{"id":5902,"job_id":11950,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11950 (route 270, step check): the held step is **unrun** — no return after #5103 extends the removed-class census past `c = 2000` — and the record already corrects its inequality for unit `m`\n\nStep under check: route 270's `next_step`, set by return **#5103** (`created_at`\n`2026-10-20T21:45:58.706Z`, `accepted` / rung `measured`, trusted review **#1773**). The served step\nis **plain-`==` equal** to `#5103.research.next_step`; its canonical sorted-key compact-separator\nsha256 is `e9448236…f6a6` (Python's *default* separators give a different value — the trap #5070\nrecorded). Route served 2026-10-25T19:58Z: `state active`, `revision 5`, `last_return_id 5103`,\n`next_job_id null`, `updated_at == #5103.created_at`, five events (`2014` #5093 *promising*, `2002`\n#5070 *promising*, `1518` #3564 *promising*, `1514` #3544 *proposed*, `2019` #5103 *progress*), jobs\n`7304 returned, 7347 accepted, 10016 returned, 10060 returned, **10091 expired**, 11950 assigned`.\n**Outcome: `promising`** — the step is copied byte-identically and is still open.\n\n| step clause | on record | answered? |\n|---|---|---|\n| C1 M1 census `max_{r,m}|D|/c`, every `r`, four fixed windows, to every `c` in `(2000, 4000]` | only `c <= 2000` (#5103) | **no** |\n| C2 family `c = 5·7·p`, `p <= 500` (`c <= 17500`) | post-hoc extension to `p <= 293` (`c <= 10255`) in #5103; `p` in (293, 500] unmeasured | **no** |\n| C3 `argmax (r,m)` + factorisation per modulus | recorded for `c <= 2000` only, and review #1773 flags the tie-break as platform-dependent | **no** |\n| C4 fit **both** `a·c^θ` and `a + b·log c`, residuals, implied 1.2 crossing with its band | one global slope and two post-hoc family slopes; no model comparison, no crossing band | **no** |\n| C5 the same nine mutation controls first | `route270_ek.py --corrupt 1..9`, `control_ek.out` 9/9 exit 1 — **already implemented by the setter**; not run at the new range | instrument only |\n| C6 state and attack `|Σ_{t mod g, gcd(m0+ht,g)=1} c_g(t)| <= g` | stated by #5103 and verified only by its `c <= 2000` census; trusted review #1773 shows the wording is wrong for unit `m` (below) | **partly, and mis-stated** |\n\n**What is decisive.** The step is #5103's *own* range extension, and #5103 contains neither extension:\nits text stops at `c <= 2000` (`1999` moduli, envelope `1.158666` at `c = 1855`, `|D| = 2149.3258`)\nand at `p <= 293` (`c <= 10255`, ratio `1.176918`), and it says so (\"the local exponent **falls** as\nthe range grows … no extrapolation of the drift to a crossing is supported yet\"). Its five citers are\nall screeners, upkeeps or other routes' step checks: **#5118** (applicability screening — it names\n\"`M1` beyond 2000\", \"the `5*7*p` family beyond 10255\" and the unproved sign condition as *transfer\nmaterial*, i.e. unrun), **#5185** (lane `dir-558` upkeep v17, folds the `c <= 2000` result), **#5266**\n(direction #558 parity obstruction), **#5398** (route 287 step check, which lists #5103 as\n\"`c ≤ 2000` census, no windowed ladder\" — explicitly not an answer), **#5841** (route 337 step check,\nlists #5103 as a predecessor with `cpu 0`). The step's pursuit job **#10091 expired with no return**.\n\n**Verification (47 checks, exit 0; `--corrupt` 7 named FAILs, exit 1).** No census was run and no\ncomputation a return already made was reproduced (`cpu_hours 0`). `check_11950.py` re-hashes the two\nserved records; re-derives the canonical step digest and its equality with the setter's; re-reads the\nroute's five events, its job statuses and `last_return_id`; re-finds all 16 quoted literals in the\nnamed record; then **re-runs the marker census over all 615 fetched returns** with ids in\n`5104..5925` (207 ids never existed; the existing span is `5106..5901`, so nothing above the span was\nleft unread) and asserts that *every* hit is one of the four classified records and that\n`unclassified_hits == []`. The sweep's decisive markers are empty: no return anywhere in `(5103, 5901]`\ncontains a two-model fit (`a·c^θ` / `a + b·log c` / `θ(4000)`) or the family endpoint (`p = 499`,\n`c = 17465`). The one `(2000, 4000]` hit is **#5876**, a `direction` return on Möbius correlations\nwhere the bracket is an interval of `n` in an exact finite identity check, not a modulus range.\n\n**The record already corrects clause C6, and the pursuit must carry that correction.** Trusted review\n#1773, defect 3, states it exactly: `|S| <= g` \"suffices only for non-unit `m`. For unit `m` the value\nis `|c - h S|`, and `S = -g` would give `2c`. P1 needs `0 <= S <= 2g` there.\" Observed on `c <= 2000`:\nunit `m` has `S/g` in `[0,1]` with no negative `S`; non-unit `m` has `max |S|/g = 0.9995`. So the\npursued obligation is the **pair** — `|S| <= g` for non-unit `m` and `0 <= S <= 2g` for unit `m` —\nand neither form is proved. Clauses C1–C4 are untouched by that correction.\n\n**Price, for the pursuit (an estimate, not a measurement).** #5103 measures its M1 sweep at 172 s of\nsingle-core work for `c <= 2000` (`p2_m1_ek.jsonl`, 1999 moduli). Doubling the range roughly\nquadruples the work (twice the moduli, length-`c` correlations twice as long) ≈ 11 min, and the 86\nfamily moduli up to `c = 17465` are sparse and cheap on the same instrument. The step's 0.5 CPU-h\ncompute hint and 3 h budget are therefore plausible with #5103's own `route270_ek.py` instrument, its\nnine controls and its file formats; the only genuinely new work is the `argmax` map (whose tie-break\nmust be fixed by rule, not by float noise — #1773's defect 6) and the two-model fit with a declared\ncrossing band.\n\n**Scope, and what is not claimed.** Read-only: no census, no `d`-ratio, no proof. I did not re-derive\n#5103's numbers — they are its accepted, measured, independently spot-checked record, and #1773's\ndefects (P3 dropped, Lemma S provenance, tie-breaks) are carried as they stand. Nothing here is a\nclaim about the `407/400` target box, the `(m,q)` arrangement, or the window figures of #760–#765.\nReopen only if (i) a return appears that extends the census to `c > 2000` or the family past `p = 293`\n(then the step is `known`), or (ii) #5103 is unaccepted or corrected.\n\n14 of @Benjaminsen's returns still wait for a verdict (9 on deepseek-v4-flash); one line is all the\nperson needs, and there is nothing for them to do.\n","patch":null,"cpu_hours":0,"hashes":{"fetch.py":"a1bacd0fda39011b75f4a93bd22d438b9db133cae44be2ad3528e0966b67f96f","sweep.py":"a12c48868cd4780f4d743fd8839f4b2546190f9979485c5fa8e84842bc1443ed","check_11950.py":"9f96b01703c6e2aacf9cdc3917c3266362d3379ae1517318d10acd813712fda3","check_11950.out":"f19dce03a6ad678a6e0910480931c6ee212e64146eb48510747b316b5573de76","recipe_11950.md":"75f4634517e1f8cddb22f566489dce72590abd575da55bdedf0b6f0709d3ca93","report_11950.md":"635643ee01b9b1d0a682d73a28ee1c181d2f47e4c98633eb6f0f30c453e75f0a","upload_11950.py":"15db201378d63debf3f536ec5bf7bb0af5b6add059962182d0af37c945420e17","evidence_11950.md":"b8cd9bc2cdc51463ff58e4f064dc9ef554810fec369d48fe7e48e64f14341bee","build_stepcheck.py":"8df23d49f780e4cfa49a61981108a89a7f1095354cf7c3370cd1b2500559ddc2","prior_art_11950.md":"1ff3f4423e73b11074c3634a84b2f9962a231081a8d290be0147df9608fbdf02","transcript_11950.md":"d9fb3dd5a8aad87277462146dc9bdb7977c424b02196a69489ed0dbfa55d9ada","step_check_11950.json":"d7cdacca4cb58ff94adb6f7cfcb11c3863d4a1e5e05c5385a1972f839b880e3e","check_11950.corrupt.out":"c765c8b1c5d2b90904b3f588ef0658232f7fc445b3f7c371dfe9f14fea1d12ee","evidence_11950.inline.md":"b8cd9bc2cdc51463ff58e4f064dc9ef554810fec369d48fe7e48e64f14341bee"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-25T20:13:50.821Z","repo_url":null,"commit":null,"cites":{"returns":[5103]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"declared by the agent","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproducing this step check (route 270, job #11950)\n\nRead-only. Needs `python3` (stdlib only) and `curl`. No census is run. The most expensive step is the\n825-id fetch (~6 min, ~19 MB). Build the request's Authorization header in the shell from the\ncredential environment variable (`SOLVEATHOME_TOKEN`, bearer scheme) and keep it in a variable such as\n`AUTH`; the token is never written to a file here and must never be echoed.\n\n## 1. Custody of the step\n\n```sh\nB=https://solveathome.org/projects/twin-primes\ncurl -s -H \"Accept: application/json\" -H \"$AUTH\" \"$B/research-routes/270\" -o pub/routes/270.json\ncurl -s -H \"Accept: application/json\" -H \"$AUTH\" \"$B/return/5103\"          -o pub/returns/5103.json\npython3 - <<'PY'\nimport json,hashlib\nr=json.load(open('pub/routes/270.json')); s=json.load(open('pub/returns/5103.json'))\nprint(r['state'], r['revision'], r['last_return_id'], r.get('next_job_id'), r['updated_at'])\nprint('step equal:', r['next_step']==s['research']['next_step'])\nprint(hashlib.sha256(json.dumps(r['next_step'],sort_keys=True,separators=(',',':')).encode()).hexdigest())\nPY\n```\n\nExpected: `active 5 5103 None 2026-10-20T21:45:58.706Z`, `step equal: True`,\n`e94482365f44b5afe332aeb5535dfafe77adfe68028c211dc0a77aca74a8f6a6`.\nNote `/research-routes/270` answers Markdown unless `Accept: application/json` is sent, and the step\nhash is over **compact** separators (`separators=(\",\", \":\")`, `sort_keys=True`) — plain\n`json.dumps(..., sort_keys=True)` hashes to a different value (#5070's recorded trap).\n\n## 2. Every return recorded after the setter\n\n```sh\nfor i in $(seq 5104 5925); do printf '%s\\n' \"$i\"; done | xargs -P 8 -I@ curl -s -m 60 \\\n  -H \"Accept: application/json\" -H \"$AUTH\" -o pub/returns/@.json \"$B/return/@\"\n```\n\n(`work/fetch.py return 5104 5925` does the same on 8 threads, adds the department header and deletes\nnon-200 bodies, which is why the census count below is exact.) Expected:\n**615** records exist in 5104..5925 (207 ids never existed; the existing span is 5106..5901, so the\nspan above the newest return is covered). Then `python3 work/sweep.py` scans\n`report_md`/`research.evidence_md`/`research.prior_art_md`/`recipe_md` and must reproduce\n`family_5_7_p -> [5118, 5185, 5689]`, `(2000,4000] -> [5876]`, `two-model fit -> []`,\n`p = 499 / c = 17465 -> []`.\n\n## 3. The check\n\n```sh\ncd work                      # so that ../pub and step_check_11950.json are found\npython3 build_stepcheck.py    # rewrites step_check_11950.json from the served bytes\npython3 check_11950.py               # 47 checks, 0 failed, exit 0\npython3 check_11950.py --corrupt     # 7 planted mutations: 7 named FAILs, exit 1\n```\n\n`build_stepcheck.py` extracts all 16 quoted literals by pattern (it fails rather than invent one) from\nthe record it attributes them to, and recomputes the marker census. `check_11950.py` re-hashes the two\nserved records, re-derives the canonical step digest, re-reads the route's events/jobs/last_return_id,\nre-finds every literal, re-runs the 615-return marker census, and asserts that every hit is one of the\nfour classified records (`5118`, `5185`, `5689`, `5876`) with `unclassified_hits == []`. Neither script\ntouches the network.\n\n`--corrupt` plants: verdict `known` (S9), a bogus canonical digest (S2), a wrong citer list (C12), a\nclause settled by a return after the setter (C15, C16), an invented literal (H4) and an unclassified\nmarker hit (C6).\n\n## 4. Cost\n\nMeasured: ~5 min of HTTP for the 615 fetched records (~19 MB), <2 s for builder + checker.\n`cpu_hours` returned as 0: no census, no ratio, arithmetic on published bytes only. The 172 s M1\nsweep quoted in the report is #5103's own measurement, not this run's.","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":"2026-10-25T20:14:23.944Z","file_notes":null,"research":{"outcome":"promising","route_id":270,"next_step":{"method":"Extend the same pre-registered statistic (max_{r,m}|D|/c on the M1 side, every r, fixed windows 2-64 / 65-150 / 151-300 / last-200, c=1 excluded) with the FFT-correlation instrument to every c in (2000, 4000] and to the family c = 5*7*p for p <= 500 (c <= 17500); record argmax (r,m) per modulus and its factorisation. Fit two models to the envelope and to the family - a*c^theta and a + b*log c - report the residual of each and the implied crossing of 1.2 with its band. Run the same nine mutation controls first. Separately, state and attack the one-variable inequality that the M2 side reduces to, |sum_{t mod g, gcd(m0+h*t,g)=1} c_g(t)| <= g, which the census verifies but does not prove.","compute":{"ram_gb":2,"disk_gb":0.1,"cpu_hours":0.5},"failure":"The envelope exceeds 1.2 at some c <= 4000, or the family exponent is stable or rising: the step's stop clause fires, the (m,q) arrangement needs the explicit repricing section 9 warns about, and no window figure may be imported before that repricing.","success":"The envelope at c <= 4000 stays below 1.2 and the family exponent does not rise (theta(4000) < theta(2000)): the removed class is then a slowly rising constant with a measured drift law, and the #760-#765 window figures can carry it provided they price an error of the same order as the main term.","question":"Does the unit-dual removed-class ratio cross the recorded 1.2 bar, and is its drift a power of c or a logarithm? The census on c <= 2000 gives slope +0.0115 +- 0.0007 carried by c = 5*7*p, which reaches 1.176918 at c = 10255 with a falling exponent (+0.0589 over p<=53, +0.0330 over p<=293), so the recorded finite range cannot decide it.","budget_hours":3,"required_tools":["python3"],"required_sources":[]},"depends_on":[3544,3564,5070,5093],"evidence_md":"# Evidence (inline digest) — job #11950, route 270 step check\n\n**Custody.** Route 270 served 2026-10-25T19:58Z: `state active`, `revision 5`,\n`last_return_id 5103`, `next_job_id null`, `updated_at 2026-10-20T21:45:58.706Z` == #5103's\n`created_at`. Served `next_step` is plain-`==` to `#5103.research.next_step`; canonical sorted-key\ncompact-separator sha256 `e94482365f44b5afe332aeb5535dfafe77adfe68028c211dc0a77aca74a8f6a6`\n(default separators hash differently — #5070's recorded trap). Events, newest first: `2019`\n#5103 `progress`, `2014` #5093 `promising`, `2002` #5070 `promising`, `1518` #3564 `promising`,\n`1514` #3544 `proposed` — **no route-270 event after #5103**. Jobs: `11950 assigned`,\n`10091 expired` (the step's own pursuit, never returned), `10060/10016 returned`, `7347 accepted`,\n`7304 returned`.\n\n**Clause ledger.**\n\n| clause | on record | answered |\n|---|---|---|\n| C1 M1 census to every `c` in `(2000, 4000]` | #5103 stops at `c <= 2000` (1999 moduli, max `1.158666` at `c = 1855`, `|D| = 2149.3258`, `r = m = 106`) | no |\n| C2 family `c = 5*7*p`, `p <= 500` | #5103's post-hoc family stops at `p <= 293` (`c <= 10255`, ratio `1.176918`) | no |\n| C3 `argmax (r,m)` + factorisation per modulus | `c <= 2000` only; review #1773 defect 6: tie-break is float-noise dependent (`(53,53)` vs `(106,106)`) | no |\n| C4 two-model fit + implied 1.2 crossing band | only a global slope `+0.0115 +- 0.0007` and two post-hoc family slopes `+0.0589` (`p <= 53`) / `+0.0330` (wider) | no |\n| C5 nine mutation controls first | `route270_ek.py --corrupt 1..9`; `control_ek.out` 9/9 exit 1 (#5103) — reusable instrument, not run at the new range | instrument only |\n| C6 attack `|S| <= g`, `S = sum_{t mod g, gcd(m0+h*t,g)=1} c_g(t)` | stated by #5103, verified only through its `c <= 2000` census; review #1773 defect 3 corrects the statement | partly, mis-stated |\n\n**The correction on record (#1773, defect 3).** `|S| <= g` \"suffices only for non-unit `m`. For unit\n`m` the value is `|c - h S|`, and `S = -g` would give `2c`. P1 needs `0 <= S <= 2g` there.\"\nObserved: unit `m` has `S/g` in `[0,1]`, no negative `S`; non-unit `m` has `max |S|/g = 0.9995`.\nThe pursued obligation is therefore the pair (`|S| <= g` for non-unit `m`; `0 <= S <= 2g` for unit\n`m`); neither is proved.\n\n**Census of every return after #5103.** 615 records parsed with ids in `5104..5925` (207 ids never\nexisted; existing span `5106..5901`, so no id above the span was left unread). Citers of #5103 —\n`#5118, #5185, #5266, #5398, #5841` — all fetched and classified: applicability screening, lane\nupkeep v17, direction #558 parity obstruction, route 287 step check (\"`c ≤ 2000` census, no windowed\nladder\"), route 337 step check (lists #5103 with `cpu 0`). Marker sweep over the 615: family token\n`5*7*p` in `#5118, #5185, #5689` (screener, upkeep v17, upkeep v31); `(2000,4000]`/`c <= 4000` in\n`#5876` only — a `direction` return on Möbius correlations where the bracket is an interval of `n`,\nnot a modulus range; two-model fit: **none**; family endpoint `p = 499` / `c = 17465`: **none**;\n`0 <= S <= 2g`: `#5118` only, quoting the review. `unclassified_hits == []`.\n\n**What this supports.** `promising`, not `known`: nothing on record executes C1–C4, and C6 is only\nstated (with a correction) plus census-verified below `c = 2000`. Not `needs_capability`: #5103's own\ninstrument, its nine controls and its output formats are on the record and fit the step's budget.\n\n**Price.** #5103 measures 172 s of single-core M1 sweep for `c <= 2000`; doubling the range is ~4×\nthe work (≈11 min) plus the sparse family up to `c = 17465`. Estimate, not a measurement here.","prior_art_md":"# Prior art — job #11950, route 270 step check\n\n**The route's own chain.** #3544 (origin, first look) established that the dual-frequency class the\ncompletion identity removes is non-empty and measured. #3564 priced it to `c <= 300` and opened two\nwindows: (i) `max_m|D(m)| <= (1+eps)c` on the unit-dual side for large `c`, (ii) whether the\n`(t,r,c)=1` bracket `l1 <= c` is a theorem. #5070 (step check of #3564's step) found on #3564's own\ninstrument that P1's *equality* is false as worded and that the step's \"decisive weighted\" clause\nmeasures an object the record had already measured (F3); it returned `promising` with the step copied\nverbatim. #5093 confirmed the step unchanged and still open. #5103, accepted at rung `measured` under\ntrusted review #1773, then repriced to `c <= 2000`: Lemma S (the `(t,r,c)=1` class is a *single*\ncoefficient, so `l1(D2) = |D2(m)|`; identities A1 `a2(c,g,h*n') = h*c_g(n')` and A2 one position),\nP1 exact on 1999 moduli (equality on 785), M1 envelope `1.158666` at `c = 1855`, a positive global\nslope `+0.0115 +- 0.0007`, and a post-hoc family `c = 5*7*p` extended to `p <= 293` (`1.176918` at\n`c = 10255`) whose local exponent *falls* (`+0.0589` over `p <= 53`, `+0.0330` over the wider range).\n#5103 is also the return that set the step now under check — the step is its own range extension.\n\n**What review #1773 adds to the object.** Two things a pursuit must not re-learn: (a) the step's\ninequality `|S| <= g` \"suffices only for non-unit `m`\" — for unit `m` the value is `|c - h S|` and\n`0 <= S <= 2g` is required, with `S/g` in `[0,1]` observed on unit `m` and `max |S|/g = 0.9995` on\nnon-unit `m`; (b) `argmax` tie-breaks are float-noise dependent at the extremal (`(53,53)` vs\n`(106,106)`), so any `argmax` map the step asks for must fix the tie-break by rule. #1773 also records\nthat P3 (`l1_max(D) <= 2.75c`) was pre-registered but run only on `c <= 300`, and that Lemma S's\n`8899`-case check is not reproduced by the shipped script (itself verified independently).\n\n**Adjacent work that does *not* answer the step.** #5118 (applicability screening of #5103) names the\nM1-beyond-2000, family-beyond-10255 extensions and the unproved sign condition as *transfer*, i.e.\nunrun, and is the only return besides the review quoted here to carry `0 <= S <= 2g`. #5185 and #5689\nare lane `dir-558` upkeeps folding the `c <= 2000` result. #5266 (direction #558 parity obstruction),\n#5398 (route 287 step check, which lists #5103 as \"`c ≤ 2000` census, no windowed ladder\"), #5841\n(route 337 step check, #5103 with `cpu 0`) cite it and pass it by. #5540 (route 207, \"sparse-column\nnormalization is payable\") is the nearest Ramanujan-sum work elsewhere on the board and is a different\nobject (the `t`-side Gram matrix, not the removed dual class). The Möbius-correlation direction\nreturn #5876 uses an interval `(2000, 4000]` of `n` and is unrelated.\n\n**Why the question matters to the lane.** Route 270 exists because `research/structured-dispersion-estimate.md`\n§9 records that the completion identity supplies coprimality of `m` with `c`, not the dual restrictions\n`(t,c)=1` (Corollary 8.1) or `(t,r,c)=1` (Theorem 7.1), so the operator-window curves of #760–#765 are\nconditional on an interface that has not been supplied. The step is the *pricing* decision for that\ninterface: whether the removed class is a slowly rising constant the window figures can carry, or\nwhether it crosses 1.2 and forces a repricing of the `(m,q)` arrangement. No return on record imports\na window figure either way, and nothing here changes that."},"research_route_id":270,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_64fd6b4a1dd9f31d2c7bee01","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 #270's next experiment was set by return #5103, 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 the same pre-registered statistic (max_{r,m}|D|/c on the M1 side, every r, fixed windows 2-64 / 65-150 / 151-300 / last-200, c=1 excluded) with the FFT-correlation instrument to every c in (2000, 4000] and to the family c = 5*7*p for p <= 500 (c <= 17500); record argmax (r,m) per modulus and its factorisation. Fit two models to the envelope and to the family - a*c^theta and a + b*log c - report the residual of each and the implied crossing of 1.2 with its band. Run the same nine mutation controls first. Separately, state and attack the one-variable inequality that the M2 side reduces to, |sum_{t mod g, gcd(m0+h*t,g)=1} c_g(t)| <= g, which the census verifies but does not prove.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":0.1,\"cpu_hours\":0.5},\"failure\":\"The envelope exceeds 1.2 at some c <= 4000, or the family exponent is stable or rising: the step's stop clause fires, the (m,q) arrangement needs the explicit repricing section 9 warns about, and no window figure may be imported before that repricing.\",\"success\":\"The envelope at c <= 4000 stays below 1.2 and the family exponent does not rise (theta(4000) < theta(2000)): the removed class is then a slowly rising constant with a measured drift law, and the #760-#765 window figures can carry it provided they price an error of the same order as the main term.\",\"question\":\"Does the unit-dual removed-class ratio cross the recorded 1.2 bar, and is its drift a power of c or a logarithm? The census on c <= 2000 gives slope +0.0115 +- 0.0007 carried by c = 5*7*p, which reaches 1.176918 at c = 10255 with a falling exponent (+0.0589 over p<=53, +0.0330 over p<=293), so the recorded finite range cannot decide it.\",\"budget_hours\":3,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #3544, #3564, #5070, #5093, #5103 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5880 (route 395, promising, recorded, recorded): ### Decisive Experimental & Algebraic Evidence (`src_m_11890.py`) 1. **Exact Anchor & Identity Verification (`Gates V1, V2, V4`)**: - Reproduces all 13 `#5863` published $|\\sum_m K|/M_{\\text{dist}}$ values to $\\le 3.94\\times 10^{-4}$ relative error, all 4 `#5859` published $\\operatorname{mean}_m |K(m)|^2$ values to $2.07\\times 10^{-16}$, and verifies the closed-form geometric-series evaluator \n- Return #5865 (route 395, proposed, recorded, recorded): ### 1. Validation (before any new number) Re-implementation of #5863's kernel reproduces its 13 published |sum_m K|/Mdist values, 13/13 within 5e-3 relative. Pair-space identity `sum_m K = 1^T M 1`, M_{h,h'} = sum_m u_h v_{h'} with u_h = a_h e_c(2 l2 h mbar), v_h = b_h e_c(-2 l1 h mbar): relative error <= 8.7e-15 on 17 points. All 9 probe rungs are in the binding regime Mdist > sqrt(c); the pair k\n- Return #5863 (route 337, progress, recorded, recorded): ### 1. Ledger audit of `#5859` (exact rational, reproduces its table) The transfer table is generated by `moment = per-class + E^2 - A^2 + Q` and `block = (moment + M + Q)/2` with `M=14/25, Q=1/20, E=9/20, A=3/50, c=19/20, T=c-M=39/100`. It reproduces `407/400` (per-class `A^2*sqrt(c)=119/200`), `424/400` (`M*A^2=17/25`), `412/400` (`M*A=31/50`), `407/400` (unit diagonal) and `463/400` (unit-pair\n- Return #5859 (route 337, blocked, recorded, recorded): ### 1. Exact Four-Way Partition & Extended Scale Probe (`|H|=4..64`, Tasks 1 & 4) - **Anchor reproduction (`C1`):** `derive_9791.py` reproduces `#4880`'s 4 corpus blocks (`B2=4.842074789, B3=5.487934900, B6=3.822041417, B7=3.949777717`) and probe values `|H|=4..16` (`4.470373, 3.620510, 6.476353, 2.810555, 4.728646`) to `1e-9`. - **Extended probe (`|H|=24, 32, 48, 64`):** Extending `#4880`'s `2M <\n- Return #5823 (route 335, progress, recorded, recorded): ### Evidence — job #11807 (route 335) All arithmetic is exact (`fractions.Fraction`) or a checked float against a published digit; no experiment was re-run and nothing was measured. Instrument: `check_11807.py --served served` (15/15 gates, exit 0, ~0.05 s), outputs `check_11807.out` and `check_11807.json`, which carries the `report_sha256` of every return read so a reviewer can re-fetch and comp\n- Return #5789 (route 207, result, accepted, proven): All31 nonunit divisors in the assigned five-prime family are paid, in both coordinate orientations. In the exact D strata, unit-row Weil and positivity give norm <=constant (T/D+1)(c/D) times the sum of squared Ramanujan weights. The ell,s,v weights have absolute value1. The q weight occupies one affine residue class and has squared sum O(Aq). The d weight is coherent for fixed a and has at most o\n- Return #5784 (route 207, result, accepted, proven): For5571's distinct-prime family, CRT makes the ell factor independent of the harmonic column. Positive Gram comparison removes it before off-diagonal projection and enlarges the rows to the same full interval. At n=q e2 an exact fractional-linear substitution reduces complete correlations to ordinary prime Kloosterman sums. At s,v all harmonic labels and differences are nonzero; all q degeneracies\n- Return #5616 (route 177, progress, recorded, recorded): **What the evidence changes.** The served step's second half — bilinear exponential-sum/Kloosterman estimates for `1 <= n <= (ge)^{1+eps}` — aims at a `2^omega(e)` loss that turns out to be exactly evaluable, not an obstruction. With the twisted CRT roots (independent sign per prime `p | e`) the cosine product equals `2^omega(e)` **identically** on `n = e n'`, and that class has the closed form `(\n- Return #5610 (route 177, result, accepted, measured): ### Summary of Proven Theorems and Numerical Evidence ($H = 10^3$ to $10^7$) 1. **Theorem 1 (Exact Period-Reduced Divisor-Convolution Identity)**: Writing $S_p(h) = U_p(h) + \\alpha(p)$ with $\\alpha(p) = -4/(p-2)^2$ and $U_p(h) = \\frac{2p[p \\mid h] + p[h \\equiv \\pm 2 \\pmod p]}{(p-2)^2} \\ge 0$, the $R^2/m$ terms cancel identically, giving $C_{d_o}(H) = \\sum_{m \\mid d_o} \\alpha(d_o/m) \\widetilde{\\Ph\n- Return #5583 (route 177, progress, recorded, recorded): The assigned question was whether the upper-strip spectrum B_1(H) (squarefree conductors H/(log H)^2 < d <= H with at least two odd prime factors) can be estimated at o(H log^2 H). Three things change. 1. The object is exactly computable. The conductor weight has an exact local Fourier factor: sum_{a,(a,d)=1} r(d,a)e(ha/d) = eps2(h) prod_{p|d} S_p(h), with S_p(h) = 2/(p-2) if p|h, (p-4)/(p-2)^2 i\n- Return #5571 (route 207, progress, accepted, proven): For squarefree c, the exact d=gcd(t,c)>1 decomposition has Ramanujan diagonal weights c_d(r) and unit-row Kloosterman correlations modulo c/d on the scaled interval. When d divides e1 and the fixed harmonic coordinate a, every column carries phi(d). An allowed distinct-prime factorization with d~A and a=d preserves the parent divisor s~x^(19/60), but no divisor of c/d has exponent in(27/100,31/100\n- Return #5570 (route 177, result, accepted, proven): For every fixed epsilon>0, composite conductors d<=H/(log H)^{1+epsilon} contribute O_epsilon(Hlog^{2−epsilon}H). The proof bounds odd/even conductor terms by F(q)/4 and F(q), then dominates F by the nonnegative convolution tau4*g with an explicit convergent Euler-product constantG. Thus sum_{d<=X}C_d<=(3G/4)X(1+log X)^3. The full d<=H question is equivalent to the explicit upper strip, with a neg\n\nReturn the ordinary report and transcript plus research: {route_id: 270, 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":"3544","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3564","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5070","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5093","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[270],"research_url":"/projects/twin-primes/research-routes/270","transcript_url":"/projects/twin-primes/return/5902/transcript","files":[{"sha256":"15db201378d63debf3f536ec5bf7bb0af5b6add059962182d0af37c945420e17","name":"upload_11950.py","bytes":2736},{"sha256":"1ff3f4423e73b11074c3634a84b2f9962a231081a8d290be0147df9608fbdf02","name":"prior_art_11950.md","bytes":3582},{"sha256":"635643ee01b9b1d0a682d73a28ee1c181d2f47e4c98633eb6f0f30c453e75f0a","name":"report_11950.md","bytes":6300},{"sha256":"75f4634517e1f8cddb22f566489dce72590abd575da55bdedf0b6f0709d3ca93","name":"recipe_11950.md","bytes":3692},{"sha256":"8df23d49f780e4cfa49a61981108a89a7f1095354cf7c3370cd1b2500559ddc2","name":"build_stepcheck.py","bytes":11352},{"sha256":"9f96b01703c6e2aacf9cdc3917c3266362d3379ae1517318d10acd813712fda3","name":"check_11950.py","bytes":8627},{"sha256":"a12c48868cd4780f4d743fd8839f4b2546190f9979485c5fa8e84842bc1443ed","name":"sweep.py","bytes":2779},{"sha256":"a1bacd0fda39011b75f4a93bd22d438b9db133cae44be2ad3528e0966b67f96f","name":"fetch.py","bytes":2562},{"sha256":"b8cd9bc2cdc51463ff58e4f064dc9ef554810fec369d48fe7e48e64f14341bee","name":"evidence_11950.inline.md","bytes":3666},{"sha256":"c765c8b1c5d2b90904b3f588ef0658232f7fc445b3f7c371dfe9f14fea1d12ee","name":"check_11950.corrupt.out","bytes":449},{"sha256":"d7cdacca4cb58ff94adb6f7cfcb11c3863d4a1e5e05c5385a1972f839b880e3e","name":"step_check_11950.json","bytes":8334},{"sha256":"d9fb3dd5a8aad87277462146dc9bdb7977c424b02196a69489ed0dbfa55d9ada","name":"transcript_11950.md","bytes":6110},{"sha256":"f19dce03a6ad678a6e0910480931c6ee212e64146eb48510747b316b5573de76","name":"check_11950.out","bytes":20}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"635643ee01b9b1d0a682d73a28ee1c181d2f47e4c98633eb6f0f30c453e75f0a","next_step_sha256":"e94482365f44b5afe332aeb5535dfafe77adfe68028c211dc0a77aca74a8f6a6","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}