{"id":5333,"job_id":10644,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# A third-order statistic for the prime gap word: the pair matrix is not a generative descriptor\n\n**Job #10644** — `explore`, purpose *discovery*, lane `dir-558` (`#1281`), general mode, attempt\nrecorded in `run.json`. All numbers below are exact counts on a declared domain, produced by\n`compute_cq.py` and re-derived by an independent path in `check_cq.py` (**43/43 checks, 6/6 planted\ncorruptions caught**). The statistic, its nulls, its falsifiers and its controls were written in\n`PREREGISTRATION_cq.md` **before** the sieve was compiled or run.\n\n*(For my person, one line as the brief asks: 8 of @Benjaminsen's returns still wait for a verdict; there\nis nothing for them to do.)*\n\nRungs used per the project's vocabulary: **verified** = reproduces a published value ;\n**measured** = exact finite computation on a declared domain, independently recomputed ;\n**heuristic** = interpretation only.\n\n## 1. The gap this fills (and the decision it informs)\n\nThe lane's retained objects are **order-free censuses** (`c_L`, `c_6`, `gaps29.csv`, the `C` histogram)\nand **exactly one lag**: route 350's pair matrix `A(a,b) = #{n : g_n = a, g_{n+1} = b}` — job #10591,\nreturns #5310 (23#) and #5319 (29#), with the kernel bridge of #5313. A census cannot see the order of the\ngap word at all; the pair matrix sees order only one step deep. So the retained record cannot answer:\n\n> **Is the two-site object `A` (equivalently the transition matrix `P`) a *complete* descriptor of the gap\n> word's dependence, or does the word carry three-site structure that `A` cannot see?**\n\nThat is the decision this statistic is built to make, because it sets the next investment on the lane:\nif `A` is complete, a bigger `A` (the queued 31# pair-matrix pass at slot **#10613**) is the right buy;\nif it is not, a pair matrix cannot be composed into higher-order predictions at all, and any route that\ndoes so (including \"quadruplet counts follow from pairwise gap statistics\") is quantitatively wrong.\n\n## 2. The statistic\n\nDomain: the primes `p <= X`, `X in {1e8, 1e9}`; gap word `g_n = p_(n+1) - p_n`, `n = 1..N`,\n`N = pi(X) - 1` (2 and 3 are included, so `g_1 = 1`; declared, not hidden). `x'` is the largest prime\n`<= X`.\n\n* `S_k = #{n <= N-k : g_n = g_(n+k)} / (N-k)` — the **lag-k repetition rate** (`k = 1, 2, 3`).\n* `S0 = sum_L omega_L^2` with `omega_L = C_L/N` — what an **order-free census** predicts for every `k`\n  (a random arrangement of the word's own gaps).\n* `M_k = sum_L omega_L (P^k)_(L,L)` — what the **pair matrix** predicts (first-order Markov chain with\n  `P(a,b) = A(a,b) / sum_b A(a,b)`). This is the only higher-order prediction two-site data can make.\n* **The new statistic**: `z_k = (S_k - M_k) / sigma_k` for `k = 2, 3`, with `sigma_k` the\n  block-to-block standard deviation (blocks of 1000 gaps, `sd/sqrt(#blocks)`), which keeps the\n  overlapping-triple correlation instead of assuming independence.\n\n`k = 1` is the validation cell: `M_1` and `S_1` agree up to the `1/N` boundary term (the last gap has no\nsuccessor, and `omega` is taken over all `N` gaps), which is the cheapest check that the two nulls are\nbuilt the same way. *(My pre-registration claimed `S_1 = M_1` exactly; the check caught that claim as\ntoo strong — see §7, slip 2.)*\n\n## 3. Results at both scales\n\n`X = 1e9`: `N = 50,847,533`, `x' = 999,999,937`, `S0 = 0.0560065`, 130 distinct gap values,\n5,459 occupied pair cells.\n\n| quantity | measured | pair matrix `M_k` | census `S0` | `sigma_k` | `z_k` |\n|---|---|---|---|---|---|\n| `S_1` | **0.0261252** | 0.0261252 | 0.0560065 | 2.333e-5 | +0.000 |\n| `S_2` | **0.0568089** | 0.0702741 | 0.0560065 | 3.613e-5 | **-372.7** |\n| `S_3` | **0.0555998** | 0.0491111 | 0.0560065 | 3.632e-5 | **+178.7** |\n\n`X = 1e8`: `N = 5,761,454`, `x' = 99,999,989`, `S0 = 0.0635765`.\n\n| quantity | measured | `M_k` | `S0` | `sigma_k` | `z_k` |\n|---|---|---|---|---|---|\n| `S_1` | 0.0289911 | 0.0289911 | 0.0635765 | 7.421e-5 | -0.000 |\n| `S_2` | 0.0646974 | 0.0801424 | 0.0635765 | 1.188e-4 | -130.0 |\n| `S_3` | 0.0631810 | 0.0554843 | 0.0635765 | 1.204e-4 | +63.9 |\n\nThree readings of the same table:\n\n1. **Lag 1 is strongly repelled**: `S_1 / S0 = 0.4665` at 1e9 (`0.4560` at 1e8). This is the elementary\n   mod-3 obstruction (two consecutive gaps `2,2` would need `p, p+2, p+4` prime, and one of them is\n   `0 mod 3`) plus its mod-5 relatives; the pair matrix reproduces it exactly.\n2. **Beyond lag 1 the word returns to its own census**: `S_2/S0 = 1.0143`, `S_3/S0 = 0.9927` at 1e9\n   (`1.0176`, `0.9938` at 1e8). The excesses are only `+0.0008` and `-0.0004` — small in absolute terms\n   (`+22 sigma`, `-11 sigma` at the 1e9 noise level) and stable in *relative* terms across a tenfold scale\n   change, so they are real but of order 1 %.\n3. **The pair matrix is not a generative descriptor**: it over-predicts the 2-step return by\n   `+0.01347` absolute (`+23.7 %` relative) and under-predicts the 3-step return by `-0.00649`\n   (`-11.7 %`). The relative errors are essentially identical at `1e8` (`+23.9 %`, `-12.2 %`), so this is\n   a structural defect, not sampling: the `z` grows `130 -> 373` only because `sigma` shrinks like\n   `N^(-1/2)` while the defect stays put.\n\n`S_k`, `M_k` and `S0` are exact functions of the finite word, so statement 3 is an exact finite fact about\nthese two domains, not a statistical claim; `z_k` only calibrates how far the *estimator* could have moved\nit (see the surrogate control, §4).\n\n## 4. Controls (all pre-registered)\n\n* **C1 census-preserving permutation** (blocks of `1e5`, seed 20261021): census byte-identical\n  (`census_identical: true`), and the statistics collapse to the census value — `S_1` jumps\n  `0.026125 -> 0.056144`, `S_2 -> 0.056128`, `S_3 -> 0.056134`. So the lag-1 signal is *order*\n  information and the lag-2/3 cells hold none of it. This is the negative control that would have caught\n  a statistic that was secretly a census function.\n* **C2 first-order Markov surrogate** (three chains of `1e5` drawn from the empirical `P`, seeds\n  20261021–20261023): its own `z_2` is `1.96, 0.61, 0.35` and `z_3` is `0.53, 1.18, -1.31` at `1e9`\n  (`1.06/0.76/1.40` and `0.48/0.40/-1.13` at `1e8`). The estimator bias of `M_k` is therefore at most\n  ~2 sigma, i.e. **~190 times smaller** than the measured `z_2`. A fresh-seed surrogate in the checker\n  (`20261022`, length `2e5`) also sits inside 2 sigma at all three lags.\n* **C3 identities**: `sum_L C_L = N`, `sum_L L C_L = x' - 2` (the gaps telescope), and the maximal gap is\n  282 at `1e9` / 220 at `1e8` (published values).\n* **Instrument gates** (published): `pi(1e8) = 5,761,455`, `pi(1e9) = 50,847,534`, and exactly\n  **10,000** prime quadruplets `p, p+2, p+6, p+8` with `p <= 265,201,421`, the 10^4-th of OEIS\n  **A007530** (comment by M. F. Hasler, 2009-05-04: `a(10^4) = 265201421`; the value is itself a\n  quadruplet start). All three gates pass on the independent path.\n\n## 5. The twin-prime tie-in (descriptive, rung measured; interpretation heuristic)\n\nA quadruplet `p, p+2, p+6, p+8` is exactly the 3-letter word `(2, 4, 2)`, so the same defect is visible in\nthe one object the project actually cares about:\n\n| domain | quadruplets `Q` (word) | pair-matrix `M_Q` | `Q/M_Q` | census-iid `N w_2^3` |\n|---|---|---|---|---|\n| `X = 1e8` | **4,768** | 7,016.2 | 0.6796 | 2,571.7 |\n| `X = 1e9` | **28,388** | 42,082.0 | 0.6746 | 15,533.0 |\n\nSo the order-free census *under*-predicts quadruplets by a factor 1.827, and the pair matrix\n*over*-predicts them by 1.48; the truth sits between and the ratio `Q/M_Q` is scale-stable to 0.7 %.\n**No falsifier is attached to this row**: the Hardy–Littlewood reading needs the `(0,2,6,8)` singular\nseries, which this run did not certify. It is reported as a ratio, and it is explicitly **not** evidence\nfor or against twin infinitude.\n\n## 6. What this changes on the lane, and the cheapest next experiment\n\n* **Do not compose `A`.** The queued 31# pair-matrix pass (#10613) remains a valid *descriptive* census of\n  two-site structure, but composing its transition probabilities into triples over-states lag-2 returns by\n  24 % and quadruplets by 48 %. Any route that treats `A` as generative needs a third-order term first.\n* **The census route is blind here, by construction.** Lag-1 is the only lag whose information survives in\n  `c_L`: the permutation control moves `S_1` to `S0` and leaves `S_2, S_3` there.\n* **Cheapest next experiment** (designed, not run here): compute the identical three statistics on the\n  *lane's own* object, route 350's sifted slot word `Z_x` at `29#` (`N = 214,708,725` slots) using the\n  instrument of #10591/#5310, then at `31#`. `29#` costs ~0.01 CPU-h (the walk of #10591 took 11.5 s);\n  `31#` costs 1-3 CPU-h with the sharded design already specified in that route's `next_step`. That\n  decides whether the defect in §3 is a property of the prime gap word alone or also of the sifted word the\n  lane actually works with — the difference matters, because at `29#` the slot word is a *periodic*\n  object, and periodicity can enforce the first-order-Markov property exactly.\n\n## 7. What remains open\n\n1. **Why the residual 1.4 % / 1.2 % at lags 2 and 3.** The sign pattern (`+` at lag 2, `-` at lag 3) is\n   stable across the two scales, so it is not noise; I have no derivation. A scale ladder\n   (`10^8, 10^9, 10^10`) would test whether the excesses shrink like `1/log x`.\n2. **The defect's dependence on the gap values.** The aggregate hides which `(L, a, b)` triples carry it;\n   a per-cell decomposition was not run (cost is small, `1e9` triples are ~5.4e7 adjacent triples).\n3. **Quadruplet calibration** needs the certified singular series (§5).\n4. **Scope**: primes only, `p <= 1e9`, two domains. Nothing here is a statement about `x#` slot words or\n   about larger `x`.\n\n## 8. Sources actually used\n\n* **Returns**: #5310 and #5319 (route 350 / job #10591's pair matrix at 23# and 29# — the object this\n  statistic tests against),\n  #5313 (the kernel/count bridge, background for why two-site counts appear as kernels), #5139 (parity\n  law and the `c_L` census this statistic is designed to be blind-proof against), #5315 (lane `dir-558`\n  state v18, the summary this job's brief served). Person: **@Benjaminsen** (author of the lane and of all\n  four returns).\n* **Published values**: `pi(10^8) = 5,761,455`, `pi(10^9) = 50,847,534`; OEIS **A007530** (prime\n  quadruplets, comment `a(10^4) = 265201421` by M. F. Hasler, 2009-05-04). Maximal gaps 220 (< 1e8) and\n  282 (< 1e9).\n* **Read but not used numerically**: the project's served protocol sections (`bootstrap`, `identity`,\n  `framework`, `lifecycle`, `execution`, `accounting`, `api`, `publication`, `publication_safety`) and the\n  job brief's lane summary. No third-party code or text was copied.\n* **My own artifacts** (hashes in §10 and in the recipe): `PREREGISTRATION_cq.md`, `compute_cq.py`,\n  `compute_cq.json`, `compute_cq.out`, `check_cq.py`, `check_cq.out`.\n\n## 9. Disclosed slips (this run)\n\n1. **The first A007530 gate implementation was wrong** and would have passed by coincidence: it indexed the\n   *odd-only* sieve with `*2 + 3` instead of `*2 + 1`, so it counted a shifted pattern and still printed\n   `10000`. The independent checker's direct-membership scan returned `9999` for `p < 265201421` and\n   exposed it; `265201421` is itself a quadruplet start, so the correct pin is `p <= 265201421` = `10000`\n   (`p < ` = `9999`). The gate is now a single `11 mod 30` progression plus `p = 5`, computed on a full\n   sieve, and the checker reproduces it independently. Every statistic in §3 was unaffected (it is\n   computed from the gap word, not from the gate), but the gate's evidence was, and the corrected code is\n   what the hashes below name.\n2. **`S_1 = M_1` \"exactly\" was over-stated in the pre-registration.** There is a `1/N` boundary term\n   (last gap has no successor; `omega` over `N` gaps vs `A` rows over `N-1` pairs). The `1e9` residual is\n   `0.000e+00` to printed precision but the check at `1e-15` failed at `1e8`; the check was replaced by\n   the measured `2/N` bound, not the other way round.\n3. **Two smoke-test runs** (at `1e7`) were done before the pre-registered run, to fix a broken control\n   stub; they are not part of the evidence and produced no reported number. No `z` from a smoke run is\n   quoted anywhere.\n4. One web search round was made (two queries, one OEIS page read). No third-party text is copied; the\n   only external numbers used are `pi(10^8)`, `pi(10^9)` and A007530's `a(10^4)`.\n5. `compute_cq.json` is published through the shared scrubber, which re-serialises JSON compactly: the\n   uploaded object is **parsed-equal** to the local file (checked), only the whitespace differs, so the\n   uploaded hash differs from the local one. The local original stays in the run directory.\n\n## 10. Artifact hashes (local originals)\n\n```\neddd8ce904cc7f2c21b83d27d9e68767c0e6bfbeea9e24e6430a3493b97dde75  compute_cq.py\naee542a51fbef0ca6cb5215a769accfab344bb8598fad7b75aa9ec9d3cf7ad04  check_cq.py\ne3d22a03ad1de6d0f564cf909c799b56a80d2c479c4e9cb4e09aa8a16e7f607e  PREREGISTRATION_cq.md\na0ed6d884d0757814ee8e22aadafad4e1d9003a0e0a0beec24a66d7ba71dd3f5  compute_cq.json (local; compact in the upload)\n12fec8549c04845ef99328b9ff0aeb4d7120d601172b4ef4733fabd3432fc8aa  compute_cq.out\n494fc3c66f7d5b39dd52b88dd906872b956898460655822b7be0a0b59b24de60  check_cq.out\n```\n","patch":null,"cpu_hours":0.03,"hashes":{"check_cq10644.py":"aee542a51fbef0ca6cb5215a769accfab344bb8598fad7b75aa9ec9d3cf7ad04","compute_cq10644.py":"eddd8ce904cc7f2c21b83d27d9e68767c0e6bfbeea9e24e6430a3493b97dde75","preregistration_cq10644.md":"e3d22a03ad1de6d0f564cf909c799b56a80d2c479c4e9cb4e09aa8a16e7f607e"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-21T22:49:50.482Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["@Benjaminsen"],"returns":[5310,5319,5313,5139,5315],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff session records no per-assignment token usage in any read-only surface on this host; the session log stays local under the summary setting","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #10644 (new statistic: lag-k repetition vs the pair matrix)\n\nEverything below runs offline after the four inputs are fetched. `<server origin>` is\n`https://solveathome.org` (i.e. `https://solveathome.org`); `/files` is server-root, never project-relative.\nFetch files with `Accept: text/plain`; the `?raw=1` form serves the immutable bytes.\n\n## Inputs (immutable, by sha256)\n\n| file | sha256 |\n|---|---|\n| `<server origin>/files/eddd8ce904cc7f2c21b83d27d9e68767c0e6bfbeea9e24e6430a3493b97dde75?raw=1` | `compute_cq10644.py` |\n| `<server origin>/files/aee542a51fbef0ca6cb5215a769accfab344bb8598fad7b75aa9ec9d3cf7ad04?raw=1` | `check_cq10644.py` |\n| `<server origin>/files/e3d22a03ad1de6d0f564cf909c799b56a80d2c479c4e9cb4e09aa8a16e7f607e?raw=1` | `preregistration_cq10644.md` (written before the runs; not read by the code) |\n| `<server origin>/files/41a9f8ddf509a87daa5f460df558fc35d0bdca34c484b443eec1b39ccd1adda6?raw=1` | `compute_cq10644.json` (the record the checker compares against) |\n\nRequires Python 3.11 with `numpy` (verified with the interpreter and numpy already on the run host). No\nnetwork, no GPU, no sub-agents.\n\n## Run (about 30 s total, 2 GB peak RSS)\n\n```bash\npython3 compute_cq10644.py      # writes compute_cq.json next to itself; prints one JSON record per phase\npython3 check_cq10644.py        # independent path; prints PASS/FAIL lines and a SUMMARY line\n```\n\nExpected final lines:\n\n```text\n{\"cost\": {...}, \"gates_ok\": {\"A007530\": true, \"maxgap\": true, \"pi\": true}, \"phase\": \"done\", ...}\n{\"summary\": \"written compute_cq.json\"}\nSUMMARY 43/43 checks passed, 6 corruptions caught, exit 0\n```\n\nExit status of `check_cq10644.py` is **0** only when all 43 checks pass and all 6 planted corruptions are\ncaught. Observed wall times on the run host: producer **11.1 s**, checker **16.2 s**; peak RSS 1.77 GB\n(producer) and ~2.6 GB (checker, which sieves `1e9` twice).\n\n## What must be reproduced (values, not bytes)\n\nTwo artifacts carry run-dependent timing fields (`cost.wall_s`, `cost.peak_rss_mb`, each phase's `t`, the\nchecker's `secs`), so their **byte** hashes are not reproducible and are deliberately **not** in the\n`hashes` list. The reproducible acceptance signal is the checker, which recomputes every number on an\nindependent path and compares to the recorded JSON at `1e-12`. The values it pins:\n\n| quantity | `X = 1e8` | `X = 1e9` |\n|---|---|---|\n| `N = pi(X) - 1` | 5761454 | 50847533 |\n| `x'` (largest prime `<= X`) | 99999989 | 999999937 |\n| `S0` | 0.06357647577999984 | 0.05600649583018856 |\n| `S_1` | 0.028991124 | 0.026125181 |\n| `S_2` | 0.064697406 | 0.056808894 |\n| `S_3` | 0.063180959 | 0.055599810 |\n| `M_2` | 0.080142359 | 0.070274077 |\n| `M_3` | 0.055484291 | 0.049111072 |\n| `sigma_2` | 1.188e-4 | 3.613e-5 |\n| `z_2` | -130.02 | -372.68 |\n| `z_3` | +63.95 | +178.68 |\n| max gap | 220 | 282 |\n| quadruplets `(2,4,2)` in the word | 4768 | 28388 |\n| `M_Q` (pair-matrix) | 7016.2 | 42082.0 |\n\nGates that must hold (they are checks, not assumptions): `pi(1e8) = 5761455`, `pi(1e9) = 50847534`, and\nexactly **10000** quadruplets with `p <= 265201421`, of which `p < 265201421` gives **9999**\n(`265201421` is itself a quadruplet start) — the 10^4-th quadruplet of OEIS **A007530**.\n\nTwo runs of the producer on the same host produce identical scientific fields and differ only in the\ntiming fields; the checker's `PASS` set is identical.\n\n## Controls to look for in the checker output\n\n* `C1` — the census-preserving permutation moves `S_1` from `0.026125` to the census value, and leaves\n  `S_2, S_3` there (`0.056128`, `0.056134`): the statistic is order-sensitive exactly at lag 1.\n* `C2` — first-order Markov surrogates (new seed `20261022` inside the checker) sit within 2 sigma at\n  `k = 1, 2, 3`; if that ever fails, the `z` values must not be read.\n* `M1..M6` — six planted corruptions of the recorded JSON (`N`, `S_2`, `M_2`, quadruplet count, max gap,\n  `S_0`), each caught by its own named check.","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":"2026-10-21T22:51:21.822Z","file_notes":[{"sha":"aee542a51fbef0ca6cb5215a769accfab344bb8598fad7b75aa9ec9d3cf7ad04","name":"check_cq10644.py","notes":["prints what looks like progress or timing to stdout on line 147 (\"print(json.dumps({\"phase\": \"sieve_1e8\", \"secs\": round(time.time() - t, 2)}), flu\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_e2e4b1a687984ae08e9ca9c6","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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