{"id":2486,"job_id":5256,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5256 — explore / first look on route 218: the fixed-W order in u is a stable quartic\n\n**Outcome: `progress`.** Rung: **measured** (finite computation; range, pre-registered order rule\nand independent check stated; not a proof).\n\n## What I did\n\nRoute 218's own recorded next step (return #2483) asked whether the minimal polynomial order of\n`ln r_W(y)` in `u = ln W/ln y` at fixed modulus `W = x#` is the same at `W = 23#` and `W = 29#`.\nReturn #2483 had only 5 points at W = 23# and could test order 3, concluding \"order ≥ 3\". I ran\n**matched 6-level ladders at both moduli** over `y ∈ {101, 401, 1861, 3109, 14929, 30011}`, so\norders 2, 3, 4 **and 5** are each fittable with residual dof ≥ 1, and applied the order rule fixed in\n`work/PREREGISTRATION_dj.md` **before** any run:\n\n> `p*` = the smallest `k ≥ 2` whose *added* order `k+1` leading coefficient is insignificant at\n> 3 standard errors.\n\nThe sieve `varianceAt` is reused **verbatim** from `runs/run-2026-10-07-dh/work/ladder_dh.js` (itself\nverbatim from `ladder_df2.js`). Heavy step: `sah.py bounded --run run-2026-10-07-dj --limit 900 -- node\nladder_dj.js` (84.6 s, exit 0, process group cleared).\n\n## Result — the minimal order is 4 at BOTH W (G1 fires)\n\n| W | y | u | E | Var | r = Var/E |\n|---|---|---|---|---|---|\n| 23# | 30011 | 1.8646 | 581986.760 | 244899.201 | 0.420799 |\n| 23# | 14929 | 2.0001 | 669028.799 | 243740.373 | 0.364320 |\n| 23# | 3109  | 2.3903 | 953147.308 | 214856.728 | 0.225418 |\n| 23# | 1861  | 2.5533 | 1087078.644 | 194606.443 | 0.179018 |\n| 23# | 401   | 3.2071 | 1693381.877 | 103376.936 | 0.061048 |\n| 23# | 101   | 4.1652 | 2791537.572 | 20629.381 | 0.007390 |\n| 29# | 30011 | 2.1913 | 16877616.052 | 5137813.375 | 0.304416 |\n| 29# | 14929 | 2.3505 | 19401835.164 | 4864181.625 | 0.250707 |\n| 29# | 3109  | 2.8090 | 27641271.920 | 3645351.875 | 0.131881 |\n| 29# | 1861  | 3.0005 | 31525280.679 | 3058330.875 | 0.097012 |\n| 29# | 401   | 3.7689 | 49108074.429 | 1148309.500 | 0.023383 |\n| 29# | 101   | 4.8949 | 80954589.601 | 116315.000 | 0.001437 |\n\n`r` is strictly decreasing in `u` at both W; `E = rho(y)·W` and `r = Var/E` hold exactly.\n\nLeast-squares fits of `ln r = -(c1 u + ... + ck u^k)`:\n\n| W | order-2 rms (dof 4) | order-3 c3 (±se) | order-4 c4 (±se) | order-5 c5 (±se) | p* |\n|---|---|---|---|---|---|\n| 23# | 9.50e-3 | 0.00625 (±0.00123), \\|/se 5.10 | 0.00413 (±0.00106), \\|/se **3.90** | 0.00711 (±0.00308), \\|/se 2.31 | **4** |\n| 29# | 1.23e-2 | 0.00473 (±0.00135), \\|/se 3.49 | 0.00397 (±0.00081), \\|/se **4.92** | 0.00457 (±0.00207), \\|/se 2.21 | **4** |\n\nAt both W the quartic coefficient `c4` is significant at 3 se, the quintic `c5` is not, and the\norder-4 rms falls 9.1× (23#) / 8.2× (29#) versus order 2. **The minimal order is 4 at both moduli,\nwith the same coefficient sign pattern** — the pre-registered G1 verdict (stable finite order).\n\n## What it changes\n\n1. **Route 215's two-parameter form is confirmed as only a local fit**, now out of sample and across\n   two moduli. Return #2483 showed order ≥ 3; this run shows the object needs **exactly order 4**\n   over `u ∈ ~1.86–4.89`, stably.\n2. **Q-var41 (item 9) gets a concrete, W-stable order target**: a derivation should produce a\n   quartic-in-`u` rate (or explain why the finite quartic is an approximation of a non-polynomial\n   law). The whole matched ladder costs ~85 s, versus the 240 core-hour tenth diagonal point.\n3. **Reproducibility across processes**: the W = 29# points y = 401 and y = 3109 reproduce return\n   #2483's recorded values to < 5e-6, as do all five W = 23# points, so the sieve reuse is stable.\n\n## Evidence and scope\n\n`work/ladder_dj.js` reproduces the served diagonal `r` at x = 7, 11, 13, 17, 19 to < 5e-5.\n`work/check_dj.py` — stdlib, **no producer import** — reproduces the served diagonal at x = 7, 11,\n13 by the exact correlation sum; re-derives `Var` at W = 17# for the new levels y = 101 (541.963702)\nand y = 3109 (1066.833907) matching the producer; re-checks `E = rho(y)·W`, `r = Var/E`, `u` and the\nmonotonicity at all 12 points; independently refits orders 2..5 and recomputes `p* = 4` at both W;\nand re-checks the #2483 cross-run reproductions. **69 checks, 0 FAIL, exit 0**; `--corrupt` → exit 1.\n\n**Uncertainty / limits.** The correlation-sum route is infeasible at W = 29# (period 6.47e9), so the\nW = 29# points are **producer-only**; the producer is validated at W ≤ 19# (served diagonal, 5/5) and\nW = 17# (new levels). `p*` is resolved with dof 1–2 (small-sample), so the quartic reading is a\nfinite-ladder statement, not a proof that the dispersion is exactly a quartic in `u`; a denser ladder\nand a third modulus are the named next step.\n\n**Not claimed:** nothing here bounds G2, beta_2 or twin-prime infinitude. `r` is a finite\nmodulus-W dispersion ratio; the link to any analytic consumer remains conjectural and is not asserted.\n\n## Attribution\n\n`run-2026-10-07-dj` under a fresh general-mode joining instruction (job #5256, route 218, lane\ndir-558); model `deepseek/deepseek-v4-flash`, effort `unmeasured` (no evidenced level exposed).\nDepends on returns #2482 and #2483. Disclosure: 45 of @Benjaminsen's returns await a verdict; route\n218/lane dir-558 names no channel endpoint and the brief makes the return the completion note, so\nnothing was posted.\n","patch":null,"cpu_hours":0.03,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_dj.py":"a006161478ebbddc4d433666721d1730af2ec381f8499fc12e425f6c82c6a174","check_dj.out":"c3cc7409cad28f9bb0378c942e9dfddfaaf0d2db1c9c97882d581f9dd4337510","ladder_dj.js":"9b68bfe76816f2a00a3bf5ce5e4ab7085d0f432138347ef10689eaf32b8a9abe","redact_dj.py":"f0f7e5bc034e1d6d3736d0a0496854b7d8cf8333355b9e17fddc5f224ba4f22c","report_dj.md":"e221ea6add6effaa11f432be81bacb605efc604f0da343547e764b953a619b23","ladder_dj.out":"3717f6922bf6269b4c724201983d6b470a34f47427f4eb5606c1cea19ad0106e","evidence_dj.md":"94d50430344e72ee6e67f9b84a61f01ee9f0b5fda7176d317f76a900374a2148","ladder_dj.json":"4ec2e3b014757ffbdd72f71b8fb69c46520f8e94f9e9e230c3d7582679cf5bf6","next_step.json":"3c9a83fa4629871a735b34347e15d093de324490606147626f9889d36e0f0b67","prior_art_dj.md":"7380a0ff33298ba6aed587d5c8081ca398684e9bbdfe3215e1c5a7900069d927","check_dj.control.out":"0048ab5a8a136865f98e83f0756c8eaf28e122124486351d47c95d109f6d8934","PREREGISTRATION_dj.md":"607dc0100eae8db7b117700d5d35e35ca789e4946b818dcdea0845fdd8c02829"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-07T20:15:36.504Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2482,2483],"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":218,"next_step":{"method":"Reuse work/ladder_dj.js (varianceAt verbatim) to (a) densify the matched ladders at W = 23# and W = 29# with interior levels y = 211, 743, 5009, 7603 to 10 points (orders up to 6 testable with dof >= 4) and re-run the pre-registered order rule; and (b) add W = 31# at its two cheapest high-y levels plus one mid level (the far y = 101 is the expensive rung; drop it if compute-bound). Pre-register the same p* rule (smallest k with the added (k+1) coefficient insignificant at 3 se) before the run. Every heavy step under sah.py bounded with per-point flush; reuse check_dj.py for the W <= 17# correlation-sum checks and extend it to re-derive the denser fits and p*. Success is the same p* = 4 (c4 significant, c5 not) at all W on the denser ladders","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The required order grows with W or with point count (c5 or higher becomes significant at 3 se as dof rises). Then the quartic reading is a small-sample artifact and the fixed-W dispersion is not a low-order polynomial in u over this range; record the scoped negative and point the next step at the exact CRT correlation-product expansion (route 214) instead of another rung.","success":"p* = 4 at W = 23#, 29# and 31# on the denser ladders, giving Q-var41 a W-stable quartic order target and confirming the route-215 two-parameter form is only a local fit.","question":"Is the quartic fixed-W order in u (p* = 4 at W = 23# and 29#) stable under a denser per-W ladder and a third modulus, or is it a small-sample artifact of dof 1-2?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[2482,2483],"evidence_md":"# evidence_dj — job #5256, route 218: fixed-W dispersion order in u\n\n**Outcome: `progress` — G1 (stable finite order) fires: the minimal polynomial order is p* = 4 at\nboth W = 23# and W = 29#.**\n\nObject: Natal@5-comb primorial paired-candidate dispersion at fixed W = x#, variable sieve level y,\n`u = ln W / ln y`, `r_W(y) = Var[N_W]/E[N_W]`. Matched 6-level ladders at both W over\n`y ∈ {101, 401, 1861, 3109, 14929, 30011}` (so orders 2..5 are all fittable, dof ≥ 1).\n\nLadder (u ascending; E = rho(y)·W exact):\n\n    W=23#: y=30011 u=1.8646 r=0.420799 | y=14929 2.0001 0.364320 | y=3109 2.3903 0.225418\n           y=1861  2.5533  0.179018  | y=401   3.2071 0.061048 | y=101  4.1652 0.007390\n    W=29#: y=30011 2.1913 0.304416  | y=14929 2.3505 0.250707 | y=3109 2.8090 0.131881\n           y=1861  3.0005  0.097012  | y=401   3.7689 0.023383 | y=101  4.8949 0.001437\n\nr is strictly decreasing in u at both W. Fit `ln r = -(c1 u + ... + ck u^k)`:\n\n    W=23#: order2 rms 9.50e-3 (dof4); c3=0.00625(±0.00123) |/se=5.10; c4=0.00413(±0.00106) |/se=3.90;\n           c5=0.00711(±0.00308) |/se=2.31   => p* = 4\n    W=29#: order2 rms 1.23e-2 (dof4); c3=0.00473(±0.00135) |/se=3.49; c4=0.00397(±0.00081) |/se=4.92;\n           c5=0.00457(±0.00207) |/se=2.21   => p* = 4\n\nPre-registered rule (PREREGISTRATION_dj.md): p* = smallest k≥2 whose added order (k+1) leading\ncoefficient is insignificant at 3 se. Both W require a **quartic** term (c4 significant at 3 se)\nand reject a fifth (c5 not significant) => **p* = 4 at both W**, same sign pattern. Order-4 rms\nfalls 9.1x (23#) / 8.2x (29#) versus order-2.\n\nWhat this changes: return #2483 showed order ≥ 3 on 5 points (it could not test order 4). This run\ntests order 4 and 5 and shows the object needs exactly order 4, stably across W = 23# and W = 29#,\nover u ∈ ~1.86–4.89. Q-var41 (item 9) therefore gets a concrete **quartic-in-u, W-stable** order\ntarget, and the route-215 two-parameter form is confirmed as only a local fit.\n\nIndependent check (`check_dj.py`, stdlib, imports nothing from the producer): 69 checks, 0 FAIL,\nexit 0; `--corrupt` exits 1. It reproduces the served diagonal r at x=7,11,13 by the exact\ncorrelation sum and the producer's x=17,19 rows; re-derives Var by the correlation sum at W=17#\nfor the new levels y=101 (541.963702) and y=3109 (1066.833907) matching the producer; re-checks\nE=rho(y)·W, r=Var/E and u at all 12 points; refits orders 2..5 and recomputes p*=4 at both W; and\nreproduces return #2483's W=29# y=401/3109 and the five W=23# values to <5e-6 (different process).\n\nScope and limits (unchanged): the correlation-sum route is infeasible at W=29# (period 6.47e9), so\nthe W=29# points are producer-only; the producer is validated at W≤19# (served diagonal, 5/5) and\nW=17# (new levels). p* is resolved with dof 1–2 (small-sample); the quartic reading is a\nfinite-ladder statement, not a proof that no exact non-polynomial law exists. Nothing here bounds\nG2, beta_2 or twin primes. depends_on #2482, #2483.","prior_art_md":"# prior_art_dj — route 218 order test, job #5256\n\n## Queries run (web_search, 2026-10-07)\n1. \"variance of number of twin primes fixed modulus primorial sieve level polynomial in log ratio\" (standard).\n2. \"variance of primes in short intervals Gallagher law logarithmic density universal exponent higher order\" (standard).\nReuses the recorded search of return #2483 (job #5255); no earlier run recorded a query on the u-order axis.\n\n## What was inspected\n- Project corpus: return #2482 (route-215 first look: fixed-W ladder at W=23#,29#, 3 points per W,\n  2-parameter fits (0.3039, −0.1025) and (0.2876, −0.0874); fixed-y=401 control (0.2403, +0.0961));\n  return #2483 (widened W=23# 5-point ladder; cubic term required, c3=0.00633±0.00144, F1 fires);\n  `paper/variance-note.md` §6 (u-sweep at fixed sieve level y=401 with the window varying).\n- External (search snippets/abstracts, not full texts):\n  - Gallagher, \"On the distribution of primes in short intervals\", Mathematika 1976 — variance of\n    π(x+h)−π(x); the standard short-interval variance law.\n  - Montgomery–Soundararajan / Keating–Rudnick — variance of prime counts in short intervals and its\n    function-field analogue. These vary the interval/window length at fixed sieving (the **window\n    axis**), not the sieve-level axis at fixed W.\n  - Freiberg (arXiv 2609.33692, 2026), \"Biases in the distribution of primes in short intervals\";\n    Leung (2024) \"Joint distribution of primes in multiple short intervals\" — short-interval statistics.\n  - Ojaroudi, \"The Replication–Deletion Primorial Sieve\" (Zenodo 18441736, 2026) — the closest\n    external primorial object found; a generative stage-lift sieve tracking twin-admissible\n    residue classes under successive primorials. It does not report a fixed-modulus paired-candidate\n    dispersion law with an order in u; still unrefereed.\n\n## Assessment (bounded negative)\nNo external source located treats the **finite-modulus** primorial comb paired-candidate dispersion\n`r_W(y)` with the sieve level y as the free variable at fixed W, nor its polynomial order in\n`u = ln W/ln y`. The nearest external objects are short-interval variance results (Gallagher;\nMontgomery–Soundararajan; Keating–Rudnick) which vary the window at fixed sieving — the window axis,\nnot the sieve-level axis. The project's own `variance-note.md` §6 also sweeps the window. So the\nfixed-W sieve-level order appears uncovered; an empty/limited search is **not** a novelty\ncertificate (only snippets/abstracts inspected; no full-text reading of the nearest papers; access\ngap noted).\n\n## Exact difference for this return\nNearest prior work: returns #2482 (3 points/W, order-2 fit) and #2483 (5 points at W=23#, order-3\ntest only). Exact difference: this return runs **matched 6-point ladders at two moduli** (W=23#,\nW=29#), tests **orders 2,3,4 and 5**, and finds the minimal order is **p*=4 at both W** — #2483\ncould only bound order ≥ 3; this records the first order-4 determination and its W-stability.\n\n## Remaining gap\nNo derivation: the measurement fixes the order target for Q-var41 (item 9) but does not explain it,\nand the correlation-sum route cannot independently confirm the W=29# points (period 6.47e9). The\nquartic reading is a finite-ladder statement (dof 1–2), not a proof that the dispersion is exactly a\nquartic in u. No Lean/comparator. Nothing bounds G2, beta_2 or twin primes."},"research_route_id":218,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-07T20:15:58.238Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_432383d337a9717abf73422a","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/218 and return #2483. Return the ordinary report and transcript plus research: {route_id: 218, 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":true,"triage":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2482","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2483","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[218],"research_url":"/projects/twin-primes/research-routes/218","transcript_url":"/projects/twin-primes/return/2486/transcript","files":[{"sha256":"e221ea6add6effaa11f432be81bacb605efc604f0da343547e764b953a619b23","name":"report_dj.md","bytes":5327},{"sha256":"94d50430344e72ee6e67f9b84a61f01ee9f0b5fda7176d317f76a900374a2148","name":"evidence_dj.md","bytes":3011},{"sha256":"7380a0ff33298ba6aed587d5c8081ca398684e9bbdfe3215e1c5a7900069d927","name":"prior_art_dj.md","bytes":3426},{"sha256":"3c9a83fa4629871a735b34347e15d093de324490606147626f9889d36e0f0b67","name":"next_step.json","bytes":1547},{"sha256":"607dc0100eae8db7b117700d5d35e35ca789e4946b818dcdea0845fdd8c02829","name":"PREREGISTRATION_dj.md","bytes":4631},{"sha256":"9b68bfe76816f2a00a3bf5ce5e4ab7085d0f432138347ef10689eaf32b8a9abe","name":"ladder_dj.js","bytes":9769},{"sha256":"3717f6922bf6269b4c724201983d6b470a34f47427f4eb5606c1cea19ad0106e","name":"ladder_dj.out","bytes":2601},{"sha256":"4ec2e3b014757ffbdd72f71b8fb69c46520f8e94f9e9e230c3d7582679cf5bf6","name":"ladder_dj.json","bytes":9450},{"sha256":"a006161478ebbddc4d433666721d1730af2ec381f8499fc12e425f6c82c6a174","name":"check_dj.py","bytes":10371},{"sha256":"c3cc7409cad28f9bb0378c942e9dfddfaaf0d2db1c9c97882d581f9dd4337510","name":"check_dj.out","bytes":4405},{"sha256":"0048ab5a8a136865f98e83f0756c8eaf28e122124486351d47c95d109f6d8934","name":"check_dj.control.out","bytes":4302},{"sha256":"f0f7e5bc034e1d6d3736d0a0496854b7d8cf8333355b9e17fddc5f224ba4f22c","name":"redact_dj.py","bytes":3311},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280}],"decided_by_author_handle":false,"reviews":[],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"elevate","note":"Route 218 order measurement: matched 6-level fixed-W ladders at W=23# and W=29# give a stable minimal polynomial order p*=4 in u=lnW/ln y (c4 significant at 3 se at both W, c5 not). This fixes a concrete W-stable quartic order target for Q-var41 (item 9) that a derivation will build on; the finite-ladder claim (dof 1-2) and the producer-only W=29# points are disclosed in the return.","decided_at":"2026-10-07T20:15:58.238Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[]}],"decision":{"status":"pending","final_rung":null,"provisional":false,"by":"elevate","note":"Route 218 order measurement: matched 6-level fixed-W ladders at W=23# and W=29# give a stable minimal polynomial order p*=4 in u=lnW/ln y (c4 significant at 3 se at both W, c5 not). This fixes a concrete W-stable quartic order target for Q-var41 (item 9) that a derivation will build on; the finite-ladder claim (dof 1-2) and the producer-only W=29# points are disclosed in the return.","decided_at":"2026-10-07T20:15:58.238Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[]},"duplicates":[],"cited_messages":[]}