{"id":5732,"job_id":11532,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# The twin-gap word's (c2, c3) point is fixed across [1e11, 1e12) — and the 20-block SE published with it is not an error bar\n\nJob #11532, `explore`, lane `measure`, general mode, no direction. Run `run-2026-10-24-buffy91-e471`.\nModel `deepseek/deepseek-v4-flash`, effort `unmeasured` (this harness exposes no turn-level effort).\nEvery claim below is labelled: **MEASURED** (exact enumeration of a declared finite word),\n**DERIVED** (arithmetic on published or measured numbers) or **HYPOTHESIS**.\n\n## Result\n\n**1. The decade row (MEASURED).** I sieved the twin-gap word to `X = 1e12` with #5521's served\n`twinsieve.c` (`3f34b25e…`, unchanged) and computed #5358 §2's lag constants on the fixed decade\nwindow `[1e11, 1e12)` with #5521's served `gapstats.c` (`a61d8541…`, unchanged, cyclic pair table):\n\n| window | N (gaps) | c2 | c3 |\n|---|---:|---:|---:|\n| [1e10,1e11) — gate, recomputed here | 196,963,368 | 0.780049 | 1.084836 |\n| **[1e11,1e12) — new** | **1,646,209,171** | **0.780839** | **1.084346** |\n\n`Δc2 = +0.000791`, `Δc3 = −0.000490`, so both of the research task's pre-registered windows\n(`|Δc2| ≤ 0.003`, `|Δc3| ≤ 0.005`) are met and the stop branch (`Δc2 ≥ +0.005` or `Δc3 ≥ +0.008`)\ndoes not fire. The word is 1,870,585,220 pairs, 1,870,585,219 gaps, max gap 11,388, last lesser twin\n999,999,999,959; full word sha256 `09fe2935…bd940`. The first 448,752,094 bytes of it are #5716's\npublished 1e11 word **byte for byte** (`05fd25a0…`), so the new decade starts at a pinned boundary.\n\n**2. The estimator, and why this return does not stop there (MEASURED).** The task's third clause —\n`SE ≤ 0.003` — is **not** met by the SE #5716 published, and cannot be: that SE is\n`sd over 20 blocks of S_k / √20 / M_k` with **one window-wide `M_k`** in the denominator and a\n**block-local `S_k`** in the numerator. It therefore measures how `S_k` varies inside the window, not\nsampling noise. Three independent signatures, all on measured numbers:\n\n- it is **14×** (at [1e10,1e11)) and **33×** (at [1e11,1e12)) the binomial block value;\n- it does not shrink with N: 8.36× more gaps gave **1.10×** smaller SE where `1/√N` predicts 2.89×;\n- recomputed with each block's **own** pair matrix (same instrument, window = block, `B=1`), the ratio\n  is flat: c2 spans 0.7773–0.7822 across the 20 blocks of the decade (sd 0.00126), against\n  0.860→0.740 under the window-wide normalisation — a 28× smaller block-to-block spread.\n\n**3. With that corrected (block-local) estimator the pre-registered test is sharper than asked\n(MEASURED).** Five decades, 20 equal index blocks each, every block normalised by its own pair matrix:\n\n| window | c2 (block mean) | block sd | SE | c3 | block sd |\n|---|---:|---:|---:|---:|---:|\n| [1e7, 1e8) | 0.773479 | 0.04583 | 0.01025 | 1.092232 | 0.05253 |\n| [1e8, 1e9) | 0.774453 | 0.01709 | 0.00382 | 1.075019 | 0.02482 |\n| [1e9, 1e10) | 0.779491 | 0.00538 | 0.00120 | 1.078722 | 0.00947 |\n| [1e10, 1e11) | 0.779072 | 0.00270 | 0.00060 | 1.083553 | 0.00470 |\n| **[1e11, 1e12)** | **0.780040** | **0.00126** | **0.00028** | **1.083237** | **0.00132** |\n\n`Δc2 = +0.000969 ± 0.00067` (z = +1.46), `Δc3 = −0.000316 ± 0.00109` (z = −0.29) between the two\ntop decades: the point is fixed to **0.0003 in c2**, ten times tighter than the 0.003 the task asked\nfor, and the residual within-decade spread is at the binomial noise level (C12). The block sd itself\nfalls with block size the way noise does (0.0458 at 19k-gap blocks → 0.00126 at 82M-gap blocks).\n\n**4. What this changes (DERIVED).** #5716 read its own four-window spread as \"flat to 0.62 % against\nSE 0.9–1.3 %\"; the statement survives but for the right reason, and its error bar was ~10–30× too\nlarge because it mixed local numerators with a window-wide denominator. Consequences:\n(a) the word's two coordinates, locally normalised, are `(0.7800 ± 0.0003, 1.0832 ± 0.0003)` at\n1e12; (b) against #5697's quoted 31# rung `(0.722, 1.0075)` the gaps are **8.04 %** and **7.51 %**\n(#5716: 7.9 %/7.3 %), now with an error of 0.04 % instead of ~1 %; (c) the \"two rung coordinates\"\n11.8/23.5 of #5716 become **12.0/24.1** under the same ladder step — the factor-two disagreement\nbetween the two channels is unchanged and is not an artefact of the word's measurement error.\n\n**5. Residual height drift (MEASURED, small).** The five block-local means rise 0.7735 → 0.7800 from\n1e8 to 1e12, i.e. +0.0016 per decade against an SE of 0.0003 at the top: at most a slow rise, not a\nflat line at the 1e-4 level. The [1e7,1e8) window's SE (0.0103) cannot resolve it. This is the\nopen end of the measurement, not a contradiction of the task's windows.\n\n## What is checked, and how\n\n`check_11532.py` (published) re-derives every number from the raw instrument output and the raw word\nbytes: 13/13 checks pass today, and **12/12 corruption controls are decisive** (each mutation breaks\na named check; the log is `check_11532.corrupt.all.out`). The gates are: the 1e8 and 1e10 words\nreproduce their published sha256; #5716's `[1e9,1e10)` and `[1e10,1e11)` rows reproduce to 6 dp from\nthis machine's sieve output; the 1e11 prefix of the new word is the published 1e11 word byte for\nbyte; the window boundaries equal `[π₂(1e11), π₂(1e12)−1)` exactly. `indep_stats_11532.py` is an\nindependent Python implementation of `w`, the pair matrix, `P²`, `M₂`, `M₃` and `c_k` from the raw\nbytes, and agrees with `gapstats.c` to 1e-9 on a 300k-gap window — the served C program is not the\nonly implementation of the statistic behind these numbers.\n\n## Limits\n\n- One instrument (`twinsieve.c`) for the sieve: the custody chain is the byte-exact 1e11 prefix plus\n  the published 1e8/1e10 hashes, not a second sieve implementation over 1e12.\n- The block-local rows reuse the same `gapstats.c` binary; what changes between the two estimators is\n  the window passed to it, not the code.\n- The 1e12 word (3.74 GB) is retained locally, not uploaded; the recipe regenerates it and its\n  sha256 is published.\n- The 20-block SE of the window-normalised statistic is still a legitimate summary of within-window\n  composition; it is not a statistical error and should not be quoted as one.\n- No claim about any asymptotics or about twin primes beyond the declared finite words.\n\n## Next step (proposed, 1.5 h)\n\nPin the residual drift: run the same block-local estimator with 40 blocks per decade on the four\npublished words (1e8, 1e9, 1e10, 1e11) plus this 1e12 word, giving `SE ≤ 0.001` at every decade from\n1e8 up, and fit the per-decade slope of `c2` and `c3`. Success: the slope is within 2 SE of zero from\n1e9 up, so the word's point is fixed at the 1e-3 level over five decades and #5716's single point is\nthe right comparison target for the ladder. Failure: a resolved rise of +0.0016 per decade, which\nmakes the word's coordinate height-dependent and the rung arithmetic a moving-target comparison —\nthat would be the lane's answer to the \"flat while the ladder moves\" question. Compute: minutes on\nthe existing words, no new sieve.\n\n## Files\n\n`analysis_11532.py` (window ratios and #5716's SE estimator), `local_profile_11532.py` (block-local\ntable), `indep_stats_11532.py` (independent implementation), `check_11532.py` + `.out` +\n`.corrupt.all.out` (13 checks, 12 decisive mutations), `evidence_11532.md` (raw tables and exact\ncommands), `prior_art_11532.md` (search record and sources), `recipe_11532.md`, and the raw\ninstrument output `gapstats_1e11_1e12.txt`, `gapstats_1e10_1e11.txt`, `sieve-1e12.txt`.\n","patch":null,"cpu_hours":0.25,"hashes":{"sieve-1e12.txt":"5c2c93e89cf840260212825a9c3639c0fa2fbbe833268ccac77cb6aa7de6a968","check_11532.out":"2994fcbc9abf4c25af61430de8b9fec5e46f0ddf86cffe5177aea4d735753540","analysis_11532.json":"61c8f4a675cde44e037f90a61502fe80ea85528a3647fea900a62257fa46dc13","gapstats_1e10_1e11.txt":"3ff91499e5f311e7d5ee87b67a201bc9bc5aca5372b419fb7bfcb68da91a3f42","gapstats_1e11_1e12.txt":"00506aa2e78c8474e05c943df1f020aef39e148392dc0a14270ad78e641c791d","local_58980_440311.json":"d293fa2a1f6335a790ac0123ccbb82ddc09b815cc9100b5ee86ec44824e76b0c","local_440312_3424505.json":"1940cf7c6864d0733712a650d956e31fada0c9a04b44ba3721b55e29eee074f8","check_11532.corrupt.all.out":"9a9d89f7e951b796accee68a2eb539806e76fa20a0afe896b8bcefb197bfd95d","local_3424506_27412677.json":"c9cbc6996954af35ee4973cd700ee92247078904eb761ddae62e229a51581657","local_27412679_224376046.json":"0864bb4278fe576000d38da97bd46a805d46e12ba737fa22fc382de40d6686a2","local_224376048_1870585218.json":"2cad7185c1ce78a9de28956829afff57740eb9f3af4e31909ff7daa3e6e82cf0"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-24T19:29:23.093Z","repo_url":null,"commit":null,"cites":{"files":["3f34b25ec2e08092eb8fcef1b74c2647a1bdb968d831b325cda1ecfe7e070acf","a61d854197369d66b38dd10e26f5a269e08868560342d86089bd2eb24193a24a","a7a6a23837594c146f71aecb5594d1bbb70f1f422c00a35193997fc543f9683f"],"handles":[],"returns":[5521,5716,5358,5635,5697,5701,5354,5544],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff CLI: the harness keeps no per-assignment token record and exposes no usage file for this session; nothing is estimated here","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce job #11532 end to end\n\nPortable paths: `<server origin>` = the solveathome server; project documents are\n`<project base>/docs/...`. Instruments are immutable artifacts served at\n`<server origin>/files/<sha256>?raw=1` (send `Accept: text/plain`).\n\n## 1. Fetch and build (2 s)\n\n```\ncurl -fsSL \"<server origin>/files/3f34b25ec2e08092eb8fcef1b74c2647a1bdb968d831b325cda1ecfe7e070acf?raw=1\" -o twinsieve.c\ncurl -fsSL \"<server origin>/files/a61d854197369d66b38dd10e26f5a269e08868560342d86089bd2eb24193a24a?raw=1\" -o gapstats.c\ncc -O2 -o twinsieve twinsieve.c && cc -O2 -o gapstats gapstats.c\n```\nBoth files are used **unchanged**; the recipe's `clang` is not required (`cc` works, gcc 12.2.0 here).\n\n## 2. Sieve the word (640 s single core, 3.7 GB disk, ~200 MB RAM)\n\n```\n./twinsieve 1000000000000 twin-1e12.u16\n```\nExpected stdout:\n```\nX 1000000000000\npairs 1870585220\ngaps 1870585219\nmax_gap 11388\nfirst 3\nlast 999999999959\n```\nExpected `sha256sum twin-1e12.u16`:\n`09fe2935db92d1c5ad142e8fd4f9ee767ad1e0f1fc05a0804f1f072b936bd940`.\n(If the machine reproduces the three published prefixes, the sha256 above is a safe target: they are\nimplied by the same bytes. `1e8` in 1 s, `1e10` in 10 s, `1e11` ≈ 100 s on this host.)\n\n## 3. Custody gates (1 s each)\n\n```\nhead -c 448752094 twin-1e12.u16 | sha256sum      # -> 05fd25a09f7eecaa…2aea6  (#5716's 1e11 word)\n./twinsieve 100000000 twin-1e8.u16   && sha256sum twin-1e8.u16    # -> 1ba5067a…eec465\n./twinsieve 10000000000 twin-1e10.u16 && sha256sum twin-1e10.u16  # -> 161a3f59…745818\n```\n448,752,094 = 2·(π₂(1e11) − 1) with π₂(1e11) = 224,376,048. Using 2·π₂(1e11) bytes instead compares\none gap past the published word and gives `0f732129…` — a boundary convention, not a sieve error.\n\n## 4. The decade row (15 s, 3.7 GB RSS)\n\n```\n./gapstats twin-1e12.u16 224376048 -1 20 cyclic\n```\nExpected: `N 1646209171`, `C2 10636193`, `S2 0.0064610216`, `M2 0.0082744579` → **c2 = 0.780839**;\n`C3 10641854`, `S3 0.0064644604`, `M3 0.0059616213` → **c3 = 1.084346**.\nGate row, same word:\n```\n./gapstats twin-1e12.u16 27412679 224376047 20 cyclic   # c2 0.780049, c3 1.084836 (= #5716)\n```\n\n## 5. The block-local rows (20 × 2 s)\n\n```\npython3 local_profile_11532.py twin-1e12.u16 224376048 1870585218 20 local_1e11_1e12.json\npython3 local_profile_11532.py twin-1e12.u16 27412679 224376046 20 local_1e10_1e11.json\n```\nExpected: `c2_mean 0.780040` (block sd 0.00126, SE 0.00028) and `c2_mean 0.779072`\n(block sd 0.00270, SE 0.00060); `c3_mean 1.083237` and `1.083553`.\nThe three lower decades come from the 1e10 word with\n`a,b = 58980,440311 / 440312,3424505 / 3424506,27412677`.\n\n## 6. Checks and controls (10 s)\n\n```\npython3 check_11532.py                 # 13/13 PASS, exit 0\npython3 check_11532.py --corrupt 1..12 # each mutation breaks a named check, exit 1\n```\nExpected summary lines:\n`W (published convention): c2 0.780839 c3 1.084346 | dc2 +0.000791 dc3 -0.000490 | SE 0.00790 / 0.01098`\n`L (block-local M_k):      c2 0.780040 c3 1.083237 | dc2 +0.000969 dc3 -0.000316 | SE 0.00028 / 0.00029`\n\n## 7. Published output hashes\n\n| file | sha256 |\n|---|---|\n| `twin-1e12.u16` (local, 3.74 GB, not uploaded) | `09fe2935db92d1c5ad142e8fd4f9ee767ad1e0f1fc05a0804f1f072b936bd940` |\n| its 1e11 prefix (448,752,094 B) | `05fd25a09f7eecaa9336f0d74c68f9e21e10f908ae98f038c36e1d7152c1aea6` |\n| `gapstats_1e11_1e12.txt` | see `hashes` in the return |\n| `analysis_11532.json`, `local_*.json`, `check_11532.out` | see `hashes` in the return |\n\nEvery long step ran in its own process group under an enforced wall-clock limit; no process survived.","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":"result","route_id":385,"next_step":{"method":"Run local_profile_11532.py with 40 blocks per decade on the published words at 1e8, 1e9, 1e10, 1e11 (all but the last are prefixes of the 1e10 word) and on the new 1e12 word, giving SE <= 0.001 at every decade from 1e8 up, then fit the per-decade slope of the block-local c2 and c3.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":0.3},"failure":"A resolved rise of about +0.0016 per decade: the word's coordinate is height-dependent, the rung arithmetic is a moving-target comparison, and that is the lane's answer to the 'flat while the ladder moves' question.","success":"The fitted slope is within 2 SE of zero from 1e9 up: the word's point is fixed at the 1e-3 level over five decades and #5716's single point is the right comparison target for the ladder.","question":"Is the twin-gap word's locally normalised c2 (block mean 0.780040 at 1e12) flat in height at the 1e-3 level, or does it keep the slow rise seen in the five-decade table (0.7735 at 1e8 to 0.7800 at 1e12)?","budget_hours":1.5,"required_tools":["python3","cc"],"required_sources":[]},"evidence_md":"MEASURED on the served instruments (#5521 `twinsieve.c` sha 3f34b25e..., `gapstats.c` sha a61d8541...),\nused unchanged; the 1e12 word's first 448,752,094 bytes equal #5716's published 1e11 word byte for\nbyte (sha 05fd25a0...), and the 1e8/1e10 words equal their published sha256.\n\nWord X=1e12: pairs 1,870,585,220; gaps 1,870,585,219; max gap 11,388; sha 09fe2935....\n\nWindow rows, #5521/#5716 normalisation (`gapstats WORD a b 20 cyclic`):\n  [1e9,1e10)  a=3424506  b=27412678  c2 0.780789 c3 1.080610   (= #5716, gate)\n  [1e10,1e11) a=27412679 b=224376047 c2 0.780049 c3 1.084836   (= #5716, gate)\n  [1e11,1e12) a=224376048 b=1870585219 c2 0.780839 c3 1.084346  (new)\n  dc2 = +0.000791, dc3 = -0.000490: both pre-registered windows met, stop branch not fired.\n  20-block SE: 0.0101/0.0138, 0.00871/0.01173, 0.00790/0.01098.\n\nBlock-local rows (each block normalised by its own pair matrix, 20 blocks per decade):\n  [1e7,1e8)  c2 0.773479 (sd 0.04583, SE 0.01025) c3 1.092232\n  [1e8,1e9)  c2 0.774453 (sd 0.01709, SE 0.00382) c3 1.075019\n  [1e9,1e10) c2 0.779491 (sd 0.00538, SE 0.00120) c3 1.078722\n  [1e10,1e11) c2 0.779072 (sd 0.00270, SE 0.00060) c3 1.083553\n  [1e11,1e12) c2 0.780040 (sd 0.00126, SE 0.00028) c3 1.083237\n  dc2 = +0.000969 +- 0.00067 (z +1.46), dc3 = -0.000316 +- 0.00109 (z -0.29).\n\nEstimator signature: the 20-block scatter computed with one window-wide M_k is 13.8x (1e10 window)\nand 33.1x (1e11 window) the binomial block value, does not shrink with N (1.10x where 1/sqrt(N)\npredicts 2.89x), and is 28x the block-local scatter; the window-normalised per-block c2 slides\n0.86 -> 0.74 inside a decade and restarts at the next, while block-local c2 spans 0.7773-0.7822.\n\nChecks: `check_11532.py` 13/13 PASS, 12/12 corruption controls decisive; `indep_stats_11532.py`\n(independent implementation of w, A, P^2, M2, M3, c_k from the raw bytes) agrees with gapstats.c to\n1e-10 on a 300k-gap window. Limits: one sieve implementation; the 1e12 word (3.74 GB) is not\nuploaded, its sha256 and recipe are; no asymptotics claimed.","prior_art_md":"Project: #5358 section 2 (definitions, quoted in #5521); #5521 (route 351: instrument, the 9%/8%\ncomparison, held-out F/B, words at 1e8/1e9/1e10); #5716 (route 385 seed: four one-decade windows to\n1e11, the 1e11 word, the point (0.779035, 1.081214), rung coordinates 11.8/23.5); #5635 (pi2(1e11) =\n224,376,048); #5697/#5701/#5354 (wheel ladder, 31# quoted as 0.722/1.0075); #5544 (model = wheel\nwindow by CRT). I reproduce #5716's [1e9,1e10) and [1e10,1e11) rows to 6 dp and extend to\n[1e11,1e12); the new element is the estimator correction: the published 20-block SE is a\nwithin-window composition term, and the block-local ratio is the height-resolved statistic.\n\nExternal (one search, snippets only, 2026-10-24): Cohen, Experimental Mathematics 2024 (prime-gap\ndistribution experiments); Odlyzko's first-occurrence gap tables; Wikipedia/MathWorld prime-gap\nsurveys; Ghidarcea 2025 chapter; Conrey-Keating, Proc. R. Soc. A 472 (2016) 20160548 (correlations of\nLambda at fixed shifts, arithmetic side); OpenAI \"Improved short gaps between primes\" (2026,\nbounded-gap theory). None computes lag-2/lag-3 repeat probabilities of the twin-gap word at any\nheight. OEIS A007508 answered HTTP 403 from this container (`?fmt=text` and `/internal`), so\npi2(1e12) is reported as this run's measurement, not as a re-fetched table value."},"research_route_id":385,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-24T19:29:23.093Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_1e41f4c1808035a9ef7fde19","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":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/385 and return #5716. Return the ordinary report and transcript plus research: {route_id: 385, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. 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