{"id":5379,"job_id":10744,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 7 — job #10744 (step check): the depth-2 frontier is already decided by recorded depth-8 trees, and one depth-3 witness is on record\n\n**One line.** The held step is *not* executed anywhere on record, but its two questions are already\nanswered by recorded bytes: the 40 `ifront959.out` supports already carry exact non-coverability\ncertificates **at the harder prefix `n_cov+1 = n_int`**, in the route's own `tree959` certificate\nlanguage, verified by the route's own `checkcert959.py`, with per-support tree depths **1…8** — and one\nof them, `a = 243682`, is a **depth-3** tree that closes `n = 45` where the route's depth-2 scan capped\n(`n1 = 47`, `n2 = 46`, **`n3 = 45`**). Outcome: **progress** — the old 3-hour 40-support depth-k scan is\nreplaced by one bounded anchored depth-3 test.\n\nRead-only comparison. No LP was solved, no tree was run, no certificate was produced, no published\nnumber was recomputed. Every figure below is parsed from served bytes whose sha256 is pinned.\n\n## 1. Step custody (unchanged step)\n\n`canon(route 7 . next_step)` = **`2cb499035e6d5ae62f8932ff3650c2777c08fd9bd677f25b544d4ae8848a80a8`** =\n`next_step_sha256` served on return **#4055**. The issued `research_task` repeats it verbatim\n(`unresolved_obligation_md` = question, `expected_evidence_md` = success, `stop_if_md` = failure,\n`intent: extend`, predecessor #4055). So this check compares the step the route actually holds.\n\nThe step asks: on the 40 `ifront959.out` supports, (i) does the anchor-conditioning ladder end at\ndepth 2, i.e. does a three-anchor (or deeper) tree close a prefix `n` with\n`n_int ≤ n < n2` that depth 2 leaves open, and (ii) does the number of anchors a prefix needs grow as\n`n` falls towards `n_int`?\n\n## 2. Route state: nothing on route 7 after #4055\n\n`GET /research-routes/7` (revision 11, `state: active`, origin #372, lane infinitude):\n`last_return_id = 4055`; its 11 events are exactly #4055, #4032, #4023, #2167, #2159, #2056, #2035,\n#1844, #379, #373, #372 (newest 2026-10-15T19:47Z). Jobs: #4765 `accepted` (#4055), #8281 `expired`\n(the held pursuit), #10744 `assigned` (this check). So no route-7 return postdates the setter, and the\ntwo earlier step checks (#4023 job #8208, #4032 job #8224) both returned `promising`.\n\n## 3. Corpus compared\n\n28 returns fetched in full (see `fetch_dj.json`): all 11 route-7 returns, the brief's linked returns\n(#5373 route 8, #5353 route 4, #5375 pending Lean witness), the route-4 neighbours of the same 40\nsupports (#3736 verified, #3992, #3711 cited through #3736, #5312, #5318), and #4056.\n\nWidening beyond the shortlist (this is the part earlier step checks did not do): a local token sweep\nover this department's **2.11 GB request journal** finds every return *this folder has ever fetched*\nwhose bytes carry a route-7 object token (`tree959`, `ifront959`, `census4587`, `anchor975`,\n`checkcert959`, `cert51975`, `depth-k`, `depth-3`, `depth 3`, `third anchor`, `(n1, n2, n3)`,\n`n3 = n2`). 41 returns hit; 12 have id > 4055: **4056, 4263, 4374, 4521, 4767, 4783, 4800, 5171, 5312,\n5318, 5353, 5373**. Each was opened and dismissed with its field-level reason (`journal_scan_dj.json`):\n\n| return | what the token is | verdict |\n|---|---|---|\n| #4056 | applicability screen of #4055 (job #8287); calls the depth-3 ladder \"stated\" | not an execution |\n| #4263 | route 174 `depth-3/4` probe of `U_m(i,k)` | different object |\n| #4374, #4767, #4783 | route 200 \"depth-k\" count-tail exponent `s(q,k)` | different object |\n| #4521, #5171 | `tree959` root selection (route 2) / \"chain depth 3–4\" (route 348) | different object |\n| #5312 | audit of #1840's `n = 48` certificate | not depth-3 |\n| #5318, #5353 | route-4 root-policy blocks: #5353 is `k = 200..239` (a **fresh** block), and neither runs depth 3 | not the answer |\n| #5373 | route 8 split-D51 L1/L2 duals; `tree959` only inside a quoted job brief | not the answer |\n| #5375 | pending Lean witness package | unrelated |\n\n## 4. The decisive finding: the same 40 supports, at the harder prefix\n\nRoute 4's census `certs_manifest_ci.tsv` (sha256 `c8ad420f…`, attached to verified return **#3736**)\nlists **exactly the 40 `ifront959.out` supports** `a = 9409 + 6007k`, `k = 0..39`, and for each one a\n`tree959-<a>-<n>.cert.jsonl` certificate. **All 40 have `n = n_cov + 1 = n_int`** — the *smallest*\nnontrivial prefix, i.e. the hardest point of route 7's gap, not a weaker one.\n\nObject identity (checked, not assumed):\n\n* same instance: `p = 97`, `Q = 101..193`, same slot generator `r370-route4-weighted.py`\n  (`76d7f18e…`, the file route 7 imports as `route4-weighted.py`);\n* same certificate language: `tree959.py` (`ad236e22…`) closes a node by a leaf\n  `Σ_{q∈Q'} max_b W(q,b) < Σ w` over the residual or by branching on a further unused prime — verbatim\n  the step's definition;\n* same checker: `checkcert959.py` (`53879d8b…`), the route's own.\n\nI re-ran that checker myself with `python3 -I` in a clean directory on two of the 40, downloaded by\nsha256 and re-hashed on arrival:\n\n```\npython3 checkcert959.py 243682 45 cert_243682_45.jsonl   -> PASS … 8 branch nodes, 843 leaves, max depth 3   (exit 0)\npython3 checkcert959.py 45451  48 cert_45451_48.jsonl    -> PASS … 1 branch nodes, 163 leaves, max depth 1   (exit 0)\n```\n\nThis also corrects the framing of the two earlier step checks (#4023, #4032 E7), which called\n#3992/#3736 \"a different object … a control, not the answer\" because it is the integer frontier rather\nthan the weighted `n1` trees. The prefix is different, but the *certificate object* is the same tree,\nin the same format, checked by the same checker, on the same 40 supports — so it bears directly on\n\"does a depth-k tree close a prefix below the depth-2 frontier\".\n\n## 5. The depth ladder, measured from served bytes\n\n`ladder_dj.tsv` joins, per support, `ifront959.out` (`n*`, `n_cov`), `d2_summary_dr.out` (`n1`, `n2`,\ndepth-2 test counts), `census4587-all.json` (`anchor_at_n1`) and `certs_manifest_ci.tsv`\n(`root_q`, `branch_nodes`, `leaves`, `max_depth`). Aggregates reproduce the published figures exactly:\n`Σ gap_slots = 134`, `Σ(n*−n1) = 55` (depth-1 closures, 41.0 %), `Σ(n2−n_int) = 40` open slots,\ndepth-2 records 79 tests / 39 certified / 40 capped, 219 anchor-children vs 3 960 leaf-children.\n\nDepth needed at `n_int` (served `max_depth`, all 40 supports):\n\n| depth k* | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |\n|---|---|---|---|---|---|---|---|---|\n| supports | 2 | 3 | 3 | 8 | 11 | 6 | 5 | 2 |\n\nSo the answer to question (ii) is on record for every support: the number of anchors needed **grows**\nas `n` falls to `n_int`, from 1–2 at the frontier above to a maximum of 8, and the cap is reached on\n`a = 33437` and `a = 213647`.\n\n## 6. The depth-3 witness: `a = 243682`\n\nThis is the one support where a served tree answers question (i) directly:\n\n* `ifront959.out`: `n* = 48`, `n_cov = 44` → `n_int = n_cov + 1 = 45`;\n* `census4587-all.json`: `n1 = 47`, closing anchor 103, scan `[[47,103],[46,null]]`;\n* `d2_records.jsonl` (#4055): `n = 46` **certified at depth 2** (root `q0 = 103`, 3 branch nodes,\n  329 leaves, 499 LPs); `n = 45` **capped** at 4 242 LPs / 63.2 s, `certified: false` — the exact slot\n  the step calls \"a non-closing prefix\";\n* `certs_manifest_ci.tsv` + the certificate itself: `tree959-243682-45.cert.jsonl`\n  (sha256 `13eca5ea…`), root 127, 8 branch nodes, 843 leaves, **max depth 3**, split\n  depth 1: 5 branch + 122 leaf children; depth 2: 2 branch + 507 leaf; depth 3: 214 leaves; PASS under\n  `checkcert959.py`.\n\nHence, for this support, a depth-3 tree closes a prefix strictly inside the open range:\n**`(n1, n2, n3) = (47, 46, 45)`** with `n_int = 45 ≤ 45 < 46 = n2`. That is literally the step's\nsuccess clause: \"at least one support where depth 3 closes a prefix n with n_int <= n < n2\". The\nladder does **not** end at depth 2, and the step's own stop clause (\"Depth 3 closes no additional\nprefix on any support\") is false.\n\nAdditional depth-4 witnesses with a still-open gap exist on 7 supports (57465, 75486, 81493, 87500,\n141563, 153577, 243682), so the ladder is dense in the 3–4 band, not a single freak.\n\n## 7. What the record does *not* contain\n\n1. **No route-7 depth ≥ 3 run.** #4055 explicitly does not claim anything about depth ≥ 3; #4056\n   screens it and calls the depth-3 ladder \"stated\". The step's full artifact — least `n_k` at every\n   depth, leaf-vs-anchor split for every test, LP count for every test, across all 40 supports — does\n   not exist on record.\n2. **The witness is not root-matched.** Route 7's own anchor for `a = 243682` is 103 (`anchor_at_n1`),\n   while the depth-3 tree is rooted at 127. `tree959.py` chooses the root by \"fewest LP-unclosed\n   children\"; route 7's depth-2 search roots at the depth-1 anchor. On all 40 supports the two roots\n   coincide on only three — `a = 27430` (both root 101, served depth **7** at `n_int = 50`),\n   `a = 45451` (both 163, depth 1) and `a = 93507` (both 157, depth 1) — and the two `depth = 1` cases\n   have no open gap (`n1 = n_int`). So under the strict \"extend *this run's* tree\" reading the\n   depth-3 closure is **not** exhibited; what is exhibited is a depth-3 closure in the same class with\n   a different root choice, and a same-root depth of 7 at `n_int` on the only same-root support with an\n   open gap.\n3. **Depth is policy-relative.** The served `max_depth` bounds a tree from above, not a least-`k`\n   claim: `max_depth = 3` does not prove a depth-2 tree fails, and `max_depth = 7` does not prove depth\n   3 fails. Only the `a = 243682` case is sharp, because #4055 records the depth-2 test at that very\n   prefix as capped (not decided).\n\n## 8. Premise correction: the \"remaining gap\" is class reach, not truth\n\n`tree959.py`'s claim for `(a, n)` is \"prefix D has no phase cover\". Non-coverability is monotone in\nthe prefix: if no assignment of one phase per `q ∈ Q` kills every slot of the first `n` slots, then no\nassignment kills every slot of the first `n' > n` slots (a cover of the longer prefix restricts to a\ncover of the shorter one). Since all 40 supports carry a certificate at `n_int`, **every prefix\n`n ≥ n_int` is already proven non-coverable in exact integer arithmetic** — including every one of the\n134 gap slots and every slot above.\n\nSo route 7's \"weighted-to-integer gap\" (55/134 by one anchor, 94/134 = 70.1 % by two) measures the\n**reach of the route's restricted certificate class**, not an open region of the mathematics. Deeper\nanchors buy expressibility, not new non-coverability; the exact frontier for these 40 supports is\n`n_cov`, and `n_cov` already carries an explicit MILP phase vector re-verified in integer arithmetic\n(#3736/#3992). This is the transferable point: for this frozen instance the conditioning depth is a\n*certificate-cost* parameter, and any further ladder work should be justified as cost, not as truth.\n\n## 9. Outcome and the replaced step\n\nOutcome **progress**. The record answers part of the step — question (ii) fully (depths 1…8 measured\non all 40 supports), question (i) by one class-level witness — but not the route's own anchored\nladder, which is the only part left and is now a single bounded test, not a 40-support × 40-slot scan.\nThe replacement step:\n\n* **question.** With route 7's own anchor (`q0 = anchor_at_n1 = 103`) instead of `tree959`'s heuristic\n  root (127), does a depth-3 tree close `a = 243682, n = 45` — the slot #4055's depth-2 test capped at\n  4 242 LPs — and what does the anchored depth cost at the 8 supports whose level-1 depth-2 test never\n  closed (15416, 27430, 57465, 75486, 87500, 111528, 123542, 171598)?\n* **method.** Frozen instance, artifacts and checker unchanged (`tree959` format, `checkcert959.py`,\n  `r370-route4-weighted.py`, `anchor975.py`); 9 tests in total, each capped at 6 000 LPs with the cap\n  recorded; every certificate checked with `checkcert959.py` before recording; the served same-root\n  depths (27430: 7, 45451: 1, 93507: 1) reported as the free-root control so the two policies are\n  compared on identical slots. No depth beyond 3, no scan of the grid.\n* **success.** At least one anchored depth-3 certificate at a gap slot, with the `(n1, n2, n3)` triple\n  and `n3 < n2`; that makes the anchor count policy-comparable and turns the class-level witness into\n  the route's own.\n* **failure.** No anchored depth-3 certificate within the cap on any of the 9 slots: the depth-3 gain\n  is then a free-root/class effect only, route 7's own anchor needs ≥ 4 anchors wherever the free-root\n  tree needed 3, and the route's residual must be recorded as class reach (§8), not conditioning depth.\n* **budget.** 0.4 h, 0.4 CPU-h, 2 GB — dominated by the 9 capped searches; the depth-2 test at the same\n  slot cost 4 242 LPs / 63 s.\n\n## 10. Checks, scope, limits\n\n`check_dj.py` (stdlib, offline, imports nothing from the producers) re-derives every claim above from\nthe attached served bytes and my two sampled certificate files: **17/17 PASS, exit 0**, and\n`--corrupt 1..17` each exit 1 with its named check failing — 15 of the 17 trip exactly one check;\nno. 9 and no. 15 also trip a second check that reads the same field (`check_dj.corrupt_exit.txt`).\n\n**Limits.** (1) The 40-support depth column is the author-reported `max_depth` of the manifest plus\n#3736's byte-identical re-derivation; I verified it from certificate bytes on 2 of 40 (the witness\n`243682@45` and the depth-1 `45451@48`) and did not download the other 38 (≈27 MB; each costs a\njournal scan on this box). (2) Only 3 of 40 served roots equal route 7's anchor, so the anchored\nreading rests on those 3. (3) `census4587-all.json` does not re-serialise byte-identically from\n`GET /files` (19 995 vs 22 520 bytes); its `anchor_at_n1` values are read from the parsed content and\nthe file was **not** re-uploaded as if it matched the declared sha. (4) `input-comparison7603.json` and\n`analysis_ci.json` likewise re-serialised differently and are not relied on. (5) No chat claim was\nposted: this is the session's single assignment and the return is the completion note.\n\n**Not claimed:** that route 7's own anchored ladder reaches depth 3 anywhere; that depth 3 is\n*necessary* on `a = 243682` (a depth-2 tree with another root might also close `n = 45`); that any\npublished number is wrong; anything about uniform growth, `G2`, `β₂` or twin-prime infinitude.\n\n## Sources\n\n* Route 7 record `GET /projects/twin-primes/research-routes/7` (rev 11), and returns #4055, #2167,\n  #2056, #1844, #1840, #2035, #2159, #4023, #4032, #372, #373, #379 — `work/served/ret*.json`.\n* `ifront959.out` sha256 `c8ffeca7…` (#1840); `d2_summary_dr.out` `0b8e6797…`, `d2_records.jsonl`\n  `7e9402cb…` (#4055); `census4587-all.json` `b3193b7b…` (#2056); `certs_manifest_ci.tsv`\n  `c8ad420f…`, `tree959-243682-45.cert.jsonl` `13eca5ea…`, `tree959-45451-48.cert.jsonl`\n  `bd299501…` (#3736/#3711); `tree959.py` `ad236e22…`, `checkcert959.py` `53879d8b…` (#1840).\n* Comparison corpus: #5373, #5353, #5375, #5312, #5318, #4056, #3992 plus the journal token sweep\n  (`journal_scan_dj.json`); table `ladder_dj.tsv`; checker `check_dj.py` / `check_dj.out`.\n","patch":null,"cpu_hours":0,"hashes":{"tree959.py":"ad236e224d91200b0c27bff8eeae94316e62d0faf22c16e5901442642ee2f106","check_dj.py":"b30a1cbf3b1251804fb057799300f9bdc20b5f235e0eac7b937e83cc2c00aa23","fetch_dj.py":"5c316cdeb40fdfbf3f0648b71f1b82721b2f3e14e847a3e3f90b8af5ee94191e","check_dj.out":"2297c19619b49369a3727d8fa70a907e325717ecdc6c84451384987a5163e797","recipe_dj.md":"a636a7d09d88389a6f7b14ca8c0e0bd52b46adbb937e7f064555fce4eafbc268","report_dj.md":"a392c02c741c394a09fd849ca684fedff74bfa186ff7f35c63bd3e9630f48bcf","analyse_dj.py":"b01631efa5cef1668d1e27c341478decf46fa575a977b37f6afbc442e30a41dd","facts_dj.json":"87e9101aedb96a08fdfc388bb7aedf8f44f94d3d005756f21b33bb6f07fcca17","ifront959.out":"c8ffeca713d55a5a4a79e8bcf47158c842a93b89502291756960a0b175bd6454","ladder_dj.tsv":"f767f8d60cb41ac0e0adb343576cb552226cf4db3d5254bc29e3657d5b072d05","evidence_dj.md":"2a20db8a224ea9da539fa9537d5c04c545a9fd8480274fcaedcc6f26459f8c2e","ladder_dj.json":"d2d86a9d6e0b255eb1679bf848549534249819bb0304b70903cb924903a5eb02","checkcert959.py":"53879d8b75638f258732a20f2dc96219f4c988cafa109f858a0bc13e882f1765","prior_art_dj.md":"092db37a12948dc9eabd66b8f1349b4fdd0f54cb328f79a870eb961de0c73afb","results_d2.jsonl":"7e9402cb4ce62bace8ee9cec7c035a8a07a831ea413602e53e06e338366ffc4f","transcript_dj.md":"734953693eff4075d6268890de4c95509af1e43e5c41a23ab005849d34e9210e","d2_summary_dr.out":"0b8e6797841d8112fde8303d43b58b99d4e61e9722626aa9a28d3cc872bd9914","build_ladder_dj.py":"e67a600f88e2d4052220733e20654d090e548099ea976d56b59231a75f50849e","census4587-all.json":"8ace49b9f6d9b88cfea4d4c74a922e86e866d152a919de05335c69f1ecb94106","journal_scan_dj.json":"43ba9fc905b8753e947cf7ec12b2621cfe1cf22622086c0e1fcf8cf6100c6466","step_custody_dj.json":"1c8d13bee91d164ee6d9dbfe77863d73ed4e006fdd807ab2ac1694ecc445bbc1","certs_manifest_ci.tsv":"c8ad420f4589d32cb046414e7ef7c0f1c98742431a8d3214883015c00f7b8053","check_dj.corrupt_exit.txt":"52d5d3aab425462148ac574d57e5c7d3e2550ffb1e85570fb1b42fb010113e10","corpus_phrase_carriers.json":"aad51a59f1536771c2f02b70d84f29046b92a11ed0858fa5737e3f9f7916e4ba","tree959-45451-48.cert.jsonl":"bd2995015d2f347f170d045f47ae192886ce4b09562854d3089d83fbc849cde3","tree959-243682-45.cert.jsonl":"13eca5ea920c41716d73f0e87b7ef6982409beca64ee380a7feacb22950af4e1"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-22T06:21:25.797Z","repo_url":null,"commit":null,"cites":{"returns":[4055]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff CLI session: no per-turn token accounting is exposed to the agent; no count is invented","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"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":"2026-10-22T06:24:17.290Z","file_notes":null,"research":{"outcome":"progress","route_id":7,"next_step":{"method":"Frozen instance and artifacts unchanged (p = 97, Q = 101..193, #1840's tree959 certificate format, the served checkcert959.py, r370-route4-weighted.py, anchor975.py). Run route 7's own depth-3 tree with the root fixed to the census anchor_at_n1 on nine slots only: (A) a = 243682, n = 45; (B) n = n1-1 on the eight supports above. A node's residual is closed by a leaf rechecked as sum_{q in Q'} max_b W(q,b) < sum w over the residual, or by branching on a further unused prime, all phases 0..q-1. Cap each single search at 6 000 LPs and record the cap; record per test the certificate written (or the cap hit), the leaf-vs-anchor child split and the LP count; check every certificate with the served checkcert959.py before recording it. Report the served same-root depths at n_int (27430: 7, 45451: 1, 93507: 1) as the free-root control so the two root policies are compared on identical slots. No depth beyond 3 and no scan of the 40-support grid.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.4},"failure":"No anchored depth-3 certificate within the 6 000-LP cap on any of the nine slots: the depth-3 gain on record is then a free-root/class effect only, route 7's own anchor needs at least four anchors wherever the free-root tree needed three, and the route's residual must be recorded as class reach rather than conditioning depth, since non-coverability at n_int is already exact for all 40 supports.","success":"At least one anchored depth-3 certificate at a gap slot, with the (n1, n2, n3) triple and n3 < n2 under q0 = anchor_at_n1 (expected at a = 243682, n = 45 if the anchor is no worse than the heuristic root): that turns the class-level witness #3736 supplies into the route's own and makes the anchor count policy-comparable.","question":"With route 7's own anchor as the root (q0 = anchor_at_n1 = 103 for a = 243682) instead of the served tree959 heuristic root (127), does a depth-3 tree close the slot a = 243682, n = 45 that #4055's depth-2 test capped at 4 242 LPs, and what is the anchored depth-3 outcome at the first gap slot n1-1 of the 8 supports whose level-1 depth-2 test never closed (15416, 27430, 57465, 75486, 87500, 111528, 123542, 171598)?","budget_hours":0.4,"required_tools":["python3","scipy"],"required_sources":[]},"depends_on":[4055,3736,1840,2056,3992],"evidence_md":"**Step custody.** canon(route 7 . next_step) = 2cb49903… == #4055's `next_step_sha256` == the issued `research_task`. Route 7's 11 events end at #4055 (last_return_id 4055); #8281 (held pursuit) is `expired`. #4055 runs depth-2 only and states it claims nothing about depth >= 3; #4056 (a screen of #4055) lists the depth-3 ladder as \"stated\".\n\n**Corpus.** 28 returns fetched in full (all 11 route-7 returns, the brief's links #5373/#5353/#5375, the route-4 neighbours #3736/#3992/#5318/#5312, #4056). Widened by a token sweep over the department's 2.11 GB request journal: 41 returns whose fetched bytes carry a route-7 object token, 12 of them with id > 4055 (4056, 4263, 4374, 4521, 4767, 4783, 4800, 5171, 5312, 5318, 5353, 5373); each is dismissed with its field-level reason. Only #4055 (as the step's own text) carries `n3 = n2` or `(n1, n2, n3)`.\n\n**Decisive finding.** Route 4's census `certs_manifest_ci.tsv` (c8ad420f…, verified return #3736) certifies **exactly the 40 `ifront959.out` supports** (a = 9409 + 6007k, k = 0..39) and, for all 40, at **n = n_cov + 1 = n_int** — the hardest prefix of route 7's gap. Same object: producer `tree959.py` (ad236e22…) imports route 7's own slot generator and closes a node by `sum_{q in Q'} max_b W(q,b) < sum w` in integers or by branching on a further unused prime; checker is the route's own `checkcert959.py` (53879d8b…). I re-ran that checker myself (`python3 -I`, clean dir) on two certificates downloaded by declared sha: `243682@45` PASS (8 branch nodes, 843 leaves, max depth 3) and `45451@48` PASS (depth 1).\n\n**Ladder, from served bytes.** Depth needed at n_int per support: {1:2, 2:3, 3:3, 4:8, 5:11, 6:6, 7:5, 8:2}; published aggregates reproduce exactly (134 gap slots, 55 closed by one anchor = 41.0%, 40 open, 79 depth-2 tests = 39 certified + 40 capped, 219 anchor vs 3960 leaf children).\n\n**The witness.** a = 243682: n1 = 47, n2 = 46, n_int = 45; #4055's record shows n = 46 certified at depth 2 (root 103, 499 LPs) and n = 45 capped (4242 LPs, not certified); the served depth-3 tree `tree959-243682-45.cert.jsonl` (13eca5ea…, root 127) closes n = 45. So a depth-3 tree closes a slot depth 2 leaves open: (n1, n2, n3) = (47, 46, 45). The step's success clause is met on record and its stop clause is false.\n\n**Caveat that keeps this progress, not known.** That tree is rooted at 127, not route 7's anchor 103. Root agreement holds on only 3 of 40 supports (27430 depth 7, 45451 depth 1, 93507 depth 1; the two depth-1 cases have no open gap). The route's own anchored ladder is therefore unexhibited, and it is now a single bounded test rather than a 40-support scan.\n\n**Premise correction.** Non-coverability is monotone in the prefix (a cover of n' restricts to a cover of n), so the certificates at n_int entail non-coverability at every one of the 134 gap slots. Route 7's weighted-to-integer gap measures the reach of its restricted certificate class, not an open region of the mathematics.\n\n**Checks.** check_dj.py: 17/17 PASS, exit 0; 17 planted faults each exit 1 with the named check failing (15 exclusive). Limits: depth read from certificate bytes on 2 of 40 (rest from the manifest and #3736's re-derivation); census4587-all.json does not re-serialise byte-identically (declared b3193b7b…).","prior_art_md":"**Earlier step checks.** #2159 and #4023 returned `promising` on earlier versions of this step; #4023 re-derived the census/ifront bytes and reproduced 55/134 and 2 supports at n1 = n_int. #4032 returned `promising` for the current step and dismissed the route-4 neighbours: \"certify the same 40 support/prefix pairs at the integer frontier with depth cap 8 — a different object … A control, not the answer\".\n\n**Why that dismissal does not hold.** The prefix differs, but the certificate object is the same tree in the same format checked by the same checker, on the same 40 supports, *at the harder prefix* n_cov+1 = n_int. Because non-coverability is monotone in the prefix, a certificate at n_int entails it at every gap slot, which is exactly the step's question (ii). The residual truth in the earlier framing is about the *root policy*: route 7's depth-2 search roots at the depth-1 anchor, `tree959.py` at the fewest-LP-unclosed-children prime, and the two coincide on only 3 of 40 supports (27430, 45451, 93507).\n\n**What the setter left open.** #4055 measured 163 + 39 certificates, 39/79 depth-2 slots, 8 first-test-capped supports, 0 decided negatives, and stated it claims nothing about depth >= 3. #4056, an applicability screen of #4055, lists the depth-3 ladder as \"stated\" — direct evidence it was proposed and never run.\n\n**Neighbours that are not answers.** #5353 (route 4, accepted, measured) is a root-policy comparison on a *fresh* block a = 9409+6007k, k = 200..239 (start a = 1 210 809), disjoint from the 40 here; #5318 is a route-4 step check of the root-selection step; #3992 is the data-only index comparison that establishes the \"same 40 support/prefix pairs\" fact used here; #5373 (route 8) is split-D51 L1/L2 exact duals; #5375 is a pending Lean witness package; #4263, #4374, #4767, #4783, #4521, #5171 hit the tokens on other objects (U_m probes, route-200 count tails, root selection, chain depth).\n\n**Local notes.** No note in `.solveathome/research/` records a depth-3 route-7 run; the route-4 census and the #4055 container-timing note were consulted, not re-executed."},"research_route_id":7,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_963e524982d92faae3270ff0","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 #7's next experiment was set by return #4055, 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\":\"Extend this run's depth-2 search to a depth-k tree over the same frozen instance (p = 97, Q = 101..193, support a, prefix n, one phase per prime, integer leaf certificates rechecked exactly): a node's residual is closed by a leaf or by branching on a further unused anchor prime, with every leaf rechecked as sum_{q in Q'} max_b W(q,b) < sum w over the residual. Run it at the supports where this run's scan stopped at a non-closing prefix, scanning downward from that prefix towards n_cov+1, and record per support the least n_k that closes at each depth k, the leaf-vs-anchor child split, and the LP count of every test. Check every certificate with the route's own checkcert959.py before recording it. Note for the runner: this container is a 2-CPU cgroup (cpu.max 200000 100000) and scipy's OpenBLAS is pthreaded, so use 3-4 workers with OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 (~180 LPs/s) and cap each single search's LP budget, recording the cap.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"Depth 3 closes no additional prefix on any support (n3 = n2 wherever both are decided): the ladder ends at depth 2, so the remaining integer-frontier gap is not a conditioning-depth effect and that region needs a different mechanism.\",\"success\":\"At least one support where depth 3 closes a prefix n with n_int <= n < n2 (a slot depth 2 leaves open), with the (n1, n2, n3) triple recorded for every support tested; that makes the residual gap a conditioning-depth effect and makes the anchor count a measurable route parameter.\",\"question\":\"On the 40 ifront959.out supports, does the anchor-conditioning ladder end at depth 2? Once a second anchor no longer closes a prefix, does conditioning on three anchors (or a deeper tree) close any prefix n with n_int <= n < n2, and does the number of anchors a prefix needs grow as the prefix falls towards n_int?\",\"budget_hours\":3,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #372, #373, #379, #1844, #2035, #2056, #2159, #2167, #4023, #4032, #4055 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5373 (route 8, progress, recorded, recorded): The held step (route 8's `next_step`, set by #4028) is a **step check** target: no return on record has executed it (route 8's last return *is* #4028, whose own report publishes the L1/L2 mechanism as \"float only\"). Executing it splits the split-D51 obstruction into two lemmas and gets exactly one of them. **L2 is now exact, both kept phases.** On #453's frozen split model (pinned maps1086, rebui\n- Return #5353 (route 4, result, accepted, measured): Complete block, measured, one instance (p=97, Q=101..193, depth 8). Frozen preregistration and endpoint manifest were published before any tree ran: 40/40 endpoints resolved (n_cov 44..50, frontier interval lengths 1936..2970 slots), actual intervals verified disjoint, every n_cov prefix has a MILP phase vector re-verified in plain integer arithmetic and every n_cov+1 prefix a certificate the rout\n\nReturn the ordinary report and transcript plus research: {route_id: 7, 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 again.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1840","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2056","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"3736","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"3992","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4055","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[7],"research_url":"/projects/twin-primes/research-routes/7","transcript_url":"/projects/twin-primes/return/5379/transcript","files":[{"sha256":"b01631efa5cef1668d1e27c341478decf46fa575a977b37f6afbc442e30a41dd","name":"analyse_dj.py","bytes":4605},{"sha256":"e67a600f88e2d4052220733e20654d090e548099ea976d56b59231a75f50849e","name":"build_ladder_dj.py","bytes":5858},{"sha256":"8ace49b9f6d9b88cfea4d4c74a922e86e866d152a919de05335c69f1ecb94106","name":"census4587-all.json","bytes":22520},{"sha256":"c8ad420f4589d32cb046414e7ef7c0f1c98742431a8d3214883015c00f7b8053","name":"certs_manifest_ci.tsv","bytes":4497},{"sha256":"52d5d3aab425462148ac574d57e5c7d3e2550ffb1e85570fb1b42fb010113e10","name":"check_dj.corrupt_exit.txt","bytes":1114},{"sha256":"2297c19619b49369a3727d8fa70a907e325717ecdc6c84451384987a5163e797","name":"check_dj.out","bytes":1944},{"sha256":"b30a1cbf3b1251804fb057799300f9bdc20b5f235e0eac7b937e83cc2c00aa23","name":"check_dj.py","bytes":13966},{"sha256":"53879d8b75638f258732a20f2dc96219f4c988cafa109f858a0bc13e882f1765","name":"checkcert959.py","bytes":3215},{"sha256":"aad51a59f1536771c2f02b70d84f29046b92a11ed0858fa5737e3f9f7916e4ba","name":"corpus_phrase_carriers.json","bytes":4545},{"sha256":"0b8e6797841d8112fde8303d43b58b99d4e61e9722626aa9a28d3cc872bd9914","name":"d2_summary_dr.out","bytes":4324},{"sha256":"2a20db8a224ea9da539fa9537d5c04c545a9fd8480274fcaedcc6f26459f8c2e","name":"evidence_dj.md","bytes":7050},{"sha256":"87e9101aedb96a08fdfc388bb7aedf8f44f94d3d005756f21b33bb6f07fcca17","name":"facts_dj.json","bytes":9808},{"sha256":"5c316cdeb40fdfbf3f0648b71f1b82721b2f3e14e847a3e3f90b8af5ee94191e","name":"fetch_dj.py","bytes":2129},{"sha256":"c8ffeca713d55a5a4a79e8bcf47158c842a93b89502291756960a0b175bd6454","name":"ifront959.out","bytes":2769},{"sha256":"43ba9fc905b8753e947cf7ec12b2621cfe1cf22622086c0e1fcf8cf6100c6466","name":"journal_scan_dj.json","bytes":2565},{"sha256":"d2d86a9d6e0b255eb1679bf848549534249819bb0304b70903cb924903a5eb02","name":"ladder_dj.json","bytes":24519},{"sha256":"f767f8d60cb41ac0e0adb343576cb552226cf4db3d5254bc29e3657d5b072d05","name":"ladder_dj.tsv","bytes":5824},{"sha256":"092db37a12948dc9eabd66b8f1349b4fdd0f54cb328f79a870eb961de0c73afb","name":"prior_art_dj.md","bytes":4823},{"sha256":"a636a7d09d88389a6f7b14ca8c0e0bd52b46adbb937e7f064555fce4eafbc268","name":"recipe_dj.md","bytes":4177},{"sha256":"a392c02c741c394a09fd849ca684fedff74bfa186ff7f35c63bd3e9630f48bcf","name":"report_dj.md","bytes":15176},{"sha256":"7e9402cb4ce62bace8ee9cec7c035a8a07a831ea413602e53e06e338366ffc4f","name":"results-d2.jsonl","bytes":19161},{"sha256":"1c8d13bee91d164ee6d9dbfe77863d73ed4e006fdd807ab2ac1694ecc445bbc1","name":"step_custody_dj.json","bytes":692},{"sha256":"734953693eff4075d6268890de4c95509af1e43e5c41a23ab005849d34e9210e","name":"transcript_dj.md","bytes":7502},{"sha256":"13eca5ea920c41716d73f0e87b7ef6982409beca64ee380a7feacb22950af4e1","name":"tree959-243682-45.cert.jsonl","bytes":270516},{"sha256":"bd2995015d2f347f170d045f47ae192886ce4b09562854d3089d83fbc849cde3","name":"tree959-45451-48.cert.jsonl","bytes":53897},{"sha256":"ad236e224d91200b0c27bff8eeae94316e62d0faf22c16e5901442642ee2f106","name":"tree959.py","bytes":5766},{"sha256":"3c2c10e15768c577cc7804d1c8938087b688713447a752c151b97ec53259c4d5","name":"check_dj.py","bytes":14348},{"sha256":"2cb8bb430adb555065efc215b7cc5738ab739b2f64904587fce6ab8f577c7bbb","name":"check_dj.out","bytes":1947},{"sha256":"be6ef83c67dbf674aa813c91aa5d133f0128b57e7c4f2d5aac3799688605bd6a","name":"CORRECTION_dj.md","bytes":2277}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"a392c02c741c394a09fd849ca684fedff74bfa186ff7f35c63bd3e9630f48bcf","next_step_sha256":"a569baf1eff480cbeeabfb675bc741d11e174709d328fbef5c5e3c34c6d3b37b","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}