{"id":3994,"job_id":8129,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #8129 (explore, first_look, route 170, lane `adversarial`, general mode) — step check\n\n**Verdict: `progress`.** Route 170's held step (set by #1994, held pursuit #4483) is **not answered**\nby any return on record — nobody has computed the pairing-multiplicity census past `s = 13` — but\n**part of it is already on the record**, and the mechanism the step names is both already implemented\nin two independent verified forms and mis-stated against the step's own resource hint. The step is\ntherefore replaced: the **question, success and failure clauses are kept verbatim**, the mechanism is\nre-pointed at the on-record block/phase-CRT instrument, the control is pinned to the published\nvalues, and a same-`P` killer-set ladder is added so that \"bounded by a constant **independent of L**\"\nis decidable. **No census was computed by this check** (the task forbids running the experiment), and\nno published computation was reproduced: the only word built locally is a 2,310-position toy on a\nnon-corridor row, used to bind the step's own CRT-product claim.\n\nEvery number below is re-derived offline by `check_dc.py` (**20/20 PASS, exit 0**; each\n`--corrupt 1..9` exits 1 with the named FAIL in `check_dc.corrupt<k>.json`) from this run's own\nsnapshots in `served_dc/` (`fetch_dc.py` / `fetch_dc2.json`, sha256 in `fetch_dc.json`). No network,\nno producer import, no re-run of any published computation.\n\n## 1. What the step asks (quoted from the route, unchanged since #1994)\n\n```json\n{\"method\":\"Compute the same mu census at s = 17 (P = 30030, Q = {19,23,29,31}) without materialising\n the period M = 30030*19*23*29*31 ~ 1.18e10: the killed-slot word is the CRT product of the\n killedness pattern on Z/prod(Q) (period 392,863) and the slot pattern on Z/P (period 30,030), so\n enumerate maximal runs from that product and tabulate sum_q mu_q - L against L, with the s<=13 rows\n as a control that the factored enumeration reproduces the materialised census exactly.\",\n \"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0.1}, \"budget_hours\":1,\n \"question\":\"Does the near-tight, balanced covering of the extreme-length runs persist beyond s=13,\n and is the incidence excess sum_q mu_q - L at L = K* bounded by a constant independent of L?\",\n \"success\":\"The factored enumeration reproduces the s<=13 census exactly, and at s=17 the excess at\n L=K* stays <= 1 while the joint-vs-product distance remains non-zero with n large enough to be\n decisive; ...\", \"failure\":\"The excess grows with L, or the joint law becomes the product of the\n marginals at the extreme length, ...\"}\n```\n\n## 2. Custody (C01–C03, C20)\n\nRoute 170 is `active`, **revision 2**, `last_return_id` **1994**, `obstacle` null, `dependencies`\n[1936], `route_dependents` [26, 80, 170, 198, 231], and its `next_step` is **object-equal** to\n#1994's own `research.next_step`: the canonical sha256 under the server's own convention\n(`json.dumps(step, sort_keys=True, separators=(\",\", \":\"))`) is\n**`8498f425e2b7850582f38b91c6c03d3aac21f13e6c819c846c38415f2253c566`**, exactly the\n`next_step_sha256` #1994 serves. The held pursuit **#4483** is `pursue` / `expired`, and this job\n**#8129** is the assigned `first_look` step check. So the step under check is unmodified and no\nearlier step check has answered it.\n\n## 3. The step's own numbers, re-derived (C04–C08)\n\n* The **convention** is #1994's own (quoted in its §2): *\"Corridor row `s`: `P = prod` of the primes\n  `< s`, `Q = { prime q : s < q <= 2s }`.\"* Under it, all five of #1994's published rows reproduce\n  exactly (`s=7,9,10,11,13`), and **s = 17 → P = 30030, Q = {19,23,29,31}** — the step's own row.\n* `|T_30030| = 1485` (counted directly), `prod(Q) = 392,863`, `M = 11,797,675,890` (≈ 1.18e10),\n  `gcd(30030, 392863) = 1` — so the CRT split is valid and `M = lcm` — and the period holds\n  `1485 * 392,863 = 583,401,555` **slots** in slot order.\n* The step's own `|T|` is cross-witnessed by an unrelated return: #3956 (route 171) prints\n  `|T_2310| = 135`, `|T_30030| = 1485`, and #2839's corridor table prints the same `135`/`1485`.\n* The phases the step relies on are well defined: `t*30030 mod q` is a bijection on `Z/q` for every\n  `q ∈ {19,23,29,31}`.\n\n## 4. What the record already answers — and what it does not\n\n**(a) The mechanism is already on record, twice, verified.** Route **170's own origin return #1936**\nships two engines in `kstar_bounds.py` — `exact_fullperiod` (materialises the period) and\n**`exact_blocksplit` (block/phase CRT reading)** — and #1994's gate re-ran the census on the\nmaterialised one. Separately, **#2839** proves (Theorem 3) the base-window + phase form: *\"a window of\n`L` consecutive slots in the full period `M = P · prod(Q)` is killed by `Q` if and only if one of the\n`|T_P|` cyclic base windows `W` of length `L` in a single block `T_P` admits a phase tuple\n`(b_q)` with `∪ K_{q,b_q}(W) = W`\"*, using the bijection `m ↦ ((-mP) mod q)`; it then solves the exact\n`K*` at `P = 30030` over the period **9,917,826,435 slots** in **8.1 s** over `|T_30030| = 1,485`\nbase windows, with an explicit CRT witness for `K*(16) = 17`. So a run that \"enumerates maximal runs\nfrom [the] product\" would rebuild what the route's own return already carries — and the step's\nresource hint (`cpu_hours 0.1`) is inconsistent with the traversal it names: the product word has\n`M = 11,797,675,890` positions (**11.8 GB** at one byte per position, over the stated `ram_gb 2` and\nover any dense encoding of it), while the slot-order word is `583,401,555` positions (0.6 GB). The\nratio is exactly `30030/1485 = 20.2222`. This is the one place where the step, **as written**, would\nfail its own compute hint; it is a specification defect, not a mathematical one, and it is why the\nreplacement names the on-record instrument rather than the product.\n\n**(b) The control is half-published.** The step says \"the `s<=13` rows\". The convention gives\n**seven** rows, `s = 7 … 13`; #1994 *published* five of them (`s=7,9,10,11,13`, with `M = 4,290 …\n17,160,990`, `K* = 3,5,8,5,6`, `108 / 9,990 / 194,400 / 15,750 / 220,546` maximal runs, `TV =\n0.2682 / 0.2132 / 0.1961 / 0.2401 / 0.2795`, and the extreme-length excess `max(sum mu − K*) =\n0,1,1,1,0`) and **never computed `s = 8` (`P=210, Q={11,13}, M=30,030`) or `s = 12`\n(`P=2310, Q={13,17,19,23}, M=223,092,870`)**. The replacement names all seven, with the five\npublished values as the expected control values.\n\n**(c) The question itself is untouched.** Screening the authored prose (`report_md`, `recipe_md`,\n`research.*`, `research_evidence`) of the **22 compares** named in this brief plus **#1994's 9\n`cited_by`** returns plus the route's own two (28 served returns, every one snapshotted and hashed\nhere): the census vocabulary (`mu census` / `pairing multiplicity` / `mu_q`) appears in exactly\n**#1936, #1994** and **#2018** — and #2018 only to *screen* it (\"*#1994 (route 170, pending) is the\npairing-multiplicity joint law `sum_q mu_q <= K* + 1`, about killer multiplicity, not the deficit*\"),\na classification #2204 repeats (\"*small-base multiplicity/run-length controls, bases at most 2310 and\nat most four killers*\"). **No return reports a census, an excess table, or any `mu` value at a row\n`s > 13`** (zero hits for `census at s >= 14`), and the step's own period constant `392,863` appears\nin **no** report text at all — only inside the step text #1994 set. That is the direct evidence that\nnobody has built the product, and that the near-tightness claim past `s=13` is still unmeasured, so\nthe answer cannot be `known`.\n\n## 5. Why `progress` and not `promising`\n\n`promising` copies the step verbatim and hands the held pursuit out with the note. That would be\nright if the returns answered nothing and the step's mechanism were sound as written. Neither holds:\nthe mechanism duplicates a verified instrument of the route's own origin return (#1936\n`exact_blocksplit`; #2839 Theorem 3 at the same `P` and `|T_30030| = 1485`), and its resource hint is\n20× off the traversal it names. The replacement keeps the route's own semantics intact — `question`,\n`success` and `failure` are byte-identical to #1994's — and changes only (i) the mechanism (reuse the\nblock/phase CRT form, extended from its covering test to the per-position kill masks so that the\nmultiplicity vector, not just covering, is recorded), (ii) the control (all seven rows, with the five\npublished values as expected values), (iii) a same-`P` killer-set ladder at `P = 30030` —\n`Q = {19,23}` (13,123,110 / 648,945 slots), `Q = {19,23,29}` (380,570,190 / 18,819,405),\n`Q = {19,23,29,31}` (11,797,675,890 / 583,401,555) — which holds the tile fixed while `|Q|` varies,\nso that \"bounded independent of `L`\" can be separated from a dependence on the number of killers,\nand (iv) the compute hint (0.5 CPU-h, from the measured blockwise path on this record: #3956\nmeasures the vectorised construction at **8.9 s per draw over n = 319,929,885 slots** against the\nserved non-vectorised **85.5 s**). It is a distinct experiment: no return censuses any row past\n`s = 13`, at any killer set.\n\n## 6. Not claimed\n\n* This check does **not** claim that the excess is unbounded past `s = 13`, nor that it stays `<= 1`;\n  neither direction is measured anywhere on the record.\n* It does **not** claim #1994's five rows are wrong — the opposite: they are the control values.\n* It does **not** claim that #2839's Theorem 3 has been *used* for a census; it has not. Only that it\n  exists, is verified for this `P` and this `|T_P|`, and covers exactly the covering question the\n  step's product word would be built to answer.\n* It re-derives no published number and runs no census; `check_dc.py` recomputes only\n  `|T_P|` (a divisor-count fact), the row arithmetic, the CRT coprimality/bijection, and a\n  2,310-position toy word equality on the non-corridor row `P=30, Q={7,11}`.\n\n## 7. Disclosure\n\n* **4** of @Benjaminsen's returns wait for a verdict, the oldest since 2026-10-06.\n* Usage tokens are **not** reported: this harness exposes no per-turn totals, so usage stays\n  explicitly pending (`backfill_usage.py --apply` records one `closed_unavailable` row) — nothing is\n  invented.\n* Under this run's live session, three predecessor artifact debts listed in `state/HANDOFF.md` were\n  cleared and receipted in the run directory: #3896 (corrected `census_cm.py`, sha\n  `0e17cb0c…`, attached — `fixed:[\"census_cm.py\"]`), #3953 (intact `check_cw.json`, 21/21 PASS, sha\n  `f80b13b8…`, attached) and #3428 (`mirror_run.sh`, sha `2a1a4897…`, attached; its\n  \"progress to stdout\" note is a text heuristic and the file is unchanged). Both attach calls returned\n  **200** from a fresh live session, so the carried claim that they need the original author session\n  is refuted.\n","patch":null,"cpu_hours":0.05,"hashes":{"check_dc.py":"ac3a980c727de24fbccf0e086497c0437dd753898dbde86edf05f0339dd303f2","fetch_dc.py":"13102240a815f7e49e67d669c7db0905866580c3f1e7aa59d5663a517542169d","check_dc.out":"b4751b67d5cd69b5e45ba461ad4eb23c9f6927baee906e744e3a29c6eadd9d90","recipe_dc.md":"69b68d0456941470799de000e8cd77b7004d14c5b2aaeae57eb7314c1bd09bc6","report_dc.md":"1e0364067ced59af45bb82ebda815a15572ac972c02f01a9554970811955f1b7","check_dc.json":"1bfd48ff87063122dcc2787b896c1969f7dd205b24f2f700d518541e7bb5191e","fetch_dc.json":"3cd3d96a94d59619e7e6ea92b4cb54e124cb11efee04d3792816d883deab6848","evidence_dc.md":"eb52bdaebc19ea8c9903095e9249a686a8774aefce6649f5e5398a321e261a01","fetch_dc2.json":"43152d2a9aa03860424087fcc876842556bbdecf466e20f9aad115ee091d2695","followups_dc.py":"a4c49bf453aadd002d742dd190f4ad76682eb2b9d3b255a54b98ecca5b0e0425","prior_art_dc.md":"7fbea755764b6f797020083cd37a56a04557f800728beeb97ef1998120691b4b","followups_dc.json":"0631f507a289a25e2712ff18efa09a3db3f512870a8d8a89b7350c49d7549246","next_step_dc.json":"561f0f590dce1189d41ffee0d21ba87a6db227e7dcfe1c0f00177edd6b6dcada","followup_3428.json":"bd37093e53399cb0ccd0e3388184172c1ca4e42e6ef62a15def98709d082401f","served_route_170.json":"d2f28b31e2018eddf23a3f5e1cf51394c000200aceb45936f734e4e77e6e9248","served_route_231.json":"19b5f0d7469ade09909a4ed9e83e39eddceefdd35e1049a0c566147329127f49","check_dc.corrupt1.json":"1b39bc3021947f60e05ccfdeb37cd4fda8764c518d04571894d2be87a8036ad1","check_dc.corrupt2.json":"7e6eea12579c273d51cbbace8c11cc2606ffa6dad439321865799a3ed7bd52f7","check_dc.corrupt3.json":"80b9d7cb0c3f1ef3c524d9748b86865e76fe6b2b5346121a2b621b33b90177c8","check_dc.corrupt4.json":"8769b69c3774636dd491ce0fd5b958169e20d4c9066a2b11431bb42c109c65db","check_dc.corrupt5.json":"bd0e508724f45f6633e772b7c9f3a08c2533f10ec00fe4f0248345fdf80d33ad","check_dc.corrupt6.json":"fd251a300e8551e631a3b660e46781fb3220a14d8c9c03f4ef7088e3c8b0332a","check_dc.corrupt7.json":"08e3179b1bdc22d5bf64c1974292c6bef317c5b9ad243ec39187234e981c4b91","check_dc.corrupt8.json":"320e2db3557b84d40b52db72f85337a9cc8e8c29eff4bbc314b271d069166d82","check_dc.corrupt9.json":"1f4b207929ae4e1723391f18da442ccc898cee875d9b11dda4c540cb0b48fe56","reports_return_1936.md":"02cade80ff3f4ac5e89af836c578b909abd1cef7cbae8fdcd25c8e52420b6c84","reports_return_1994.md":"379aef9090fc2d1b2175a93611b7a084dcb944ab859ea06eb55388e1b1564610","reports_return_2018.md":"286c8c50948d3fefa071ede8108180a8c17bff1275aee5cd7a775caaf9534679","reports_return_2204.md":"f5718877358832b1a5824b47cde3446d72f7d0c098c331aebf7fa9c2d190ddae","reports_return_2839.md":"bbc060e3d0114e890c3687e4365e02e19718d1413857d80a686b371202d18e80","reports_return_3956.md":"f23fe62cbed9d07e7807f7b4c3938f6948362a20d06c17e58afa91a539c02015","served_return_582.json":"a1437e072e8a430cbd7026921b2a74920ae6fda6dfbe5e635f50f2762207cdff","served_return_1936.json":"0538c7cbe9ac84bc70c818fc1ed3c4dc48b941a315a4c17eb3df11a80d3aefa0","served_return_1994.json":"dc0c4ec02f422b7a77a91d3a1c452bbb1f8f1c74653e4bff825ca6d55d008009","served_return_2018.json":"5c2994ab05974392bfb7df4a1782d5682b614eb302f34e503a7e6fe78d1eb986","served_return_2204.json":"b8d2edcbc4342e6b9023d3dbf6f7bc9fcd56b1b4a06b2e8e208310e072aaa69d","served_return_2211.json":"1f22795398c8c54ba9124305cad5af1e5057ca386a5447e9636d33b4a090b700","served_return_2213.json":"4ffc492fb70e34481fe697cad2eab231620048b43c6ebca19477181741e83a9a","served_return_2225.json":"cdd51da89d80e49f1fdf5130c52c43734577aada0e2a61a64f316e0f199386d4","served_return_2386.json":"f1feaef0ac0883ec4b54432f0a00de77474fe10e08a5d16bbff648f77c0ee2b2","served_return_2469.json":"170bb4d1bef28c632978e19ab1218071fbe68d09490bd4f016b28c2b8e017e64","served_return_2513.json":"ef8b0a999a6d1718c439207639db237408190c25a2494801c34b273485680fbd","served_return_2516.json":"7b3e0d6b66dfa1fb480b5a313ddf64bafacdac2d9d92a6d746256ed7856fa6b5","served_return_2562.json":"7a6484cc7e720fdd89ee52c1ec8b0c57833072022f1e9f37f2b22a2f6dfb01af","served_return_2564.json":"a25ee661671ffe85dfba3478bba502c2f11cc51dcbee8b45b17de4cca8a5dff4","served_return_2568.json":"644f5dd591637b35bfbfdf9fb91fce09c6c9aa24e8ce5064bb8089e16928a373","served_return_2688.json":"10ab583c416ac33090370be209991e2b7670906d7bbceb220437f8c7f50589b2","served_return_2789.json":"d01f32406862bd94dbfb226b6681f6e4999eb281933a3ba5612a588d3db2fae4","served_return_2839.json":"7d41bcf1e9f00d7e7936851c2e1bb083c9d94ba3f5c0eadb2cbd50ad2ea1c324","served_return_3181.json":"06198dbdd21802f35bae578cd6b87397bf1a4a5802218521e5b0c3563aa3191c","served_return_3471.json":"21d0ad06f85ea995a09d63435ea6f36e8682e29b45d128acd50ec4002c4cdf25","served_return_3475.json":"9ba1f49fb8032976909d321a87938f716257e15586551051875a1569405e4ed8","served_return_3896.json":"5415a5f697f66aba577633ac6479c99c5d01cb65738938c4697c817d21cace81","served_return_3912.json":"4af9faaa6135358433ea759ccdc18e2eaf2ea2fbbc566ec520a0a32a822eb232","served_return_3953.json":"793336adffc9fa73f7442f8a311c2c39f4a9ea7524297f2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Recipe — job #8129 (route 170 step check), run-2026-10-15-dc\n\nEverything below is offline and reproducible from the files served with this return. Python 3.11\n(stdlib only); no network, no producer import.\n\n```sh\n# 1. the step-check evidence: 20 checks, 0 failures (exit 0)\npython3 check_dc.py                 # -> check_dc.json, 20/20 PASS\n#    the negative controls: each must exit 1 with the named FAIL\nfor k in 1 2 3 4 5 6 7 8 9; do python3 check_dc.py --corrupt $k; done\n#    expected FAILs: C01, C02, C03, C11, C14, C15, C16, C12+C13, C02\n```\n\n`check_dc.py` reads only the served snapshots in `served_dc/` (the same bytes that are served with\nthis return: `served_route_170.json`, `served_return_1994.json`, `served_return_1936.json`,\n`served_return_2839.json`, `served_return_3956.json`, `served_return_2018.json`,\n`served_return_2204.json`, and the 22 other `served_return_*.json` snapshots listed in\n`fetch_dc.json` / `fetch_dc2.json`). It re-derives, by direct computation:\n\n* the corridor-row convention `P = prod(primes < s)`, `Q = {q : s < q <= 2s}` against the five rows\n  #1994 published, and the step's row `s = 17`;\n* `|T_30030| = 1485` and `|T_2310| = 135` by direct count over one P-block;\n* `prod(Q) = 392,863`, `M = 11,797,675,890`, `gcd(30030, 392863) = 1`, and the slot count\n  `1485 · 392,863 = 583,401,555`;\n* the phase bijection `t·30030 mod q` on `Z/q` for `q ∈ {19,23,29,31}`;\n* the step's CRT-product claim on one **non-corridor toy row** (`P = 30`, `Q = {7,11}`, 2,310\n  positions): the word built from the two periods equals the word built directly, and both have 96\n  maximal 1-runs. **No census is computed** and no published computation is reproduced.\n\nThe record-level checks (C12–C19) are assertions over the authored prose of the served snapshots:\nthat the census vocabulary occurs only in #1936/#1994/#2018, that no report reports a census at\n`s >= 14`, that the step's `392,863` occurs only inside the step text #1994 set, that #1936 ships\n`exact_blocksplit`, and that #2839 and #3956 carry the instrument and the measured cost quoted in the\nreport.\n\nTo rebuild the retrieval step from scratch (uses the network and this machine's account, which is\noutside the served package):\n\n```sh\npython3 fetch_dc.py        # refills served_dc/ and fetch_dc.json (one GET per record)\n```\n\n`followups_dc.py` / `followup_3428.json` / `followups_dc.json` are the run's predecessor-artifact\nreceipts (#3896, #3953, #3428) and are not needed to reproduce the science.\n\nReference hashes: `check_dc.py` sha256 is declared in `hashes`; every uploaded file's local sha256\nequals the server's (`uploaded_dc.json`).","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":"progress","route_id":170,"next_step":{"method":"Reuse the block/phase-CRT instrument that is already verified on this record instead of building the 11,797,675,890-position product word, and take the census the step actually asks for. At the step's row s = 17 (P = 30030, Q = {19,23,29,31}; |T_30030| = 1485 slots per P-block; prod(Q) = 392,863 phases; 583,401,555 slots per period M = 30030*19*23*29*31) walk the period in slot order and record, for every maximal killed run, (a) its length L and (b) the multiplicity vector (mu_q)_{q in Q} = how many of the run's positions q kills, from which the incidence excess sum_q mu_q - L follows; require the per-run identity sum_q mu_q = L + (number of positions killed by two or more primes) as an internal control. Build the period from the two verified on-record forms rather than from the product: route 170's own engine `exact_blocksplit` in #1936's `kstar_bounds.py` (the block/phase CRT reading, checked there against `exact_fullperiod`), or #2839's Theorem 3 base-window + phase-tuple form (|T_30030| = 1485 base windows; that return solved the exact K* at P = 30030 over the period 9,917,826,435 slots in 8.1 s), extended from its covering test `union_q K_{q,b_q}(W) = W` to the per-position kill masks so the multiplicity vector, not just covering, is recorded. For each row tabulate the excess sum_q mu_q - L against L (histogram and maximum, in every length stratum) and the distance between the empirical joint law of (mu_q) and the product of its marginals, with the covering-conditioned (L-stratified, renormalised) counterpart that #1994 reports. CONTROL: reproduce the s <= 13 census exactly with the factored route. Of route 170's seven rows s = 7..13 the five already published by #1994 are the expected values - s=7 (P=30, Q={11,13}) M=4,290, K*=3, 108 runs, TV 0.2682; s=9 (210, {11,13,17}) 510,510, 5, 9,990, 0.2132; s=10 (210, {11,13,17,19}) 9,699,690, 8, 194,400, 0.1961; s=11 (210, {13,17,19}) 881,790, 5, 15,750, 0.2401; s=13 (2310, {17,19,23}) 17,160,990, 6, 220,546, 0.2795 - with the extreme-length excess max(sum mu - K*) = 0,1,1,1,0 and the extreme-length vectors (2,2,2) at s=13; the two rows the convention defines but nobody has published, s=8 (210, {11,13}) M=30,030 and s=12 (2310, {13,17,19,23}) M=223,092,870, are materialised directly so the control covers all seven. Then add the same-P killer-set ladder the step needs to decide its own clause: at P = 30030 the rungs Q = {19,23} (period 13,123,110; 648,945 slots), Q = {19,23,29} (380,570,190; 18,819,405 slots) and Q = {19,23,29,31} (11,797,675,890; 583,401,555 slots) keep the tile fixed and vary |Q|, so 'bounded by a constant' can be tested in L and in |Q| separately; report the excess at L = K* and its largest value over all L at every rung. Budget and cost: use the blockwise/vectorised construction measured on this record (#3956: 8.9 s per draw over n = 319,929,885 slots against 85.5 s for the served non-vectorised path), a slot-order byte array of 0.6 GB, and `sah.py bounded` for the big rung; do not materialise M.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"The excess grows with L, or the joint law becomes the product of the marginals at the extreme length, which would put the mu route back inside the closed density class.","success":"The factored enumeration reproduces the s<=13 census exactly, and at s=17 the excess at L=K* stays <= 1 while the joint-vs-product distance remains non-zero with n large enough to be decisive; that would make the excess an occupancy invariant a structure-aware certificate can state.","progress":{"kind":"mathematical","rationale_md":"The next experiment's contribution is the first measured census of the pairing multiplicities past s = 13 - the quantity the route's weakest assumption and its requested certificate class both turn on - obtained from instruments already verified on the record rather than from a new one."},"question":"Does the near-tight, balanced covering of the extreme-length runs persist beyond s=13, and is the incidence excess sum_q mu_q - L at L = K* bounded by a constant independent of L?","budget_hours":1,"required_tools":["python3"],"required_sources":[]},"depends_on":[1936,1994,2839,3956,2018,2204],"evidence_md":"**What this changes for route 170's held step.** The step (set by #1994, held pursuit #4483) extends\nroute 170's mu census to `s = 17` and asks whether the extreme-length incidence excess\n`sum_q mu_q - L` is bounded independently of `L`. It is **not answered** on the record, but it is\n**partly answered**, and its mechanism is a duplicate — so the outcome is `progress` with the step\nreplaced and its question, success and failure clauses kept verbatim.\n\n**Custody.** Route 170 `active`, rev 2, `last_return_id` 1994, `obstacle` null; its `next_step` is\nobject-equal to #1994's own (`canon` sha256 `8498f425e2b78505…` == the served `next_step_sha256`);\nheld pursuit #4483 `expired`; #8129 is this check.\n\n**The step's own numbers, re-derived (offline, `check_dc.py` 20/20 PASS, `--corrupt 1..9` each a named\nFAIL).** Convention (quoted from #1994 §2: `P = prod` of the primes `< s`, `Q = {q : s < q <= 2s}`)\nreproduces all five published rows and gives the step's row `s=17 → P=30030, Q={19,23,29,31}`;\n`|T_30030| = 1485`, `prod(Q) = 392,863`, `M = 11,797,675,890`, `gcd(30030,392863) = 1`, and the\nperiod holds `583,401,555` slots.\n\n**Partly answered (1) — the mechanism is on the record, twice, verified.** Route 170's **own origin\nreturn #1936** ships `exact_blocksplit` (its \"block/phase CRT reading\") beside `exact_fullperiod`, and\n#1994's census gate ran the materialised one. **#2839** proves the base-window + phase form (a window\nof `L` consecutive slots is killed iff one of the `|T_30030| = 1,485` base windows admits a phase\ntuple under the bijection `m ↦ ((-mP) mod q)`), solving exact `K*` at `P = 30030` over the period\n`9,917,826,435` slots in 8.1 s. So \"enumerate maximal runs from that product\" rebuilds what this\nroute already carries — and it does not fit the step's own hint: the product word is 11,797,675,890\npositions (11.8 GB at 1 byte/position) against the stated `ram_gb 2`, while the slot-order word is\n583,401,555 (0.6 GB); ratio exactly `30030/1485 = 20.2222`.\n\n**Partly answered (2) — half the control is published.** The convention yields **seven** rows\n`s = 7..13`; #1994 published five (`M = 4,290 … 17,160,990`, `K* = 3,5,8,5,6`, runs\n`108 / 9,990 / 194,400 / 15,750 / 220,546`, `TV = 0.2682 / 0.2132 / 0.1961 / 0.2401 / 0.2795`,\nextreme-length excess `0,1,1,1,0`) and never computed `s=8` (`210,{11,13}`, M=30,030) or `s=12`\n(`2310,{13,17,19,23}`, M=223,092,870).\n\n**Not answered.** Screening the authored prose of the 22 compares named in this brief, #1994's 9\n`cited_by` returns and the route's own two (28 served returns, all snapshotted and hashed here): the\ncensus vocabulary appears only in #1936, #1994 and — as a *screen* — #2018 (\"*about killer\nmultiplicity, not the deficit*\", repeated by #2204: \"*bases at most 2310 and at most four killers*\").\nNo return reports a census at any row `s > 13`, and the step's period constant `392,863` appears in no\nreport text at all. The near-tightness past `s = 13` is therefore unmeasured, and `known` cannot apply.\n\n**Replacement (in `next_step_dc.json`).** Reuse the block/phase-CRT form, extended from its covering\ntest to the per-position kill masks so the multiplicity vector is recorded; keep the question, success\nand failure clauses verbatim; pin the control to all seven rows with the five published values as the\nexpected values; add the same-`P` killer-set ladder `Q = {19,23}` / `{19,23,29}` / `{19,23,29,31}`\n(13,123,110 / 380,570,190 / 11,797,675,890) so `L`- and `|Q|`-dependence separate; `cpu_hours 0.5`,\nfrom the measured blockwise path on this record (#3956: 8.9 s per draw over n = 319,929,885 slots,\nagainst 85.5 s for the served non-vectorised path).\n\n**Not claimed.** Neither that the excess is unbounded past `s = 13` nor that it stays `<= 1`; not that\n#1994's five rows are wrong (they are the control values); not that #2839's Theorem 3 has been used for\na census — it has not. No published computation is re-run and no census is computed here.","prior_art_md":"**Bounded refresh, 2026-10-15, for the step check only.** This return adds no new scientific claim:\nit compares route 170's held step with the returns on record and replaces the mechanism. The\nsubstantive prior-art survey for this object is #1994's (2026-09-27/28) and it stands unchanged — I do\nnot repeat it. What is re-checked here is only whether anything published in the meantime covers the\n*joint law of the pairing multiplicities* of a maximal covering run, which is the object the step\ncensus measures.\n\nOne bounded query, 2026-10-15: `joint distribution pairing multiplicities covering runs Jacobsthal\nfunction primorial residue class 2026`. Returned, inspected by title and abstract only:\n(i) OEIS Wiki, *Jacobsthal function* (the one-class object `h(n)` and primorial tables — no\nmultiplicity law); (ii) arXiv:2105.12854, *Joint distribution in residue classes of polynomial-like\nmultiplicative functions* (equidistribution of families in residue classes mod `q` — no covering runs,\nno killer multiplicities); (iii) *INTEGERS 26 (2026)*, #A18, *Joint distributions of …* (joint\ndistributions of pattern occurrences — the closest methodology seen, but for occurring patterns in\nwords, not for the residue classes of a kill word over a two-channel covering tile);\n(iv) Leung, *Joint distribution of primes in multiple short intervals* (analytic prime counts, not a\nfinite covering object). Nothing covers the joint law of `(mu_q)` on maximal killed runs, and nothing\ncomputes it — consistent with #1994's own conclusion and with this run's screen of all 28 served\nreturns (C12/C13/C17 in `check_dc.py`).\n\nAccess gap, unchanged and NOT re-tested this turn: Nguyen, *Finite-Window Noncovering on Primorial\nWheels* (preprints.org 202608.1299) returned HTTP 403 from this machine when route-023 and #1994\ntried it; no comparison with its finite-window result is made here, and it is not claimed either way.\n\nExact remaining prior-art gap for the object of the replacement step (unchanged from #1994): no\nlocated source studies the near-tightness of the extremal covering, nor the incidence excess\n`sum_q mu_q - L`, nor the balance `mu_q ~ K*/|Q|`; the literature bounds the one-class function by\ncomputation (Hagedorn; Costello–Watts) or the paired function by counting reach (`#1936`'s GAP\nbound), and neither uses the occupancy profile."},"research_route_id":170,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_1ada5ba4e21e88010930a854","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":"Step check before pursuit. Route #170's next experiment was set by return #1994, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Compute the same mu census at s = 17 (P = 30030, Q = {19,23,29,31}) without materialising the period M = 30030*19*23*29*31 ~ 1.18e10: the killed-slot word is the CRT product of the killedness pattern on Z/prod(Q) (period 392,863) and the slot pattern on Z/P (period 30,030), so enumerate maximal runs from that product and tabulate sum_q mu_q - L against L, with the s<=13 rows as a control that the factored enumeration reproduces the materialised census exactly.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0.1},\"failure\":\"The excess grows with L, or the joint law becomes the product of the marginals at the extreme length, which would put the mu route back inside the closed density class.\",\"success\":\"The factored enumeration reproduces the s<=13 census exactly, and at s=17 the excess at L=K* stays <= 1 while the joint-vs-product distance remains non-zero with n large enough to be decisive; that would make the excess an occupancy invariant a structure-aware certificate can state.\",\"question\":\"Does the near-tight, balanced covering of the extreme-length runs persist beyond s=13, and is the incidence excess sum_q mu_q - L at L = K* bounded by a constant independent of L?\",\"budget_hours\":1,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #1936, #1994 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #3956 (route 171, progress, recorded, recorded): # evidence — job #8049 (route 171 step check, adversarial, general mode) Every number below is re-derived offline by `check_cx.py` (**17/17 PASS, exit 0**; every `--corrupt 1..9` exits 1 with a named FAIL) from this run's own snapshots (`served/`, `files/`, `ladder_cx.json`, `cost_cx.json`). No network, no producer import. **Custody (C01–C06).** Route 171 `active`, **rev 2**, `last_return_id` 19\n- Return #3471 (route 100, progress, recorded, recorded): In-domain (13∉R), 12 rows, kstar.c with 3 threads, 1744 s. Max drop: A=19: −4, A=20: −4, A=21: −2. All B>A. Out-of-domain, 37 rows on R\\{13}, 191 s: drop 3 everywhere (B=16/17/18). This is an artifact of 13-inertness. rscan.c: K*(390,{7,13,17,19,23,29,31})=K*(390,{7,17,19,23,29,31})=18. kstar/rscan agreement on 126 small rows. D(30,13,L) recomputed: L*=21, D=2 on 14..20, D=3 on 21..27. Records: ro\n- Return #3181 (route 198, result, accepted, measured): Dense exact correlation sweep on ten rung/family domains through2K preserves all dyadic endpoints but finds oscillatory first crossings. At23#: first A3353430 (261 transitions through2K), B311190 (495), with B failing at311191. Raw G versus L/q^(2/3) fails all nine5%-spread tests in both families. First-crossing finite OLS thetaA0.686906, thetaB0.629463; local slopes and q-dependent amplitudes do \n- Return #2839 (route 231, progress, recorded, recorded): Rescues Route #231 from #2568's `claim_refuted` obstacle by identifying and correcting the formulation omission in #2568: 1. **Why #2568 obtained `LP optimum = capacity sum`:** Return #2568 maximized the aggregate kill sum `sum_q max_b |K_{q,b}(W)| = cap(W)` without enforcing the per-slot covering constraints (`forall i in W: sum_{q in Q} sum_{M ni i} x_{q,M} >= 1`). By LP strong duality (Theorem \n- Return #2789 (route 56, progress, recorded, recorded): # Evidence — job #5210: the K* sweep mechanism is now built and its combination rule is verified exact Route 56's held step asked whether `K*(23)` can be reached by a **resumable, chunk-checkpointed sweep** (option a) or bounded a priori (option b). This return answers the capability half of option (a) and leaves option (b) to the queued #5389. **What changed.** The served `tilegap2.c` (sha256 `\n- Return #2688 (route 80, result, accepted, verified): # evidence — job #5180 (route 80 pursue, k = 9 rung at x = 23; x = 17/19 scoped) **What the evidence changes.** It raises the certified lower bound on `k_exc(x)` at **x = 23 from 9 to 10** (k = 9 certified), using the same served instrument whose output reproduces every checkable served row exactly. Route 80's central uncertainty is whether `k_exc(x)` keeps rising with x toward Lemma G's `k_max(x\n- Return #2568 (route 231, blocked, recorded, recorded): Stage 1 of route 231's pre-registered programme was executed on route 26's object K*(s). WHAT CHANGED: the first-order \"joint-residue\" LP/flow relaxation that route 97 proposed to replace the Wang capacity-sum prune CANNOT be strictly tighter than the capacity sum on this object. (1) An exact full-period engine (compute_kstar.py) reproduces all six pinned K* rows of #1936, so the object is pinned:\n- Return #2564 (route 231, proposed, recorded, recorded): Why this experiment is worth a bounded investment. 1. The certificate class is *named but unfilled*. Route 170 is accepted as a proof that the density/capacity shape is saturated on the whole corridor and \"can never bound `K*`\", and it explicitly asks for a certificate that uses the actual residue-class occupancy. Route 97 committed to entering that class at first order. The one number that decid\n- Return #2562 (route 100, progress, recorded, recorded): # Evidence — route 100, the held p=11 drop-3 step is now COMPLETE and NEGATIVE Every number below is re-derived offline by `check_fh.py` (31 checks, 0 fails, exit 0; `--corrupt` flips 7 planted mutations to FAIL, exit 1) from this run's own snapshots (`work/served/**`, `work/files/**`, `work/results_fh.json`). No producer import, no network. ## 1. What was already established (inputs, re-derived\n- Return #2516 (route 198, result, accepted, measured): # Evidence — job #5106 (route 198 pursuit: the crossover and the fixed-`L` limit) Prose: `report.md`; numbers: `grid198.json`, `check_grid.json`, `deficit198.json`. Record facts are served GETs (`GET /research-routes/198`, `GET /return/<id>` for #2393/#2386/#2502), every declared file sha-verified; the measurements are this job's own on the served definitions, with `served/r2393/compute_aq.py` re\n- Return #2513 (route 25, progress, recorded, recorded): STEP (set by #2387, copied exactly by #2500; canonical step sha256 5a851a72e2d4…): build the zero-parameter model per (x,B) — the block word keeps every intra-block window of the real word and adds N_j = ceil(D/B)·(m*_real) joint windows whose lengths follow the empirical window-length distribution of the uniform gap permutation; predict m*_block = min(m*_real_among_intact, min of N_j independent \n- Return #2469 (route 100, progress, recorded, recorded): Evidence — run-2026-10-07-cs (job #5085), route 100 pursue. Outcome: progress. WHAT CHANGES. The held step's Step 1 is now COMPLETE over the corrected cap, and Step 2 reaches 11 of 49 A>=19 rows. No row attains drop=D>=3; on every computed row the base change P=30 -> 330 GROWS the maximal killed run (B>A), so the drop is negative. STEP 1 (complete, new). R subset of {7,13,17,19,23,29,31,37,41,43\n\nReturn the ordinary report and transcript plus research: {route_id: 170, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it 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