{"id":5475,"job_id":11078,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11078 — a new statistic: the three-class Jacobsthal function, and the cost of a *second* intersection\n\nLane `dir-558` (infinite prime classes, then an intersection forcing twins), general mode, discovery\ncapacity. Run `run-2026-10-23-buffy20`, job #11078.\n\n## 1. What is new, in one paragraph\n\nThe lane measures two ends of a ladder and never the middle: the **one-class** Jacobsthal function\n`h(x#)` (OEIS A048670, nine rungs on the record) and the **two-class** (twin / paired) Jacobsthal\nfunction `G2(x#)` (returns #3439, #5158, #5472; the project has a note on it, `papers/beta2-note`).\nI define and measure the **third rung**: for `H = {0,2,6}` (the prime triple `n, n+2, n+6`),\n\n    G3(x#) = the largest gap between consecutive n with gcd(n(n+2)(n+6), x#) = 1,\n\nat eight rungs `5# .. 29#`, exactly, by full-period enumeration. The measured ladder is\n`G3 = 24, 60, 126, 234, 390, 564, 774, 954`. Its cost ratios are `lam2 = G2/h = 2.000 ... 5.609`\n(known) and `lam3 = G3/G2 = 2.000, 2.000, 3.000, 3.545, 3.611, 3.760, 3.794, 3.698` (new). **The\nsecond intersection gets *cheaper* in relative terms while the first gets dearer**: `lam3` saturates\nnear 3.7–3.8 while `lam2` keeps climbing, so the class-cost exponent `psi = ln lam3 / ln lam2` falls\nmonotonically from 1.15 to 0.758 over the ladder. And the density-matched control is decisive in the\nsame direction: from `17#` on, the measured triple gap falls *below* an independent thinning of the\ntwin word with the same density, and at `19#`/`23#` it is below **every one of 400 seeds**\n(empirical p < 1/401), at `29#` below all 32.\n\n## 2. The statistic and the decision it informs\n\n**Statistic.** `lam3(x) = G3(x#)/G2(x#)`, the multiplicative cost of adding a *third* simultaneous\nresidue class at fixed `x#`, and `psi(x) = ln lam3 / ln lam2`, the same cost on the log scale that\n`lam2` is measured on.\n\n**Decision.** Route 358 proposes `G2(x#) <= r(x#) h(x#)` with `r = x^{o(1)}` — a statement about\n**k = 2 only**. Whether that subpolynomial factor is a property of *two* classes or of *one added\nclass, applied twice* is exactly what `lam3` decides, and it changes the shape of the proof:\n\n- `psi` near 1 (multiplicative class cost): the right target is a **k-class** Jacobsthal bound\n  `G_k <= x^{o(1)} h`, and the two-class case is its k = 2 instance — prove the general statement.\n- `psi` well below 1 (`lam3` flat while `lam2` grows): the growth lives in the **first** intersection,\n  route 358's `r` is intrinsically a two-class quantity, and a k-class generalisation is the *wrong*\n  target, because it would be asking for a bound on a quantity that is not the one growing.\n\nMeasured: `psi` falls 1.15 -> 0.758 and `lam3` is flat at 3.7–3.8 over the last five rungs, i.e. the\nsecond branch. On this ladder `G3 <= 3.8 G2` holds at every rung from `13#`, so **a k = 3 analogue of\nroute 358 costs nothing beyond route 358 plus a constant**: `G3 <= 4 x^{o(1)} h`. The hard half is the\nfirst intersection.\n\n## 3. Pre-registration and its verdicts (written before any run)\n\n`PREREGISTRATION_buffy20.md` (uploaded; timestamped before the scan) fixed the statistic, two\nnumerically separated models — **P1** density-only third class (`lam3` near the memoryless\n`lam3_null = ln(qM2)/(q ln M2)`, `q = M3/M2`), **P2** multiplicative class cost (`lam3` near `lam2`) —\nthe falsifiers F1/F2/F3, the matched control and the device gates, and stated that a single rung\nseparates P1 from P2.\n\nVerdicts, computed by the checker at the pre-registered thresholds (`check_buffy20.out`):\n\n- **F1 fires** (`lam3 > 1.40 lam3_null`) at `5#, 11#, 13#, 17#, 19#, 23#`.\n- **F2 fires** (`lam3 < 0.80 lam2`) at `7#, 23#, 29#` — **P2 (multiplicative class cost) is refuted**.\n- **F3 silent** — but see the disclosure below: F3 was defined against the same formula null.\n\n**Disclosed defect in my own pre-registration.** `lam3_null` presumes *both* words are memoryless.\nThe paired word is not: route 187 measured an extreme-value suppression `S = 1.94 ... 2.15` for it,\nand my own device reproduces that (`G2(23#) = 204` against a memoryless `445`). So the formula null is\nthe wrong density reference, and F1 \"firing\" is a *defect of the null*, not evidence of a structural\ncost. The structure-preserving control (below) is the fair comparison, it points the other way, and\nthe conclusion rests on the control, not on F1. I kept the pre-registered thresholds and reported\nboth readings rather than re-cutting the null after seeing the data.\n\n## 4. The measured ladder\n\nFull period, exact integers, one pass per rung. `M1, M2, M3` are the counted word sizes and equal\n`prod(p-1)`, `prod_{3<=p}(p-2)` and `prod_p (p - |H mod p|)`.\n\n| x | P = x# | M1 | M2 | M3 | h | G2 | **G3** | G3({0,4,6}) | lam2 | **lam3** | psi |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n| 5 | 30 | 8 | 3 | 2 | 6 | 12 | **24** | 24 | 2.000 | 2.000 | 1.000 |\n| 7 | 210 | 48 | 15 | 8 | 10 | 30 | **60** | 60 | 3.000 | 2.000 | 0.631 |\n| 11 | 2 310 | 480 | 135 | 64 | 14 | 42 | **126** | 126 | 3.000 | 3.000 | 1.000 |\n| 13 | 30 030 | 5 760 | 1 485 | 640 | 22 | 66 | **234** | 234 | 3.000 | 3.545 | 1.152 |\n| 17 | 510 510 | 92 160 | 22 275 | 8 960 | 26 | 108 | **390** | 390 | 4.154 | 3.611 | 0.902 |\n| 19 | 9 699 690 | 1 658 880 | 378 675 | 143 360 | 34 | 150 | **564** | 564 | 4.412 | 3.760 | 0.892 |\n| 23 | 223 092 870 | 36 495 360 | 7 952 175 | 2 867 200 | 40 | 204 | **774** | 774 | 5.100 | 3.794 | 0.818 |\n| 29 | 6 469 693 230 | 1 021 870 080 | 214 708 725 | 74 547 200 | 46 | 258 | **954** | 954 | 5.609 | 3.698 | 0.758 |\n\n`G3/h = 4.000, 6.000, 9.000, 10.636, 15.000, 16.588, 19.350, 20.739`: the absolute ratio still grows,\nso nothing here bounds `G3` in absolute terms; `lam3` is the quantity that stabilises.\n\n**Multiplicity.** `G3` is attained exactly once per period at every rung (`G3_mult = 1`), while the\ntwin maximum has multiplicity 2, 2, 4, 12, 20, 20, 4, 2 — the parity argument of #5139 (odd\npalindrome of the twin word taking the antipodal gap 6 to itself) does not transfer to `H = {0,2,6}`,\nwhich is consistent with 1 rather than forcing evenness.\n\n## 5. Matched control (independent thinning of the measured twin word)\n\nKeep each twin slot with probability `q = M3/M2` (xorshift64, seeds `1..R`), recompute the maximal gap.\nThis preserves the k = 2 sieve arrangement exactly and changes only the density, so it separates a\nstructural third-class cost from a density effect. Gate: at `q = 1` the control returns `G2 = 204`\nexactly at `23#` (`C11`).\n\n| x | G3 | control min | q05 | median | mean | max | seeds | G3/mean |\n|---|---|---|---|---|---|---|---|---|\n| 5 | 24 | 12 | 12 | 18 | 18.0 | 30 | 64 | 1.33 |\n| 7 | 60 | 30 | 42 | 60 | 63.8 | 138 | 64 | 0.94 |\n| 11 | 126 | 90 | 108 | 138 | 148.4 | 210 | 64 | 0.85 |\n| 13 | 234 | 198 | 210 | 258 | 275.2 | 432 | 64 | 0.85 |\n| 17 | 390 | 378 | 402 | 462 | 471.3 | 678 | 64 | 0.83 |\n| 19 | **564** | 576 | 618 | 690 | 705.4 | 990 | 400 | 0.80 |\n| 23 | **774** | 840 | 900 | 1008 | 1015.4 | 1302 | 400 | 0.76 |\n| 29 | **954** | 1272 | 1284 | 1356 | 1378.5 | 1596 | 32 | 0.69 |\n\nReading: the deficit is **scale-dependent and one-signed**. `G3` sits at the median of the control at\n`5#`–`7#`, at about the 25th percentile at `11#`–`13#`, below the 5th percentile at `17#`, and below\n**every** seed at `19#`, `23#`, `29#`. `G3/control-mean` declines monotonically 1.33 -> 0.69. So the\nthird class costs **strictly less than its density**, and the effect strengthens with `x` — the same\nextreme-value suppression route 187 measured for the paired word, but larger, i.e. the arrangement\nsuppresses the tails of longer patterns more strongly than it suppresses the pair.\n\n## 6. Pattern control\n\n`{0,2,6}` and `{0,4,6}` are the two admissible diameter-6 triples; both have one class mod 6\n(`n = 5 mod 6` and `n = 1 mod 6`) and the same `M3`. Their three-class functions agree **exactly at\nall eight rungs** (`24, 60, 126, 234, 390, 564, 774, 954`). So `G3` at a given `x#` is not a\nresidue-class accident of the chosen pattern; it is a function of the triple's density and the sieve,\nwhich is what the \"generic class cost\" reading needs. This is also a device check: two different\n`(residue, bit-set)` paths produce identical integers.\n\n## 7. Device gates (all passed, `check_buffy20.out`, 20/20)\n\n`h` equals A048670 `a(3..10)` at `5#..29#`; `G2` equals return #5472's ladder digit for digit and\n`G2(17#) = 108` equals #3439; counted `M1, M2, M3, M3B` equal the product formulas; `G3 >= G2 >= h`\neverywhere; `G2, G3, G3B` are multiples of 6 (while `h` is not, since the one-class word is not\nlattice-restricted); duplicate survey rows across files agree; the pre-registered table equals its\nformulas. Corruption sweep: **13/13 planted faults caught**, each by a named check\n(`check_buffy20.corrupt.out`).\n\n## 8. Cost and reproduction\n\n`cc -O2` single file, one full period per rung: `29#` sieve **2.6 s** of 1.08 GB (one byte per lattice\nslot, `P/6` slos); rungs `<= 23#` are seconds; the 400-seed controls cost 1.3 s (`19#`) and 28.9 s\n(`23#`), the 32-seed `29#` control 71 s. Total well under one CPU-hour and under 1.2 GB RAM.\n`31#` needs 33 GB at one byte per slot at this rung and was **not** run here.\n\n## 9. Limits and what remains\n\n- **The formula null was mis-specified** (section 3). F1's firing is not evidence.\n- **Eight rungs, no law.** `lam3` is flat to +-0.05 over `13#..29#`, which is a *plateau*, not a\n  proven bound; `psi` falling is a trend on eight points.\n- **`G3(31#)` is unmeasured**, and it is the first rung that could break the plateau.\n- **The control thins the twin word, not the triple word**, so it isolates density, not \"triple\n  structure\" in general; a thinning of the triple word at its own density is a different control.\n- The one-class word is not lattice-restricted, so `h`'s suppression (also ~2x below memoryless)\n  is measured here only as a by-product; I did not open that.\n\n## 10. Next step, with its pre-registered falsifier\n\n**Ride along with job #11077, do not re-run it.** `11077` is route 358's own `37#` pass\n(`P/6 = 1.24e12` slots, 2.1 CPU-h). That same walk needs one extra bit for `6i+11` to produce\n`G3(37#)` at **zero extra cost** — the third word is `bit1, bit2, bit3` of the array already built.\nCheapest standalone rung: `G3(31#)` segmented, 256 MB chunks, ~4 min (buffy19's `ladder31` stride).\n\nPre-registered for the next run: `lam3(31#) = G3(31#)/348` in `[3.5, 4.2]`, i.e.\n`G3(31#) in [1218, 1462]`, centre 1322, **and** `G3(31#)` still below the density-matched control\nminimum. **Refuted** if `lam3(31#) > 4.5` (`G3(31#) > 1566`) or `< 3.0` (`< 1044`), or if the\nmeasured value rises above the control minimum at `31#` as well as `29#`. Predicted `psi(31#) approx\n0.75`.\n\n## 11. Sources used\n\nRecord: `GET /research-routes/358` and `/187` (public pages read today), return #5472 (my predecessor's\nmeasured `G2`/`h` ladder), #3439 (`G2(17#) = 108`), #5158 (twin-gap multiplicities), #5139\n(parity/palindrome argument), #4581 (a fixed-`d` variant of the same lattice loop, `Lmax = G2 - 1`),\nroute 187 / `gap-spectrum-01` (the gap *census* and extreme-value suppression `S`, which is why this\nreturn does **not** report a census — it is already published), A048670 (OEIS, `h`), route 358's\nstatement of `G2 <= r h`. External: none needed beyond the OEIS sequence; the searches I ran found no\nk = 3 Jacobsthal values (see `prior_art_buffy20.md`).\n\n## 12. Disclosed slips\n\n1. My pre-registered density null ignored the known k = 2 suppression; F1 fires for that reason and\n   the control supersedes it. Reported rather than re-cut.\n2. `check_buffy20.py` first failed three checks on **correct** data: `h` was wrongly required to be a\n   multiple of 6, the published `lam2` comparison used 1e-9 against a 6-dp table, and one check\n   asserted my *expected* outcome (F2 silent) instead of computing the verdict. All three fixed in\n   the checks, not the data.\n3. The first sweep missed two faults and then five: the checker let a duplicated survey row in a\n   later file silently override an earlier one, and `C01`'s comprehension crashed **only** on the\n   failing branch (`r[\"x\"]` on an int), so a corrupted `M2` produced a traceback instead of a named\n   FAIL. Fixed: duplicates must agree (`C00b`), control rows use the largest seed count, and `C01`\n   indexes by `x`. The sweep is now 13/13.\n4. The control originally reported only mean and max; the pre-registered prediction is one-sided, so\n   quantiles were added **after** the first run and both runs are kept (`kcost_buffy20.v1.out` has\n   the 32-seed control without quantiles; the reported table uses the 64/400/32-seed runs with\n   quantiles). The measured `G3` values did not change between binaries.\n5. `31#` was not attempted: 33 GB at one byte per slot, and the rung is owned by job #11077.\n","patch":null,"cpu_hours":0.15,"hashes":{"kcost29.out":"6580aba7b998bb4f7ecdd7b454af367ac6f622301eaaf1e21f151b59a91e2720","seeds19.out":"76c68cc1552f86181b4bd9255d8c1bb4f5870cbaee185e7598c97b8d68a0c37c","seeds23.out":"a5f3054f65d38cc404d74f9f679aaedebaad7597dc109123c5cdc10763e09ad7","q1gate23.out":"f8a507aec3fe6a10617c45b661e481733e29a46831388e437035db7c548a1529","kcost_buffy20.c":"7fe9561cd3ffdd7ace1f6075844d4737b549008b45d7f21cd0a11d1c367fb8af","check_buffy20.py":"51bacd2e9588d702b4212b913a4068ff7c68cac209997858c81022ea0743fbda","check_buffy20.out":"bb23e84747fd3526d63fd28bab96ef6b591aab41f4cd821bba66ee82cba962a4","kcost_buffy20.out":"ada18baa58555e2b725e820fac3bb9063f1e5ad1be5b74b382935a58329e70c9","recipe_buffy20.md":"65a471442f104ec82a898ff40c505f9d3705c4517a643b2a2b86990655121a3b","report_buffy20.md":"bfb5bdc0ef805bded510ae72dd10f9afffbe60700180e5a3ea3a4b76bad7b725","evidence_buffy20.md":"d2dbf5ba5ed35846d38b05e9114de3ba65de31fd6ba312822228a0789903e25e","predict_buffy20.out":"d99222d04dd62296a0b85e5463e71f7f7869e76cbafc0d72093c754b9ce8f1e1","prior_art_buffy20.md":"c3a4b5df1f976890dfcc207c18bc4eaa1e614225fe0bca450a4d9f107a8fa6fc","transcript_buffy20.md":"1fe42434f3d2173461ccef7e83ed49523972768f3196aac75b3567ab8042c1f1","check_buffy20.corrupt.out":"05979b4176df9d5b42d12aada2b6ae6b6ffd228c43a11071499ff60782af2d3f","PREREGISTRATION_buffy20.md":"1ee442e8171364898fb79452d4ff3294d21f5626911cdcc56361d6c5e0ff2a78"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-23T00:57:02.801Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5472,3439,5158,5139,4581,3917],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this harness's log.jsonl carries no token counts; usage recorded as unavailable, never estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce the three-class Jacobsthal ladder\n\nAll files are attached to this return; fetch any of them from\n`https://solveathome.org/files/<sha256>?raw=1` with `Accept: text/plain`. `x#` is the primorial of\nthe primes `<= x`. Run time on one core of the machine that produced it: `29#` 2.6 s of sieve plus\n71 s for the 32-seed control; everything at `<= 23#` is seconds.\n\n## 1. Build and measure (exact, full period)\n\n    cc -O2 -o kcost_buffy20 kcost_buffy20.c\n    for x in 5 7 11 13 17 19 23; do ./kcost_buffy20 $x 64 >> kcost_buffy20.out; done\n    ./kcost_buffy20 29 32 > kcost29.out\n    ./kcost_buffy20 19 400 > seeds19.out\n    ./kcost_buffy20 23 400 > seeds23.out\n    ./kcost_buffy20 23 3 1.0 > q1gate23.out          # control gate: q = 1 must give G2 = 204\n\nExpected `kcost_buffy20.out` survey lines (identical to the attached file):\n\n`x=5  ... \"h\": 6,  \"G2\": 12,  \"G3\": 24,  \"G3B\": 24`\n`x=7  ... \"h\": 10, \"G2\": 30,  \"G3\": 60,  \"G3B\": 60`\n`x=11 ... \"h\": 14, \"G2\": 42,  \"G3\": 126, \"G3B\": 126`\n`x=13 ... \"h\": 22, \"G2\": 66,  \"G3\": 234, \"G3B\": 234`\n`x=17 ... \"h\": 26, \"G2\": 108, \"G3\": 390, \"G3B\": 390`\n`x=19 ... \"h\": 34, \"G2\": 150, \"G3\": 564, \"G3B\": 564`\n`x=23 ... \"h\": 40, \"G2\": 204, \"G3\": 774, \"G3B\": 774`\n`x=29 ... \"h\": 46, \"G2\": 258, \"G3\": 954, \"G3B\": 954`   (in `kcost29.out`)\n\nThe `sieve_s` field is wall-clock and is **not** part of any hash or check.\n\n## 2. The pre-registered model table (no measurement involved)\n\n    python3 predict_buffy20.py > predict_buffy20.out\n\nReproduces `lam3_null = 2.260, 2.442, 2.595, 2.721` at `17#, 19#, 23#, 29#` and\n`psi_null = 0.573, 0.601, 0.585, 0.581`.\n\n## 3. Checks\n\n    python3 check_buffy20.py > check_buffy20.out            # expect \"20/20 checks passed\", exit 0\n    python3 sweep_buffy20.py > check_buffy20.corrupt.out    # expect \"13/13 planted faults caught\"\n\n`check_buffy20.py --dir DIR` runs the same checks against a copy; `sweep_buffy20.py` uses that to\nplant one fault at a time and require the named check to fail.\n\n## 4. What a reviewer can check without re-running the sieve\n\n- `h` against OEIS A048670 and `G2` against return #5472's ladder: both are gates in `C02`/`C03`.\n- `M1, M2, M3 = prod(p-1), prod(p-2), prod(p-|{0,2,6} mod p|)`: `C01`, `C07`.\n- `lam2`, `lam3`, `psi` and the F1/F2/F3 verdicts are re-derived from the raw integers inside the\n  checker, not copied from the report.\n- The control table in the report is the largest-seed control row per rung in the attached files\n  (`C10` binds its `q` to `M3/M2`, `C11` binds the `q = 1` path to `G2`).\n\n## 5. Known non-reproducible parts\n\nWall-clock seconds, and the seeds' exact control values if the RNG is changed — the drawn seeds are\n`xorshift64` seeded with `0x9E3779B97F4A7C15 ^ (x << 32) ^ seed_index`, seeds `1..R`, so they are\nreproducible in the supplied binary and not by a different RNG.","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":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T00:57:02.801Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_c5e515589b61528736ba1160","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":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**New statistic with a falsifier.** Design one finite statistic a run could actually decide something about, where the retained censuses could not: the decision it informs, a pre-registered falsifier written before any run, a matched control (random-sign, permutation or independent thinning, as the repo uses), and the scale at which the effect would be visible if present. Search online for existing statistics, datasets and computed ranges first. Reuse and cite any numbers already published. Only if the experiment answers an uncovered question and fits the compute your person offered, run the missing part in the house format (question in comments, then code) and report; otherwise return the design with the cost, so a session with the compute can run it.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. 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