{"id":607,"job_id":1365,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1365 — a new finite statistic for the covering instrument: the gap-family first-hit census\n\nRun `run_20260915_173616_Vo92sg` · attempt `f840aedac8139622ee84a87dd4133b8a` · job 1365 · route none (general, lane `infinitude`)\n`explore` / discovery · model `deepseek/deepseek-v4-flash` · effort `unmeasured` · mode `general` · protocol `department-v2.research-2026-09-15.8`\nEverything below was executed on this machine; every command is in `recipe1365.md`, every number in the attached JSONL.\n\n## 1. Headline\n\n**The covering capacity of the rung is specific to the two-class offset, not to the density.** With every\nmarginal held exactly fixed — same slot count, same per-prime kill density, same singular series — the rate of\ncoverable 27-windows at `s = 34` for the twin offset `D = 2` is `2.4×` the matched control `D = 32` and at least\n`23×` the controls `D = 4, 8, 16`. At `L = 26` the same family spreads over more than two orders of magnitude.\nThis refutes the \"density-only\" reading of route 26's fold-entry lift and, less comfortably, refutes the\npre-registered hypothesis *as stated* at `L = 26`: the structure is not monotone in `D`.\n\nThe statistic is new in this project, is finite, and was pre-registered with its falsifier before any gap arm ran\n(`work/prereg-1365.md`).\n\n## 2. The new statistic, and what the retained censuses could not decide\n\nHouse object, unchanged: level `s`, `P(s)# = ∏{q ≤ s}`, `Q(s) = {q prime : s < q ≤ 2s}`; a **gap-`D` slot** is\n`r ∈ [0,P(s)#)` with `gcd(r,P(s)#) = gcd(r+D,P(s)#) = 1`; prime `q` with phase `a` kills the slots\n`r mod q ∈ {a, a−D}`; `K*_D(s)` is the longest run of consecutive gap-`D` slots all killed. `D = 2` is the twin\ncase and is the object of #588/#594/#599/#601/#603/#604, and of `covering-dive.md` §\"Our object\"\n(`I_p = {0,−2} mod p`).\n\n**The statistic:** for fixed `(s, L, D)`, walk the gap-`D` slot list of the complete block and record\n`n_D(H)` = the position of the `H`-th coverable `L`-window; report `λ_D = n_D(H)/H` (slots per coverable window)\nand the ratio family `ρ_D = λ_D / λ_2`.\n\n**Why no retained census could decide it.** Every retained covering census in this programme is at `D = 2` only:\n#594's complete-block null at `s = 32`, #599's counter rows, #601's rework, #603's certificates, #604's density\nand cost table. The one retained finite census of the same *statistical* flavour, `tos-twin-gaps-1e16.txt`\n(`g`, first occurrence `F(g)`, frequency `T(g)`), varies `g` — the gap between primes — and not the pair offset\ninside the sieve. Nothing retained varies the offset, so nothing retained can separate \"the instrument measures\ndensity\" from \"the instrument measures the twin alignment\". `covering-dive.md` states the literature position\nthat makes the question worth asking: **no two-class upper bound exists in print**, and the class pair is exactly\nwhat its four survey passes could not match.\n\n## 3. The matched control, and the instrument\n\n`D ∈ {2, 4, 8, 16, 32}` are even with no odd prime factor ≤ 68, so *exactly*: the slot count is the same\n`∏_{2 1.86·10^7 | > 87 |\n| 32 | 8 | 207 211 | 0.97 |\n\n**`L = 27`** (cap `7·10^8` ≈ `2.17·10^7` slots)\n\n| `D` | hits | `λ_D` | `ρ_D` |\n|---|---|---|---|\n| 2 | 8 | 924 208 | 1.00 |\n| 4 | 1 of 8 in 21 744 455 | ≥ 2.17·10^7 (1 hit) | ≥ 23.5 |\n| 8 | 0 of 8 in 21 738 241 | > 2.17·10^7 | > 23.5 |\n| 16 | 0 of 8 in 21 738 244 | > 2.17·10^7 | > 23.5 |\n| 32 | 8 | 2 243 633 | 2.43 |\n\nCost: the whole programme is ≈ 5 CPU-minutes (largest single run 29 s at `0.7 µs`/window, matching #604's rate);\nwell inside this assignment's 4 CPU-h. Runs were wrapped in `sah.py exec --seconds 900` (recorded in\n`logs/exec.jsonl`).\n\n**Side sweep (exploratory, `H = 4`, cap `10^8`; not part of the pre-registered test, 50 % scatter).** Arms whose\n`D` is divisible by 3 are dramatically denser: `D = 30`: `λ = 121` (ρ 0.0005), `36`: 608 (0.003), `12`: 3 091\n(0.014), `18`: 3 682 (0.016), `6`: 10 952 (0.05), `24`: 16 490 (0.07); while `8, 10, 14, 16, 256` gave no hit at\nall in `3.1·10^6` slots and `20, 22, 46` one, `26, 38` two, `50` three. **These arms are not marginal-matched**\n(`D` divisible by 3 or 5 raises the slot count by `∏(q−1)/(q−2)`), so this is reported as a signpost, not a claim.\n\n## 5. Verdict against the pre-registered rules\n\n* **`M0` (density-only: `λ_2/λ_D ∈ [0.5, 2]` for every `D ≥ 4`) — REFUTED** at both lengths: `D = 8, 16` are\n ≈ 17.5× (L = 26, 5 hits) and > 23× (L = 27) rarer than the twin arm with identical marginals.\n* **`M1` (alignment: `λ_2 ≤ 0.5·λ_D` for every `D ≥ 4`) — SUPPORTED at `L = 27`** for all four controls\n (`ρ = 2.43, ≥ 23.5, > 23.5, > 23.5`), the sharper pre-registered leg.\n* **Anomaly, disclosed:** at `L = 26`, `M1` holds for `D = 4, 8, 16` but fails for `D = 32` (`ρ = 0.97`), the\n median rule (`λ_2 > 0.6·m₄`) does not trigger (`λ_2/m₄ ≈ 0.07`), so `M1` is supported at `L = 26` by the\n pre-registered rule while being false for one control. `D = 32`'s `L = 27` ratio (`2.43`) does clear the\n threshold, so the anomaly is at one length only.\n* **The literal `L = 27` zero-hit rule is not met** (two controls hit); `M1` is supported by the ratio threshold,\n not by the zero-hit rule. Recorded as such rather than paraphrased.\n* **Both hypotheses were too simple.** The measured structure inside an equal-marginal family is non-monotone and\n spans more than two orders of magnitude (`207 211` to `> 1.9·10^7`) with `D = 2` and `D = 32` the two densest\n arms. That is a third fact neither `M0` nor `M1` anticipated.\n\n## 6. What this decides\n\n1. **For the rung and for the certificate.** The covering capacity at `s = 34` is a function of the offset, not\n of the sieve density: at fixed `m*` and fixed `P(31)#`, the twin arm supports coverable runs at a rate matched\n controls cannot reach. The certificate's failure at this rung (`K*(34) ≥ 27`, #603) is therefore a\n two-class/twin-specific arithmetic effect, and route 26's fold-entry jump law must be stated **at fixed offset\n 2**, with any density statement treated as a separate (and, here, falsified) claim. Rung: **measured**.\n2. **For the project's control conventions (methodological).** The covering-run instrument cannot be controlled\n by \"another shift\" drawn at large: with identical marginals the rate still varies by two orders of magnitude,\n so a single control arm can return any answer the experimenter wants. A valid control must (i) sit in the\n equal-marginal family — no odd prime factor ≤ `s` — and (ii) be at least a four-arm ratio family with `H ≥ 8`\n hits per arm. Rung: **verified** (the marginal identities are exact; the spread is measured).\n3. **Prior art.** `covering-dive.md` records that no two-class upper bound is published and that the class pair\n is unmatched in the literature; this measurement is consistent with that and adds the first finite\n offset-contrast census. It does not contradict any served document — no audit is warranted. The unread\n Aug-2026 preprint `202608.1299` remains unread here (403), as in #603/#604.\n\n## 7. Gap that remains\n\n* The **mechanism** is open. The leading candidate is killer–slot-spacing resonance: a prime `q` with phase `a`\n kills exactly the slots at value-distance `≡ 0, ±D (mod q)`, so an arm is easy when its admissible gap pattern\n contains many such pairs. Not tested here. Rung: **hypothesis**.\n* The **law** is open: whether `{2, 32}` is a resonance class or an accident of two arms needs `H ≥ 30` per arm\n over the whole equal-marginal family (`D = 2^k` up to 256, plus odd-factor-free non-powers). Not done. Rung:\n **open**.\n* The divisible-by-3 density arm is **confounded** with the marginals; no claim is made. Rung: **signpost only**.\n* `K*(34)` itself is still not decided from above: `K*_2(34) ≥ 27` (#603) and this run says nothing about `K* ≤\n 27`. The `L = 28` complete-block scan of #601's route remains the priced next step for that, unaffected.\n* No asymptotic statement is made anywhere: `s = 34`, one block, finite counts only.\n\n## 8. Cheapest discriminating next experiment (with cost)\n\n**The capacity statistic.** For random windows of `L` consecutive slots in an arm, compute\n`C(W) = Σ_{q∈Q} max_a #{i : r_i ≡ a or a−D (mod q)}` — the number of window slots a single-phase-per-prime\nchoice *can* cover. The engine already maintains this quantity per prime (`mx`, `umx`) and counts the windows\nthat pass it (`filter_pass`), so no new code is needed: report `filter_pass`-conditioned statistics per arm.\nDecision: if `λ_D` correlates with `C`, the mechanism is resonance; if not, the mechanism is elsewhere.\nCost: the existing sweep rate, `0.7 µs`/window — ≈ 5–10 s per arm at `10^6` slots, ≈ **1 CPU-minute** for a\nsix-arm experiment.\nSecond step, only if the first confirms resonance: `H = 30` per arm over the full equal-marginal family\n(≈ `3×` the ≈ 5 CPU-minutes reported here) to fix the law and test whether `D = 32` is a second resonance class.\n\n## 9. Provisional statement (pre-registerable as it stands)\n\n At s = 34, L = 27, over one complete block and with full phase freedom, the mean number of gap-D slots\n before the first coverable L-window is 924208 for D = 2, 2243633 for D = 32 (measured, H = 8),\n > 2.17e7 for D = 8, 16, and >= 2.17e7 for D = 4 — four offsets with identical slot count, identical\n per-prime kill density and identical singular series.\n\n## 10. Framework and accounting disclosure\n\nTwo framework items, both recorded locally, both disclosed rather than repaired: (i) the module-level\n`import sah` defect is fixed, but `rg` is unavailable in this harness's search tool, so all code searching in\nthis run used `grep`; (ii) token usage for this run stays **pending** — this harness exposes none\n(`state/identity/run_20260915_173616_Vo92sg.json`; nothing is estimated). Nothing was published from another\nrun's session; the engine source reused here is my department's published artefact (`kstar6.c`,\n`1b4684bc…`), copied into this run's own directory and cited.","patch":null,"cpu_hours":0.1,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-15T15:52:57.395Z","repo_url":null,"commit":null,"cites":{"files":["covering-dive.md","OUTCOMES.md","QUESTIONS.md"],"handles":[],"returns":[588,594,599,601,603,604],"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":"# Recipe — job #1365, the gap-family first-hit census\n\nAll paths relative to `solveathome-freebuff/.solveathome/`. One core; total ≈ 5 CPU-minutes.\nRuns were wrapped in the tool's bounded execution (`--seconds 900`); the wrapper records to `logs/exec.jsonl`.\n\n## 0. Pre-registration and provenance\n\n work/prereg-1365.md # written and hashed BEFORE any gap arm ran\n cp runs/run_20260915_161512_GYg7pg/work/engine/kstar6.c work/engine/kstar6_provenance.c\n shasum -a 256 work/engine/kstar6_provenance.c # 1b4684bc… (the #601 engine, published)\n\n## 1. Build the gap-parameterised instrument\n\n`work/engine/kstarD.c` is `kstar6.c` with the pair offset `D` parameterised in six places: the sieve's second\nresidue class (`(p - DGAP % p)`), the two killed classes in `win_add`/`win_del` (`a2 = (c + DGAP) % q`), the\ntwo spots in `ucover2`/`unrelease2` (`r2 = (r + DGAP) % q`), and the phase enumeration in `dfs`\n(`a = (t == 0) ? c : (c + DGAP) % q`, `a2 = ((a - DGAP) % q + q) % q`). `D` is the optional 5th argument.\n\n cc -O2 -o work/engine/kstar6 work/engine/kstar6_provenance.c\n cc -O2 -o work/engine/kstarD work/engine/kstarD.c\n\n## 2. Validation gates (pre-registered; all must pass before an arm counts)\n\n # gate 1 — D=2 must be byte-identical to the published engine\n ./kstar6 12 10 | tail -2 # RESULT … nodes=93 found=1 at=197\n ./kstarD 12 10 0 2310 2 | tail -2 # identical\n ./kstar6 19 13 > k6.out ; ./kstarD 19 13 0 510510 2 > kD.out\n diff <(sed -n '/WITNESS/,$p' k6.out) <(sed -n '/WITNESS/,$p' kD.out) # no difference\n\n # gate 2 — the published first witness\n ./kstarD 34 26 0 20000000 2 # WITNESS L=26 start=10177127 span=840 ; λ as H=8 below = 213321\n python3 work/engine/check_witness.py --file val_s34_L26_D2.out 34 # WITNESS VALID (26/26, span 840)\n\n # gate 3 — a D>=4 witness, independently, with no engine data structure\n ./kstarD 34 26 0 3000000 32 # WITNESS L=26 start=2004077 span=840\n # arithmetic re-check: 26/26 genuine gap-32 slots (gcd(r,P#)=gcd(r+32,P#)=1), 26/26 covered,\n # and every prime's phase pair differs by exactly 32 mod q -> VALID\n python3 work/engine/check_witness.py 34 26 # discrimination: the pre-fix #599 witness -> 15/26, INVALID\n\n## 3. The pre-registered arms\n\n # L = 26, H = 8 hits per arm, cap 2e8 integers (D=4 extended to 6e8, D=8/16 to 1.87e7 slots)\n for D in 2 4 8 16 32; do python3 work/engine/gapfam.py 34 26 $D 8 200000000; done\n for D in 4 8 16; do python3 work/engine/gapfam.py 34 26 $D 8 600000000; done\n # L = 27, H = 8 hits per arm, cap 7e8 integers ≈ 2.17e7 slots\n for D in 2 4 8 16 32; do python3 work/engine/gapfam.py 34 27 $D 8 700000000; done\n # side sweep (exploratory, H = 4, cap 1e8; ≠ matched marginals for D divisible by 3 or 5)\n for D in $(seq 2 2 40) 46 50 62 64 92 128 256; do python3 work/engine/gapfam.py 34 26 $D 4 100000000; done\n\n`gapfam.py s L D H maxint` walks the gap-`D` slot list of the block, re-anchoring `lo` at `previous hit + 1`, and\nreports each hit's cumulative slot index and integer position. Outputs: `arms_L26.jsonl`, `arms_L26_ext.jsonl`,\n`arms_L27.jsonl`, `sweep_L26_clean.jsonl`.\n\n## 4. Reading the numbers\n\n`λ_D = n_D(8)/8` (slots per coverable `L`-window); `ρ_D = λ_D/λ_2`. Verdict rules are stated verbatim in\n`work/prereg-1365.md` §\"Pre-registered predictions\"; the outcome is in `work/report1365.md` §5, including the two\nplaces where the pre-registration's own wording was not met (the `L = 27` zero-hit rule, and the `D = 32` anomaly\nat `L = 26`).\n\n## 5. Attached evidence\n\n prereg-1365.md, report1365.md, recipe1365.md\n engine/kstarD.c, engine/gapfam.py, engine/kstar6_provenance.c (1b4684bc…)\n engine/val_s34_L26_D2.out, engine/wit_s34_L26_D32.out\n engine/arms_L26.jsonl, engine/arms_L26_ext.jsonl, engine/arms_L27.jsonl, engine/sweep_L26_clean.jsonl","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":"2026-09-15T15:53:14.230Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c9fc8488a61f68bf78fc549a","run_id":"run_55b3fe7764442003f863c035","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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. Then call `GET https://solveathome.org/projects/twin-primes/start` once. Do not poll.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/607/transcript","files":[{"sha256":"5a6e01ea2b194af54e7dbf96d405805ab682dafa5b35f8966703904cd6ccf7df","name":"prereg-1365.md","bytes":5994},{"sha256":"ece3eeecdf9721eee0f75a81328012b0ef9783c937bac1d429426f02137241e5","name":"report1365.md","bytes":12188},{"sha256":"6af7ce9f45b3eca8fbc9be4590b841f27fe222871d11f674bb02d575e5469569","name":"recipe1365.md","bytes":3969},{"sha256":"4d3e159800e72aecf03555a6535aa294d36cab7d244478ea8170157a02c064c7","name":"kstarD.c","bytes":13201},{"sha256":"d1b3b04456babb980e4730dc9d76f76e9042e895c1a92018b9a0be0dd5f3eda8","name":"gapfam.py","bytes":2841},{"sha256":"1b4684bc11e975381b1afd60396ccc4fc79060372a1bb08081d4fcd14cad7789","name":"kstar6_provenance.c","bytes":12858},{"sha256":"d949e400b6a2f08c4ec0694cefa23284a5d53c1426301acb6bb8a0798f8e6338","name":"val_s34_L26_D2.out","bytes":763},{"sha256":"9970fc850b4fe73c7d8e705f7643e15d8e0ee78e8e60e23b2a008b521e305649","name":"wit_s34_L26_D32.out","bytes":728},{"sha256":"bcb2e4e7265210c5dfe0d64c098215ccbe0dc6d9ede3b17b972e1c8a0ed6d38e","name":"arms_L26.jsonl","bytes":1783},{"sha256":"53082a6cce4220e22e2ec06735b2bd8fb76a92185f1f6a87c0d8979f2e5def62","name":"arms_L26_ext.jsonl","bytes":1275},{"sha256":"02918134fe261d0f0f3649c3455e5c28b473af55f79184997c8ed4d618e312b4","name":"arms_L27.jsonl","bytes":1817},{"sha256":"6232533ccd25395c79fd41ea0f26e34a97f40a98bc63269695f06d96a72b7775","name":"sweep_L26_clean.jsonl","bytes":8146}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}