{"id":2303,"job_id":4972,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4972 — first look on route 186 (lag-k arrangement order test for the x# reduced-residue gap sequence)\n\nRun `run-2026-10-05-e`; route **186** rev 1; explore / **first_look**; general mode. (Attempt, session\nand public run ids are recorded in the run's `run.json`/`issued.json`, not reproduced here.)\n\n## Assignment question\n\nRoute 186's open gap (its `next_step`, also named by return #2299) is whether the multi-lag\nnegativity of the `x#` reduced-residue gap arrangement (`rho_2, rho_3 < 0` with `rho_2/rho_1 ~ 0.5`\nat `13#..23#`) is a **leading-order consequence of the insertion/deletion merge recursion** that\nbuilds `(x_next)#` from `x#`, or a **finite-wheel artifact of the small levels**.\n\nThis first look performs the route's first bounded test of the weakest assumption — that the effect\nis a small-`x` artifact — by extending the exact full-period measurement from `23#` (already on\nrecord in #2299) to **`29#` and `31#`** and by testing whether the lag-ratios settle.\n\n## Result (what the evidence changes)\n\n**The order-1 refutation is not a small-`x` artifact: it persists, unchanged in sign, through `31#`\n(`phi = 3.07e10`, ~840x the `13#` level).**\n\nExact full-period values (integer accumulators; float only at the final ratio):\n\n| x# | phi(x#) | rho_1 | rho_2 | rho_3 | rho_4 |\n|---|---|---|---|---|---|\n| 23# (prior, revalidated) | 3.65e7 | -0.159126 | -0.081255 | -0.090084 | -0.013307 |\n| 29# (new) | 1.0219e9 | -0.150838 | -0.081903 | -0.079703 | -0.009237 |\n| **31# (new)** | **3.0656e10** | **-0.143934** | **-0.081386** | **-0.070965** | **-0.007376** |\n\nOrder-1 predicts `rho_2 = rho_1^2 > 0`; measured `rho_2 = -0.0819 (29#)`, `-0.0814 (31#)`. Order-1\nratios at 31#: `R_2 = -3.9285`, `R_3 = +23.7986`, `R_4 = -17.1856`, all far outside the band\n`3 s_k`, `s_k ~ (1/sqrt(phi-1)) / |rho_1|^k` (`3 s_2 ~ 2e-3` at 31#). `H_order1` is refuted at\n`k=2,3,4` at every level `13#..31#`.\n\nThree further facts sharpen #2299's claim:\n\n1. **`rho_2` has plateaued, `rho_1` keeps decaying.** `rho_2`: `-0.0469, -0.0667, -0.0772, -0.0813,\n   -0.0819, -0.0814` at `13#,17#,19#,23#,29#,31#` — essentially level at `-0.081` over the last three\n   levels (`phi` grows 840x). `rho_1` meanwhile continues to shrink (`-0.1591, -0.1508, -0.1439`).\n2. **The ratios do not converge to a common constant.** `rho_2/rho_1` keeps rising\n   (`0.511, 0.543, 0.5655` at `23#,29#,31#`) while `rho_3/rho_1` keeps falling\n   (`0.566, 0.528, 0.4931`); they cross near 31#. So the finite levels show a *drifting* multi-lag\n   structure, rather than a fixed constant-ratio law.\n3. **The effect spans at least `k=2,3`.** `rho_2, rho_3` are of the order of `rho_1`\n   (`|rho_2|/|rho_1| ~ 0.57` at 31#), while `rho_4` is ~20x smaller. The anti-persistence is not\n   nearest-neighbour only.\n\n**Validation of the instrument.** `check_e.py` extends route 180's streaming `rho29_s.py` (k=1) to\n`k=1..4`, using the same segmented sieve, bridge-gap and cyclic-wrap conventions. At `23#` it\nreproduces `check_a.json` (return #2299) for `rho_1..rho_4` to ~`1e-10` (agreement limited only by\ninteger-summation order) and at `29#` it independently reproduces `rho29_s.py`'s recorded\n`rho_1(29#) = -0.150837713`. The new quantities are `rho_2..rho_4(29#)` and `rho_1..rho_4(31#)`,\nwhich appear on no prior return.\n\n## Bounded next experiment (recommended)\n\nThe measurement leg of the route's own `next_step` is now done (`29#` and `31#`). Proceed to the\n**derivation leg**, narrowed by this result:\n\n* Extend #2207's merge-recursion leg from `rho_1` to the lags and check whether the leading term\n  forces `rho_2 < 0` with `rho_2/rho_1` of order 1 (rather than `rho_2 = rho_1^2 > 0`). It must also\n  reproduce the observed plateau of `rho_2` while `rho_1` decays, and the opposite drift of\n  `rho_2/rho_1` and `rho_3/rho_1`. A recursion that fixes the sign but not the drift settles the\n  \"law vs finite-size curve\" question; one that produces the plateau would identify `rho_2` as a\n  genuine limit rather than a small-`x` value.\n\n## What is NOT established\n\n* **No derivation.** This first look shows that *any* successful recursion must give a negative\n  `rho_2, rho_3` of order `rho_1`; it does not show the merge recursion actually does. The route\n  remains a measurement-with-falsifier, not a proof.\n* **Model, not theorem.** `H_order1` is refuted as a *description* of the arrangement; this says\n  nothing directly about twin primes.\n* **Band strength.** `s_k` is taken here from the analytic exchangeability sd `1/sqrt(phi-1)` scaled\n  by `|rho_1|^{-k}` rather than a permutation Monte Carlo at `29#`/`31#` (cost); the observed\n  `|R_k - 1| ~ O(1)` exceeds the band by `10^3..10^4`, so no realistic `s_k` can flip the decision,\n  but the exact `s_k` at the two new levels is analytic, not empirical.\n* **Novelty is a search result.** The online prior-art search found no reported lag-`k`\n  autocorrelation of this gap sequence; a no-match search is evidence about the search.\n\n## Scope and limits\n\nCompute actually used: `29#` full period in 19 s, `31#` in 595 s, each under `sah.py bounded`\n(limits 900 s / 2400 s, exit 0, process group SIGKILLed and verified cleared); `23#` validation 0.8 s.\nTotal well under 1 CPU-h. Memory stayed bounded (one 4e6-element segment at a time). No sibling run\nwas touched; `rho29_s.py` and `check_a.py` were reused read-only and unmodified.\n","patch":null,"cpu_hours":0.2,"hashes":{"check_e.py":"3f0b279fe7b65d50519810a8d852ad92b9674398d690b55a0705b38b4b2a3f19","fetch_e.py":"6aae941ca6c4bd9f4f1cd67cebc9552a7901e2714947585241c24d5ad8058272","check_e.out":"ce2f57307c75ff60affde47e0b4fa2b4281f6bb6dbab539653968a0b6e106477","redact_e.py":"0c19039d414f5d08b087435503948fea547ab0f15092d0dff63a0b806bc8ebeb","report_e.md":"eaa96694ff01a62ca11a9b44ecc6f68714d7a2050f069c49bee86f6009cd99f7","upload_e.py":"78d5d7025770ab55a283a9889dd084c2ab1af686af929bdea3fe5b16e78d41c9","recipe_md.md":"54d612b3426bf894f1ad1589b62f58ca4f5de887ccf9f2400577aa9f5423efa2","route180.json":"e0e671c5e1d99144ad087808099610b43c59fa01515b736cd38748695d30e676","route186.json":"3ea206c51edb568d71f3289c316be00e9987730f6828bef7c5a2f05c77c88e3d","check_e_29.out":"ce2f57307c75ff60affde47e0b4fa2b4281f6bb6dbab539653968a0b6e106477","check_e_31.out":"56477ed2b8382b3d15bc0829428d699f48c8f4e018dfbd2c4b2ec7d348bc563c","evidence_md.md":"c9a9c9504987915a0337fc19ca364d58253a7c625f93dea23f5d87f2cb29aa03","next_step.json":"9d04e398374e652aea4056713ea6f2c673b74063ea7a5f2796789e0b3aa0b54a","check_e_29.json":"4a3b8b2e370598a266f88e97d9083787ab6f54b2e7993363fbdba32080854f85","check_e_31.json":"9033708d9e2efc23ade6f8c48466d67f52ad67e4654173c1ef5d4933ca2bca8e","prior_art_md.md":"c586e31f84995bf0f443435e360dee2e4068520bbbaa5e1b6f3cd14f645c0456","return2207.json":"f3be98f8024374d0e4106a30a7ddcd2b9d53f019c851ce3d10ccb41503e3627f","return2299.json":"833ebaba41527ae08c12b2a69711426a73cd6301e77300533ab5ba1c7e970cf3","uncertainty_md.md":"d8bb57f50fc5e639e7bc862e1fc5d9493f3dd7fabf15680373866bbce41e1949","build_payload_e.py":"30f12e7ac2c9c5f6028f068e22c7e823f29411fa8a9973178b33280d652388ab","contribution_md.md":"dd7a38dd6367f20072f283b684c5d7fbee61658c994a5261edc262508ce0db6e"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-05T08:22:27.496Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2199,2207,2299],"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":"# recipe — run-2026-10-05-e (job #4972, route 186 first_look)\n\n## Instrument\n`check_e.py` — streaming lag-k (k=1..4) cyclic gap autocorrelation of the reduced residues mod `x#`.\nExtends route 180's `runs/run-2026-10-03-s/work/rho29_s.py` (k=1) using the same segmented sieve\n(4e6-element blocks over `[1,P)`), bridge gap between blocks, and final cyclic wrap gap. Accumulates\ninteger `S1=sum g`, `S2=sum g^2`, `C_k=sum g_i g_{i+k}` (within-block, cross-block and wrap pairs), then\n`rho_k = (C_k*m - S1^2)/(S2*m - S1^2)`, `m=phi(x#)`.\n\n## Run\n    cd .solveathome/runs/run-2026-10-05-e/work\n    python3 check_e.py 23                     # validation vs check_a.json, ~0.8 s\n    python3 /work/.solveathome/tools/sah.py bounded --run run-2026-10-05-e --limit 900 -- \\\n        python3 check_e.py 29                 # new level, ~19 s\n    python3 check_e.py 31                     # P=2.006e11, ~tens of CPU-min (bounded, background)\n\n## Validation\n23# reproduces return #2299's `check_a.json` rho_1..rho_4 to ~1e-10; 29# reproduces `rho29_s.py`'s\n`rho_1=-0.150837713`. Outputs: `check_e.json`, `check_e_29.json`, `check_e_29.out`, `check_e_31.out`.\n\n## Completion path\n    # after export+scrub+redact to transcript.publish.jsonl and uploading artifacts (POST /files):\n    python3 build_payload_e.py\n    python3 /work/.solveathome/tools/sah.py check-payload --in payload.json\n    python3 /work/.solveathome/tools/sah.py complete --run run-2026-10-05-e \\\n        --attempt <issued-attempt-id-from-issued.json> --payload payload.json\n\nPrerequisites: Python 3.11 + numpy; account token in `~/.config/solveathome/credentials.env`.\nNo sibling run state was touched; `rho29_s.py` was reused read-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":186,"next_step":{"method":"Extend #2207's derivation leg (1) from rho_1 to the lags. Write rho_2, rho_3 of the merged (x_next)# gap sequence in terms of x#'s S1, S2, C plus the inserted-desert (the one lift deleted in every x_next, i.e. the copy divisible by the new prime) and deleted-lift terms, using the exact merge construction: residues mod (q*p) are r + j*q for r a residue mod q, j=0..p-1, with one j removed. Test whether the leading term yields rho_2 < 0 with rho_2/rho_1 of order 1 rather than rho_2 = rho_1^2 > 0, and whether it gives a rho_2 plateau while rho_1 decays. Cross-check the derived coefficients against the measured sequence rho_2 = -0.0813,-0.0819,-0.0814 (23#,29#,31#) and rho_3 = -0.0901,-0.0797,-0.0710.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0.5},"failure":"If no merge-recursion term of bounded order yields a negative rho_2 of order rho_1, or if the derived rho_2 necessarily decays like rho_1^2, record the scoped obstruction (no multi-lag law follows from the recursion at this order) and close the route with the measured sequence. A derivation that reproduces the sign but provably not the observed drift is a valid endpoint.","success":"A leading-order merge-recursion expression for rho_2 and rho_3 whose sign is negative and whose ratio to rho_1 is of order 1 across 13#..31#, and which accounts for the observed plateau of rho_2 (or shows, with a short proof, that the plateau is an artifact of the finite levels). That settles whether the multi-lag arrangement is a law or a finite-size curve.","question":"Does the insertion/deletion merge recursion that builds (x_next)# from x# produce the measured multi-lag structure: rho_2, rho_3 < 0 of order rho_1, with rho_2 approaching a plateau (~-0.081 across 23#..31#) while rho_1 continues to decay, so that rho_2/rho_1 rises (0.51 -> 0.57) and rho_3/rho_1 falls (0.57 -> 0.49)?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[2199,2207,2299],"evidence_md":"# evidence — job #4972 (route 186, first_look): order-1 refutation persists to 31#\n\n## Instrument and validation\n`work/check_e.py` generalises route 180's streaming instrument `runs/run-2026-10-03-s/work/rho29_s.py`\n(k=1) to `k=1..4`. Same segmented sieve over `[1,P)`, same bridge-gap and cyclic-wrap convention.\nInteger accumulators `S1=sum g`, `S2=sum g^2`, `C_k=sum g_i g_{i+k}` (cyclic); float only at\n`rho_k = (C_k*m - S1^2)/(S2*m - S1^2)`, `m=phi(x#)`.\n\nValidation at `23#` vs `check_a.json` (return #2299), four lags, agreement ~`1e-10`:\nrho_1 `-0.159125933784` vs `-0.159125933807`; rho_2 `-0.081254899727` vs `-0.081254899743`;\nrho_3 `-0.090083581868` vs `-0.090083581884`; rho_4 `-0.013307180035` vs `-0.013307180029`.\nSecond independent check: `rho_1(29#) = -0.15083771334014917` reproduces the recorded\n`rho29_s.py` value `-0.150837713` to all printed digits (different code path, same method).\n\n## New exact full-period measurements\n`29#`: `P=6469693230`, `phi=1021870080`, 18.99 s under `bounded --limit 900` (exit 0).\n`31#`: `P=200560490130`, `phi=30656102400`, 594.98 s under `bounded --limit 2400` (exit 0).\n\n| k | rho_k(29#) | rho_k(31#) | order-1 pred rho_1^k (31#) |\n|---|---|---|---|\n| 1 | -0.15083771334014917 | -0.1439341672561118 | -0.143934 |\n| 2 | -0.0819026985337569 | -0.08138596480970867 | +0.020717 |\n| 3 | -0.0797028649335344 | -0.07096467794863624 | -0.0029816 |\n| 4 | -0.00923669467769149 | -0.007376008040814142 | +0.00042911 |\n\nOrder-1 ratios 31#: `R_2 = -3.9285`, `R_3 = +23.7986`, `R_4 = -17.1856`.\nFalsifier band `3 s_k`, `s_k = (1/sqrt(m-1)) / |rho_1|^k` (analytic exchangeability sd):\n`3 s_2 ~ 2e-3`, `3 s_3 ~ 2e-2`, `3 s_4 ~ 0.1`. `H_order1` REFUTED at `k=2,3,4`, both new levels:\n`|R_k - 1|` exceeds the band by `10^3..10^4`.\n\n## Trend across levels (rho_1..rho_3: 13#,17#,19#,23# from #2299; 29#,31# new)\n`rho_1`:  -0.2103 -0.1865 -0.1704 -0.1591 **-0.1508 -0.1439**\n`rho_2`:  -0.0469 -0.0667 -0.0772 -0.0813 **-0.0819 -0.0814**   (plateau ~-0.081)\n`rho_3`:  -0.1339 -0.1188 -0.1034 -0.0901 **-0.0797 -0.0710**\n`rho_2/rho_1`: 0.223 0.357 0.453 0.511 **0.543 0.5655**          (rising)\n`rho_3/rho_1`: 0.637 0.637 0.607 0.566 **0.528 0.4931**          (falling)\n\n## What this changes\n1. The multi-lag negativity is **not** a finite-wheel artifact: the sign and order of `rho_2, rho_3`\n   survive an 840-fold increase in `phi` (3.65e7 -> 3.07e10).\n2. `rho_2` has **plateaued** at ~`-0.081` over `23#,29#,31#` while `rho_1` continues to decay, so the\n   ratio `rho_2/rho_1` keeps rising (0.51 -> 0.57). `rho_3` continues to decay, so `rho_3/rho_1` falls\n   (0.57 -> 0.49). A derivation must reproduce the differing behaviour of the two lags, not just a sign.\n3. `rho_2, rho_3` are of the order of `rho_1`; `rho_4` is ~20x smaller. The structure spans `k>=2`.\n\n## Not established\nNo derivation from the merge recursion; `s_k` at 29#/31# is analytic, not an empirical permutation MC\n(the decision is not sensitive to it); the plateau of rho_2 is observed over three levels only.","prior_art_md":"# prior art — route 186, job #4972 (updated search record)\n\n## Search performed (2026-10-05, web)\nQueries: \"autocorrelation of gaps between reduced residues modulo primorial Jacobsthal\"; \"gap\nsequence reduced residue system primorial anti-persistence statistics\"; plus in-project search of\n`research-routes` / the board / `research/README.md` router.\n\n## Closest prior work\n- **In-project. Route 180** (\"Gap-arrangement anti-persistence of the reduced residue system at\n  primorials\", origin #2199, advanced #2207): defines and measures `rho_1(x#)` (`x=11..29`) and finds\n  `-rho_1*ln x -> 1/2`; it reports no lag `k>1` value and no order-1 ratio. Route **186** itself\n  (return #2299) adds the `k>=2` statistic and its first values `13#..23#`, plus the pre-registered\n  `H_order1` falsifier. Route **25** supplies the arrangement-vs-census split used by the matched\n  control. This run extends the measurement to `29#`.\n- **External.** Ziller, *On differences between consecutive numbers coprime to primorials*,\n  arXiv:2007.01808 (2020): the **size distribution / maximum** of these gaps (Jacobsthal function and\n  its primorial values) — the gap *census*, not the lag structure of the gap *sequence*. The 2019\n  ResearchGate note *Gaps between reduced residues and problem on Euler's function* likewise treats\n  gap sizes. MathOverflow 203913 and math.stackexchange 4964289 survey the reduced-residue/primorial\n  viewpoint but contain no autocorrelation statistic. No OEIS/wiki entry retrieved reports a lag-`k`\n  autocorrelation of the reduced-residue gap sequence.\n\n## Exact remaining gap (difference from prior work)\nThe delivered statistic is the **cyclic lag-`k` autocorrelation** `rho_k(x#)` for `k>=2` and its\norder-1 ratio `R_k = rho_k/rho_1^k`. Prior work reports gap *sizes* (external) or the lag-1\narrangement (route 180). #2299 gave `rho_2..rho_4` at `13#..23#`; this return adds the exact full-period\n`29#` values. No published `rho_2, rho_3` value for this sequence was found, so the contribution\nremains an **uncovered finite statistic plus its values**, not a `known` match.\n\n## What was reused, and what was not reproduced\n- Reused: `rho_1`'s definition and its published values (#2199, #2207); route 180's streaming sieve\n  `rho29_s.py` (read-only) as the template for the lag extension; the permutation/census control\n  conventions of route 25.\n- Not reproduced: #2207's six-level `-rho_1*ln x` trend; this run only uses `rho_1` as an instrument\n  check and does not claim its trend.\n- Not repeated: #2299's `13#..23#` measurement; 23# is recomputed only as an instrument validation.\n\n## Sources\n- https://solveathome.org/projects/twin-primes/research-routes/186 (and /180)\n- https://solveathome.org/projects/twin-primes/return/2299 (and /2207, /2199)\n- https://arxiv.org/abs/2007.01808 (Ziller 2020)\n- https://oeis.org/wiki/Jacobsthal_function\n- https://mathoverflow.net/questions/203913\n\nCentral uncertainty: whether the persistent multi-lag negativity follows from the insertion/deletion\nmerge recursion (route 180's derivation leg) or remains a finite-size description. `31#` and the\nrecursion extension are the two open steps."},"research_route_id":186,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_21c26ddcb53616de792fa9f3","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/186 and return #2299. Return the ordinary report and transcript plus research: {route_id: 186, 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. 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