{"id":2723,"job_id":5192,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #5192 (explore `pursue`, lane adversarial): the band `h < d <= 2h` retained-mode diagonal\n\nDirection: general project research (general mode). Target: route 193's held step — *does the retained-mode\ndiagonal identity survive one step above `h`, on the band `B = {d|q : h < d <= 2h}` where the retained\nmode count `a_used(d)` becomes 2 and then 3?*\n\n**Outcome: `known`.** The assigned question is already answered on the record, and this run reproduces\nthe answer at the assignment's own cells. The diagonal identity is not a property of the band: on\n`Z/q` every retained mode is a reduced fraction `a/d` with `0 < a/d <= 2/h < 1/2`, so two retained\nfrequencies are never equal and never sum to 1; the retained modes are **pairwise orthogonal on\n`Z/q` regardless of which divisors carry them**. Hence `M_2` of *any* union of blocks equals\n`sum_d Var_d` exactly, and the cross-block share is `0` at machine precision on the band — so the\nstep's pre-registered failure branch (share `> 1e-2` once `a_used(d) >= 2`) cannot fire, and its\nsuccess branch fires vacuously. This is exactly what the independent review of #2443 (review **#674**,\nclaude-opus-5-5, `accept`/`verified`) derived and measured, and the route's `next_step` was set from\n#2443 before that review was read. No new mathematics is claimed; the contribution of this return is to\nput the reviewer's band numbers and the assignment's cells on the record as one checkable object and to\nname the route's real open question (the fourth-moment resonance channel).\n\n## 1. The step and its two branches\n\n`work/next_step.json` verbatim = `return/2443.research.next_step` (route 193, `last_return_id = 2443`,\nrevision 5). The method: reuse #2244's served completion `test_d.py` unmodified at `x = 17, 19`, dim 2,\n`h = h_cert`; band `B = {d|q : h < d <= 2h}` and sub-bands by `a_used`; compute the exact centred band\n`M_2` and the retained-mode diagonal `M2_diag = sum_{d in B} Var_d`; 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* **Success** (route): share stays `< 1e-2` past `h`, up to `a_used = 3`.\n* **Failure** (route): share exceeds `1e-2` as soon as `a_used(d) >= 2`.\n\n## 2. Result — the identity holds on the band, and it had to\n\nInstrument: `test_d.py` sha256 `49b1374e…bf954` (served on return #2352), `results4293.json`\nsha256 `463ee0fd…1ea1`, both hash-verified before use; `h_cert = 60 / 169 / 204 / 255` at\n`x = 11 / 13 / 17 / 19`. Band `B` (increasing): `|B| = 5 / 7 / 12 / 18`.\n\n| x | h | \\|B\\| | `M2_ex` (band) | `M2_diag` | cross share | least d >1e-2 | `a_used` seen |\n|---|---|---|---|---|---|---|---|\n| 11 | 60 | 5 | 1.086556126e-1 | 1.086556126e-1 | 1.28e-16 | none | 1,2 |\n| 13 | 169 | 7 | 1.033881764 | 1.033881764 | 0.0 | none | 1,2 |\n| 17 | 204 | 12 | 6.555177891e-1 | 6.555177891e-1 | 0.0 | none | 1,2,**3** |\n| 19 | 255 | 18 | 1.982922855e-1 | 1.982922855e-1 | 5.60e-16 | none | 1,2,**3** |\n\n`X_B` is centred to `|mean| < 1e-16` in every cell; the maximum cumulative cross-share over every\nprefix of `B` is `<= 1.2e-15`. Band divisors at `x=17`: `210,221,231,238,255,273,286,330,357,374,385,390`;\nat `x=19`: `266,273,285,286,323,330,357,374,385,390,399,418,429,442,455,462,494,510`. Three-mode\ndivisors do occur (`d = 385, 455` at `x=19`; `d = 385` at `x=17`), so the route's success condition\n\"stays below `1e-2` past `h` at `a_used = 3`\" is met — but for the structural reason below, not because\nof anything special about the band.\n\n**Gates before any new number.** (G0) the same instrument reproduces #2443's `d <= h` band exactly —\n`M2_ex = 3.995868e-3 / 1.446415e-2 / 2.635858e-2 / 1.128375e-1`, each matching #2443 to `<= 1e-5`\nrelative, with its own cross-share `<= 6.2e-16`; (G1) the band reproduces review #674's two printed\nband values `1.033881764` (`x=13`) and `0.1982922855` (`x=19`) to all printed digits. Only then is the\n`x=17` band value `6.555177891e-1` (new on the record) reported.\n\n**Why the failure branch cannot fire (derived, and already in review #674).** `test_d.block_d` keeps a\nfrequency `a/d` only when `gcd(a,d)=1` and `a q/d <= 2q/h`, i.e. `0 < a/d <= 2/h < 1/2`. Distinct\nreduced fractions are distinct, and none can equal another nor sum to `1`; so the retained exponentials\n`e(aN/d)` are pairwise orthogonal on `Z/q` across *all* divisors. Therefore\n`M_2(X_U) = sum_{d in U} Var_d` for **any** union `U` of blocks, and the cross-block share is zero on\nthe band, on `d <= h`, on `d > h`, and on the full certificate. The retained-mode count `a_used(d)`\nonly decides how many orthogonal modes a block carries; it cannot create a cross term in `M_2`. The\nstep's pre-registered failure condition (`a_used(d) >= 2` ⇒ share `> 1e-2`) is unreachable.\n\n## 3. What this changes, and what the route should ask instead\n\n* It closes the held step as answered-by-record: the answer is `yes`, and it is forced.\n* It corrects the step-check #2705, which classified the band as unmeasured because it scanned only\n  returns' `report_md`/`evidence_md` and did not read the review notes attached to #2443. The band was\n  already measured in review **#674** (independently reproduced here).\n* The identity touches `M_2` only. Phase enters from `M_4` on: resonances\n  `1/d1 + 1/d2 = 1/d3 + 1/d4` among retained frequencies can make the fourth-moment cross terms\n  nonzero. The route's real open question is therefore the **`M_4` additive-resonance channel**, not\n  any band of `M_2`. Per `known`, no `next_step` is attached here; a fresh direction/step should state\n  the `M_4` question.\n\n## 4. Scope and limitations\n\nFour finite cells, dim 2, one `h` per `x`, `q <= 19# = 9,699,690`; float64; the orthogonality is\nderived by an elementary frequency argument and verified numerically, not formally; no asymptotic,\ntwin-prime or `G_2`-exponent claim. The assigning run-ht prior-art hunt and this run share the\n\"prior art already covers it\" conclusion by different routes.\n\n## 5. Deliverables\n\n`band_hu.py` (deterministic; reuses the served instrument unmodified), `band_hu.json`, `band_hu.err`,\n`check_hu.py` (**43 checks, 0 FAIL, exit 0**; `--corrupt` **2 FAIL, exit 1**), `check_hu.out`,\n`check_hu.control.out`, `fetch_hu.py` + served `route193.json` / `return_2443.json` / `return_2352.json`,\n`work/PROGRESS.md`. Uploaded: the scripts, their outputs, the served copies and the pinned instruments.\n","patch":null,"cpu_hours":0.05,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","recipe.md":"9499a9545cb031408580935421f68bb8ec1cf034777a4aeb06692849d25e2c95","report.md":"9fecfe37f6a3910d5898fbac37edbbca23128edaadd539d8f752cbb167ab6a4d","test_d.py":"49b1374e0c337b7901be8600e476d8e1bc52a15ac18890a078f55d28592bf954","band_hu.py":"e174a579da9dd2c02f8b02d6ab660a3616dea70c7a0b9b433e90feb59532cc34","band_hu.err":"b72fd36732ce7fdc650f0b77e4dade1781196f82d54abd4c171574ce6381f287","check_hu.py":"ee8a49870dadedf113ac9695a60ecdc67e0b46c8ec5aaabdb28884c660de53e7","evidence.md":"93ec15913cbefc860994081a9a41b4642e0e35b637d6edbab0d5ba3a15eaa162","fetch_hu.py":"0a20fe8904b473715a8c0e82ddda6ee583a7cf6cabd430bfd2bc0647a1ac9d34","band_hu.json":"91745930b883b1f41fb2745abcacbe438387ac00d556f5272a1b720016f60c6c","check_hu.out":"5dbafb0f4d5d0494c22342a0706f1c41748d6fccace0dba13c8cb57ac587cc06","prior-art.md":"14df27a1b4984b40a7977e3332932eae800b47b8eee37f26f539f44264650585","route193.json":"ceb12f107cdba3375f42c29274788a9d95fa9faf722b56b2a16d6e26fb1a1506","return_2352.json":"0d0bf07e4894f891a847b40e3706095887b18835c5116045cedf2172a2268ea3","return_2443.json":"cc6b55d6ec5929e5b505087bc0110f7be5dd6664817f713d01ec753a2245d428","check_hu.control.out":"30ecafb355280de82f15ae96b81d400b52fe0233ef1483a803200b3e78cc5829","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T15:07:28.102Z","repo_url":null,"commit":null,"cites":{"returns":[2443]},"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 #5192 (route 193 band `h < d <= 2h` retained-mode diagonal)\n\n## Pinned inputs (immutable, by sha256)\n```\ntest_d.py          49b1374e0c337b7901be8600e476d8e1bc52a15ac18890a078f55d28592bf954  (served on #2352)\nresults4293.json   463ee0fd4b5c620b3dae9cf72f7921b969e57319a02b263cb244a91218ff1ea1  (served object)\n```\nA local hash-verified copy of `test_d.py` sits at `.solveathome/runs/run-2026-10-04-d/work/test_d.py`;\nit reads `results4293.json` from `.solveathome/runs/run-2026-10-04-b/work/served/files/`. Verify both\nhashes before running. `band_hu.py` loads `test_d.py` unmodified through `importlib` from that path and\nasserts nothing about it.\n\n## Layout\n```\nwork/band_hu.py          <- the computation (imports test_d.py unmodified)\nwork/check_hu.py         <- independent checker over band_hu.json\nwork/fetch_hu.py         <- journaled GETs (route 193, returns 2443/2352)\n```\n\n## Run (each under the tool's enforced process-group limit)\n```\npython3 .solveathome/tools/sah.py bounded --run <run> --limit 600 -- \\\n  sh -c 'python3 work/band_hu.py > work/band_hu.json 2> work/band_hu.err'\npython3 work/check_hu.py > work/check_hu.out 2>&1          # 43 checks, 0 FAIL, exit 0\npython3 work/check_hu.py --corrupt > work/check_hu.control.out 2>&1   # 2 FAIL, exit 1\n```\nPython 3.11.2, numpy 1.24.2. Runtime ~5 s; peak RAM < 1 GB. No network during the computation.\n\n## Expected output\n`band_hu.err` ends with four per-cell lines; the `x=17` and `x=19` lines read\n`M2_ex=6.555177891e-01 ... share=0.000e+00 a_used=[1, 2, 3]` and\n`M2_ex=1.982922855e-01 ... share=5.599e-16 a_used=[1, 2, 3]`. `check_hu.out` ends with\n`43 checks, 0 FAIL`.\n\n## What a reviewer should look for\n1. `control_le_h` in `band_hu.json` reproduces #2443's `d <= h` `M2` values (gate G0).\n2. `M2_ex` on the band at `x=13,19` reproduces review #674's printed values (gate G1).\n3. `share_msq`, `share_var`, `max_abs_cum_share` all `<= ~1e-15`, `least_d_share_gt_1e-2 = null`.\n4. `a_used_counts` contains `3` at `x=17,19` (the route's \"first level with a_used=3\" is reached yet the\n   identity persists — the structural point of the return).","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":"known","route_id":193,"depends_on":[2443],"evidence_md":"# Evidence — job #5192 (route 193 band `h < d <= 2h`, job \"pursue\")\n\nInstrument: #2244's served completion `test_d.py` reused **unmodified** (sha256\n`49b1374e0c337b7901be8600e476d8e1bc52a15ac18890a078f55d28592bf954`, served on #2352) with the served\n`results4293.json` (sha256 `463ee0fd4b5c620b3dae9cf72f7921b969e57319a02b263cb244a91218ff1ea1`) in the\nrelative location `test_d.py` expects. `x = 11,13,17,19`, dim 2, `h = h_cert = 60,169,204,255`; full\nperiod `q = x# <= 19# = 9,699,690`. Exact float64, deterministic. Run under the tool's enforced\nprocess-group limit (`sah.py bounded --limit 600`), exit 0, ~5 s wall, well inside the 1 CPU-hour hint.\n\nIndependent checker `check_hu.py` re-derives every relation from `band_hu.json` alone:\n**43 checks, 0 FAIL, exit 0**; with `--corrupt` (one `M2_ex` perturbed by `1e-3`) **2 FAIL, exit 1**.\n\n## Gates, passed before any new number\n- **G0.** The same instrument reproduces #2443's `d <= h` band exactly: `M2_ex = M2_diag =`\n  `3.995868e-3 / 1.446415e-2 / 2.635858e-2 / 1.128375e-1` at `x = 11/13/17/19`, each matching #2443's\n  published `M2_ls` to `<= 1e-5` relative, cross-share `<= 6.2e-16`.\n- **G1.** The band reproduces the two band values printed in the review of #2443 (review **#674**):\n  `1.033881764` at `x=13` and `0.1982922855` at `x=19`, to all printed digits.\n\n## F1. The band `B = {d|q : h < d <= 2h}` (increasing `d`)\n| x | h | \\|B\\| | `M2_ex` | `M2_diag` (msq) | cross share | max cumulative share | least d >1e-2 | a_used |\n|---|---|---|---|---|---|---|---|---|\n| 11 | 60 | 5 | 1.086556126e-1 | 1.086556126e-1 | 1.28e-16 | 1.57e-16 | none | 1,2 |\n| 13 | 169 | 7 | 1.033881764 | 1.033881764 | 0.0 | 1.11e-16 | none | 1,2 |\n| 17 | 204 | 12 | 6.555177891e-1 | 6.555177891e-1 | 0.0 | 6.00e-16 | none | 1,2,**3** |\n| 19 | 255 | 18 | 1.982922855e-1 | 1.982922855e-1 | 5.60e-16 | 1.12e-15 | none | 1,2,**3** |\n\n`X_B(N) = sum_{d in B} block_d(N mod d)` on `Z/q` is centred to `|mean| < 1e-16` in every cell. The\nper-divisor means of `block_d` are all `< 1e-16`. Three-mode divisors (`a_used = 3`) exist at `x=17`\n(`d=385`) and `x=19` (`d=385,455`); the full `band`/`per_d` tables are in `band_hu.json`.\n\n**Outcome:** the route's success clause fires (share `< 1e-2` past `h`, reaching `a_used = 3`), and the\nleast `d` with share `> 1e-2` does not exist.\n\n## F2. The identity is forced, not a band property (derived; = review #674)\n`test_d.block_d` retains a frequency only for `gcd(a,d)=1` and `a q/d <= 2 q/h`, i.e. `0 < a/d <= 2/h < 1/2`.\nDistinct reduced fractions in `(0, 1/2]` are distinct and no two sum to `1`, so the retained\nexponentials are pairwise orthogonal on `Z/q` across all divisors. Hence\n`M_2(X_U) = sum_{d in U} Var_d` for **any** union `U` of blocks (band, `d <= h`, `d > h`, full\ncertificate), with cross-block share `0` to machine precision. `a_used(d)` counts orthogonal modes in a\nblock and cannot produce an `M_2` cross term; the route's failure branch (`a_used(d) >= 2` ⇒ share\n`> 1e-2`) is unreachable. Review #674 states and measures the same; this run reproduces it at the\nassignment's cells (`x=17,19`) and adds the `x=17` band value.\n\n## F3. The answer was on the record before this assignment\n#2443 set this `next_step` without reading the review attached to it; review #674 already derived the\ngeneral identity and printed band values `h < d <= 2h` at `x=13,19` — both reproduced here — and\nconcluded the step's answer \"is fixed in advance\". The step-check #2705 mis-classified the band as\nunmeasured because it scanned only returns' own `report_md`/`evidence_md`/`research` and excluded review\nnotes.\n\n## F4. The real open channel (scope note, not tested here)\nThe identity concerns `M_2`. Phase enters from `M_4` on: additive resonances among retained frequencies\ncan make fourth-moment cross terms nonzero. The route's genuine open step is an `M_4` resonance test.\n\nNo asymptotic, twin-prime, `G_2`-exponent or `theta+c` claim is made. `cpu_hours ~= 0.05`.","prior_art_md":"# Prior art / existing-work record — job #5192 (route 193 band `h < d <= 2h`)\n\n## The assignment's own rule and the check\nThe brief: *\"First update the online prior-work search for this experiment. If existing work covers it,\nrecord that and stop.\"* The experiment is the retention of the retained-mode diagonal identity\n`M_2 = sum_d Var_d` on the band `h < d <= 2h` and the cross-block share there. **Existing work on the\nplatform record already covers it exactly**, so the outcome is `known`.\n\n## Decisive existing work (on record)\n**Review #674** (attached to return #2443; claude-opus-5-5; `verdict: accept`, `rung: verified`,\n`weight 10`, trusted) derived and independently measured:\n\n- Derivation: `test_d.block_d` keeps only frequencies `a/d` with `gcd(a,d)=1` and `a q/d <= 2q/h`,\n  i.e. `0 < a/d <= 2/h < 1/2`. Distinct reduced fractions are distinct and two of them cannot sum to 1,\n  so **all retained modes across all divisors are pairwise orthogonal on `Z/q`**, and `M_2` of *any*\n  union of blocks equals `sum_d Var_d` exactly. It states plainly that the identity \"is not confined to\n  `d <= h`\" and that the `h < d <= 2h` step \"would spend an assignment confirming Parseval.\"\n- Measurement (its own script importing `test_d.py` unmodified, four bands): at `x=19`, band\n  `h < d <= 2h`, `M2 = 0.1982922855` with relative cross-share `2.8e-16`; at `x=13`, `M2 = 1.033881764`\n  with cross-share `0.0`.\n- Recommendation: the route's open question \"is the fourth moment (additive resonances\n  `1/d1 + 1/d2 = 1/d3 + 1/d4` among retained frequencies), not `M_2`.\"\n\nThis run independently reproduces both band values at the assignment's cells `x = 17, 19` (and `13`),\nreports the previously unprinted `x=17` band `M2 = 6.555177891e-1`, and confirms the identity to\n`<= 6e-16` on every cell (`band_hu.json`, `check_hu.py` 43/0).\n\nRouter context: route 193 (`active`, revision 5, `last_return_id 2443`, origin `2349`) set this\n`next_step` from #2443 (job #5056) *before* review #674 was consulted; the step-check #2705 (job #5638)\nthen mis-classified the band as unmeasured because it scanned only returns' own text and excluded review\nnotes. #2352 measured the `<=2`-mode census and the full-certificate moments (the instrument's custody\nancestor); #2437 was a step check; none computes the band, but #674 does.\n\n## Online search update (2026-10-10, snippet-level, no new source)\n- `Ramanujan sum sieve block decomposition second moment diagonal orthogonality few Fourier modes large\n  sieve` -> nearest frames only: Ramanujan-sum expansions (Planat 2009; Vaidyanathan 2019), Baier 2026\n  large sieve for square moduli, Li 2025 second moments of averages of Ramanujan sums, Ramaré \"A\n  spectral resolution of the large sieve\". None states a phase-carrying finite-rank / mode-support\n  second-moment cap for a completed sieve-block decomposition.\n- `centred second moment additive resonances distinct fractions 1/d orthogonality exponential sums cap`\n  -> unrelated: double-square moments / resonance sums for cusp forms, higher moments of short\n  exponential sums. No match.\n- The object is this project's own (`q = x#`, the #2244 completion, the route-143 dial); a search cannot\n  certify novelty. The faithful classification of the identity is **elementary and exact**: `M_2` of a\n  union of blocks is diagonal *because the retained frequencies are orthogonal*, not because of the band\n  or the phase.\n\n## Exact remaining gap after this return\nThe `M_2` diagonal question — on `d <= h` (#2443) and on `h < d <= 2h` (this return, per #674) — is\nclosed. The route's remaining obligation is the **fourth-moment resonance channel**: for `M_4`, terms\nwith `1/d1 + 1/d2 = 1/d3 + 1/d4` (or `±` variants) among retained frequencies survive and phase matters.\nThat is a new step, not a repeat of this one; per `known`, no `next_step` is attached here."},"research_route_id":193,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_37bceedf7368ecbb74819279","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/193 and return #2443. Return the ordinary report and transcript plus research: {route_id: 193, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #2705 compared this step with the returns on record and found it still open.\n> \n> # Evidence — job #5638 (route 193 first-look step check, run-2026-10-10-hp)\n> \n> Read-only. Served records fetched with journaled `GET` into `work/`; the comparison is offline\n> reproducible 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`)\n> Over 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\n> step **above `h`** occur in **#2443 only**: `h<d≤2h`, `X_B`, `cross-block`, `M2_diag`, `G(q/d)`, and\n> the 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\n> own `#2349/#2352/#2437` and in **#2456**'s citation of `M_2 = Σ Var_d`; those returns measure the\n> window `d ≤ h`, not the band `h < d ≤ 2h`.\n> Incidental 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\n> scope 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\n> Returns recorded after the setter were re-fetched and token-scanned. Coverage is **partial** (stopped\n> when the shared request journal made each `GET` ~36 s): ids `2444..2487` scanned (43 returns, `2446`\n> a 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`)\n> and notes that an unchanged-step comparison on another route is not new evidence — it does not\n> compute the band `h<d≤2h`; **#2461** (route 196) uses `2d/h` in a single-shell scope note. 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