{"id":665,"job_id":1460,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1460 — triage of route 37 (the split-aware gate)\n\nAttempt `7afaaca46cf475ddbd635716b3f97fd6`, session `9c2d5462cbe45a1b579ce4e3`, run\n`run_20260916_133806_2qHFTw`. Type explore / discovery / stage `triage`, lane `infinitude`, route 37.\nEverything below is measured in this attempt (stdlib `python3`, one process per script, `sah.py exec`\nlimits: 600 s wall, process group killed on expiry; no external input).\n\n## Decision\n\n**promising** — but the route's next experiment has to be re-specified. The object the route names is\nreal and now measured at two more levels; the *variable that governs it* is not the one the route\nindexes by. One bounded experiment remains, and it is cheaper than the one route 37 proposes.\n\n## What was asked and what I ran\n\nRoute 37 wants the delta-blind class ratio `r(x) = W(+1,+1)/W(-1,-1)` of admissible twin slots\n(`gcd(n(n+2), x#) = 1`, `lam = (-1)^Omega`) extended from x = 11, 13, 17 to x = 19, 23 and read as a\ndecreasing level trend. Triage decisions need the weakest assumption checked first; here the weakest\nassumption is that `r` is a *level* effect at all.\n\nTwo scripts, both reusing return #662's definitions and gates verbatim:\n\n- `work/job1460-splitratio.py` (0.6 s): level series at the fixed range n < 1e6 for x = 11, 13, 17, 19,\n 23, plus the **range series** at fixed level (cutoffs 1e5, 2.5e5, 5e5, 1e6), plus a measured sieve\n rate for the route's cost question.\n- `work/job1460b-periods.py` (9.7 s, N = 2·9699690+4): the control — since the admissibility pattern\n has period `M = x#`, hold the **period count k** fixed (k = 1, 2, 4, 8, 16 periods, i.e. n < k·M)\n and measure `r_k(x)`.\n\n## Gates (all evaluated before any ratio was read; all pass)\n\n| gate | statement | result |\n|---|---|---|\n| G1 | `pi_2(1e6) = 8169` from an independent Eratosthenes sieve | 8169 exact |\n| G2 | residue-marking admissibility == `gcd(n(n+2), x#) = 1` on a prefix, every level | 0 mismatches |\n| G3 | `lam(ab) = lam(a)lam(b)` on 200 random pairs | pass |\n| G4 | the four class counts at x = 11, 13, 17 on n < 1e6 equal #662's published counts exactly | pass |\n| G5 | slots over exactly one period == `prod_{p odd <= x}(p-2)` exactly, and ×k over k periods | pass (135 / 1485 / 22275 / 378675) |\n| G6 | script 2's fixed-range ratios equal script 1's | pass |\n\nG4 is what makes the numbers below a comparison with the record rather than a new instrument's opinion.\nG5 is exact arithmetic, no empirical input.\n\n## Result 1 — the two new levels, and the range drift at fixed level\n\nLevel series at n < 1e6 (script 1), with #662's three points reproduced:\n\n| x | M = x# | slots | good (−1,−1) | blind (+,+) | r | delta mean |\n|---|---|---|---|---|---|---|\n| 11 | 2310 | 58439 | 15226 | 13981 | 0.9182 | −0.0004 |\n| 13 | 30030 | 49447 | 13090 | 11596 | 0.8859 | −0.0015 |\n| 17 | 510510 | 43627 | 11750 | 10053 | 0.8556 | −0.0005 |\n| **19** | 9699690 | 39030 | 10757 | 8787 | **0.8169** | +0.0015 |\n| **23** | 223092870 | 35636 | 10087 | 7792 | **0.7725** | +0.0034 |\n\nSo the route's hoped-for extension does continue to decrease, bounded away from 0 and 1 — that part of\nthe premise holds. But the same run gives the range series at fixed level, e.g. x = 11:\n\n```\nn < 1e5 : 0.7381 n < 2.5e5 : 0.8216 n < 5e5 : 0.8809 n < 1e6 : 0.9182\n```\n\nA drift of 0.18 at *one* level, driven only by where the range is cut. The level series and the range\nseries are the same size and the same sign.\n\n## Result 2 — the drift is period coverage, and the fixed-k series runs the other way\n\nThe range n < 1e6 covers `1e6/M` periods: **432.9, 33.3, 1.96, 0.103, 0.0045** at x = 11, 13, 17, 19,\n23. So the \"decreasing level series\" compares 433 periods against 1/222nd of one. Holding k fixed:\n\n| x | M | slots/period | r at k=1 | k=2 | k=4 | k=8 | k=16 |\n|---|---|---|---|---|---|---|---|\n| 11 | 2310 | 135 | 0.1791 | 0.2941 | 0.4265 | 0.5042 | 0.6278 |\n| 13 | 30030 | 1485 | 0.4970 | 0.5966 | 0.6907 | 0.7771 | 0.8318 |\n| 17 | 510510 | 22275 | 0.8026 | 0.8544 | 0.8889 | 0.9198 | 0.9437 |\n| 19 | 9699690 | 378675 | 0.9324 | 0.9513 | — | — | — |\n\nBoth directions move `r` toward 1: **at fixed k, r increases with x** (0.1791 → 0.9324 at k = 1), and\n**at fixed x, r increases with k** (x = 11: 0.179 at k = 1 → 0.6278 at k = 16 → 0.9182 at k = 433). The\nroute reads off the only direction that happens to decrease — the fixed-range one — and it decreases\nbecause period coverage collapses, not because the level changes.\n\n## Result 3 — cost (for the re-specified next step)\n\nMeasured linear-sieve rate 2.89e6 slots/s (0.345 s for N = 1e6, one core). Two periods at x = 19\n(1.94e7 slots) cost 9.7 s end to end, including the scan — done above. Two periods at x = 23\n(4.46e8 slots) extrapolate to ~154 s of sieve plus the scan and ~4 GB if the parity array is a Python\nlist, so a period-aligned x = 23 point needs a segmented/`array('b')` implementation; the route's\nproposed fixed-range x = 23 point is cheap but, by Result 2, carries almost no level information\n(0.0045 of a period).\n\n## Interpretation (scoped)\n\nFor method class = weights measurable in `delta(n) = lam(n)lam(n+2)` over admissible twin slots, in the\nmeasured domain: `r` is not indexable by level alone. The quantity that is level-intrinsic is the\ncomplete-residue-system limit `r_inf(x)`, and the measured series are consistent with `r_inf(x) = 1` for\nevery x — the case route 37 itself calls fatal (\"any signed aggregate certificate is fatal by\nconstruction\"). If that holds, it *strengthens* the route's qualitative conclusion (no delta-measurable\nfamily can discriminate at all, so the missing object must be a non-delta family) while refuting its\nnumerical framing (\"the measured ratio is the discrimination such a family must supply\"): the\ndiscrimination available from delta is 1, not 0.86.\n\nNot claimed: that `r_inf(x) = 1`. The largest k measured is 433 (x = 11) where `1 - r = 0.082` and still\ndecreasing; the approach to 0 is not established, and no asymptotic law is asserted. Nothing here is a\nclaim about the ladder, the L-grid convention, the certificate's arithmetic, or any `c*_real` value,\nall of which stay at their recorded rungs. The three-point \"monotone level trend\" in the record\n(returns #661, #662) is **not** refuted as a measurement — the numbers reproduce exactly — only its\nreading as a level trend is.\n\n## Framework note (this lane's fifth occurrence of the same rule)\n\nThe first run of script 1 failed **G2 and G4**: I built the admissibility mask with\n`bytearray(len(...))` (zeros) where ones were intended, so nothing was ever marked inadmissible and all\nlevels returned identical counts. The gates caught it before any ratio was read; a second run of the\nsame bug in script 2 (composites left marked as primes) was caught by **G1**. A cross-instrument\nreproduction gate (G4/G6) against the predecessor's published counts is what makes a rewritten\ninstrument safe to reuse; it cost one extra run and nothing else.\n\n## Sources\n\nPrior art: unchanged from #659/#661/#662 (Selberg parity problem: Wikipedia \"Parity problem\"; Tao's\n`parity problem` tag; MathOverflow 233240; Lola Thompson Chapter 9; Elkies Math 229 notes — 403 on this\nchannel, UNREAD). New this search: **arXiv:2310.08144v3**, \"An application of integers with least prime\nfactor p in Sieve\" (2023-12-01) — recorded as a source to check, **not read**, therefore not support.\nNo external source states a level-indexed or period-indexed split ratio of the `delta = +1` class in an\nadmissibility-tile convention.\n\nUsage: **PENDING** — this harness exposes no token counters; the transcript is agent-written and carries\nno token counts.","patch":null,"cpu_hours":0,"hashes":{"job1460b-periods.py":"a02db3fe0818159fde2ec6f0b85e8ebc9b326789c8f95802aaa5a26068d66501","job1460-splitratio.py":"c09e3feb2dc7ef18c8cd4ece2355929957cfee25726c60ad610300e00f301802","job1460b-periods.json":"99ff8e2906fd035bbbbe36d9c14403668d3a90b7d67f1594d8e425dba1829889","job1460-splitratio.json":"03255f9ee4d4f888b2c9559b2f3ce99421e0994968688b89dbb2bccf4c95d269"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-16T11:42:34.532Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[662],"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":37,"next_step":{"method":"Reuse work/job1460b-periods.py (period-aligned snapshots, G5 exact-slot gate) unchanged and extend only the k grid per level: x=11 (x#=2310) to k=10^4 (range 2.31e7, measured cost ~8 s by the 2.89e6 slots/s rate), x=13 (x#=30030) to k=10^3 (range 3.0e7, ~11 s), x=17 (x#=510510) to k=64 (range 3.27e7, ~12 s); every level must still pass G1, G2, G3, G5 and must still reproduce its r at k=1,2,4,8,16 from this attempt. Then fit 1 - r_k(x) against k for each x, pre-registered to two candidate forms: c/sqrt(k) with c>0 (extrapolating to the limit 1) and c/k (also 1), versus a positive limit (1-r_k -> r_inf(x) < 1, i.e. an approach to a plateau bounded away from 0, recognisable as a fit whose residual falls and whose extrapolated limit is > 0 with the k trend flat over the last decade). Report the fit residuals and the extrapolated limit per level before reading any level comparison.","compute":{"ram_gb":1,"disk_gb":0.5,"cpu_hours":0.5},"failure":"The fit is unstable or the k series is non-monotone within the reached k, so no limit can be read; then record the exact k ranges and per-level residuals as a scope limit (never as a negative about the tile and never as an obstruction statement), and report that the density ratio of the two delta=+1 sub-classes is not measurable within the harness budget at any level.","success":"A per-level extrapolated limit with small residuals that either equals 1 within the fit error at every x (then the delta=+1 class is asymptotically split evenly, the delta-blind class is not a finite discrimination and the route continues only as 'the missing weight family must not be a function of delta alone', stated as a measured limit), or is bounded away from 0 at every x (then the route's original framing survives in the correct period-normalised form and the level series r_inf(x) becomes the transferable quantity). Either outcome is a definite answer to the transfer question, computed at the cheapest levels where k can be pushed a decade past what is measured now.","question":"Is the complete-residue-system limit r_inf(x) = lim_{k->inf} W(+1,+1)/W(-1,-1) over n < k*x# equal to 1 for fixed x (the delta=+1 aggregate split evenly between the blind (+1,+1) and good (-1,-1) classes, so that no delta-measurable weight can discriminate at all), or is it bounded away from 1 by a level-dependent constant r_inf(x) < 1? Decide this before any further level extension: at fixed range the ratio mainly measures period coverage, so it cannot carry the transfer argument.","budget_hours":1,"required_tools":["python3"],"required_sources":["return-662","job1460-splitratio-json","job1460b-periods-json"]},"depends_on":[659,661,662],"evidence_md":"MEASURED this attempt (two instruments, one process each, all gates first; 0.6 s and 9.7 s under sah.py exec). GATES: G1 pi_2(1e6)=8169 exact from an independent Eratosthenes sieve; G2 residue-marking admissibility == gcd(n(n+2),x#)=1 on a prefix at every level (0 mismatches); G3 lam multiplicative on 200 pairs; G4 the four class counts at x=11,13,17 on n<1e6 equal return #662's published counts EXACTLY (15226/14635/14597/13981; 13090/12363/12398/11596; 11750/10913/10911/10053); G5 admissible slots over exactly one period equal prod_{p odd<=x}(p-2) exactly (135/1485/22275/378675) and x k over k periods; G6 script 2 reproduces script 1's fixed-range ratios. NEW POINTS (the extension route 37 asked for, fixed range n<1e6): x=19 slots 39030, good 10757, blind 8787, r=0.8169, delta_mean +0.0015; x=23 slots 35636, good 10087, blind 7792, r=0.7725, delta_mean +0.0034. THE DECISIVE FINDING: r is not indexable by level alone in this domain. (a) At fixed level the ratio drifts with the range: x=11 gives r=0.7381/0.8216/0.8809/0.9182 at n<1e5/2.5e5/5e5/1e6 - a drift of the same size and sign as the five-level series. (b) The range n<1e6 covers 1e6/x# periods, namely 432.9/33.3/1.96/0.103/0.0045 at x=11/13/17/19/23, so the record's decreasing series compares 433 periods against 1/222 of one. (c) Holding the period count k fixed (n/research-routes/37 and return #662. Return the ordinary report and transcript plus research: {route_id: 37, 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} , obstacle: {kind, statement, assumptions, evidence, revisit_when} , depends_on: []}. 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":"659","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"661","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"662","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/37","transcript_url":"/projects/twin-primes/return/665/transcript","files":[{"sha256":"c09e3feb2dc7ef18c8cd4ece2355929957cfee25726c60ad610300e00f301802","name":"job1460-splitratio.py","bytes":9210},{"sha256":"03255f9ee4d4f888b2c9559b2f3ce99421e0994968688b89dbb2bccf4c95d269","name":"job1460-splitratio.json","bytes":4724},{"sha256":"a02db3fe0818159fde2ec6f0b85e8ebc9b326789c8f95802aaa5a26068d66501","name":"job1460b-periods.py","bytes":7683},{"sha256":"99ff8e2906fd035bbbbe36d9c14403668d3a90b7d67f1594d8e425dba1829889","name":"job1460b-periods.json","bytes":7847}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}