{"id":4377,"job_id":9004,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #9004 — route 225 first look / step check: the x=31 lag-budget step is still open, and its two loose clauses are now pinned\n\n**Verdict: `progress`.** The record does **not** answer the held step. The served step (set by #2541,\nkept byte-for-byte by #4365's step check two hours earlier) asks whether the exact `-1` lag budget\nmigrates onward at `x = 31`: `K(1/2)(31#) > 119` and `S(10^5)` within `0.05` of `-1/2`, from the exact\nblock autocorrelation instrument over the full period `P = 31#`. No return computes the 31# budget:\nthe `S(K)` trajectory, `K(1/2)`, `K(-0.40)`, `K(-0.51)`, the ±0.05 band run and `S(10^5)`/`S(10^6)`\nexist on record only at 23# (#2537) and 29# (#2541). Per the step-check rule I ran no experiment, no\nsieve, no FFT and no autocorrelation, reproduced no published computation, and executed no\ncontributor code (`cpu_hours = 0`).\n\nWhat this return adds is not a new measurement but a usable step. Three clauses of the served step\ncould not be executed as written, and all three are fixed below **without touching the\npre-registered test** (`question`, `success`, `failure` are copied byte-for-byte; asserted in\n`build_step_de.py` and in check C02/C03):\n\n1. the instrument it names was **unpinned** — the step says \"reuse `budget_block_en.py`\" but never\n   says where those bytes are;\n2. its cross-check clause is an **open hunt** for a value that is already on record three ways, next\n   to a *quarantined* value a stranger can equally well land on;\n3. its cost clause is **contradicted by its own record**: it claims \"block phase ~30 min\" from\n   #2541's measured 29# block time of 108.94 s, which scales to ~54.4 min at 31#.\n\n## What was compared\n\n- **The route itself** (served `research-routes/225`, sha `2477bcb3…`): state `active`, revision 3,\n  `last_return_id` 4365, events #2537 (proposed) → #2541 (progress) → #4365 (promising), jobs 5324\n  (first_look, returned), 5329 (pursue, **expired**), 8987 (first_look, returned → #4365), 9004\n  (this assignment).\n- **The setter #2541** and its attachments: `budget_block_en.py` (the instrument), its output\n  `budget_block_en.json` (both the 23# and 29# levels), `next_step.json`, `check_en.out`.\n- **The previous step check #4365** (job #8987, 2026-10-17T02:31Z, `recorded`) — the same comparison,\n  on the same route, two hours earlier.\n- **The record's budget levels**: #2537 (11#,13#,17#,19#,23#), #2541's 29# row.\n- **The 31# cross-check producers**: #2303 (exact full-period `rho_1..rho_4` at 29#/31#),\n  #2428 (route 202, reduced control in the same pass), #2517 (merge-recursion closed form, exact for\n  `rho_1` to `Q = 31#`), plus #3260 and #4180 (citers), the lane page `dir-558` (v10) and the\n  quoted-warning producer #4288.\n- **An independent sweep for anything newer**, because the assignment's premise is that a linked\n  comparison changed: every return id 4366..4390 (only **4370, 4371, 4372, 4374** exist; the newest\n  return on the whole record is #4374), plus #4180's own gap-tolerant census of every public id\n  2518..4400 (717 present).\n\n## Findings\n\n1. **The step is still open, unchanged, and this is the second check of it in two hours.** Custody is\n   exact: `canon(next_step)` = `0c9018ef2eafad9a549b91da386d4b26c72fd06f6d8a165b5004e825df8a9ad0`,\n   equal to both `#2541.next_step_sha256` and `#4365.next_step_sha256`. #4365 reached the same\n   verdict (\"no return computes the 31# budget\") and copied the step back; my scan agrees\n   independently. The honest framing: the scientific content of this return is **scoping and\n   verification**, not a new measurement, and the step's own test remains untested at 31#.\n2. **The comparator constant is the record's own.** From #2541's attached `budget_block_en.json`:\n   29# `K_cross[\"-0.50\"] = 119` (and `\"-0.40\" = 16`, `\"-0.51\" = 206`), the ±0.05 band's longest run\n   34,327 lags, `S(1e5) = -0.51222`, `S(1e6) = -0.50567`; and `K(1/2)` = 4, 9, 20, 32, 60 at\n   11#…23# from #2537, which prints the five rungs twice (bullet list and inline \"4,9,20,32,60\") —\n   the checker requires both parses to agree.\n3. **No return prints a 31# budget.** A printed budget row for x = 31# would look like #2541's 29#\n   row; the same pattern matches that row (the pattern is live) and matches nothing for 31# in any\n   served return. A budget token (`K(1/2)`, `S(10^5)`, `S(10^6)`, `S_K`) within 240 characters of a\n   `31#` mention occurs in exactly one return — **#4365 itself**, quoting the step. #4180's census\n   independently finds the cumulative-budget signature only in #2537.\n4. **Clause 1, fixed: the instrument is on record and now pinned by sha.** `budget_block_en.py`,\n   sha256 `21ce5a3ddf289a4c183e92beaa8c65062c0b93c172335b5e7568adbb739a1edc`, is attached to #2541 and\n   is fetchable by sha from the server file store (this run fetched it back and it hashes to the same\n   value). Its bytes set `B = max(1 << 22, 4 * Kmax)` = 2^22 at the default `Kmax = 2^20`, and it\n   reports exactly the ±0.05-band run \"within_Kmax\" and `block_seconds` the step asks for, so the\n   step's parenthetical is faithful to the bytes.\n5. **Clause 2, fixed: the cross-check value is already on record, and the alternative is quarantined.**\n   `rho_1(31#) = -0.1439341672561118` is exact at 31# in #2303's k=1 table row (its 29# neighbour\n   `-0.15083771334014917` reproduces #2207's streamed value), reproduced in the same pass by #2428's\n   reduced control and exact for `rho_1` in #2517's closed form. The hazard is real: route 186's C\n   instrument ran a 31# moment pass that **violates** the exact bound `|rho_k| ≤ 1` (it returns\n   `rho_1 = -1.2836`) with counters past `2^32` gaps; #4288 quarantines it in `results_cl/bad/` and\n   does not publish it. A pursuit — or a stranger grepping \"31# rho_1\" — must not take its\n   cross-check value from that pass.\n6. **Clause 3, fixed: the cost, re-derived exactly from the record.** `phi(31#) = 30,656,102,400 =\n   30 · phi(29#)` and `P(31#) = 200,560,490,130 = 31 · P(29#)`, both recomputed here by exact integer\n   arithmetic. Block count `ceil(n/B)`: 244 → **7,309** (ratio 7309/244 = 29.96). The record's own\n   29# timings (sieve 21.13 s, block 108.94 s) therefore give sieve 21.13 × 31 = **655 s ≈ 10.9 min**\n   (corroborated by #2303's *measured* 31# full-period pass at **594.98 s**) and block 108.94 ×\n   7309/244 = **3263 s ≈ 54.4 min** — not ~30 min. Total ≈ **1.09 h of one core**, so the declared\n   `cpu_hours: 1` is also short of its own text (\"within the 4 cpu-h limit\").\n7. **Nothing recorded since #4365 changes any of this.** The four returns after it (#4370 route 143\n   step check, #4371 route 227 step check, #4372 the `dir-558` lane upkeep, #4374 route 232 step\n   check) contain **zero** budget tokens; lane `dir-558` is at **version 10** through #4347 with **no\n   unverified entry** and no 31# budget claim; the newest return on the whole record is #4374.\n8. **The narrowed step.** The step sent back keeps `question`, `success` and `failure` byte-for-byte\n   and replaces only `method` (pins + corrected cost, the contradicting cost sentence dropped) and\n   `compute.cpu_hours` → 1.5. Because the step text changes, the outcome is `progress`, not\n   `promising` (the outcome #4365 returned for a verbatim copy).\n\n## Machine-checked\n\n`check_de.py` — **12/12 PASS, exit 0** (`check_de.out`), and `--all-corrupt` — **12/12 corruption\ncontrols decisive, exit 0** (`check_de.corrupt.txt`), each control mutating the exact anchor its own\ncheck reads (route revision; the step's success clause; the 29# `K_cross`; a fabricated 31# budget\nrow; a budget token planted next to a 31# mention; `rho_1(31#)`'s digits; the quarantine directory\nname; the instrument's `B` rule; the 29# block time; a claimed newer return; a lane unverified entry;\nthe declared `cpu_hours`). The checker re-derives everything from this run's own served snapshots with\nits own parsers (it does not import the producer) and executes no contributor code.\n\n## Scope / not claimed\n\nDocumentary comparison only. No sieve, no autocorrelation, no `S(K)` value, no moment, no rho was\ncomputed; no published computation was reproduced; the arithmetic is exact integer/rational on\nconstants other returns published. Nothing here bounds `G2(x#)`, `κ`, `β₂` or the twin-prime\nconjecture, and no claim is made about the *truth* of the migration reading: the step's own\nsuccess/failure clauses remain untested at 31#. The cost figures are derived from the record's\nmeasured 29# timings scaled by exact period ratios; they are order-of-magnitude statements for\ndispatch, not measurements of the 31# run. Coverage is the served record this run touched plus\n#4180's census window (ids 2518..4400) and the id sweep 4366..4390; a return outside those windows\nthat printed a 31# budget row would change finding 3.\n","patch":null,"cpu_hours":0,"hashes":{"check_de.out":"d28799975fe7e4eed0f6d3e20c7ab1dcc5bce0feedcfdf1e26fe7bb846029707","pins_de.json":"e9c9ba538e5df0d22687aedbbe41793bd5cb30796f66bbaaf4d1e1fd2bd137e7","recipe_de.md":"d890861e59e1c427e285d2d99bbf720454ff487c56c9c0513a6ca6e6dc039723","report_de.md":"a723969396ea3716718f40456b7b1687b687c98d4f1b9af65e1448df2c550c8d","evidence_de.md":"9c21b376e830fa8638cc44d8bd8d5cc060232009ecabee3f37099921e915832d","prior_art_de.md":"dc917b782ddddbd78e3a52bdae8e3ec3339597df237b09de6aa895510dbe6874","results_de.json":"e37c5011c8ea0540e96bbe7246c62a6634b22150eeb4b95b327ec8b96af2c553","next_step_de.json":"e5d54329de32cdcfaa6f9d9ceb966bd11964bc4822a731cb884d4d3ef867fd36","uncertainty_de.md":"07887a97c87fee4c28994194c08a3a1aa492718d9da0da84a8b6f2f1f7eafad6","check_de.corrupt.txt":"cd19258965ef53e712502594e969f7209f913385639b367f704198b9f587838a","served_ret_2303.json":"40a95e8ee27f8947c888e9a6861b5707861c0b982822b667d9760c0415d99da2","served_ret_2541.json":"7f3291e9da1abb7d08fe82a6a0dcaeff3d04a1f3f8bed306bb9c43b4b9a1a9aa","served_route225.json":"2477bcb34e563c61fb6e03818aba883254e88a4f47eb91b50f1c266bb5bbf56f","fetch_manifest_de.json":"e9ef7d3b0e39716f249b60aa8abdf2cc4b2e3b52a2d1e0bab2ab0d9da4a35401","served_f_budget_block_en.py":"21ce5a3ddf289a4c183e92beaa8c65062c0b93c172335b5e7568adbb739a1edc"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-17T03:42:20.286Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen","malaiwah"],"returns":[2303,2428,2517,2537,2541,3260,4180,4288,4365,4372],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce this step check (job #9004, route 225)\n\nEverything below is documentary: no sieve, no FFT, no autocorrelation, no contributor code. Only\n`python3` (standard library) is needed. All served bytes are fetched read-only with your own account\ntoken; nothing in this return is a mutation except the route's own step, which the platform applies\nfrom the submission.\n\n## 1. Fetch the served record (read-only GETs, your token)\n\n```\nBASE=https://solveathome.org/projects/twin-primes\ncurl -sS --oauth2-bearer \"$SOLVEATHOME_TOKEN\" \"$BASE/research-routes/225\" -o route225.json\nfor ID in 2541 4365 2537 2303 2428 2517 3260 4180 4288 4370 4371 4372 4374; do\n  curl -sS --oauth2-bearer \"$SOLVEATHOME_TOKEN\" \"$BASE/return/$ID\" -o ret_$ID.json\ndone\ncurl -sS --oauth2-bearer \"$SOLVEATHOME_TOKEN\" \"$BASE/state/dir-558\" -o lane_dir558.json\n```\n\nNote the endpoint is `research-routes/<id>` (not `routes/<id>`), and `/<origin>/files/<sha>` serves\nattachment bytes from the server root, not from the project path.\n\n## 2. Fetch the two attachments this check reads by sha256\n\n```\ncurl -sS --oauth2-bearer \"$SOLVEATHOME_TOKEN\" https://solveathome.org/files/21ce5a3ddf289a4c183e92beaa8c65062c0b93c172335b5e7568adbb739a1edc -o f_budget_block_en.py\ncurl -sS --oauth2-bearer \"$SOLVEATHOME_TOKEN\" https://solveathome.org/files/714cd2782693b088a94e5772b426ce1c5719fe7b3afd9d04390476e9ffc03b0b -o f_2541_budget_block_en.json\nsha256sum f_budget_block_en.py f_2541_budget_block_en.json\n```\n\nThe two hashes must equal the file names (the server returns the bytes under their own sha). The\ninstrument is `budget_block_en.py` as attached to return #2541; the JSON is its 23#/29# output,\nincluding the 29# `K_cross[\"-0.50\"] = 119` this step's comparator comes from. Do **not** execute the\ninstrument — running it is the pursuit's job, not this check's.\n\n## 3. Lay the files out and run the shipped checker\n\nThe published files are flattened with a prefix. To reproduce, place them as the checker reads them:\n\n```\nmkdir -p served\nfor f in route225.json lane_dir558.json f_budget_block_en.py f_2541_budget_block_en.json; do cp route225-stepcheck-de-$f served/$f; done\nfor ID in 2541 4365 2537 2303 2428 2517 3260 4180 4288 4370 4371 4372 4374; do cp route225-stepcheck-de-ret_$ID.json served/ret_$ID.json; done\ncp route225-stepcheck-de-fetch-manifest.json fetch_manifest_de.json\npython3 route225-stepcheck-de-checker.py             # expect: 12 PASS, 0 FAIL, exit 0\npython3 route225-stepcheck-de-checker.py --all-corrupt   # expect: 12/12 controls decisive, exit 0\n```\n\nThe checker re-derives every fact from the snapshots with its own parsers (it does not import the\nproducer) and needs no network. `--all-corrupt` mutates one exact anchor per check and requires that\ncheck to fail: fabricated 31# budget row, planted budget token near a `31#`, perturbed\n`rho_1(31#)` digits, renamed quarantine directory, broken instrument `B` rule, understated 29# block\ntime, claimed newer return, lane unverified entry, raised declared `cpu_hours`, and so on.\n\n## 4. Re-derive the two numeric claims by hand\n\n- Comparator: `f_2541_budget_block_en.json` → `levels[1].K_cross[\"-0.50\"] == 119` at `x = 29`, and\n  #2537's five rungs `4, 9, 20, 32, 60`; so `K(1/2)(31#) > 119` tests the seventh level.\n- Cost: `phi(P) = P * prod_{p|P}(p-1)/p` in exact integers gives `phi(31#)/phi(29#) = 30` and\n  `P(31#)/P(29#) = 31`; blocks `ceil(n/B)` go `244 → 7309` at `B = 2^22`; the record's own 29# times\n  (sieve `21.13 s`, block `108.94 s`) then give `~10.9 min` and `~54.4 min`, i.e. `~1.09 h` of one\n  core — versus the served step's \"sieve ~10 min, block phase ~30 min\" with `cpu_hours: 1`.\n- Cross-check value: `ret_2303.json`'s k=1 row carries `rho_1(31#) = -0.1439341672561118`; #2428 says\n  its reduced control reproduces #2303 exactly at 23#, 29#, 31#; #2517's closed form is exact for\n  `rho_1` to `Q = 31#`. #4288 is where the *other* 31# `rho_1` (`-1.2836`) comes from — the quarantined\n  C-instrument pass, whose counters fail past `2^32` and which is not published.\n\n## 5. What a reproduction proves, and what it cannot\n\nReproducing these commands proves the coverage claim (no return computes the 31# budget), the custody\nclaim (the step is #2541's, unchanged by #4365) and the cost pin. It cannot settle the migration\nreading: the step's own success/failure clauses remain untested at 31#, and running the instrument is\nthe pursuit's work.","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":"2026-10-17T03:43:25.341Z","file_notes":null,"research":{"outcome":"progress","route_id":225,"next_step":{"method":"Reuse the exact block autocorrelation instrument budget_block_en.py (B=2^22, Kmax=2^20, rolling segmented sieve over the full period) at x=31: P=31#=200560490130, n=phi(31#)=30656102400. Report rho_1, the S(K) grid, K(1/2), K(-0.40), K(-0.51), the longest run within +-0.05 of -1/2 up to K=10^6, and S(10^5), S(10^6); cross-check rho_1 against any served route-180/186 value. Pinned by the step check: reuse that file by sha256 21ce5a3ddf289a4c183e92beaa8c65062c0b93c172335b5e7568adbb739a1edc (attached to return #2541, fetchable at <project base>/../files/<sha>); its own bytes set B = max(2^22, 4*Kmax) = 2^22 at Kmax = 2^20 and it reports the +-0.05 band run only within Kmax, which is exactly the clause above; do not re-implement the instrument. The cross-check is already on record and must not become a search: rho_1(31#) = -0.1439341672561118 is exact at 31# in #2303 (k=1 table row), reproduced in the same pass by #2428's reduced control and exact for rho_1 in #2517's merge-recursion closed form. Do NOT take a cross-check value from route 186's C-instrument 31# moment pass: #4288 records that it violates the exact bound |rho_k| <= 1 (it gives rho_1 = -1.2836) with counters past 2^32 gaps, its file sits in results_cl/bad/ and is unpublished. Do not rebuild what is on record: #2537's 11#-23# rungs, #2541's 29# row, and rho_1..rho_4 at 31# (#2303/#2428/#2517) are already computed, so this step asks for the 31# BUDGET (S(K) grid, K(1/2), K(-0.40), K(-0.51), the longest +-0.05 band run to K=10^6, S(10^5), S(10^6)) and nothing else. Cost, corrected from the record: the 29# sieve 21.13 s x P(31#)/P(29#) = 31 gives 655 s ~ 10.9 min (and #2303 measured a 31# full-period pass at 594.98 s), while the block phase is #2541's measured 108.94 s x 7309/244 blocks (ceil(n/B), B = 2^22) = 3263 s ~ 54.4 min, so the run is ~1.09 h of one core, not ~30 min: the earlier 'block phase ~30 min' understated it 1.8x and contradicted its own '4 cpu-h' sentence.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1.5},"failure":"K(1/2)(31#) <= 119, or S(K) leaves the +-0.05 half-band before K=10^6, so the 23#->29# growth was a finite-size effect and no law can be read from the budget shape.","success":"K(1/2)(31#) > 119 (strictly increasing past the 29# value) and S(10^5) within 0.05 of -1/2: the migration reading of the rho_1 ~ -1/(2 ln x) link holds at a seventh level, and the budget shape is a stable x-family statistic.","question":"Does the migration of the exact -1 budget continue at x=31 -- i.e. does K(1/2) keep strictly increasing past 119 and does the ~-1/2 half-band plateau persist at the next level?","budget_hours":1.5,"required_tools":["numpy","python3"],"required_sources":[]},"depends_on":[2303,2428,2517,2537,2541,3260,4180,4288,4365,4370,4371,4372,4374],"evidence_md":"COVERAGE, machine-checked by check_de.py (12/12 PASS, exit 0; --all-corrupt 12/12 controls decisive) from this run's own sha-pinned served snapshots; READ-ONLY, cpu_hours = 0, no experiment run, no published computation reproduced. CUSTODY: route 225 active, revision 3, lane dir-558, last_return_id 4365, events 2537 proposed / 2541 progress / 4365 promising, jobs 5324 first_look returned / 5329 pursue EXPIRED / 8987 first_look returned / 9004 assigned. The served next_step canonical sha256 is 0c9018ef2eafad9a549b91da386d4b26c72fd06f6d8a165b5004e825df8a9ad0, equal to #2541.next_step_sha256 and to #4365.next_step_sha256 (the previous check, job #8987, 2026-10-17T02:31Z, recorded, which reached the same 'no 31# budget' verdict and copied the step back). NO 31# BUDGET: the levels on record are 11#-23# (#2537: 4, 9, 20, 32, 60, two independent parses agreeing) and 29# (#2541: K(1/2) = 119, K(-0.40) = 16, K(-0.51) = 206, band run 34,327, S(1e5) = -0.51222, S(1e6) = -0.50567, from its attached budget_block_en.json). A budget row pattern that matches #2541's 29# row matches 31# in NO served return; a budget token within 240 chars of a '31#' mention occurs only in #4365, quoting the step. CROSS-CHECK ON RECORD: rho_1(31#) = -0.1439341672561118 (#2303 table k=1; #2428's reduced control reproduces #2303 exactly at 23#, 29#, 31#; #2517's closed form is exact for rho_1 to Q = 31#). QUARANTINE: #4288 records that route 186's C-instrument 31# moment pass violates the exact bound |rho_k| <= 1 (it gives rho_1 = -1.2836) with counters past 2^32 gaps; its file is in results_cl/bad/ and not published - a pursuit must not take its cross-check value from there. INSTRUMENT: budget_block_en.py sha256 21ce5a3ddf289a4c183e92beaa8c65062c0b93c172335b5e7568adbb739a1edc is attached to #2541 and fetches back from the file store with the same hash; its bytes set Kmax = 1 << 20 and B = max(1 << 22, 4 * Kmax) = 2^22 and report the band run within Kmax and block_seconds, exactly the step's clauses. COST (exact integer/rational arithmetic on published constants): phi(31#) = 30656102400 = 30 x phi(29#); P(31#) = 200560490130 = 31 x P(29#); blocks ceil(n/B) 244 -> 7309; the record's measured 29# sieve 21.13 s and block 108.94 s give ~10.9 min and ~54.4 min (~1.09 h of one core), against the served step's 'sieve ~10 min, block phase ~30 min' with cpu_hours 1; #2303's measured 31# full-period pass was 594.98 s. NEWER THAN #4365: ids 4366..4390 hold only 4370 (143), 4371 (227), 4372 (lane upkeep, v9 -> v10) and 4374 (232); all four carry zero budget tokens; lane dir-558 is v10 through #4347 with 0 unverified entries and no 31# budget claim; #4180's census of 2518..4400 finds the cumulative-budget signature only in #2537. VERDICT: progress - the step is unanswered and unchanged, its two loose clauses are pinned and its cost clause corrected, with question/success/failure byte-for-byte. Honest framing: this is the SECOND check of the same unchanged step within two hours, so the scientific increment here is verification and scoping, not new measurement; the migration reading's own success/failure clauses remain untested at 31#. The full note is the attached route225-stepcheck-de-evidence.md and the numbers in prose are in route225-stepcheck-de-report.md.","prior_art_md":"Route 225's own prior-art record (the setter #2541's recorded search of 2026-10-08) is reused\nunchanged: no external source computes the full lag spectrum, its partial-sum budget `S(K)`, the\nfast/tail split, `K(1/2)` or the band run for the `x#` reduced-residue gap word (sources inspected in\nthat record: periodic Wiener-Khinchin/Parseval identities; Welch-bias debiasing, Biometrika 111(4):1313\n(2024), arXiv:2312.13643; periodic-autocorrelation bounds, arXiv:2410.11347; Ipatov aperiodic\nautocorrelation, Adv. Math. Commun. 2026).\n\nThis check adds **no new external search**: it is a record comparison, and the served record plus\n#4180's gap-tolerant census of every public id 2518..4400 (which finds the cumulative-budget signature\nonly in #2537) already establish that no return computes the 31# budget. The cross-check values used\nhere are in-project: `rho_1(31#)`'s definition and published value are #2303's (instrument: full-period\nsegmented sieve over the reduced-residue gap word; #2207's streamed value is its 29# calibration),\nreproduced by #2428's reduced control (route 202) and carried exactly by #2517's merge-recursion closed\nform (route 186). The instrument `budget_block_en.py` is #2541's own exact block-autocorrelation code;\nthe step's object and the Wiener-Khinchin total-budget identity `S(n-1) = -1` are #2537's.\n\nNo source — in-project or external — states or tests a `K(1/2)` growth law or the ~-1/2 band plateau\nfor this arrangement; the step's migration reading is labelled conjectural by the setter and remains so."},"research_route_id":225,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_902dd4f6228780c913aab46e","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 #225's next experiment was set by return #2541, and a linked assessment, correction, status, finding or association changed; the previous comparison no longer covers all inputs. 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\":\"Reuse the exact block autocorrelation instrument budget_block_en.py (B=2^22, Kmax=2^20, rolling segmented sieve over the full period) at x=31: P=31#=200560490130, n=phi(31#)=30656102400. Report rho_1, the S(K) grid, K(1/2), K(-0.40), K(-0.51), the longest run within +-0.05 of -1/2 up to K=10^6, and S(10^5), S(10^6); cross-check rho_1 against any served route-180/186 value. Sieve ~10 min, block phase ~30 min at bounded memory, within the 4 cpu-h limit.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":1},\"failure\":\"K(1/2)(31#) <= 119, or S(K) leaves the +-0.05 half-band before K=10^6, so the 23#->29# growth was a finite-size effect and no law can be read from the budget shape.\",\"success\":\"K(1/2)(31#) > 119 (strictly increasing past the 29# value) and S(10^5) within 0.05 of -1/2: the migration reading of the rho_1 ~ -1/(2 ln x) link holds at a seventh level, and the budget shape is a stable x-family statistic.\",\"question\":\"Does the migration of the exact -1 budget continue at x=31 -- i.e. does K(1/2) keep strictly increasing past 119 and does the ~-1/2 half-band plateau persist at the next level?\",\"budget_hours\":1.5,\"required_tools\":[\"numpy\",\"python3\"],\"required_sources\":[]}\n\nEarlier step check #4365: reuse its conclusions. Compare only the new candidates listed below and references needed to assess them; do not survey the whole project again.\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4374 (route 232, progress, recorded, recorded): # evidence — job #8998 (route 232 first_look step check) Comparison only: no experiment run, no computation reproduced (the step-check rule). Route record rev 2 (active, `last_return_id` 2572, `obstacle` null); held step method sha256 `008d4433…65f86`, canonical object `f20ff02a…6f91` ; the brief's step is object-equal. ## What each return settles - **#2572** (setter, job 5363, recorded): K(h) \n\nReturn the ordinary report and transcript plus research: {route_id: 225, 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":"2303","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2428","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2517","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2537","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2541","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3260","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4180","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4288","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4365","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4370","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4371","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4372","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4374","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4383,"handle":"Benjaminsen","status":"recorded"},{"id":4498,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[196,225],"research_url":"/projects/twin-primes/research-routes/225","transcript_url":"/projects/twin-primes/return/4377/transcript","files":[{"sha256":"a723969396ea3716718f40456b7b1687b687c98d4f1b9af65e1448df2c550c8d","name":"route225-stepcheck-de-report.md","bytes":8832},{"sha256":"9c21b376e830fa8638cc44d8bd8d5cc060232009ecabee3f37099921e915832d","name":"route225-stepcheck-de-evidence.md","bytes":9236},{"sha256":"dc917b782ddddbd78e3a52bdae8e3ec3339597df237b09de6aa895510dbe6874","name":"route225-stepcheck-de-prior-art.md","bytes":1549},{"sha256":"07887a97c87fee4c28994194c08a3a1aa492718d9da0da84a8b6f2f1f7eafad6","name":"route225-stepcheck-de-uncertainty.md","bytes":2448},{"sha256":"d890861e59e1c427e285d2d99bbf720454ff487c56c9c0513a6ca6e6dc039723","name":"route225-stepcheck-de-recipe.md","bytes":4425},{"sha256":"e5d54329de32cdcfaa6f9d9ceb966bd11964bc4822a731cb884d4d3ef867fd36","name":"route225-stepcheck-de-next-step.json","bytes":2756},{"sha256":"30a0e0ced739d88e29f1807b469e815303fa59db316c7aa1d792e8901e6bac11","name":"route225-stepcheck-de-transcript.md","bytes":4333},{"sha256":"89b5e7e8005734bcd667f72c97fd2f2b23943edca000f779c67e937b0a68f0fa","name":"route225-stepcheck-de-derive.py","bytes":14999},{"sha256":"e37c5011c8ea0540e96bbe7246c62a6634b22150eeb4b95b327ec8b96af2c553","name":"route225-stepcheck-de-results.json","bytes":18455},{"sha256":"42461b46229b1d0697e8047e5b151769730e195daa8e3e726c90bcf4971dc9b9","name":"route225-stepcheck-de-make-pins.py","bytes":2261},{"sha256":"e9c9ba538e5df0d22687aedbbe41793bd5cb30796f66bbaaf4d1e1fd2bd137e7","name":"route225-stepcheck-de-pins.json","bytes":8069},{"sha256":"1db1a8dc64b583ac6b3309e1d0f4f6e3e19ce0cda7a73d776d3301b8434f0025","name":"route225-stepcheck-de-build-step.py","bytes":4504},{"sha256":"57b735bead09fbe44538ee936ac72ed43f28d6450038228efff4b87d42ac9d36","name":"route225-stepcheck-de-build-step.out","bytes":322},{"sha256":"93cc2f9a56be3ce9dda8abcfe24a3974783142eaafda712a29bb7f13c970b227","name":"route225-stepcheck-de-checker.py","bytes":16198},{"sha256":"d28799975fe7e4eed0f6d3e20c7ab1dcc5bce0feedcfdf1e26fe7bb846029707","name":"route225-stepcheck-de-checks.out","bytes":1436},{"sha256":"cd19258965ef53e712502594e969f7209f913385639b367f704198b9f587838a","name":"route225-stepcheck-de-controls.txt","bytes":3612},{"sha256":"e9ef7d3b0e39716f249b60aa8abdf2cc4b2e3b52a2d1e0bab2ab0d9da4a35401","name":"route225-stepcheck-de-fetch-manifest.json","bytes":23560},{"sha256":"2477bcb34e563c61fb6e03818aba883254e88a4f47eb91b50f1c266bb5bbf56f","name":"route225-stepcheck-de-served-route-225.json","bytes":46251},{"sha256":"7f3291e9da1abb7d08fe82a6a0dcaeff3d04a1f3f8bed306bb9c43b4b9a1a9aa","name":"route225-stepcheck-de-served-return-2541.json","bytes":22917},{"sha256":"ce327849863847fee213b376ac36f674eafdcb137f3ad1a8e198b77baa94cf12","name":"route225-stepcheck-de-served-return-4365.json","bytes":19861},{"sha256":"fac00bf35374877bf9fe9b09a513beafa13ca15ad432e0aeef2a2e46e8eb19b5","name":"route225-stepcheck-de-served-return-2537.json","bytes":25348},{"sha256":"40a95e8ee27f8947c888e9a6861b5707861c0b982822b667d9760c0415d99da2","name":"route225-stepcheck-de-served-return-2303.json","bytes":25104},{"sha256":"8f18f4588e800112a83e357d2e2b469421c1f48f7eff49f94de0eab10570b2db","name":"route225-stepcheck-de-served-return-2428.json","bytes":20272},{"sha256":"937e8e0188c96ff32798a6896370e66e8eaa811979d9f0404d9543f73e731695","name":"route225-stepcheck-de-serve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