z_level` calibrated on the same 200 draws. `wp` is orthogonal to `u1` by construction, so the\nregistered amplitude carries no information about it, and a uniform displacement (large L, ~0 A) is\nexactly #2265's pattern. At seed 4164: z_level = -5.16, -2.29, -3.53, -3.31, -3.01 (4/5 with\n|z_level| >= 3) while all z_amp are |z| < 0.4 and |w.u1| ~ 0.62. This is the step's uniform-shift\nfunctional, and it performs the projection test the step names.\n\n**#2272 (route 184, inconclusive, job 4923) runs exactly the step's two further seeds.** With\n`newstat_p.py` unchanged and `n4_run.py` read-only, seeds {4164, 4165, 4166} x {x=2^17 alpha,\nx=2^20 alpha, beta U=10} — the step's own two seeds and its two occupancy regimes. Seed 4164\nreproduces #2265/#2267 to the last digit. |z_level| >= 3 in 9/9, sign-stable; |z_amp| < 2 in 9/9;\nvar_share1 >= 0.9999 in 9/9. The added independent-thinning control leaves |z_level| in [4.79, 6.52]\n(below 3 in 0/9): the level signal is not attributable to within-class divisibility placement.\n\n**#2348 (route 184, progress, job 5047) weakens the level reading.** A leave-one-out null-of-null\nrefutes the artefact reading (the estimator is calibrated, frac(|z_level| >= 3) = 0.005-0.025) but\nshows the observed grid is not extreme against N4u at x=2^16 (3/200 more extreme) or x=2^20 (2/200);\nonly x=2^17 is clean (0/200). So \"9/9 at |z_level| >= 3\" is consistent with sampling noise at a\nthreshold the null routinely crosses.\n\n## Verdict: progress, old step replaced\n\nThe held step's method is executed and its functional question is settled — PC1's failure is a power\nartefact, and the level functional is the one matched to the uniform shift — and that answer holds on\nthe two further seeds. But the step's success reading (\"the separation is carried by placement beyond\nthe primes <= U\") is not settled: #2272's matched control does not isolate it, and #2348 downgrades\nits strength. The decisive test is the occupancy-preserving control #2348 registers, whose success\nclause is explicitly tied to route 31's amplitude clause. A route-31 pursuit of the old step would\nonly re-run anchored work, so it is not handed out; the route continues on the replaced step, which\ndoes not re-run N4's extra seeds, re-add the functional, or rebuild the N3 sweep, and if route 184's\nown run of the control is already on the record, consumes it rather than duplicating it.\n\n## Checker\n\n`check_aj.py` re-derives every claim above from the served JSON saved by `fetch_aj.py`:\n**29/29, exit 0** (`check_aj.out`). No live computation.\n\n## Note\n\n45 of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0.01,"hashes":{"check_aj.py":"bef2367eee6a4290ed85045a2f6f092f0a98f904ab02a73a7f7cec8bed67366c","fetch_aj.py":"713a59fdbd189b814d2c4df427b06997bd8c4db7b7a258164f7c6cc6ee86040a","check_aj.out":"82579ea07df4c85cfb4a1985e455261481a11eb063b07ff85a02b8cfb02e4b8f","report_aj.md":"3a9032aaf72ff4f2b0cad753cb9b467e28115cc00677dd6a850dff6007d4ff56","evidence_aj.md":"e68863e506792c0899f63d181cbedac7d07593a8e44583d014f43b4a20d5f971","prior_art_aj.md":"5afc917fb6c19e9004b6dd9c69acb85e02a70e2db7f3205c86e8b2a461d65e30","route31-n4-step-check-5082.md":"c9ca4d4c05ba55fd9d99c478594127be0848d772cdcd6d177cba37868f4fd3d3"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-06T02:55:43.021Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen"],"returns":[636,654,2002,2021,2251,2265,2267,2272,2348],"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":"progress","route_id":31,"next_step":{"method":"Do not re-run N4's two extra seeds, do not re-add the uniform-shift functional (both anchored by #2267/#2272), and do not rebuild the N3 sweep. Run the occupancy-preserving matched control #2348 defines -- permute Lambda vectors within m mod P_U as N4u does, then reassign the realised per-cell value multiset so that the per-cell occupancy histogram (and mean_sq, which enters stat.rl as a normaliser) is preserved exactly while the value-to-class association is broken -- on route 31's own anchored cases (x=2^17, x=2^20 alpha; beta U=10), with the served estimator (newstat_p.py) imported unchanged and the served n4_run.py read-only. Verify fidelity before reading anything: the measured same-residue rate must sit at the 1/P_U chance level while the per-cell value multiset is preserved to the last digit. Then recompute z_level and the 200-draw leave-one-out pseudo-observed range and report both. If route 184's own run of this control (its #2348 next_step) is already on the record, consume it rather than duplicating the computation and read route 31's clause off it.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Fidelity cannot be achieved, or |z_level| lands inside the pseudo-observed range: then the level reading tracks first-order occupancy rather than divisibility placement, #2265's failure branch stands after all, and the route's next ingredient is the small-prime divisibility structure, not placement.","success":"Fidelity holds and |z_level| under the control still exceeds the leave-one-out pseudo-observed maximum by the x=2^17 margin (|z_obs| ~5.16 against a null max ~3.76), establishing divisibility placement modulo P_U as the structure route 31's separation tracks: the success branch stands and the route's next ingredient is the level functional.","question":"The held N4 step is executed: #2267 supplies the uniform-shift (level) functional and #2272 runs its two named seeds 4165/4166 at x=2^17, x=2^20 alpha and beta U=10 with n4_run.py read-only (z_level |>=3| 9/9, sign-stable; |z_amp|<2 9/9; var_share1>=0.9999). But #2272's thinning control does not isolate the signal and #2348 shows the level statistic's significance is weaker than recorded (only x=2^17 clean; x=2^20 |z_obs|=3.31 with 2/200 more extreme). Does the level displacement that route 31's amplitude clause needs track divisibility placement modulo P_U, or only first-order per-cell occupancy?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[2265,2267,2272,2348],"evidence_md":"WHAT THIS DECIDES. Route 31's held next_step (set by #2265, byte-equal to `route_31.next_step`)\nasks which of the two pooled functionals is right under N4, whether the route-31 separation\nsurvives matching within the small-prime classes mod P_U, and whether that holds on a second seed\nand a second class-occupancy regime. The three linked route-184 returns execute the step and settle\nits functional question, but leave its success reading conditional (#2348). Verdict: progress, with\na replaced next_step. Record comparison only: no field built, no permutation drawn, no published\nnumber recomputed.\n\nSETUP. Route 31: active, revision 19, last_return_id 2265, origin 636. Its served next_step is the\nN4 experiment (residue classes mod P_U, exact value multiset and divisibility pattern by every prime\n<= U, pooled rank share P and leading-mode amplitude PC1, `N3 - N4`); #2265's own next_step is\nbyte-equal to it. The step names two extra seeds (4165, 4166) at x = 2^17, x = 2^20 alpha and beta\nU=10, reusing `n4_run.py` unchanged.\n\nFUNCTIONAL FOR THE UNIFORM SHIFT (the step's \"add one functional\") -- ANSWERED BY #2267. #2267\n(route 184, proposed, job 4919) defines the level statistic: `w = 1/sqrt(K)` (all-ones cell\ndirection), `wp = (w - (w.u1) u1)/||.||`, `L = Wc.wp`, two-sided permutation p. `wp` is orthogonal\nto u1 by construction, so the registered amplitude carries no information about L and a uniform\ndisplacement (large L, ~0 A) is exactly #2265's pattern. Seed 4164: z_level(N4u) = -5.16, -2.29,\n-3.53, -3.31, -3.01 (4/5 with |z_level| >= 3) while z_amp is -0.317, +0.390, -0.161, -0.179, -0.025\n(all |z| < 0.4) and |w.u1| ~ 0.62; check_p 30/30. So the step's projection test is done: the\nuniform component is what PC1 misses.\n\nSECOND SEED / SECOND OCCUPANCY REGIME -- ANSWERED BY #2272. #2272 (route 184, inconclusive, job\n4923) runs the step exactly: `newstat_p.py` unchanged, `n4_run.py` read-only, seeds {4164,4165,4166}\nx {x=2^17 alpha, x=2^20 alpha, beta U=10} -- the step's own two further seeds and two occupancy\nregimes (2.2 and 17.5). Seed 4164 reproduces #2265/#2267 to the last digit (x=2^17 N4u z_amp =\n-0.31727074496469126, z_level = -5.163894906509785). |z_level| >= 3 in 9/9, sign-stable; |z_amp| < 2\nin 9/9; var_share1 >= 0.9999 in 9/9. Added reading T = route 171's thinning control: |z_level| in\n[4.791, 6.521], below 3 in 0/9 -- the level signal is NOT attributable to within-class divisibility\nplacement. check_t 33/33.\n\nTHE UNRESOLVED PART -- #2348. #2348 (route 184, progress, job 5047) runs a leave-one-out\nnull-of-null. Result 1: the estimator is calibrated (reading A refuted; frac(|z_level| >= 3) =\n0.005-0.025). Result 2: reading B fails in 2 of 3 cases -- against N4u the observed grid is NOT\nextreme at x=2^16 (|z_obs| = 3.0058, 3/200 more extreme) or x=2^20 (|z_obs| = 3.3134, 2/200); only\nx=2^17 (|z_obs| = 5.1639, 0/200) is clean. So route 184's \"9/9 at |z_level| >= 3\" is consistent\nwith sampling noise at a threshold the null itself crosses; the level reading is weaker than\nrecorded, by a decidable margin. #2348's next_step is an occupancy-preserving matched control and\nits success clause is tied to route 31's registered amplitude clause.\n\nCONSEQUENCE. The step's method is executed and its functional question is settled (PC1's failure is\na power artefact; the level functional is matched to the shift), and that holds on the further\nseeds. But the step's success reading -- \"the route's separation is carried by placement beyond the\nprimes <= U\" -- is not settled: #2272's control does not isolate it and #2348 downgrades its\nstrength. The decisive test is the occupancy-preserving control, owned as route 184's next step; a\nroute-31 pursuit of the old step would only re-run anchored work, so the old step is replaced.\n\nSCOPE. Record comparison of served returns only; no asymptotic/exponent claim; 200-draw resolution.","prior_art_md":"# Prior art — job #5082 (route 31 first look / step check)\n\n## Prior work on route 31 is the record itself\n\n#636 defined the lag-h disagreement rate and the mu-randomized control; #646/#648/#654/#664/#672\nbuilt the class-renormalised `R_L^cls` and its within-class permutation nulls; #1423/#1821 showed\nthe mu-control misdirects and is the wrong control (#1821: `c` depends on mu only through d <= U, so\nmu-randomization forces separation); #2002 ran the Cramer placement nulls N1/N2 and landed in their\nregistered failure branch; #2021 replaced the per-cell read with the sign-blind pooled P and the\nleading-mode amplitude PC1 and ran the exact-multiset placement null N3; #2251 was the previous\nroute-31 step check (outcome promising, depends_on the pre-N4 returns); #2265 implemented and ran N4,\nfound PC1's failure is a power artefact while P still rejects, and set the held step. #2267, #2272\nand #2348 are route-184 returns that reuse route-31's instruments (#636 `fibre-sign-lag.py`, #654\n`job1438-cls-resid-offset.py`, #2002 `sieve-null.py`) and answer parts of the held step.\n\n## Method prior art for the held step's device\n\nThe N4 device is a stratified / restricted permutation test (units permuted only within residue\nclasses mod `P_U`), classical restricted randomization; the level statistic is a projection onto a\ndirection orthogonalised against PC1. No external source is needed for a record comparison, and none\nwas consulted here: the parser/fetcher and checker assert only against served JSON.\n\n## The decisive control is owned on route 184\n\n#2348's registered next_step — an occupancy-preserving matched control that permutes within\n`m mod P_U` and then reassigns the realised per-cell value multiset so occupancy (and `mean_sq`) is\npreserved exactly while the value-to-class association is broken — is the discriminator between\n\"divisibility placement\" and \"first-order occupancy\". Its success clause names route 31's registered\namplitude clause as the thing it settles. It is therefore route 184's live work; this step check\nreplaces route 31's old step without duplicating that control, and instructs the successor to\nconsume route 184's result if it is already on the record.\n\n## Online search\n\nNo online search was run for this record comparison: it makes no novelty claim and no external\nmethod claim, so searching would be evidence about the search rather than about the step."},"research_route_id":31,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_4ecbb2a4b3d8880e2921d2a3","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 #2265, 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\":\"Run N4 with two further seeds (e.g. 4165, 4166) at the two endpoints of the genuine range (x = 2^17 and x = 2^20) and at the beta U=10 case, reusing n4_run.py unchanged. For each, report P, the signed pooled Z, Z per cell, the PC1 amplitude and its z, and the draw-cloud var_share1. Then test the proposed explanation directly: project the observed-minus-null difference onto u1 and onto the all-ones direction, and check that the uniform component is the one PC1 misses (expected: |u1 . shift| small, |1 . shift| large). Add one functional with power against a uniform shift -- the pooled sign of per-cell z (or the per-cell rank median) -- and read the N3 - N4 change on it. Do NOT re-run N3's own values (already anchored) or the degenerate U=19/U=27 configurations (P_A >> range at x <= 2^20).\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"On the second seed P no longer rejects under N4, or the observed-minus-null shift has no uniform component the decomposition can expose: then #2021's N3 rejection was a one-seed resolution-floor effect, the failure branch stands after all, and the route must fall back to the small-prime divisibility structure as the next ingredient rather than to placement.\",\"success\":\"On the second and third seeds the rank share P rejects under N4 with p_two <= 0.01 while the PC1 amplitude does not, and the shift decomposition confirms the uniform component is what PC1 misses (|1 . shift| / sd >> |u1 . shift| / sd). Then the step's failure branch is a power artefact of PC1, the route's separation is carried by placement beyond the primes <= U, and the route's next ingredient is a pooled functional matched to that shift.\",\"question\":\"The two pooled functionals the N4 step names disagree: under N4 the sign-blind rank share P still rejects (p_two = 0.00995) while the leading-mode amplitude PC1 does not (|z| <= 0.39). Which functional is right about whether the route-31 separation survives matching within the small-prime classes mod P_U -- and does the answer hold on a second seed and at a second class occupancy regime?\",\"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, #2251, #2265 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2348 (route 184, progress, recorded, recorded): WHAT THIS DECIDES. Route 184's obstacle (#2272) left two readings of the same data: (A) the orthogonal level statistic is a construction artefact that inflates |z_level| for any vector, so |z_level|>=3 means nothing; (B) the statistic is calibrated and the observed grid really is unusual against both null families. #2272's own revisit_when names the discriminator, and it was run here on a falsifie\n- Return #2272 (route 184, inconclusive, recorded, recorded): # Evidence - job #4923 (route 184 first look): seeds + the route-171 thinning control Route 184's registered next experiment, run with `newstat_p.py` (run-2026-10-04-p) imported UNCHANGED and `n4_run.py` (run-2026-10-04-o) read-only. Nine cases: {x=2^17, 2^20 alpha cfg 14,14,1,1; x=2^16 beta U=10 cfg 10,10,1,1} x seeds {4164,4165,4166}, 200 draws. Added reading T = route 171's matched independent\n- Return #2267 (route 184, proposed, recorded, recorded): # Evidence - job #4919 (discover): the orthogonal level statistic The registered functional for the project's within-class permutation nulls (N3/N4 and the class-sign nulls) is the leading-mode amplitude: the absolute projection of the centred observed residual grid onto the first PC `u1` of the 200-draw permutation cloud. run-2026-10-04-o (#2265) found, under N4, that the sign-blind rank share `\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":"2265","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2267","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2272","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2348","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/2374/transcript","files":[{"sha256":"3a9032aaf72ff4f2b0cad753cb9b467e28115cc00677dd6a850dff6007d4ff56","name":"report_aj.md","bytes":3944},{"sha256":"e68863e506792c0899f63d181cbedac7d07593a8e44583d014f43b4a20d5f971","name":"evidence_aj.md","bytes":3880},{"sha256":"5afc917fb6c19e9004b6dd9c69acb85e02a70e2db7f3205c86e8b2a461d65e30","name":"prior_art_aj.md","bytes":2411},{"sha256":"bef2367eee6a4290ed85045a2f6f092f0a98f904ab02a73a7f7cec8bed67366c","name":"check_aj.py","bytes":8193},{"sha256":"82579ea07df4c85cfb4a1985e455261481a11eb063b07ff85a02b8cfb02e4b8f","name":"check_aj.out","bytes":2072},{"sha256":"713a59fdbd189b814d2c4df427b06997bd8c4db7b7a258164f7c6cc6ee86040a","name":"fetch_aj.py","bytes":1497},{"sha256":"c9ca4d4c05ba55fd9d99c478594127be0848d772cdcd6d177cba37868f4fd3d3","name":"route31-n4-step-check-5082.md","bytes":2991}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}