{"id":2705,"job_id":5638,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #5638 (explore `first_look`, lane adversarial): route 193 step check\n\n**Step check before pursuit. Read-only.** No experiment was run and no computation was reproduced;\n`cpu_hours = 0`. Every claim below is re-derived offline from the served copies in this directory by\n`check_hp.py` (stdlib only).\n\n**Verdict: `promising`.** No return on record answers route 193's held step; the step is copied\nexactly into `next_step.json` and the held pursuit goes out with this note.\n\n## Step identity and custody (exact)\n\n- `route193.json` (`GET /research-routes/193`): `id = 193`, `state = active`, `revision = 4`,\n  `last_return_id = 2443`, `origin_return_id = 2349`,\n  `dependencies = [{2244, recorded}, {2349, recorded}, {2352, recorded}]`, `obstacle = null`.\n- Events (4): `#2349 proposed`, `#2352 promising`, `#2437 promising`, **`#2443 result`**.\n- Canonical (sorted-key, compact, UTF-8) SHA-256 of `route193.next_step` is\n  `382661eb0ed5bff7e4053d6b1270b8dbd0f3a16634a831fa582d6c413c0a1e78`, and it is **byte-identical**\n  (`==`) to `return/2443.research.next_step`. `#2443` is job **#5056** (`route_id = 193`,\n  `outcome = result`, `status = accepted`, `final_rung = verified`). `next_step.json` is that object\n  verbatim; `promising` copies it exactly.\n- `route193.last_return_id == 2443` ⇒ **no route-193 return postdates the setter**. The route's own\n  returns are exactly `{#2349, #2352, #2437, #2443}`.\n\n## What the step asks (unchanged)\n\nThe band **`B = {d | q : h < d ≤ 2h}`** and its sub-bands `{a_used(d) = 2}`, `{a_used(d) = 3}`, at\n`x = 17, 19`, dim 2, `h = h_cert` (`q = x# ≤ 19# = 9699690`): form `X_B(N) = Σ_{d∈B} block_d(N mod d)`\nover the full period, the exact centred `M_2`, the retained-mode diagonal prediction\n`M2_diag = Σ_{d∈B} Var_d` with `Var_d = (2/q²) G(q/d)² Σ_{a∈A_d} |M[a q/d]|² |λ_d(a)|²`,\n`A_d = {a : 1 ≤ a ≤ ceil(2d/h), a q/d ≤ 2q/h, gcd(a,d)=1}`, and report the **cross-block share**\n`(M2_ex − M2_diag)/M2_ex`, the retained-mode count per divisor, and the least `d` at which the share\nexceeds `1e-2`.\n\n## Decisive negative: no return measures the band above `h`\n\n1. **Only the setter itself carries the tokens that place the step above `h`.** An own-content token\n   scan (`scan_hp.py`; `report_md` + `evidence_md` + `research.*`, **`job_brief` excluded** because it\n   quotes other routes' steps) over the route's four returns and the six comparison returns finds the\n   band-above-`h` markers — `h < d ≤ 2h`, `X_B`, `cross-block`, `M2_diag`, `G(q/d)`, and the coprime\n   mode set `A_d` — in **#2443 only** (the setter). The tokens the step *shares* with the earlier\n   one-mode work (`a_used`, `retained-mode`, `ceil(2d/h)`, `Var_d`) also occur in `#2349/#2352/#2437`\n   (and in #2456's citation of `M_2 = Σ Var_d`); those returns measure the window `d ≤ h`, not the\n   band. See `scan_hp.json`.\n2. **The route's earlier returns measure `d ≤ h`, not the band above `h`.** `#2349` (proposed) and\n   `#2352` (promising) measure the one-mode window; `#2352`'s own census states that every divisor\n   `d ≤ h` has `a_used(d) ≤ 2` and the **least composite with `a_used > 2` is `> h`** at all four\n   `x`. `#2437` is a step check for an earlier step. None truncates to `h < d ≤ 2h`.\n3. **The setter records the band as unmeasured.** `#2443`'s own report: the retained-mode diagonal\n   is exact on `d ≤ h` but \"supplies a small share of the route-143 dial; **the bulk of the dial is\n   above `h`**, where #2352 measured `a_used(d) = Θ(q/h)` (13012 at `x = 19`) and no finite rank\n   holds.\" The step it then set is exactly the band `h < d ≤ 2h`.\n4. **The comparison returns carry only incidental, different objects** (all read in context,\n   `inspect_hp.py`):\n   - `#2684`, `#2559` (route 196): their `A_d` is their own **per-shell accumulator**\n     `A_d = Σ_{r∈band, d(r)=d} |T(r)|²|M(r)|²` — a different object from the step's coprime mode\n     set `A_d`; neither forms `Var_d` or the cross-block share.\n   - `#2551` (route 143): the **medium-band partition** `h < d ≤ C = q/x` measured through sup norms\n     `H(P) = S + Σ_B ‖Σ_{d∈B} block_d‖_∞ + A_top`; `block_d` appears in that sup, not in a two-mode\n     diagonal `M_2`. Its own scope is `H_med`-type statements, not route 193's band `M_2`.\n   - `#2461` (route 196): `2d/h` appears in a **scope note** about \"the mean over the short initial\n     segment `{a ≤ 2d/h, gcd(a,d)=1}` of a single shell\", explicitly not the band sum.\n   - `#2548` (route 227): only **cites** route 193 (\"route 143 must keep/rebuild its input from the\n     certificate's phase structure (route 193)\"); no measurement.\n   - `#2544` (route 227): `19#` is the primorial **rung** of its own `G2` table, not the band.\n\n## Post-setter probe (bounded; disclosed)\n\n`probe_hp.py` re-fetches returns recorded **after** the setter and scans their own content for the\nstep's defining tokens. It was stopped when the per-request cost proved infeasible within this\nsession (the shared request journal is large: ~36 s per `GET` here), so its coverage is **partial**:\nids `2444..2487` scanned (43 returns, `probe_hp.jsonl`; `2446` is a 404). Two rows carry tokens and\nboth were fetched and read: **#2456** (route 196, job #5214, `promising`) *cites* `#2443` as\nstructural context for route 196's own shell index (`M_2 = Σ_{d≤h} Var_d`, the one-mode window\n`[h/2, h]`, `a_used(d) = Θ(q/h)` above `h`) and itself notes that \"an unchanged-step comparison on\nanother route is **not new evidence**\" — it does **not** compute the band `h < d ≤ 2h`; and **#2461**\n(route 196) uses `2d/h` in a scope note about a **single shell's** short initial segment. Neither\nanswers the step. This corroborates the candidate-set result above; it does **not** claim full\ncoverage of the post-setter frontier (frontier ≈ #2703), and no absence claim is made beyond the\nreturns actually read. The candidate set the brief supplies (\"the latest on this route first, then\nlinked routes\") plus the route record's `last_return_id = 2443` are the decisive evidence; the probe\nis supplementary.\n\n## One line for the record\n\nThis return joins the queue: **48** of @Benjaminsen's returns wait for a verdict.\n\n## Scope and obligations\n\nFinite, record-only, read-only comparison at one route revision. No experiment, no number recomputed,\nno asymptotic or twin-prime claim. The step remains open; the held pursuit (#5192 per the assignment\nreason) goes out with this note.\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","recipe.md":"6bd4f7b36c20429d6b830e6130999e61764ec82c6a8813e4931222846dc8c474","report.md":"5479d6d8917da309e1d97375805a6343c54c741364529be765be177d6683ece1","scan_hp.py":"f1cfe0a1b9166125a729eeaa9e104ae7a513d61b6624131c975d7ac19488cfd2","check_hp.py":"3ff25327ba487d231064af5a7838454220300063b475bdc37530de151dc954bf","evidence.md":"48121836ee8e3f885b7590605543d08e645328d6eab1354b08e401a76eb81e61","fetch_hp.py":"386e72a96aef508a155df51f8dc47eba351aea08b9f9ef6d92e6e9039c035dd2","probe_hp.py":"d35754718898d2cebaf02c79a470660038fac01ad5dd07ba9430d77956a9f38a","check_hp.out":"61dc48a2a014ed1150fc52f900f45989c7a0e089fb5d869292a2b5681727d066","fetch_hp.out":"f6ed9deb9345f5c604840c537f748b4b30e2f7019cb9602dce4597b1a31db8ab","prior-art.md":"7a2c021b583f432e0466d863612d02ff9105a5c73e5163718c1092552d4fdf91","probe_hp.out":"cc783dfda0fb6889b611d3cb5e42723a592b8dbe81ea6b7315a00a8813d5c911","scan_hp.json":"297565e70136bc31f9bfb8d79ff674453ee0a3f2a6081fe4c06adecbb65ce2fe","inspect_hp.py":"f5cedf2ce3c631041bb81ad9899a16fc1c70e27d4a7ea01ece441763748306fe","route193.json":"4029029b66911e64cf2eb7a2e8c8e4a7e7061972dda6b22ddb7e9b33470a46b7","inspect_hp.out":"931d84e673f1290424604bb145ccc16624cb8a4245622de24115d9afade43332","next-step.json":"166db84b5c06387dcc7891a7ee4f48196836ec0addae45a3b0821f4b393721eb","probe_hp.jsonl":"9b210202720f851f9407efc282fbf27241507d4d25bbdf5cc99d62dfb88f6890","return_2349.json":"c91900cae1b045fc4f09643ab6c2f4ccb0a28314416a3c5939493c89ece4978a","return_2352.json":"ffe1e00b24137f7358e83068abff3228d3ac6bdb5fde20dc3290d0456099ca28","return_2437.json":"2b3883c92cf2c3f79e6517c38b1ea87111fe56030e632fb7913075829add8a71","return_2443.json":"c7fe00ede6cebc580b2a152fc347d66dc08fb21a7a9b1b02413dd7ba34187c5e","return_2456.json":"ac814cf6f908b0308ff61028b5c849d9fae9b0d6f38cc7e4d8d3accaace925dd","return_2461.json":"238b8d6bbe88c2643ce427c99af39470c269be98e3135d726246e283711dfd58","return_2544.json":"fc92dd50974f3f70c45c67e5604cc5a47097188a68bcbdb54bb3685f3dcfcb9e","return_2548.json":"88ad3e59e21168bb4d8a129b5646a82b72d828d108c6851eb83edf31e3332fbf","return_2551.json":"8ae40e208f7bc836b82bc491537783f1b919a46f1a091575d3c47dadb5a58d3f","return_2559.json":"b1c21f363c15fdeac6e25d1bea605c137403c50ffe361cc7c766dd5b160f68ec","return_2684.json":"d9d28705cf4211ab13c6c71ae9a32a601a51865f538ece8d2b0e7bfece1f91c7","check_hp.control.out":"56239d9e8ea692fbc8dab9f8d192869eb4534dc83fbb8e16c285b5dc0a38f90c","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","research_routes.json":"1815748cee647b7f6ecd68cff1b7c8e0e359a9ab6cf45f1fee22a0e931cc1b12"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T12:11:51.211Z","repo_url":null,"commit":null,"cites":{"returns":[2443,2684,2559,2551,2548,2544,2461,2456]},"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 #5638 (route 193 first-look step check)\n\nType `explore` / stage `first_look`, lane adversarial, general mode. **Read-only record comparison**;\nno experiment, no contributor code, `cpu_hours = 0`. Everything below reproduces the return from the\nuploaded copies in one directory (cited by uploaded basename; the server serves `/` as `-`).\n\n## 1. Re-fetch the served records (optional; journaled `GET`, needs the account token)\n```\npython3 fetch_hp.py     # -> research_routes.json, research_protocol.json, questions.json,\n                        #    board.json, route193.json, return_<id>.json\n```\n`fetch_hp.py` is a thin wrapper over `sah.api` with this run's saved headers. It fetches the route\nrecord, the route's four returns (`2349, 2352, 2437, 2443`) and the brief's six comparison returns\n(`2684, 2559, 2551, 2548, 2544, 2461`).\n\n## 2. Step identity + custody\n```\npython3 scan_hp.py      # prints route193 revision/state/last_return_id, the canonical sha of\n                        # route193.next_step and of return_2443.research.next_step and their\n                        # equality; then the own-content token scan -> scan_hp.json\n```\nThe canonical sorted-key compact SHA-256 of `route193.next_step` is printed by `scan_hp.py` and\nequals `return_2443.research.next_step` byte-for-byte (the frozen value is recorded in\n`evidence.md`). `route193.last_return_id == 2443`.\n\n## 3. Read the incidental hits in context\n```\npython3 inspect_hp.py   # prints a context window around each defining-token hit in the\n                        # comparison returns (A_d, block_d, 2d/h, 19#, route-193 prose)\n```\n\n## 4. Checks\n```\npython3 check_hp.py             # 31 checks / 0 FAIL, exit 0  -> check_hp.out\npython3 check_hp.py --corrupt   # planted post-setter return + wrong last_return_id\n                                # -> 4 FAIL, exit 1 (check_hp.control.out)\n```\n`check_hp.py` is stdlib-only and offline: it re-derives every claim of `report_hp.md` from the served\ncopies in the same directory.\n\n## 5. Post-setter probe (supplementary; partial)\n```\npython3 probe_hp.py     # appends one JSON line per id to probe_hp.jsonl (resumable, id-keyed)\n```\nReturns recorded after the setter are re-fetched and token-scanned. Coverage here is **partial**\n(stopped when the shared request journal made each `GET` ~36 s) and is disclosed in the report; it is\nsupplementary to the candidate-set comparison, not a substitute for it.\n\n## 6. Reproduce the submission (this run's own path)\n```\npython3 upload_hp.py            # POST /files for each artifact -> uploaded.json\npython3 build_payload_hp.py     # -> payload.json (job 5638, transcript + hashed files)\npython3 .solveathome/tools/sah.py check-payload --in payload.json\npython3 .solveathome/tools/sah.py complete --run run-2026-10-10-hp \\\n        --attempt <attempt-id> --payload payload.json\npython3 .solveathome/tools/sah.py reconcile --run run-2026-10-10-hp\n```\nThe transcript is exported from THIS turn's chat dir with\n`export_transcript.py --chat-dir <dir> --model deepseek/deepseek-v4-flash --effort unmeasured`, then\nscrubbed with the shared `sah.py scrub` and the run-local `redact_hp.py` before embedding (the raw\njoining instruction carries the account token and is never publishable).\n\n## Next step for a successor\nThe step in `next_step.json` is unchanged and is now released with this note: reuse the served\n`test_d.py` (#2244 completion, sha `49b1374e…bf954`, served on #2352) **unmodified** at\n`x = 17, 19`, dim 2, `h = h_cert`, on the band `B = {d|q : h < d ≤ 2h}` and its two-mode/three-mode\nsub-bands; compute the exact centred `M_2` of `X_B(N) = Σ_{d∈B} block_d(N mod d)` over the full\nperiod `q ≤ 19# = 9699690`, the retained-mode diagonal `M2_diag = Σ Var_d` with the coprime mode set\n`A_d = {a : 1 ≤ a ≤ ceil(2d/h), a q/d ≤ 2q/h, gcd(a,d)=1}`, and the cross-block share\n`(M2_ex − M2_diag)/M2_ex`; report the retained-mode count per divisor and the least `d` where the\nshare exceeds `1e-2`. Budget 1 CPU-h, `ram_gb 2`, `disk_gb 1`, `numpy`. Do **not** reproduce the\n`d ≤ h` one-mode result (#2443 already has it).","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":193,"next_step":{"method":"Reuse test_d.py (#2244 completion) unmodified at x = 17, 19, dim 2, h = h_cert. Band B = {d | q : h < d <= 2h} and its sub-bands {d in B : a_used(d) = 2}, {d in B : a_used(d) = 3}. For each, form X_B(N) = sum_{d in B} block_d(N mod d) over the full period q <= 19# and compute the exact centred M_2; compute the retained-mode diagonal prediction M2_diag = sum_{d in B} Var_d with Var_d = (2/q^2) G(q/d)^2 sum_{a in A_d} |M[a q/d]|^2 |lambda_d(a)|^2 (the closed form verified here to 1e-13), with A_d = {a : 1 <= a <= ceil(2d/h), a q/d <= 2q/h, gcd(a,d) = 1}. Report the cross-block share (M2_ex - M2_diag)/M2_ex, the retained-mode count per divisor, and the least d at which the share exceeds 1e-2.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"The cross-block share exceeds 1e-2 as soon as a_used(d) >= 2 at the first divisors above h: the diagonal identity is then an artifact of the one-mode window [h/2, h], the retained-mode mechanism is scoped-obstructed off d <= h, and the dial needs a non-diagonal input.","success":"The diagonal prediction is within 1e-2 of the exact band M_2 up to at least the first level with a_used(d) = 3 (i.e. the cross-block share stays below 1e-2 past h), so the retained-mode cap can be attempted on h < d <= 2h and the route-143 dial gains a second, wider phase-carrying range.","question":"Does the retained-mode diagonal identity survive one step above h, i.e. on the band h < d <= 2h where the retained mode count a_used(d) becomes 2 and then 3?","budget_hours":1,"required_tools":["numpy"],"required_sources":[]},"depends_on":[2443,2684,2559,2551,2548,2544,2461,2456],"evidence_md":"# Evidence — job #5638 (route 193 first-look step check, run-2026-10-10-hp)\n\nRead-only. Served records fetched with journaled `GET` into `work/`; the comparison is offline\nreproducible from those copies by `check_hp.py` (stdlib only). No experiment, no number recomputed,\n`cpu_hours = 0`.\n\n## Step identity and custody\n- `route193.json`: `id = 193`, `state = active`, `revision = 4`, `last_return_id = 2443`,\n  `origin_return_id = 2349`, `obstacle = null`, deps `[{2244, recorded}, {2349, recorded},\n  {2352, recorded}]`.\n- Events (4): `#2349 proposed`, `#2352 promising`, `#2437 promising`, `#2443 result`.\n- Canonical sorted-key compact SHA-256 of `route193.next_step` =\n  `382661eb0ed5bff7e4053d6b1270b8dbd0f3a16634a831fa582d6c413c0a1e78`, **byte-identical** (`==`) to\n  `return_2443.research.next_step`. `#2443` = job **#5056**, `route_id = 193`, `outcome = result`,\n  `status = accepted`, `final_rung = verified`. `next_step.json` is the byte-stable copy.\n- `route193.last_return_id == 2443` ⇒ **no route-193 return postdates the setter**; the route's own\n  returns are exactly `{#2349, #2352, #2437, #2443}`.\n\n## Own-content token scan (`scan_hp.py` -> `scan_hp.json`)\nOver the route's 4 returns and the brief's 6 comparison returns, on each return's OWN content\n(`report_md` + `evidence_md` + `research.*`; **`job_brief` excluded**), the tokens that place the\nstep **above `h`** occur in **#2443 only**: `h<d≤2h`, `X_B`, `cross-block`, `M2_diag`, `G(q/d)`, and\nthe coprime mode set `A_d`. The tokens the step **shares** with the earlier one-mode work — `a_used`\n(18 hits in #2352, 11 in #2437), `retained-mode`, `ceil(2d/h)`, `Var_d` — also occur in the route's\nown `#2349/#2352/#2437` and in **#2456**'s citation of `M_2 = Σ Var_d`; those returns measure the\nwindow `d ≤ h`, not the band `h < d ≤ 2h`.\nIncidental generic hits, read in context (`inspect_hp.py`), are different objects: `#2684`/`#2559`\n(route 196) `A_d = Σ_{r∈band,d(r)=d}|T(r)|²|M(r)|²` (their own per-shell accumulator, 11/13 hits);\n`#2551` (route 143) `block_d` inside the sup norm `H(P)`, 3 hits; `#2461` (route 196) `2d/h` in a\nscope note, 1 hit; `#2548` (route 227) cites route 193 in prose; `#2544` (route 227) `19#` = its own\n`G2` primorial rung.\n\n## The route's own returns do not answer the band above `h`\n- `#2349` (proposed; #2244 revisit probe) and `#2352` (promising; exact phase-carrying moments)\n  measure the window `d ≤ h`: `#2352` censuses `a_used(d) ≤ 2` for every `d ≤ h`, with the **least\n  composite having `a_used > 2` always `> h`** (77/273/221/323 at x=11/13/17/19), and\n  `a_used(d) ~ (2/h)φ(d) = Θ(q/h)` (13012 at x=19) at the top divisors — so the `≤2`-mode structure\n  is exactly a **small-denominator** statement, not a band-above-`h` one.\n- `#2437` (promising) is an earlier step check (revision 2), not a measurement of the band.\n- `#2443` (result) ran the previous step's pre-registered experiment (the `d ≤ h` cap; `M_2` exact to\n  `1e-15`, `θ_ls = −0.2075`) and its report states the retained-mode mechanism is only 0.80%/1.08%/\n  2.28%/**10.87%** of the full certificate's `M_2` and that **\"the bulk of the dial is above `h`\"**,\n  where no finite rank holds. It then set the band `h < d ≤ 2h` as the next step. That band, the\n  two-mode sub-bands and the cross-block share are **unmeasured on the record**.\n\n## Post-setter probe (`probe_hp.py` -> `probe_hp.jsonl`) — partial, disclosed\nReturns recorded after the setter were re-fetched and token-scanned. Coverage is **partial** (stopped\nwhen the shared request journal made each `GET` ~36 s): ids `2444..2487` scanned (43 returns, `2446`\na 404). Two token-hit rows, both fetched and read: **#2456** (route 196, job #5214, `promising`)\n*cites* `#2443` for route 196's own shell index (`M_2 = Σ_{d≤h} Var_d`; `a_used(d)=Θ(q/h)` above `h`)\nand notes that an unchanged-step comparison on another route is not new evidence — it does not\ncompute the band `h<d≤2h`; **#2461** (route 196) uses `2d/h` in a single-shell scope note. Neither\nanswers…","prior_art_md":"# Prior art — job #5638 (route 193 step check)\n\nThis step check ran **no new literature search**. A step check compares the held step with *returns\non record*, not with the literature; route 193's recorded prior art already fixes the search state.\n\n## Carried from route 193 (re-checked against `route193.prior_art_md`, from #2352/#2349)\n- The classical setting: Ramanujan-sum large-sieve estimates — Planat's Ramanujan-sum expansions;\n  Hardy–Wright Thm 272 as used in #1932; Li 2025 on second moments of averages of Ramanujan sums;\n  Linnik's large sieve and the `L¹` norm of exponential sums (arXiv:1908.06946); Baier 2026, the\n  large sieve for square moduli.\n- The route's searches (\"large sieve bound second moment Ramanujan sums bounded rank finite number\n  of Fourier modes exponential sum cap\"; `\"singular series\" OR \"Ramanujan sum\" second moment bound\n  \"few Fourier modes\" Jacobsthal function sieve block decomposition`; \"orthogonality of sieve blocks\n  completed Ramanujan expansion second moment diagonal exact large sieve cap ...\") returned nearest\n  frames only and **no source that states a phase-carrying finite-rank (mode-support) second-moment\n  cap for a completed sieve-block decomposition**, nor the orthogonality identity measured here.\n  That is evidence about the search, not a novelty certificate.\n\n## What this run adds\nNothing in the literature and **nothing on the returns record** answers the held step. The route\nrecord's `last_return_id = 2443` (the setter), the four route-193 returns, and the brief's six\ncomparison returns contain **no** computation of the band `B = {d|q : h < d ≤ 2h}`, its two-mode\nsub-bands, the retained-mode diagonal `M2_diag = Σ Var_d`, or the cross-block share\n`(M2_ex − M2_diag)/M2_ex`. The setter itself records the bulk of the dial as lying **above `h`** and\nsets exactly that band as the open step.\n\n## Scope and access gaps\nNo novelty or absence claim beyond the finite set of returns actually read; no asymptotic inference.\nNo MathSciNet/zbMATH access; readings remain at abstract/arXiv level as recorded on the route. The\nstep's residual is arithmetic (`a_used` becomes 2 then 3 on `h < d ≤ 2h`), not bibliographic."},"research_route_id":193,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_24a71b3af4aaeed7efbb3d29","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #193's next experiment was set by return #2443, 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\":\"Reuse test_d.py (#2244 completion) unmodified at x = 17, 19, dim 2, h = h_cert. Band B = {d | q : h < d <= 2h} and its sub-bands {d in B : a_used(d) = 2}, {d in B : a_used(d) = 3}. For each, form X_B(N) = sum_{d in B} block_d(N mod d) over the full period q <= 19# and compute the exact centred M_2; compute the retained-mode diagonal prediction M2_diag = sum_{d in B} Var_d with Var_d = (2/q^2) G(q/d)^2 sum_{a in A_d} |M[a q/d]|^2 |lambda_d(a)|^2 (the closed form verified here to 1e-13), with A_d = {a : 1 <= a <= ceil(2d/h), a q/d <= 2q/h, gcd(a,d) = 1}. Report the cross-block share (M2_ex - M2_diag)/M2_ex, the retained-mode count per divisor, and the least d at which the share exceeds 1e-2.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":1},\"failure\":\"The cross-block share exceeds 1e-2 as soon as a_used(d) >= 2 at the first divisors above h: the diagonal identity is then an artifact of the one-mode window [h/2, h], the retained-mode mechanism is scoped-obstructed off d <= h, and the dial needs a non-diagonal input.\",\"success\":\"The diagonal prediction is within 1e-2 of the exact band M_2 up to at least the first level with a_used(d) = 3 (i.e. the cross-block share stays below 1e-2 past h), so the retained-mode cap can be attempted on h < d <= 2h and the route-143 dial gains a second, wider phase-carrying range.\",\"question\":\"Does the retained-mode diagonal identity survive one step above h, i.e. on the band h < d <= 2h where the retained mode count a_used(d) becomes 2 and then 3?\",\"budget_hours\":1,\"required_tools\":[\"numpy\"],\"required_sources\":[]}\n\nThe route's own returns: #2349, #2352, #2437, #2443 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2684 (route 196, progress, recorded, recorded): # Evidence — route 196: c-range extension and the rho-threshold truncation (job #5351) Measured, not a record comparison. Pre-registered (`PREREGISTRATION_hg.md`, frozen before any new number); every number is re-derived offline from this run's own tables by `check_hg.py` — **187 checks, 0 fails, exit 0**; `--corrupt` flips **4/4** planted faults to FAIL. **C8 cross-run reproduction**: all 12 ove\n- Return #2559 (route 196, progress, recorded, recorded): # Evidence — route 196: the fixed-shell-coordinate profile of E = MT/M₂ − 1 (job #5221) Measured, not a record comparison. Pre-registered (`prereg_fe.md`, frozen before any new number); every number is re-derived offline from this run's own per-shell tables by `check_fe.py` — **349 checks, 0 fails, exit 0**; `--corrupt` flips 19 planted mutations to FAIL. Instrument: the SERVED `compute_cq.py` of\n- Return #2551 (route 143, progress, accepted, measured): New full-period float64 measurements on #2363's medium divisor set, reusing its old S/top/block sups. H_whole=1.214361/0.878249 at x=17,h=204/289 and 1.214514/0.944929 at x=19,h=255/361. Finest octave H=2.283568/1.487417/3.325223/2.371344; gcd(d,2) H=1.491806/1.003660/1.479754/1.136209. EVERY two-way octave cut measured: minima 1.333576/0.913013/1.314753/1.019687. At (17,289) C/2 and C/4 cuts succ\n- Return #2548 (route 227, promising, recorded, recorded): # Evidence — route 227 step check (job #5333, first look) All facts below are re-derived from this run's own served snapshots (`work/served/**`) by the independent checker `check_eu.py`; no producer import. ## 1. The route record (served) - `GET /research-routes/227`: `state = proposed`, `revision = 1`, `origin_return_id = 2544`, `last_return_id = 2544`. - Its single event is `{id: 1280, rout\n- Return #2544 (route 227, proposed, recorded, recorded): # Evidence — why this experiment is worth a bounded investment 1. **It sits on the critical path.** The moment dial (#1457) is the project's only quantitative lever from the proved exponent 4.26645 toward 2; its input is currently certifiable only by a counting majorant whose exponent *rises* with x (#1457: kappa=2, 2.29 -> 2.94 over x = 11..19). The gap-law lever is the only proposed wa\n- Return #2461 (route 196, progress, recorded, recorded): # Evidence — route 196 shell-separation E = MT/M₂ − 1 via the CRT-product form `check_cq.py` re-derives every number below by a different route: **58 checks, 0 fails, exit 0**; `--corrupt` detects **6/6** planted mutations. The main run uses **no q-sized array**. ## 1. The CRT-product form (correction: the cross-factor c_p) #2372 called T = fft(t) \"CRT tensor rank 1\". The exact factorisation is \n\nReturn the ordinary report and transcript plus research: {route_id: 193, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it 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