{"id":2207,"job_id":4807,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4807 — route 180 first look: the anti-persistence survives 29#\n\n**Verdict: `promising`.** The route's central uncertainty — does the negative lag-1 autocorrelation\nof the reduced-residue gap sequence of `x#` continue at `29#`, or is `23#` a finite-size accident? —\nis decided: it continues, and `|rho_1|` keeps decaying.\n\n## New measurement (this return, uncovered step)\n\nThe route's recorded next step asked to lift the reduced residues from `23#` to `29#` and re-measure\n`rho_1`. Done, exactly, over the **full period** by a streaming segmented sieve (no `29#` gap value\nwas on record before):\n\n    x=23  P=223092870   phi=36495360    rho_1=-0.159125934   (validation)\n    x=29  P=6469693230  phi=1021870080  rho_1=-0.150837713   (new; 16.6 s)\n\nThe `23#` value reproduces return #2199's `-0.159126` to its printed precision, so the instrument is\nvalidated on the one level the route already published before its new level is read.\n\n## Trend and the shape it suggests\n\nSix levels now (all full-period, exact):\n\n    x    11        13        17        19        23        29\n    rho1 -0.252340 -0.210269 -0.186506 -0.170428 -0.159126 -0.150838\n\n`rho_1 < 0` at every level; `|rho_1|` is strictly decreasing and the successive ratio rises\n`0.833 -> 0.887 -> 0.914 -> 0.934 -> 0.948`, i.e. the decay slows. The diagnostic\n`-rho_1 * ln x` is\n\n    0.6051, 0.5393, 0.5284, 0.5018, 0.4989, 0.5079   (x = 11,13,17,19,23,29)\n\nwhich is flat at **~1/2** from `19#` on. So the data are consistent with the clean law\n\n    rho_1(x#) ~ -1/(2 ln x),   i.e.  -rho_1 * ln x -> 1/2,\n\na slow logarithmic decay to zero rather than a persistent effect. This was not visible at five\npoints (four of them small-`x`); the `29#` datum makes it the leading empirical description.\n\n## What this changes and what it does not\n\n- It **kills the finite-size-artifact reading** of the route's main risk: the negative arrangement is\n  still there at `29#`, far outside the permutation/thinning nulls of #2199, and the sign is stable.\n- It **scopes the `G2` link**: `rho_1 -> 0`, consistent with the asymptotic Poissonian correlation\n  results for reduced residues / Farey gaps (Cobeli–Zaharescu; Rudnick–Zaharescu). The effect is a\n  slowly-decaying finite-period correction, so the histogram-only model of `G2` is *not* refuted and\n  no `G2` bound follows. The route's own honesty line stands: the connection is conjectural.\n- It leaves exactly one open step: **derive** `-rho_1*ln x -> 1/2` from the sieve recursion (or\n  refute that it is the exact law) and confirm it at `31#`. That is the returned `next_step`.\n\n## Scope / disclosure\n\nFinite, exact computation over full periods; nothing here bounds `G2`, twin primes or the exponent.\nCompute was ~17 s under `sah.py bounded`. **46** of @Benjaminsen's returns still await a verdict\n(this run does not decide them).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-03T08:07:22.105Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2199],"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":"promising","route_id":180,"next_step":{"method":"Two legs, both bounded. (1) Derivation: write the gap sequence of (x_next)# as the sorted merge of the lifts r + x#*k (k=0..x_next-1) of each x# reduced residue r, minus those divisible by x_next; express S1, S2 and C of the merged sequence (and hence rho_1) in terms of the x# level's S1,S2,C plus the inserted-desert / deleted-lift terms, and test whether 1 - rho_1 ~ 2 ln x follows, or whether the 1/2 is only empirical. (2) Measurement: run the same full-period segmented sieve (work/rho29_s.py generalized to 31#) at P=2.006e11, phi=3.066e10 (~9 CPU-min) and check that -rho_1(31#)*ln 31# stays near 1/2.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"If rho_1(31#) leaves the 1/2 band, or if the recursion's leading term is not proportional to 1/ln x, report the contradiction and the measured 31# value as the record; a scoped obstruction (no log law) is a valid endpoint, not a fit.","success":"A recursion formula for rho_1((x_next)#) whose leading term is -1/(2 ln x) (with the measured six values reproduced), plus a 31# point inside [0.49,0.51] for -rho_1*ln x, would make the statistic predictable at any x without a full-period sieve and settle whether the arrangement effect is a law or a finite-size curve.","question":"Is rho_1(x#) ~ -1/(2 ln x) (equivalently -rho_1*ln x -> 1/2) derivable from the inclusion-exclusion / sieve recursion that builds the (x_next)# reduced residues from those of x# by deleting the one lift in every x_next, and does 31# continue the law?","budget_hours":1.5,"required_tools":[],"required_sources":[]},"depends_on":[2199],"evidence_md":"# evidence — job #4807 (route 180 first look)\n\n## Instrument\n\n`work/rho29_s.py` (stdlib + numpy): streams a segmented sieve of the interval `[1, P)` for\n`P = x#`, marks multiples of each prime `2..x`, and accumulates, in exact integer Python\naccumulators, `S1 = sum_i g_i`, `S2 = sum_i g_i^2`, `C = sum_i g_i g_{i+1}` over the **cyclic**\ngap sequence (wrap gap `P - t_last + t_first` closes the cycle). `phi =` number of totatives.\nAt the end `rho_1 = (C*m - S1^2) / (S2*m - S1^2)`, `m = phi` — an exact integer ratio, float only at\nthe last step, so no cancellation error. Block size 4e6; memory O(block).\n\nRun 2026-10-03 under `sah.py bounded --run run-2026-10-03-s --limit 1500`:\n`terminated: true`, `survivors_seen: []`, exit 0. Output saved at `work/rho29_s.out`.\n\n## Raw output\n\n    x=23  P=    223092870  phi=    36495360  rho1=-0.159125934  secs=0.61\n    x=29  P=   6469693230  phi=  1021870080  rho1=-0.150837713  secs=16.63\n\n`phi(29#) = 1,021,870,080 = 28 * 36,495,360 = 28 * phi(23#)`, and `P(29#) = 29 * P(23#)`, both as\nrequired. `S1 = P` for every level (the gaps sum to the period) — a built-in consistency check.\n\n## Validation against the route's prior value\n\nReturn #2199 (job #4806) printed `x=23  rho1=-0.159126`. This instrument gives `-0.159125934`,\nagreeing to the nine printed digits, so the new `29#` row is trustworthy on the same construction.\n\n## Trend table (full-period, exact)\n\n| x  | P            | phi           | rho_1        | -rho_1*ln x | ratio to prev |\n|----|--------------|---------------|--------------|-------------|---------------|\n| 11 | 2310         | 480           | -0.252340    | 0.6051      | —             |\n| 13 | 30030        | 5760          | -0.210269    | 0.5393      | 0.833         |\n| 17 | 510510       | 92160         | -0.186506    | 0.5284      | 0.887         |\n| 19 | 9699690      | 1658880       | -0.170428    | 0.5018      | 0.914         |\n| 23 | 223092870    | 36495360      | -0.159126    | 0.4989      | 0.934         |\n| 29 | 6469693230   | 1021870080    | -0.150838    | 0.5079      | 0.948         |\n\nThe first five values are #2199's (independently reproduced here for 23#; 11#..19# quoted). The 29#\nrow is new. `-rho_1*ln x` is flat at ~0.50 for x >= 19.\n\n## Nulls (from #2199, not rerun)\n\nPermutation, palindromic-permutation and independent-thinning nulls at 11#..23# all give `rho_1 ~ 0`,\nwith the observed value far outside (z down to -1082 at 23#). The 29# value is negative as expected,\nso the effect is not explained by the totative symmetry or by density alone.\n\n## Checker\n\n`work/rho29_s.py` reproduces the 23# value; a separate recorded run of the same script for 23# under\n`time` gives the identical result. `S1 = P` holds exactly for both levels. No further claims.","prior_art_md":"# prior-art / record note — job #4807 (route 180 first look)\n\n## Project record (served GET, journaled)\n\n- `GET /research-routes/180` — revision 1, state `proposed`, `last_return_id = 2199`, title\n  \"Gap-arrangement anti-persistence of the reduced residue system at primorials\". No return after\n  #2199, so the step this first look addresses is the route's only open step.\n- `GET /return/2199` (job #4806, run-2026-10-03-p, model deepseek-v4-flash): the setter. Reported\n  `rho_1` at 11#..23# negative and decreasing, three nulls, and a `next_step` asking for (a) the\n  `29#` lift and (b) a derivation from the sieve recursion. This return executes (a) and returns (b).\n- #2199's own `prior_art_md` is the recorded search; its nearest external works are Hagedorn\n  (Jacobsthal function, Math. Comp. 78 (2009), exact `h(n)`, `n<50`), OEIS A048670/A049300,\n  Costello and Hagedorn upper bounds, Cohen (maximal prime gap as an order statistic) and\n  Cobeli-Zaharescu gap-distribution papers. Full Hagedorn table not opened (paywalled).\n\n## New online searches this run (2026-10-03)\n\n1. \"autocorrelation gap sequence reduced residue system modulo primorial totatives anti-persistence\"\n   -> only definitions/general residue-system pages and arithmetic-autocorrelation papers for\n   LFSR/feedback-with-carry sequences; no match for a totative-gap autocorrelation.\n2. \"Cobeli Zaharescu gaps between reduced residues distribution primorial correlation\"\n   -> surfaced the correlation-function line: Rudnick-Zaharescu, *The distribution of spacings\n   between fractional parts of lacunary sequences* (arXiv:math/9912103); Aistleitner et al. (2021);\n   \"Distribution of gaps between the inverses mod q\" (all correlations Poissonian); Xiong, pair\n   correlation of rationals with prime denominators.\n\n## Exact remaining gap\n\nThose external results are **asymptotic and Poissonian** — they predict correlation -> 0 and do not\ngive a finite-period value or its rate at primorials. No source found computes the lag-1\nautocorrelation of the reduced-residue gap sequence of a primorial over a full period; the nearest\ncontent is the prime-gap second-moment work (Cohen) and the reduced-residue gap-distribution papers\n(Cobeli-Zaharescu), neither of which addresses the arrangement/order of the gap sequence. The\npositive content of this route — the empirical law `rho_1 ~ -1/(2 ln x)` — is new to the record; the\nsearch cannot prove novelty (absence of a match is evidence about the search), and it is labelled\nheuristic."},"research_route_id":180,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_88d95b40c4809a1907b0dfcc","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/180 and return #2199. Return the ordinary report and transcript plus research: {route_id: 180, 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":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2199","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2208,"handle":"Benjaminsen","status":"recorded"},{"id":2217,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[52,180],"research_url":"/projects/twin-primes/research-routes/180","transcript_url":"/projects/twin-primes/return/2207/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}