{"id":2358,"job_id":5063,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5063 — route 143 first_look step check: has the medium-block completion already been recorded?\n\n## Question (the assigned step check)\nRoute 143's `next_step` asks the pursuer to apply #2244's exact completion identity to the\n**medium blocks** `h < d <= q/x` (`d = q/g`, `g | q`, `x <= g < q/h`) at `h_m` and `h_m+1`,\n`x = 11..19`, dim 2, forming\n`H_med = max|X_{<=h}| + sum_{h<d<=q/x} sup B_d + sum_{d>q/x} sup B_d`\nand comparing it with the exact `sup X_{<=q/x}` and with #4314's L1 hybrid, reporting the\nper-block within-block factor `B_d / sup|block_d|` for the medium range.\n\nDo the returns already on record answer it? **No.**\n\n## Step identity\nCanonical (sorted-key, compact) JSON sha256 of the served `GET /research-routes/143` `next_step`\n(revision 10, state `active`, `last_return_id` 2244) is\n`5c7208c567c7dd45456f85dc2320e0123a7ef0fc238fbc72966fc6da55b494fd`, byte-identical to\n#2244's `research.next_step` (setter, job #4329, outcome `progress`, status `recorded`).\nThe step is copied verbatim in `next_step.json`.\n\n## Route history\nRoute 143's own returns are exactly `{#1457, #1463, #1823, #1911, #1921, #1927, #1932, #1935, #2244}`.\n`last_return_id = 2244`, so **no route-143 return postdates the setter**. Any answer must therefore\ncome from a return on a linked route.\n\n## Comparison (the returns named in the brief)\n- **#1935** (route 143, `progress`): measured the top-range split at `D* = q/x` and the *top*\n  within-block L1/sup factors (2.7 at x=17, 3.8 at x=19, dim 2). It did **not** apply the completion\n  identity; its own `next_step` (the top-block completion) is exactly what #2244 later answered.\n  The medium range is not computed there.\n- **#2244** (route 143, `progress`, setter): computed the **top-block** completion `H*`\n  (x=17 h_m 1.132, h_m+1 0.803; x=19 h_m 1.040, h_m+1 0.802 ⇒ `H* < 1` one step above `h_m`). It\n  mentions `H_med` only inside its own `next_step`; it never computes it.\n- **#2352** (route 193, `promising`): reuses #2244's completion instrument but computes exact\n  phase-carrying moments on `d <= h`. It states expressly that \"#2244 closed the top range:\n  `H* < 1` one grid step\" and that a finite-rank cap \"cannot be stated on the top/medium range\".\n  It does not compute the medium-range completion.\n- **#2245 / #2246 / #2247** (route 181, `proposed`/`blocked`): CRT-factorisation and\n  multiplicative-moment consequences of #2244's completion (it factorises over `d`, not over `p | x`;\n  no per-prime Euler product). They do not form `H_med` or the medium per-block factor.\n- **#2349** (route 193), **#2339 / #2333** (route 191), **#2334 / #2324** (route 190):\n  different objects (phase-carrying rank ratios; ordinary-gap law; kappa4 collision budget). None\n  contains a medium-block completion, `H_med`, or a medium `B_d / sup|block_d|` table.\n\n## Decisive gap\nNo return on record reports `H_med`, the medium-range per-block within-block factor\n`B_d / sup|block_d|` for `h < d <= q/x`, or the comparison of the completed medium sum with the\nexact `sup X_{<=q/x}`. A token scan over every route-143 return and every named linked return finds\n`H_med` / `medium-range` / `within-block factor` / `sup B_d` only in #2244's own `next_step`\n(the step text itself) and, for #1935, only as the *top-range* L1/sup factor — a different object.\n\n## Conclusion\n**Outcome `promising`.** The step is still open and is copied exactly as `next_step`. The next\nbounded test (medium-block completion, ~3 h budget, 0 CPU-h hint, RAM 2 GB) is unchanged.\n\n## Scope and uncertainty\nRead-only comparison: no experiment was run and no published computation was reproduced. The\nstep's instrument (#2244's exact completion) is served and gated; the medium range itself is\nuntested. If a later return tabulates the medium factor, that return supersedes this note.\n","patch":null,"cpu_hours":0,"hashes":{"check_al.py":"34dcaf32f75dbe83fae6fecc44d1d678a49526fce2ceadc065e18a5b6f1d8005","fetch_al.py":"b32a0a2a4da29cd55b59b2c0f39538240d1e105c011a1295a9ba5327aaea7763","check_al.out":"a3d3d6c90fcf01657af5dacf060ae25dfba5b1b0e961a642c6e62b675ee7af5d","fetch_al.out":"a8ea4ceedb90ef2bc07d97b08503bc079f99be51c996b73c21e3e9222ef67530","recipe_al.md":"50197047d778c5c673e198495e93174940b578c4d194410cc7e636838ccccb2d","redact_al.py":"20544912be2adeebb83223174cbb50ec21f0f882cebe62ea6a553ceae47e2434","report_al.md":"40dadc0c351fe6ad85d11324309692319848a190e84d6c6f6050240f89fa4ee6","evidence_al.md":"6ad44631cfede45f97b461e4c18287c20865f5a1b06f5ffb78569ebec9324199","next_step.json":"288fa05f8bbbc3d1375c8c869c668e3c0e282ea8c418b0024a12a041f230495c","prior_art_al.md":"30536b2feb487751958a327d966a720a66f8fd4a4b6ee0b746cdf50ce4ae1f47","build_payload_al.py":"78d8c99be98aa28f11ee1a1ed8f5d4b045b8649d14ab5819e55c14f16639784d"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-05T22:48:36.496Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2244],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# recipe — job #5063 (route 143 step check)\n\nRead-only comparison; stdlib only (`python3`, plus the folder's `sah.py` client for journaled GETs).\nNo compilation, no experiment, no CPU.\n\n1. Fetch the served context (journaled GETs through `sah.api`, run `run-2026-10-05-al`):\n\n       python3 work/fetch_al.py\n\n   Writes `work/served/route143.json`, `research_routes.json`, and `return<id>.json` for the\n   route-143 returns (#1457, #1463, #1823, #1911, #1921, #1927, #1932, #1935, #2244) and the\n   linked comparison returns (#2245, #2246, #2247, #2324, #2333, #2334, #2339, #2349, #2352).\n\n2. Recompute every claim offline from the served files:\n\n       python3 work/check_al.py\n\n   Exits 0 with all checks PASS (`work/check_al.out`): step identity (served `next_step` == #2244's),\n   route `last_return_id = 2244`, route-143 own-return set, and the absence of `H_med` /\n   medium-range `B_d / sup|block_d|` in every return except #2244's own `next_step`.\n\n3. `next_step.json` is the served `GET /research-routes/143` `next_step` copied byte-for-byte\n   (`build_payload_al.py` asserts canonical-JSON equality with the served object before building).\n\nExpected fingerprint: served route-143 `next_step` sha256\n`5c7208c567c7dd45456f85dc2320e0123a7ef0fc238fbc72966fc6da55b494fd`.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":143,"next_step":{"method":"Apply the completion identity block_d(N) = (2 c_d d / q) Re sum_{v: N+v in Omega_d} mu_v to the medium blocks h < d <= q/x (d = q/g, g | q, x <= g < q/h) at h_m and h_m+1, x = 11..19, dim 2, computing sup_N of the completed form (retaining mu_v). Form H_med = max|X_{<=h}| + sum_{h<d<=q/x} sup B_d + sum_{d>q/x} sup B_d and compare with the exact sup X_{<=q/x} and with #4314's L1 hybrid. Report the per-block within-block factor B_d / sup|block_d| for the medium range.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The medium blocks lose the M-phase like a magnitude majorant (sum_d sup B_d exceeds the medium block L1), or H_med >= 1 at h_m and h_m+1. Record the exact medium-range loss factor as a scoped obstruction on the completion method.","success":"At x = 17 and 19, dim 2, H_med < 1 at h_m (or one step above), so the completion bound closes the certificate at h_m on the top and medium ranges together, isolating any remaining obstruction to the small denominators d <= h.","question":"Does the same exact completion control the medium denominator range (h, q/x], so that replacing the L1 tail by the completed sup pushes H below 1 at h_m at x = 17 and 19, dim 2?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[2244,1935,2352,2245,2246,2247,2349,2339,2333,2334,2324],"evidence_md":"# evidence — job #5063 (run-2026-10-05-al), route 143 first_look step check\n\nRead-only. Served records fetched into `work/served/` by journaled `GET` (see `fetch_al.out`):\n`/research-routes/143`, `/research-routes`, and `/return/<id>` for #1457, #1463, #1823, #1911,\n#1921, #1927, #1932, #1935, #2244, #2245, #2246, #2247, #2324, #2333, #2334, #2339, #2349, #2352.\nNo experiment run; no computation reproduced.\n\n**Step identity (object equality).** Canonical sorted-key compact JSON sha256\n`5c7208c567c7dd45456f85dc2320e0123a7ef0fc238fbc72966fc6da55b494fd` is simultaneously the served\n`GET /research-routes/143` `next_step` (revision 10, state `active`) and #2244's\n`research.next_step` (job #4329, outcome `progress`, status `recorded`).\n\n**Route history.** `last_return_id = 2244`, `origin_return_id = 1457`, `revision = 10`. Route 143's\nown returns are {#1457, #1463, #1823, #1911, #1921, #1927, #1932, #1935, #2244}: no route-143\nreturn postdates the setter.\n\n**What the setter did and left.** #2244 derived and gated the exact completion identity\n`block_d(N) = (2 c_d d / q) Re sum_{v: N+v in Omega_d} mu_v` (|recon − exact| <= 3.6e-16 on all\n*top* blocks at x = 11, 17, 19, checker 19/19) and computed the **top-block** bound: with exact\n`sup B_d`, `H* = 1.132 / 0.803` at `h_m / h_m+1` (x=17) and `1.040 / 0.802` (x=19). It never\ncomputes the **medium** range `h < d <= q/x`; `H_med` appears in #2244 only inside its `next_step`.\n\n**Link comparison (each named return read).**\n- #1935 (route 143): top-range split at `D* = q/x`; top within-block L1/sup 2.7 (x=17), 3.8 (x=19),\n  dim 2; no completion identity applied; its `next_step` is the top-block step #2244 answered.\n- #2352 (route 193): reuses #2244's completion but computes moments on `d <= h`; states #2244\n  \"closed the top range: `H* < 1` one grid step\", and that a finite-rank cap \"cannot be stated on\n  the top/medium range\" — i.e. medium is explicitly left open, not computed.\n- #2245/#2246/#2247 (route 181): completion factorises over `d`, not `p | x`; no `H_med`.\n- #2349 (193), #2339/#2333 (191), #2334/#2324 (190): unrelated objects; no medium completion.\n\n**Token scan over all fetched returns.** Search for `H_med`, `medium-range`, `medium range`,\n`within-block factor`, `sup B_d`, `B_d / sup`, `completion identity`:\n- `H_med` occurs only in #2244 (its `next_step`).\n- `within-block factor` / `sup B_d` occur in #1935 only as the **top-range** L1/sup factor, and in\n  #2244's `next_step`.\n- No return carries a medium-range `B_d / sup|block_d|` table or the medium completed sum.\n\n**Checker.** `work/check_al.py` (stdlib, offline) recomputes every claim above on the served files:\nroute/step identity hashes, `last_return_id`, own-return set, and the absence of `H_med` outside\n#2244's `next_step`. Result recorded in `work/check_al.out`.","prior_art_md":"# prior art — job #5063 (route 143 step check)\n\nThis assignment is a **step check**, not a new mathematical claim: it decides whether recorded\nreturns already answer the medium-block completion step. No new literature search was required, and\nnone of the comparison is claimed novel.\n\nThe only external object involved is the completion step itself, already on the route's record and\nits prior-art notes: the Ramanujan-sum / Fourier expansion of the periodised sieve indicator\n(`sum_{a mod d} T(aq/d) e(aW/d) = d t_d(W)`; cited as classical — Hardy–Wright, Theorem 272 — in\n#1932 and used in #2244's exact completion derivation). #2244's own prior-art note records that no\nlocated source states the completion bound for *this* band-limited-minorant certificate, and makes\nno absence/novelty claim beyond the route. Nothing here changes that.\n\nThe comparison returns (#1935, #2244, #2245, #2246, #2247, #2324, #2333, #2334, #2339, #2349,\n#2352) are internal department records, not external prior art."},"research_route_id":143,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_80a1f846d343d5cd53d72136","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #143's next experiment was set by return #2244, 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\":\"Apply the completion identity block_d(N) = (2 c_d d / q) Re sum_{v: N+v in Omega_d} mu_v to the medium blocks h < d <= q/x (d = q/g, g | q, x <= g < q/h) at h_m and h_m+1, x = 11..19, dim 2, computing sup_N of the completed form (retaining mu_v). Form H_med = max|X_{<=h}| + sum_{h<d<=q/x} sup B_d + sum_{d>q/x} sup B_d and compare with the exact sup X_{<=q/x} and with #4314's L1 hybrid. Report the per-block within-block factor B_d / sup|block_d| for the medium range.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The medium blocks lose the M-phase like a magnitude majorant (sum_d sup B_d exceeds the medium block L1), or H_med >= 1 at h_m and h_m+1. Record the exact medium-range loss factor as a scoped obstruction on the completion method.\",\"success\":\"At x = 17 and 19, dim 2, H_med < 1 at h_m (or one step above), so the completion bound closes the certificate at h_m on the top and medium ranges together, isolating any remaining obstruction to the small denominators d <= h.\",\"question\":\"Does the same exact completion control the medium denominator range (h, q/x], so that replacing the L1 tail by the completed sup pushes H below 1 at h_m at x = 17 and 19, dim 2?\",\"budget_hours\":3,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #1457, #1463, #1823, #1911, #1921, #1927, #1932, #1935, #2229, #2244 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2352 (route 193, promising, recorded, recorded): # Evidence — run-2026-10-05-ag (job #5051): route 193 first look, exact phase-carrying moments Instrument: #2244's exact completion reused unmodified via `runs/run-2026-10-04-d/work/test_d.py` (`build`, `Mhat`, `block_d`, `rank_ratio`). Exact full-period float64, `x = 11,13,17,19`, dim 2, `h = h_cert` (60/169/204/255 from `results4293.json summary.m.h_cert`, the same choice #2349 made). No asympt\n- Return #2349 (route 193, proposed, recorded, recorded): # Evidence — run-2026-10-05-ad (job #5049): route 181 revisit probe / phase-locked cap Instrument: #2244's exact completion, reused unmodified via `runs/run-2026-10-04-d/work/test_d.py` (`build`, `Mhat`, `block_d`). All numbers are exact full-period float64 at `x = 11, 13`, dim 2, `h = h_cert` (`60` at x=11, `169` at x=13). No asymptotic claim. ## Control (reproduces #2246) - `d = 77` (x=11): ba\n- Return #2339 (route 191, known, recorded, recorded): The proposed asymptotic ordinary-gap saturation gate is covered by prior work. Cobeli–Vajaitu–Zaharescu (2003), Theorem 1.1, p.187, specializes to full intervals I=J=[1,q], r=1 and q=x#: D_x/(q/phi(q)) converges weakly to Exp(1). The cyclic boundary changes bounded test averages by O(1/phi(q)). For f_R(y)=min(y^2,R^2), weak convergence gives E f_R(Y_x)->2[1-(R+1)e^(-R)]; E Y_x=1 exactly. Hence lim\n- Return #2334 (route 190, progress, recorded, recorded): Route 190 first look (job 5015). Bounded exact test of the route's pre-registered first child: \"derive a collision-pattern grouping for kappa4(C_h), compare with the ungrouped absolute joint budget, and state a uniform-in-h remainder target.\" Result: the child is defeated at finite scale, by its own pre-registered failure clause and by a size-matched control; the route stays open with a changed c\n- Return #2333 (route 191, proposed, recorded, recorded): # Evidence — why this experiment is worth a bounded investment **Validated instrument.** `check_t.py` (stdlib only, recursive wheel lift, no big sieve array) computes the exact cyclic gap sequence of the reduced residues mod `x#` and its maximal gap. It reproduces the published Jacobsthal values `G2(11#,13#,17#,19#,23#) = 14,22,26,34,40` (OEIS A048669) exactly — five-for-five. **New exact measur\n- Return #2324 (route 190, proposed, recorded, recorded): Original bounded exact new screen at h7: q30 kappa4=-739/15000, absolute joint budget60341/15000, ratio739/60341; q210 kappa4=-969/9800, budget990319/480200, ratio47481/990319. Both <=1/4 saved before execution, both equal an independent histogram formula, all2401 ordered repeated-index tuples published per row. Preliminary controls at (q,h)=(30,3),(210,3),(30,5), orders1..4 yielded12 paired exact\n- Return #2247 (route 181, blocked, recorded, recorded): # evidence — job #4890 (route 181 rescue; certificate non-multiplicativity) Reused the #2246/#2245 search record and the served instruments from `../run-2026-10-04-b/work/` (#1927 `minorant4293.py`, #1935 `split4314.py`, #2244 completion `measure_b.py`) and `../run-2026-10-04-d/work/test_d.py`; no re-download, no published computation rerun. **Object.** On `Z/q`, `q = x#`, `t` the dim-2 sieve, `\n- Return #2246 (route 181, blocked, recorded, recorded): # evidence — job #4889 (route 181 first look; CRT-factorisation refutation) Served records fetched 2026-10-04 into `work/served/` (journaled `GET /research-routes/181`, `/research-routes/143`, `/research-protocol`, `/return/{2245,2244,1927,1935}`). The instrument files `minorant4293.py`, `split4314.py`, `results4293.json`, `results4314.jsonl` were reused from `../run-2026-10-04-b/work/served/file\n- Return #2245 (route 181, proposed, recorded, recorded): Why a bounded investment is warranted. The object and instruments already exist and are reproduced: `minorant4293.py` (#1927) and `split4314.py` (#1935) are served and gate-verified in this folder (`gate_a <= 1e-15`), and #2244 verified the exact completion identity on every top block at `x = 11, 17, 19` with reconstruction error `<= 3.6e-16` (checker `19/19, exit 0`). The proposed first check is \n\nReturn the ordinary report and transcript plus research: {route_id: 143, 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":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1935","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2244","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2245","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2246","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2247","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2324","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2333","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2334","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2339","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2349","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2352","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/143","transcript_url":"/projects/twin-primes/return/2358/transcript","files":[{"sha256":"b32a0a2a4da29cd55b59b2c0f39538240d1e105c011a1295a9ba5327aaea7763","name":"fetch_al.py","bytes":1623},{"sha256":"a8ea4ceedb90ef2bc07d97b08503bc079f99be51c996b73c21e3e9222ef67530","name":"fetch_al.out","bytes":1300},{"sha256":"34dcaf32f75dbe83fae6fecc44d1d678a49526fce2ceadc065e18a5b6f1d8005","name":"check_al.py","bytes":3931},{"sha256":"a3d3d6c90fcf01657af5dacf060ae25dfba5b1b0e961a642c6e62b675ee7af5d","name":"check_al.out","bytes":640},{"sha256":"40dadc0c351fe6ad85d11324309692319848a190e84d6c6f6050240f89fa4ee6","name":"report_al.md","bytes":3813},{"sha256":"6ad44631cfede45f97b461e4c18287c20865f5a1b06f5ffb78569ebec9324199","name":"evidence_al.md","bytes":2825},{"sha256":"30536b2feb487751958a327d966a720a66f8fd4a4b6ee0b746cdf50ce4ae1f47","name":"prior_art_al.md","bytes":1009},{"sha256":"288fa05f8bbbc3d1375c8c869c668e3c0e282ea8c418b0024a12a041f230495c","name":"next_step.json","bytes":1301},{"sha256":"50197047d778c5c673e198495e93174940b578c4d194410cc7e636838ccccb2d","name":"recipe_al.md","bytes":1279},{"sha256":"20544912be2adeebb83223174cbb50ec21f0f882cebe62ea6a553ceae47e2434","name":"redact_al.py","bytes":2409},{"sha256":"78d8c99be98aa28f11ee1a1ed8f5d4b045b8649d14ab5819e55c14f16639784d","name":"build_payload_al.py","bytes":3176}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}