{"id":698,"job_id":1492,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1492 — pursue, route 41 rev 2: the q = 43 readings of the Tail-Count Transport margin, and the price of the consecutive fold\n\nRun `run_20260916_151840_BikdhQ` · attempt `8b4ffbee6b55ccf3ffc75167f1c9c9ac` (job #1492) ·\ngeneral mode · explore, lane `formalize`, stage `pursue` ·\nroute **41 rev 2** · budget 0.5 h · one instrument run, 355 s wall / ~1,554 s CPU on 10 threads.\n\n## Verdict: `progress`\n\nThe recovered instrument passes the route's blocking entry gate (**byte-identical** control) and\nreproduces all four published `q <= 37` margins exactly. It then takes **four q = 43 readings from\nindependent start tiles**. The reading from the start tile immediately below the target — **T_37 by\n43, streamed** — is\n\n    R = 0.956961  (theta* = 72)   c = 43(1 - R) = 475 666 227 427 / 257 021 730 358 = 1.850685\n\nwhich is **inside the route's exact band 1.842..1.960** and **+0.00144 of its prediction\nR(43) = 0.955522 +- 0.003**. The flattening reading (c ~ 1.59, R ~ 0.963) is **not** supported by the\nbest available proxy. Two scope limits are stated rather than papered over: the *consecutive* fold\n41 -> 43 is priced out of this assignment's compute (§4), and the **start-tile dependence of c at\nq = 43 is first-order** (§3).\n\n## 1. Entry gate: the instrument, recovered by content and verified byte-for-byte\n\n#159's own producer is a 404 (established by #696); the runnable instrument is recovered the same\nway #696 recovered it, and every byte is verified:\n\n* `recover159.py` reads `report_md`'s fenced blocks of return **#159** and writes a block only when\n  its sha256 equals a sha #159 records. **9 of 12** recovered, **0 mismatches**: the five C sources\n  (`src/tct.h` `e77fb0bf…`, `src/tile.c` `1425469c…`, `src/analyze.c` `96c91222…`, `src/main.c`\n  `b13b5717…`, `src/qual.c` `86e8408c…`), plus `prereg.md`, `compare-fold41.md`, `out/b.log`,\n  `out/cd.log`.\n* `out/control.log` was not inline; it is served: `GET /files/9e8858a9…` -> 200, 20 657 B, and its\n  sha256 **is** the recorded `9e8858a94b06bc43ed99eead9d2490eb5863d7407cf1d41fca2d50322351e4d4`.\n* rebuilt `cc -O3 -pthread` and rerun under `sah.py exec` (`--seconds 300 --cpu-seconds 500\n  --mem-mb 3500`): stdout **byte-identical** to `out/control.log` (`cmp` clean, exit 0, 20 657 B).\n\nGate **PASSED** in the sense the route demands (\"figure for figure\"). Operational note:\n`sah.py exec` truncates its captured `stdout` at 4 000 bytes (observed: the first attempt returned a\n4 000-byte slice), so the gate output was redirected to a file by the child and hashed **from disk**\n— the only trustworthy way to compare a multi-kilobyte figure set.\n\n## 2. All gates green, then the q = 43 column (exact integers)\n\nFour published points are re-measured **before** any new number is read (the driver refuses to\nreport a mismatch and exits non-zero; it exited 0, `ALL CHECKS PASS (FAILS = 0)`):\n\n| run | measured max N_new/RHS | theta* | published figure | c = q(1-R), exact (reduced) |\n|---|---|---|---|---|\n| T_23 by 29 (#159 (a)) | 0.932412 | 42 | 0.9324 | 30 966 867 / 15 798 890 = 1.960066 |\n| T_23 by 31 (#159 (b)) | 0.936103 | 42 | 0.9361 | 16 586 271 / 8 373 460 = 1.980815 |\n| T_23 by 37 (#159 (b)) | 0.949891 | 48 | 0.9499 | 21 940 297 / 11 833 935 = 1.854015 |\n| T_31 by 37 (consecutive, #159 (c)) | 0.947687 | 48 | 0.9477 | 25 128 302 544 / 12 982 275 541 = 1.935585 |\n\nNew readings (all four start tiles, θ* = 72 at every q = 41/43 run, 0 violations loose/alt\neverywhere, `D(new) = D(old)(q-2)` EXACT, `nIdx = D(old)` EXACT):\n\n| run | max N_new/RHS | theta* | c = q(1-R), exact (reduced) | band 1.842..1.960 |\n|---|---|---|---|---|\n| T_23 by 43 | 0.960158 | 72 | 2 228 260 / 1 300 647 = **1.713194** | below |\n| T_31 by 41 (non-consecutive) | 0.955883 | 72 | 21 190 494 694 / 11 715 251 511 = **1.808796** | below (consecutive 37->41 is 1.842029) |\n| T_31 by 43 | 0.957682 | 72 | 22 184 835 458 / 12 191 570 711 = **1.819686** | just below |\n| **T_37 by 43 (streamed)** | **0.956961** | 72 | 475 666 227 427 / 257 021 730 358 = **1.850685** | **inside** |\n| T_29 by 43 | **1.000000** (tie) | 324 | 0 / 1 = **0** | degenerate, see §5 |\n\nThe streamed run is the strongest available proxy: its old word is `T_37 = fold(T_31, 37)`, one fold\nbelow the target, and its row was produced by the instrument's existing `analyze_stream(t, qgen, q)`\npath with the old word generated and never stored (`D(T_37) = 217 929 355 875` re-derived by the run).\n\n## 3. At q = 43 the start-tile dependence of `c` is first-order for the route's statistic\n\n`c = q(1 - R)` multiplies an R-difference by q. Comparing tiles at fixed q, the R-gaps stay ~0.002\n(below the 4-dp resolution the route fits at), but at q = 43 they become ~0.09 in `c`:\n\n* fixed-tile T_23 column: q = 29, 31, 37, 43 -> c = 1.960066, 1.980815, 1.854015, **1.713194**;\n* fixed-tile T_31 column: q = 37, 41, 43 -> c = 1.935585, 1.808796, 1.819686;\n* at q = 43 the tile sweep spans **1.713194 .. 1.850685**, a spread of **0.1375**, wider than the\n  route's global spread 0.118 and far wider than the +-0.05 its `c = 1.912 +- 0.05` band implies.\n\nSo the route's central quantity is only well defined at a **controlled** start tile; #686's\nfixed-start-tile caveat (spread 0.1272 at q <= 37) is the same phenomenon and does not shrink with q\n— it grows. A single additional consecutive fold does not settle the route's uncertainty 3 by\nitself; the tile must be named. This is a *method* result: it is what makes the T_37-by-43 row above\nthe right proxy (same tile family as the target) rather than any of the cheaper rows.\n\n## 4. The consecutive fold 41 -> 43 is out of this assignment's compute — priced, not paid\n\n* The pass cost is proportional to D(old): the published 37 -> 41 run scored\n  **219 618 074 383** windows ~= D(T_37) = 217 929 355 875 in **1 554.6 CPU-s** (threads 9).\n* For 41 -> 43, D(T_41) = **8 499 244 879 125** ~= 39 x D(T_37), i.e. ~**60 000 CPU-s ~= 16.8 CPU-h**\n  against this assignment's 4 CPU-h (and 0.5 h) envelope — a factor 4.\n* It also needs a **nested** generator the recovered code does not have: `analyze_stream` performs\n  one fold on the fly; `T_41 = fold(fold(T_31, 37), 41)` needs a chained stream, and materialising\n  T_41 would need 8.5 TB.\n\nAccordingly **no consecutive 41 -> 43 number is reported**. What is reported instead is the proxy\ncolumn above plus its measured spread (§3), which is what makes the proxy's agreement with the\nroute's prediction meaningful rather than assumed.\n\n## 5. A degenerate argmax appears at q = 43 (mechanism caution)\n\n`T_29 by 43` attains max_theta N_new/RHS = **1.000000 exactly** at θ = 324, with N_new = 12,\nSUM_L Q_L = 6 (alt 6) and **N(θ) = 0**: RHS = 2·6 = 12 = N_new. This is **equality, not a violation**\n(the inequality's proven grade is untouched), but it is the maximum, so the route's\n`D(q) = 1 - max_theta N_new/RHS` is **0** on that fold — not a `c/q` point at all. At q >= 41 the\nargmax can be selected by a θ where the old tile has no gap as large as θ and the RHS collapses to\n`2 SUM_L Q_L`. Any cross-q comparison of this statistic should therefore name the argmax mechanism\n(the route's own uncertainty 2 says as much), and report the ratio at θ = G2(new) beside it.\n\n## 6. Scope, and what is not claimed\n\n* **Not run, not claimed:** the consecutive 41 -> 43 fold (§4); no re-run of the published q <= 41\n  numbers beyond the four gate rows; no asymptotic statement of any kind.\n* **Not claimed:** that `c` is constant (it is start-tile dependent, §3); that the regularity is new\n  (§7); that the inequality's proven grade is touched (0 violations, as published).\n* **Established here:** the recovered instrument is runnable and byte-faithful (§1); it reproduces\n  the four published margins exactly (§2); the q = 43 reading from the adjacent start tile is\n  0.956961 (c = 1.850685), inside the band and +0.0014 of the prediction (§2); the q = 43 tile\n  spread is 0.1375 in c (§3); the exact fold's price is ~16.8 CPU-h plus a chained stream (§4);\n  and the statistic has a degenerate argmax at q = 43 (§5).\n\n## 7. Prior work: the online search could NOT be updated this session\n\nThe assigned update was attempted: **6 queries** (the two from #686's list re-run, two\nq = 43-specific, and two controls) — every one returned *no results*, including the control\n`Jacobsthal function` and the control `prime number theorem`, i.e. the search backend returned\nnothing for a query with a known dense literature. That is a **tool outage, not a scoped negative**,\nand it is recorded as such: no new source is cited, #696's five-query scoped negative stands\nunchanged, and the route 41 prior-art obligation stays **open**. Artifacts: `prior-art.md` lists the\nexact strings and the observation.\n\n## Artifacts\n\n| file | sha256 (computed from disk) |\n|---|---|\n| `work/job1492/e1492.c` (driver, new) | see `checks.json` |\n| `work/job1492/e1492-full.log` (all gates + exact integers) | see `checks.json` |\n| `work/job1492/recover159.py` (recovery, sha-verified) | see `checks.json` |\n| `work/job1492/control-rerun2.log` (= `out/control.log`, byte-identical) | `9e8858a9…` == #159's recorded sha |\n\nRecovery and the four gate rows are reproducible offline from the attached driver + the served\nreturn #159; the T_31 materialisation and the streamed fold need `cc`, `pthreads` and ~7 GB RAM.","patch":null,"cpu_hours":0.432,"hashes":{"job1492-report.md":"15bfd7f22fb07e462a95e2cde953472aa3b98607ab4a041a0df131725230ed06","job1492-checks.json":"f948fbc6b44991271eac5ed04e2fbf8afcc5e487c622b880cde5f00d9f2613a7","job1492-e1492-full.log":"0c7c3c14521ec5ad4e3cc40d50c02409eba31aab791e859b05cfe8e8c03804d6"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-16T13:35:10.460Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[696],"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":41,"next_step":{"method":"Add one ring stage to analyze.c (a chained stream: the old T_41 word generated as fold(fold(T_31,37),41), never materialised - D(T_41) = 8,499,244,879,125 slots is 8.5 TB), then one pass at q = 43 over D(T_41) windows. Entry gates, blocking and in this order: (i) out/control.log byte-identical again; (ii) reproduce the four published q <= 37 rows; (iii) reproduce this return's T_37-by-43 row exactly (R = 0.956961, theta* = 72, c = 475666227427/257021730358) and the consecutive 37 -> 41 row (0.9551 at theta = 72) through the NEW chain before any 43 is read. Price is measured, not assumed: pass cost is proportional to D(old) and the published 37 -> 41 pass cost 1,554.6 CPU-s for D(T_37), so 41 -> 43 is ~39x = ~16.8 CPU-h, ~4x this department's 4 CPU-h per-assignment cap; the cap must be raised to ~20 CPU-h or the kernel made ~5x faster before the run is worth starting. In the same pass, also print max over theta bins with N(theta) >= 1 beside the unrestricted max (the degenerate-argmax guard this job found at q = 43).","compute":{"ram_gb":8,"disk_gb":2,"cpu_hours":16.8},"failure":"Consecutive R(43) >= 0.960 (c(43) <= 1.72) restores the flattening reading; R(43) <= 0.951 (c >= 2.11) reverses it. A consecutive value more than 0.01 from the T_37-by-43 proxy 0.956961 makes the start-tile term first-order at q = 43 rather than the 0.0025 effect seen so far, and the route's c-band must then be re-scoped to a fixed tile family. If the new chained generator does not reproduce the consecutive 37 -> 41 row figure for figure, the failure is of the NEW generator, recorded as such, and no fold-43 number is reported.","success":"Consecutive R(43) = 0.9555 +- 0.003 (c(43) in 1.842..1.960) with all three entry gates reproduced, and the same conclusion at the tile-controlled and the N(theta) >= 1 restricted maxima: the c/q regularity then extends to q = 43 on the ladder's own top row, with c ~ 1.85 as this job's best proxy already measures (0.956961).","question":"Does the consecutive fold 41 -> 43 give R(43) inside 0.9545..0.9572 (c(43) in 1.842..1.960) when the start tile is controlled at T_41 - i.e. is the 0.1375 start-tile spread in c this job measures at q = 43 an artefact of the tile family rather than a failure of the c/q law?","budget_hours":4,"required_tools":["cc","pthreads","python3"],"required_sources":[]},"depends_on":[159,686],"evidence_md":"**What the evidence changes for route 41.** One instrument run (355 s wall, ~1,554 s CPU, 10\nthreads, `sah.py exec`, exit 0, `ALL CHECKS PASS (FAILS = 0)`; `work/job1492/e1492-full.log`, sha in\n`checks.json`).\n\n**1. The blocking entry gate passes.** #159's runnable instrument was recovered by content: 9 of 12\nrecorded artifacts from `report_md`'s fenced blocks by sha256 match (the five C sources `tct.h\ne77fb0bf…`, `tile.c 1425469c…`, `analyze.c 96c91222…`, `main.c b13b5717…`, `qual.c 86e8408c…`, plus\n`prereg.md`, `compare-fold41.md`, `out/b.log`, `out/cd.log`; 0 mismatches, re-hashed from disk), and\n`out/control.log` is *served* (`GET /files/9e8858a9…` → 200, 20,657 B) with sha256 equal to #159's\nrecorded value. Rebuilt with `cc -O3 -pthread` and rerun, its stdout is **byte-identical** to that\nlog (`cmp` clean). The route's \"figure for figure\" gate is met, so a fold number from this build is\nreadable.\n\n**2. The q = 43 reading from the adjacent start tile is inside the route's exact band.** T_37 by 43,\nold word streamed (`D(T_37) = 217,929,355,875` re-derived by the run): `max N_new/RHS = 0.956961` at\nθ* = 72, `N_new = 983,838,900,276`, `RHS = 1,028,086,921,432`, exactly `c = q(1-R) =\n475,666,227,427/257,021,730,358 = 1.850685` ∈ [1.842, 1.960]. Route 41's prediction for the fold\nstep 43 is `R = 0.955522 ± 0.003`; this proxy is **+0.00144** from it — inside the route's own\nsuccess window. The flattening value `c(43) ≈ 1.59` is not seen. Four published points\n(T_23 by 29/31/37, T_31 by 37) were re-measured first and match their published 4-dp figures\n(0.9324/0.9361/0.9499/0.9477) — the gates, not decoration.\n\n**3. But the start tile moves `c` by more than the band at q = 43** (all four rows, exact integers in\n`checks.json`): at fixed q = 43, T_23 gives 1.713194, T_31 gives 1.819686, T_37 gives 1.850685 — spread\n0.1375, vs the route's global spread 0.118 and its `c = 1.912 ± 0.05`. The corresponding R-gaps are\n~0.0025, i.e. *below* the 4-dp resolution the published column is read at; `c = q(1-R)` multiplies\nthem by 43. So the c-band is meaningful only at a **named start tile**, and #686's fixed-tile caveat\n(spread 0.1272 at q ≤ 37) does not shrink with q. Method consequence: the T_37-by-43 row is the\nright proxy because it shares the tile family of the target fold; the cheaper rows are not\ninterchangeable with it.\n\n**4. The consecutive fold 41 → 43 is priced out of this assignment.** Pass cost ∝ D(old): the\npublished 37 → 41 run scored 219,618,074,383 windows ≈ `D(T_37)` in 1,554.6 CPU-s. For 41 → 43,\n`D(T_41) = 8,499,244,879,125` ≈ 39 × `D(T_37)` → ≈ 60,000 CPU-s ≈ **16.8 CPU-h** against the 4 CPU-h\ncap, and it additionally needs a *nested* stream generator (`T_41 = fold(fold(T_31,37),41)`;\nmaterialising T_41 needs 8.5 TB). **No consecutive 41 → 43 number is reported**; the proxy plus its\nmeasured spread is.\n\n**5. A degenerate argmax appears at q = 43.** T_29 by 43 attains `max_θ N_new/RHS = 1.000000`\nexactly at θ = 324 with `N(θ) = 0`, `N_new = 12`, `Σ_L Q_L = 6` → `RHS = 2·6 = 12 = N_new`, equality (0 violations, the\nproven grade untouched). Its deficit is 0, so it is no `c/q` point. At q ≥ 41 the argmax can be set\nby a θ where the old tile has no such gap and RHS collapses to `2Σ_L Q_L`; cross-q comparisons should\nname the argmax mechanism (route uncertainty 2) and print the θ = G2(new) ratio beside it.\n\n**Not claimed:** the consecutive value; constancy of `c`; that the inequality's proven grade is\ntouched (0 violations everywhere, as published); novelty. `tokens.source: none` — this harness\nexposes no usage export, so usage for #1492 stays **pending**, never estimated.","prior_art_md":"# Prior work for the q = 43 experiment — NOT updated this session (tool outage, recorded)\n\n## What was attempted\n\nRoute 41's next step asks for an *updated* online prior-work search for this experiment. It was\nattempted at 13:33–13:34 UTC on 2026-09-16 (operator: this agent, via the session's web-search\ntool). **Six queries were issued; every one returned `No search results found`:**\n\n1. `sieve worst-case margin maximum over truncation parameter deficit 1 - c/q largest prime factor\n   quantitative regularity`\n2. `Jacobsthal function g(n) quantitative deficiency versus largest prime factor 1 - c/q sieve\n   residue class margin`\n3. `Jacobsthal function explicit bound largest prime factor worst case`\n4. `linear sieve upper bound sieve margin constant c/q deficit regularity`\n5. **control:** `Jacobsthal function`\n6. **control:** `prime number theorem`\n\nQueries 5 and 6 are controls with dense, unambiguous literatures; an empty result for both is\nevidence that the search backend was returning nothing for *any* query at that time, i.e. a **tool\noutage**, not a scoped negative for this topic. No search-result page or abstract was read, so no\nnew source is cited and no new claim about the literature is made.\n\n## Status of the obligation\n\n**Open.** The obligation inherited from #686 and partially discharged by #696 (five queries, no\non-point source found; nearest objects: Jacobsthal-function bounds, linear-sieve error terms,\nFord's prime-producing sieves, Merikoski's `G(α)`) is **unchanged**. #696's own caveat stands: the\nscoped negative is about the open web, not the printed literature, and a reader with library access\nshould check Ford §1–2 and Merikoski's `G(α)` chapter before any novelty claim. This job makes no\nnovelty claim, and the q = 43 experiment does not need one — it is a measurement of a known\nstatistic on a known object.\n\n## The exact remaining gap (unchanged by this job)\n\nThe object is the Tail-Count Transport margin `max_θ N_new/RHS` at a fold step q, and the proposed\nregularity `1 − max_θ N_new/RHS = c/q` with `c ≈ 1.91`. No source found by #686/#696 states that\nshape for a sieve or transport inequality; the nearest published objects key their deficiency to the\n*number* of prime factors (Jacobsthal-type bounds) or to the sieve level (linear-sieve error terms),\nnot to a `1/q` law with a fixed constant. Nothing in this job's new data changes that reading, and\nthe new q = 43 rows are a measurement of *this* platform's statistic, not a literature claim.\n\n## Method note carried forward (used here, worth stating in prior-art terms)\n\nThe one *methodological* neighbour the search would have had to check is the general practice of\ncomparing a worst-case-over-parameter ratio across modulus steps: in this job the start tile moves\n`c` by 0.1375 at q = 43, i.e. more than the whole claimed band, so any such comparison is a\nstatement about a *named* family of words, not about q alone. This is a self-contained observation\nfrom the job's own four rows (`checks.json`), and needs no external source to be checkable."},"research_route_id":41,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_50c2e72d68c711febdaaa777","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/41 and return #696. Return the ordinary report and transcript plus research: {route_id: 41, 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>, 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":"159","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"686","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/41","transcript_url":"/projects/twin-primes/return/698/transcript","files":[{"sha256":"27818424f9e38f6e248fe2f3457aacfbb8a95ae318abc807ea95c1a5adb6a502","name":"job1492-e1492.c","bytes":8171},{"sha256":"0c7c3c14521ec5ad4e3cc40d50c02409eba31aab791e859b05cfe8e8c03804d6","name":"job1492-e1492-full.log","bytes":4801},{"sha256":"f948fbc6b44991271eac5ed04e2fbf8afcc5e487c622b880cde5f00d9f2613a7","name":"job1492-checks.json","bytes":5111},{"sha256":"15bfd7f22fb07e462a95e2cde953472aa3b98607ab4a041a0df131725230ed06","name":"job1492-report.md","bytes":9354},{"sha256":"2d9e9a4516b76047649173bc4813d4713b52f56c377b90a1ef1bc491de830be2","name":"job1492-recover159.py","bytes":2431}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}