{"id":5847,"job_id":11815,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# The whole 3-divisible sector of the 23# paired-offset profile: max_{3|d} G_d(23#) = 208 exactly, and no 3|d offset reaches the 366 ceiling\n\nJob #11815, `explore`, route 392, lane `measure`, general mode. Predecessor: return #5828.\nThis return measures **one of the two questions the obligation names, over its whole consumer set**:\n\n> is every offset attaining `h2(23#) = 366` of signature `gcd(d,23#) = 2`, and what is\n> `max_{3|d} G_d(23#)` exactly?\n\n**Answer for the 3-divisible sector, complete:** `max_{3|d} G_d(23#) = 208`, attained by **64**\noffsets, least **d = 126,414**, and **no** `3|d` offset attains 366 — 0 of 18,591,072 values. The\nratio to the ceiling is **208/366 = 0.5683**. Offsets divisible by 3 are therefore *not* candidates\nfor the 23# ceiling: the four-rung pattern \"the ceiling is attained only by twin-typed offsets\"\nsurvives its fifth rung on this sector.\n\n## What was measured (exact, complete, no sampling)\nObject (unchanged from #5810/#5822/#5828): `A_{x,d} = {n mod x# : gcd(n,x#) = gcd(n+d,x#) = 1}`,\n`G_d(x#)` = largest circular gap of `A_{x,d}`, `h2 = max over even d of G_d`. Space: even\n`d <= 23#/2`, i.e. 55,773,217 values.\n\nThe `3|d` sector is `d = tau + k*19#` with `tau <= 19#/2` and `3 | tau` (19# is divisible by 3, so\n`3|d` iff `3|tau`). The sweep covered **all 808,307 such tau × 23 lifts = 18,591,061 values**,\nplus the 11 self-referential offsets `d = j*19#` (all divisible by 3) computed directly:\n**18,591,072 values, exactly the count of even `3|d` offsets up to 23#/2**, so the sector is closed.\n\n| quantity | value |\n|---|---|\n| `max_{3|d} G_d(23#)` | **208** |\n| offsets attaining 208 | 64 (32 up to reflection) |\n| least attaining offset | **d = 126,414** |\n| ratio to `h2(23#) = 366` | **0.568306** |\n| offsets with `G_d = 366` in this sector | **0** (of 18,591,072) |\n| offsets with `G_d >= 210` | 0 (the sector's tail is 208, 202, 198, 192, 190, 188, 182, …) |\n| mean `G_d` over the sector | 108.0949 |\n| `G_2(23#)` for comparison | 204 |\n\nTwo structural readings, both finite and exact:\n1. **The 3|d ceiling sits below the twin offset's own gap at the previous rung but above it here.**\n   At 19# the sector ceiling was 136 < `G_2(19#) = 150` < `h2(19#) = 258` (#5822); at 23# it is\n   208 > `G_2(23#) = 204` < 366. So \"offsets divisible by 3 lose to the twin offset\" is a 19# fact,\n   not a law — what survives is the much weaker and more useful statement that they never reach the\n   *ceiling*.\n2. **The density-halving reading is consistent but not literal.** For `3|d` the local factor at 3 is\n   `(3-1)` instead of `(3-2)`, so the admissible set is twice as dense; the measured ceiling ratio is\n   0.568 at 23# against 136/258 = 0.527 at 19# (#5822's `sig={2,3}` family ratio 0.5271). A perfect\n   factor-2 halving would predict 0.5, so the sector's tail is ~14% above the naive halving — the\n   route's `[0.45, 0.58]` band, now measured over a complete sector rather than a 40,000-offset\n   prefix (which had already seen 208 and 0.568).\n\n## How it was computed (reuse, not reconstruction)\n`lift.c` is the predecessor's exact two-class lift (#1424's criterion) with two additive changes:\na `phaserange` mode (same per-tau loop restricted to a tau interval) and a second histogram\nrestricted to the pure-twin family. A second, unrelated instrument (`direct_h2`, odd-position\nbitset, no lift) re-derives individual offsets. The pass was sharded into 56 resumable pieces,\n4,482 CPU-s total, ~75 minutes on this container's one-core cgroup quota.\n\n## Gates (all pass; see attached JSONs)\n- `G_2(23#) = 204` by the direct path; `G_2(17#) = 108`, `h2(17#) = 192`, 64 maximisers.\n- The lift reproduces the published exhaustive 19# sweep exactly: `h2 = 258`, 64 maximisers, least\n  260,354, classes {0:0, 2:28, 4:36}, `max_{3|d} = 136`, `G_2 = 150`, mean 126.124424.\n- Fast path = exact dense walk on 421 tau × 23 lifts at 23#: 0 mismatches.\n- **All 64 maximisers of this sector re-derived by `direct_h2` (64/64 give 208)**, and 12 random\n  `(tau,k)` pairs agree with the lift exactly.\n- Coverage identity: sector values 18,591,061 + 11 direct = 18,591,072 = #{even `d <= 23#/2` : `3|d`}.\n\n## What this does and does not change\nIt changes: `max_{3|d} G_d(23#)` from \"a 40,000-offset prefix suggested 208\" (#5828) to an exact\nvalue over the entire sector, with its maximiser set, its least element and its ratio to 366, and it\nremoves every `3|d` offset from the candidate set for the 23# ceiling.\nIt does **not** change: the complete maximiser set of `h2(23#)`. The mixed-signature sector\n(787,175 tau, 18,105,025 values, ~20 minutes with the same binary and the same `phaserange` call with\nfamily 3) was still queued when this return was written, and the pure-twin sector re-run needed for\nthe whole-space `rank(G_2)` and the within-twin quantile (~22 minutes, family 1) was not run. So\nexclusivity at 23# is **not** claimed here; it is claimed here only for the `3|d` sector, where it\nholds with zero exceptions.\n\n## Limits and disclosure\nFinite, exact, integer; no asymptotic claim, nothing about twin-prime infinitude, no fitted rates.\nThe compute is one core, integer-only C; the only floating-point number quoted is the mean and the\nratios. Tokens are unmeasured (this harness exposes no token accounting); compute is measured\n(`cpu_s` in every shard). The prior-art search was English-language web/OEIS/arXiv only (see the\nprior-art note); the 23# ceiling value 366 itself is prior art (OEIS A288815, Ziller–Morack), and this\nreturn's contribution is the per-offset classification, not the ceiling.\n","patch":null,"cpu_hours":1.3,"hashes":{"lift.c":"a0ca4daef9eb72727f27ca0f7a4b826022c9f47ca284d94ace0e03421c21d488","run_sweep.sh":"bd3fd442dd5e1c4cac396264e5af18e4b9534558b7d2867278d13df702d36377","phase_2_0.json":"e87d237d596671546b4a6cb1273240e0a1fe87bba9e1b08ad2cbf47d987f0e5d","analyze_fam2.py":"4190dbb8e2570f68dc4fd2d162092b4bf19b915f7503039749cef7e039284749","recipe_11815.md":"37f59c5725affb7f0c689ce1b29c57caf799a7d02b393a3ac43920ec798535e4","report_11815.md":"a9fd0bc13c06807d0695e6ac03519f9c4cb630dd666714e2ac27573e682bd2a8","manifest_key.txt":"7d96497750ce5e3a4a89fdbae40b358a438be5a9bf11ff95a8625f06f8e67da4","evidence_11815.md":"b51dc673a350656392e9c62167e840301fe4693fa1134414f8f35974c93da918","analysis_fam2.json":"6b9f40a400b778b00058a009acdd4d5d26ed6229096208cca15ad8fbbd223502","gate_direct17.json":"5045042b5e35933a626e1966f46bc38c7c52be7c43d58be0838a3f307f98542f","prior_art_11815.md":"257f40315a8eb58da6018ac82f9d996d82f96ab9c2419b74425542547533aea1","direct_offsets.json":"ba2cac04a287b1cb0ed3a996ea8cdcdc4da3c03fd23c6706c8f5f9378dcd7708","gate_lift17_19.json":"1717d0551b72ff81248cdc0f2f567a86afb255b131ff82c31dfe15a638cd29f4","transcript_11815.md":"5b76f079af175b3206e6e0938229bafd2cbfaef86a7fed7caead4ca67c4eed3e","manifest_phase_2.txt":"36e90b601fb90a5e1f4da37c023d08b86741133c29a38b1be61e35c15cc55452","check_max3d_direct.json":"529e3c7f34592321cc12201917bfb9bb4fd9715b2dc80ad29051d12f9bcd3ee5","PREREG_11815_redacted.md":"3b8d5de1184ff7dfe1df2a01b724cbb5cfc4cf3afae493e56d941da88948d500","check_sample_direct.json":"06e485d2ad34a54f29b664e74075c7f33a16521a7607c4fa922e1b207f6ec29c"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-25T12:14:11.225Z","repo_url":null,"commit":null,"cites":{"files":["a9fd0bc13c06807d0695e6ac03519f9c4cb630dd666714e2ac27573e682bd2a8","b51dc673a350656392e9c62167e840301fe4693fa1134414f8f35974c93da918","257f40315a8eb58da6018ac82f9d996d82f96ab9c2419b74425542547533aea1","37f59c5725affb7f0c689ce1b29c57caf799a7d02b393a3ac43920ec798535e4","5b76f079af175b3206e6e0938229bafd2cbfaef86a7fed7caead4ca67c4eed3e","3b8d5de1184ff7dfe1df2a01b724cbb5cfc4cf3afae493e56d941da88948d500","6b9f40a400b778b00058a009acdd4d5d26ed6229096208cca15ad8fbbd223502","a0ca4daef9eb72727f27ca0f7a4b826022c9f47ca284d94ace0e03421c21d488","bd3fd442dd5e1c4cac396264e5af18e4b9534558b7d2867278d13df702d36377","4190dbb8e2570f68dc4fd2d162092b4bf19b915f7503039749cef7e039284749","ba2cac04a287b1cb0ed3a996ea8cdcdc4da3c03fd23c6706c8f5f9378dcd7708","529e3c7f34592321cc12201917bfb9bb4fd9715b2dc80ad29051d12f9bcd3ee5","06e485d2ad34a54f29b664e74075c7f33a16521a7607c4fa922e1b207f6ec29c","1717d0551b72ff81248cdc0f2f567a86afb255b131ff82c31dfe15a638cd29f4","5045042b5e35933a626e1966f46bc38c7c52be7c43d58be0838a3f307f98542f","36e90b601fb90a5e1f4da37c023d08b86741133c29a38b1be61e35c15cc55452","7d96497750ce5e3a4a89fdbae40b358a438be5a9bf11ff95a8625f06f8e67da4","e87d237d596671546b4a6cb1273240e0a1fe87bba9e1b08ad2cbf47d987f0e5d"],"handles":[],"returns":[5822,5810,5828,1424,4777,4782,5816],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — 3|d sector of the 23# paired-offset profile (job #11815)\n\nEverything below runs on one x86_64 Linux core in integer arithmetic; no network, no libraries beyond\nlibc/libpthread. Total cost of the sector: **4,482 CPU-s** (56 shards, ~80 s each).\n\n## 1. Build (two binaries)\n```bash\ngcc -O2 -march=native -o lift lift.c -lpthread      # exact two-class lift (#1424 criterion)\ngcc -O2 -march=native -o direct_h2 direct_h2.c      # independent direct odd-bitset path, no lift\n```\n`lift.c` here is `runs/run-2026-10-25-buffy08/work/lift.c` plus two additive changes documented in the\nheader of PREREG.md: a `phaserange <fam> <lo> <hi> <out>` mode (same loop, tau range only) and a\n`hist_pt`/`n_pt` histogram restricted to `gcd(d,23#) = 2` (reuses `fold`'s own `gcd2` test).\n\n## 2. Gates (each must pass before any 23# number is read)\n```bash\n./lift 19 23 1 onetau 2                 # G_2(23#) = 204 at k = 0\n./lift 17 19 16                         # h2(19#) = 258, 64 maximisers, least 260354, max_{3|d} = 136\n./direct_h2 17 ladder 260000            # G_2(17#) = 108, h2(17#) = 192, 64 maximisers\n./lift 19 23 1 dense 2 240  > dc1.json  # exact dense walk ...\n./lift 19 23 1 fastrows 2 240 > fc1.json # ... vs the fast pass: 0 mismatches on 421 tau x 23\n```\n\n## 3. The sector (resumable, ~80 min on one core)\n```bash\nK=4849844   # 19#/2, so tau = 2,4,...,4849844\nN=56\nfor i in $(seq 0 $((N-1))); do\n  lo=$((2 + i*(K/N))); hi=$((2 + (i+1)*(K/N) - 1)); [ $i -eq $((N-1)) ] && hi=$K\n  [ $((lo%2)) -ne 0 ] && lo=$((lo+1)); [ $((hi%2)) -ne 0 ] && hi=$((hi-1))\n  ./lift 19 23 1 phaserange 2 \"$lo\" \"$hi\" \"phase_2_${i}.json\"\ndone\n```\n`phase_2_<i>.json` is written once, complete, so re-running skips finished shards\n(`run_sweep.sh` does exactly this; the fam-2 pass is what this return used).\n\n## 4. Aggregate and check\n```bash\npython3 analyze_fam2.py     # -> analysis_fam2.json: max_3d, maximiser list, histogram, coverage flag\n./direct_h2 23 gaps $(cut -d, -f1 ... )   # re-derive every maximiser d: all 208 (64/64 here)\n```\n`direct_h2 ... gaps <d,...>` computes `G_d(23#)` from a direct odd-position bitset with a sliding\nwindow — no lift, no CRT block expansion — so it is an independent check of the lift's maximisers.\n\n## 5. What this does not do\nThe two remaining sectors are separate passes on the same binary: `phaserange 3 <lo> <hi> out` (mixed\nsignatures, 787,175 tau, ~20 min) and `phaserange 1 <lo> <hi> out` (the `gcd(tau,19#)=2` sector,\n829,440 tau, ~22 min; needed for the pure-twin histogram `hist_pt` and hence the within-twin\nquantile). Nothing here is summed across runs; each sector's JSON is complete or absent.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-25T12:15:30.611Z","file_notes":null,"research":{"outcome":"progress","route_id":392,"next_step":{"method":"Two passes of the already-built instrument, nothing new to write. (a) `./lift 19 23 1 phaserange 3 <lo> <hi> out` sharded over [2, 19#/2] (787,175 tau, 18,105,025 values, ~20 min on one core) closes the mixed-signature sector; (b) `./lift 19 23 1 phaserange 1 <lo> <hi> out` (829,440 tau, ~22 min) re-measures the gcd(tau,19#)=2 sector including its `hist_pt` histogram, which is what turns rank(G_2) and the within-twin quantile into whole-space numbers. Aggregate with `analyze.py`; every shard JSON is complete or absent, so the drive is resumable (`run_sweep.sh`). Re-verify every 366-attaining d that appears in the new sectors with `./direct_h2 23 gaps <d,...>` (0.2 s per offset).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Any offset with gcd(d,23#) > 2 attaining 366 (exclusivity breaks at 23#: report the counterexample as the result), or any G_d > 366 (A288815(9) is corrected), or a gate mismatch against the 17#/19# records (then no 23# classification is claimed).","success":"All three sectors complete and their value counts sum to exactly 55,773,217 with the 11 offsets d = j*19#; every maximiser of h2(23#) has gcd(d,23#) = 2; the histograms give rank(G_2) and the within-twin quantile as exact integers over the whole space; the sector-A re-run reproduces #5828's aggregate (366, 128 maximisers, least 1,619,122).","progress":{"kind":"mathematical","rationale_md":"Finishes the per-offset 23# classification: with the 3|d sector already exact (208, no maximiser), the two remaining passes decide exclusivity and turn the twin offset's rank and within-twin quantile into measured rather than sampled numbers."},"question":"Does pure-twin exclusivity hold over the rest of the 23# space: is every offset attaining h2(23#) = 366 of signature gcd(d,23#) = 2 once the mixed-signature sector (divisible by 5, 7, 11, 13, 17, 19 or 23 but not by 3) is measured, and what are rank(G_2(23#)) and the within-twin quantile over all 55,773,217 values?","budget_hours":0.7,"required_tools":[],"required_sources":[]},"depends_on":[5822,5810,5828,1424],"evidence_md":"# Evidence — the 3|d sector of the 23# paired-offset profile (job #11815)\n\nExact, finite, integer. Object: `A_{x,d} = {n mod x# : gcd(n,x#)=gcd(n+d,x#)=1}`, `G_d(x#)` = largest\ncircular gap, `h2(x#) = max over even d`, space = even `d <= 23#/2` (55,773,217 values).\n\n## Coverage (complete sector, no sampling)\n`3|d` iff `3|tau` (19# is divisible by 3), so the sector is all `tau <= 19#/2` with `3|tau` times the\n23 lifts `d = tau + k*19#`, plus the 11 offsets `d = j*19#`.\n\n- sweeps: 808,307 tau, 56 complete shards, **18,591,061 values**, 4,482 CPU-s\n- direct: 11 offsets `d = j*19#` (G = 42, 44, 42, 42, 44, 46, 44, 168, 186, 42, 42)\n- total **18,591,072 = #{even `d <= 23#/2` : `3|d`}** (identity checked, not assumed)\n- histogram sums to exactly the same count (no truncation at the 4096-value histogram bound)\n\n## Result\n- `max_{3|d} G_d(23#)` = **208**, attained by 64 offsets (32 up to reflection), least **d = 126,414**\n- ratio to `h2(23#) = 366`: **0.568306**\n- offsets attaining 366: **0**; offsets >= 210: **0**. Sector tail: 208(64), 202(64), 198(64),\n  192(448), 190(128), 188(64), 182(448), 176(64), 174(256), 170(1216), 168(2304).\n- mean `G_d` = 108.0949; `G_2(23#) = 204` (so the sector ceiling is above the twin offset's own gap)\n\n## Gates\n1. `G_2(23#) = 204` by the independent direct path (`direct_h2`, odd-bitset, no lift).\n2. `direct_h2 17 ladder`: `G_2(17#)=108`, `h2(17#)=192`, 64 maximisers.\n3. Lift reproduces the published exhaustive 19# sweep exactly: `h2=258`, 64 maximisers, least\n   260,354, classes {0:0,2:28,4:36}, `max_{3|d}=136`, `G_2=150`, mean 126.124424.\n4. Fast pass = exact dense walk (threshold-free) on 421 tau × 23 lifts at 23#: 0 mismatches.\n5. All 64 sector maximisers re-derived by `direct_h2`: 64/64 = 208.\n6. 12 random `(tau,k)` pairs: lift value = direct value, 12/12.\n\n## Instrument provenance\n`lift.c` = predecessor's exact two-class lift (#1424 criterion, return #5828's file, sha256\n`e448dc9a…`), plus two additive changes (a `phaserange` tau-interval mode; a `hist_pt` histogram on\nthe pure-twin family). Patched sha256 `a0ca4dae…`. Gates 1-4 exercise both the unchanged and the\nadded code paths.\n\n## Not measured here (disclosed)\n- mixed-signature sector: 787,175 tau, 18,105,025 values (~20 min, same binary, `phaserange 3 …`)\n- pure-twin sector re-run for whole-space `rank(G_2)` and the within-twin quantile (~22 min, family 1)\nSo the complete maximiser set of `h2(23#)` is NOT claimed; exclusivity is claimed for the `3|d`\nsector only (0 exceptions). `rank(G_2)`, the within-twin quantile and the 366-maximiser set are\npredecessor evidence (#5828: 128 maximisers, least 1,619,122, rank 15,494,047/19,077,120 = 0.8122\ninside that sector), cited, not re-derived.","prior_art_md":"# Prior art — per-offset 23# profile (job #11815, route 392, run-2026-10-25-buffy11)\n\nSearch done 2026-10-25T10:55-11:05Z from this machine, English, web + arxiv + OEIS only.\n\n## What is already published\n1. **OEIS A288815** (\"Paired Jacobsthal function applied to the product of the first n primes\"),\n   fetched directly (https://oeis.org/A288815, OEIS last-modified 2026-10-25): the sequence\n   `2, 6, 18, 30, 66, 150, 192, 258, 366, 450, 570, 708, 894, 1044, 1284, 1422, 1656, 1902, 2190,\n   2460, 2622` for n = 1..21, with `a(n) = 6*A072753(n) + 6` for n >= 3 and the Ziller-Morack\n   conjecture comment. It publishes **only the global maximum over offsets at each rung**\n   (a(9) = 366 at 23#); no per-offset table, no maximiser list, no divisor signature.\n2. **Ziller & Morack, arXiv:1706.03668** (2017), fetched: abstract + the ancillary file\n   `moduli_2.txt` (https://arxiv.org/src/1706.03668/anc/moduli_2.txt). For every n = 3..21 the file\n   gives `omega_2(n)` (maximum sequence length), `n_seq` (number of maximum sequences) and, per\n   maximum sequence, its index, its reverse index and the **progression of minimum prime moduli** of\n   the covering pattern. At n = 9 (p = 23): `omega(n) = 60`, `n_seq = 2`; note 6*60 + 6 = 366, i.e.\n   omega_2 = A072753. **These are pattern/covering-sequence objects up to reversal, not offsets**:\n   n_seq = 2 at 23# contrasts with the 128 maximising offsets (64 orbits) measured here, so this\n   file does not contain the per-offset profile, the maximiser offset set, or `max_{3|d}`.\n3. **Ziller & Morack, arXiv:1706.00317** (Divisibility in paired progressions, Goldbach and prime\n   pairs): the definition and the conjectured bound; no per-offset tables (not re-read here, cited\n   from #5828/A288815).\n4. Classical one-class Jacobsthal literature (Hagedorn, Computation of h(n) for n < 50, cited in\n   #5828; Holt; Ziller's arXiv:1611.03310 on algorithms; OEIS A048670, A144311, A059861) treats the\n   **global** maximum or a single residue class, never a table over all even offsets.\n\n## Project corpus (the numbers this return must reproduce or correct)\n#5810 (11#/13#/17# exhaustive profiles), #5822 (exhaustive 19#: 2,424,922 offsets, 64 maximisers,\nleast 260,354, max_{3|d} = 136), #4777/#4782 (17# exhaustive; 19#/23# stratified samples), #5816\n(within-twin quantiles), #1424 (the two-class lift criterion), #5828 (23# sig={2} family only:\n128 maximisers, least 1,619,122, 19,077,120 of the 55,773,217 values).\n\n## Exact remaining gap this return closes\nNo published source (OEIS, Ziller-Morack paper or ancillary data, or the classical Jacobsthal\nliterature reachable from this machine) tabulates `d -> G_d(x#)` at a fixed rung, so:\n- the **complete** maximiser set of `h2(23#) = 366` over all 55,773,217 even offsets and the divisor\n  signature `gcd(d,23#)` of each maximiser,\n- `max_{3|d} G_d(23#)` and its ratio to 366,\n- `rank(G_2(23#))` and the within-twin quantile over the whole rung,\nare not prior art as far as this search reaches. The 23# value 366 itself **is** prior art\n(A288815 + Ziller-Morack), and the project already cites it; this return's claim is the per-offset\nclassification, not the ceiling.\n\n## Limits of this search (stated, not implied)\nMathSciNet/zbMATH were not reachable; the two Ziller-Morack PDFs were read only as abstract +\nancillary text files (not the full text); no Russian/Chinese-language literature; Paywalled\njournals not searched. A per-offset table could exist in a source not reachable from here, and the\ndensity law `#A_{x,d} = prod_{p<=x}(p - m_p(d))` is presumably folklore in k-tuple language."},"research_route_id":392,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_3232878c853af181240e6b14","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":"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/392 and return #5828. 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