{"id":4370,"job_id":8989,"problem_id":1,"lane_id":4,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Step check, route #143 (job #8989): the moment-dial step for the medium aggregates is still open\n\n**Verdict: `promising`.** Route 143's held step (set by #2551; served route record revision 14, `last_return_id` 2551, `obstacle` null) asks whether finite higher counting moments `U_k(B)`, k = 1,2,4,8,16,32,64, certify the two phase-retaining medium aggregates at `h_m+1` without using their exact maximum: the C/2 two-band partition at (17,289) and the whole-medium band at (19,361). Per the step-check rule I did not run the experiment and did not reproduce any computation a return already made; I compared the returns on record. No return on this route or on the linked routes computes `U_k` for either aggregate or determines the least tested k with `S + sum_B U_k(B) + A_top < 1`, and the setter's own boundary statement marks the higher counting moment as a different input. The step is still open exactly as written; the held pursuit goes out with this note.\n\nTwo of this handle's returns wait for a trusted verdict; nothing for the person to do.\n\n## What was compared\n\n- **Route record** `research-routes/143`: `state active`, `revision 14`, `last_return_id 2551`, `origin_return_id 1457`, `obstacle null`. Its `next_step` is object-equal to this brief's step (canonical sorted-key JSON sha256 `ee48e908cf2eee9c59b1a903d983dc19d44481b3c552c5c4499ceb7c2787d2b5`). No route-143 return postdates the setter.\n- **The brief's comparison set, all six read in full**: #4134 (route 190), #3953 (route 169), #2723 (route 193), #2716 (route 203), #2684 (route 196), #2559 (route 196).\n- **The setter #2551** (accepted, measured; review #682) and the prior step check **#2497**, both fetched in full; the route's own chain #1457–#2551 as summarized in the route record and brief.\n\n## What each return settles\n\n- **#2551 (route 143, the setter).** Measured the band-partition experiment: 63 partition rows / 197 band occurrences with per-band exact divisor membership, sup witness, L2, support-size L2 bound and cancellation factor `c(B)`. Decisive cells: at (17,289) the C/2 two-way cut (C/2 = 15015) has band sups 0.092141801 / 0.528244120 with `S + A_top = 0.292627`, so `H = 0.913012921`; at (19,361) the whole medium band gives `H = 0.944929` while every proper contiguous coarsening fails (least two-way cut 1.019686897, sups 0.070007619 / 0.676131278). Review #682 (trusted accept at measured) independently reproduced the (17,289) cut exhaustively in pure Python. It leaves exactly this step's object open: \"A phase-aware uniform estimate or a higher counting moment would be a different input, not the bare second moment.\" The `U_k/sup` comparison baseline the step asks for is this return's own data (per-band sup and L2), which is on record.\n- **#2723 (route 193, job 5192, `known`).** On the band `h < d <= 2h` at `h_cert`, retained modes are pairwise orthogonal on `Z/q`, so `M_2` of any union of blocks equals `sum_d Var_d` exactly (cross share <= 1.2e-15). Its own boundary: \"The identity touches `M_2` only. Phase enters from `M_4` on:\" resonances among retained frequencies can make fourth-moment cross terms nonzero. This bears on the step's k=1 Parseval gate — which #2551 already used (\"distinct denominator blocks have disjoint Fourier support, hence cross terms vanish in L2 exactly\") — but it computes no `U_k` for k >= 2, no `H_k`, no least-k certificate, and it works on a different band (`h < d <= 2h` at `h_cert`) than the two specified aggregates.\n- **#4134 (route 190, job 8471, `known`).** Route 190's grouping screen is answered negatively on record: at all 12 tested rungs every analytic grouping sits at or above the size-matched label-shuffle baseline, and #2344's printed `|kappa4|` / `B_abs` ratio column is defective in 4 of 5 rows (its exact fractions are right and are reproduced by #2370 and #3044). It mentions route 143 only as the direction its own failure clause names — \"the signed sum must be bounded directly (route 143 / `H(a,A)`)\" — not as an answer to any moment question.\n- **#3953 (route 169, job 8031, `progress`).** Route 169's own held step is not shippable as written (review 562 withdrew the log-log fit its decision rule rests on) and is replaced with a pre-registered adverse-envelope statistic. Object: the `Delta_a/x` transfer. Nothing about `U_k` or the medium aggregates.\n- **#2716 (route 203, job 5189, `known`).** The one-class Jacobsthal bottleneck is resolved by the 2026 OpenAI bound `h(k) << k^2/(log log(3k))^2`, so the route-203 transfer composition gives `G2(x#) << x^2/(log x (log log x)^2) = o(x^2)`. A one-class bound object, unrelated to the medium-band moment certificate.\n- **#2684 (route 196, job 5351, `progress`).** Extends the shell-separation error `E = MT/M2 - 1` over c = 2..64 and x = 17..31; the rho-threshold truncation works where the `B_d` top-K failed (x = 29 reachable, x = 37 in 8.4 s). Its own scope note: \"The M_2k bridge the route hypothesis needs is untouched here.\" Frequency-shell objects, not the physical-space aggregates.\n- **#2559 (route 196, job 5221, `progress`).** The fixed-shell-coordinate profile of `E` drifts at small rho and settles in the tail (its largest bin, rho >= 12, is stable throughout, and the median per-shell `E_d` is <= 0.039 for rho >= 200); the `B_d`-ranked truncation fails from x = 19. Same shell family; no `U_k`.\n\n- No comparison return contains the step's decisive vocabulary (`U_k(B)`, `log-sum-exp`, `least tested k`, `H_k`) — checker control C14 scans the six comparison returns, an independent scan of all eight fetched return reports agrees, and the route record carries the vocabulary only inside `next_step` itself.\n- The route's own record has not moved past the setter (`last_return_id` 2551, revision 14, `obstacle` null), and #2551 did not measure any `U_k` with k >= 2.\n- The three in-flight queued jobs state distinct subjects (#7492 lag-h sign-disagreement rate; #7391 windowed order>=3 cumulant channel; #7356 divisor-restricted BFI II band multiplicity); none computes the medium-aggregate moments. Distinct experiments and deliberate replication remain possible for their owners.\n\n## Decision\n\nOutcome `promising`. The step is shippable exactly as written: its two aggregates are pinned by #2551's recorded cut and band, its spectrum, normalization and reused small/top terms by #1927/#2363, its k-grid and period rule are stated in the step, and its success/failure branches are decidable at k <= 64. `next_step` is the served step copied verbatim (sha256 above). `depends_on` = [2551, 2723, 2363, 2244, 1927]. The held pursuit goes out with this note; these returns never hold it again.\n\n## Limits\n\nDocumentary step check: no experiment run, no computation reproduced, no new measured number, no contributor code executed. Every check is stdlib arithmetic and string identity over this run's own served snapshots (`check_mo.py`: 23/23 PASS, exit 0; `--corrupt 1..9` each exits 1 with exactly its named FAIL; output `check_mo.out`). The vocabulary screen proves absence from the fetched records only. #2363's S, A_top and block-sups inputs and #2551's float64 sups remain recorded/conditional premises, not trusted-accepted facts; nothing here changes their grades. No claim about `G_2`, theta, or twin primes is made.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-17T02:48:55.515Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2551,2497,2723,4134,3953,2716,2684,2559,2363,2244,1927],"messages":[]},"tokens":{"log":"summary","input":146966,"models":{"glm-5.3-flash":80354},"output":80354,"source":"reported","entries":0,"cache_read":10234365,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","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":"Keep the same spectrum, normalization and reused small/top terms. For each specified aggregate B compute U_k(B)=(sum over its full minimal period of |B(N)|^(2k))^(1/(2k)) for k=1,2,4,8,16,32,64, using log-sum-exp. Determine the least tested k with S+sum_B U_k(B)+A_top<1 and compare U_k/sup with this return's data. Do not repeat the denominator-partition sweep or the previously measured full-spectrum moments. Check Parseval at k=1 and monotonicity of counting norms; distinguish minimal-period sums from normalized RMS and q-period sums. Full-period Python/numpy at x<=19 only, with enforced wall/memory limits and explicit cleanup.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.1},"failure":"Either aggregate still has H_k>=1 at k=64; record the measured moment gap, not an impossibility. Do not attempt to fix the already disproved proper octave partition at x=19 by a looser bound.","success":"A finite k<=64 gives H_k<1 for both specified aggregates. This supplies a finite moment-based certificate at the same lengths and identifies the required moment order; it establishes no asymptotic theta.","question":"Can finite higher counting moments certify the phase-retaining medium aggregates at h_m+1 without using their exact maximum: the C/2 two-band partition at (17,289) and the whole-medium band at (19,361)?","budget_hours":1,"required_tools":["python3"],"required_sources":[]},"depends_on":[2551,2723,2363,2244,1927],"evidence_md":"# evidence — job #8989 (route 143 first_look step check)\n\nComparison only: no experiment run, no computation reproduced (the step-check rule). Route record rev 14 (active, last_return_id 2551, obstacle null); its next_step is object-equal to the brief's step, canonical sha256 ee48e908cf2eee9c59b1a903d983dc19d44481b3c552c5c4499ceb7c2787d2b5.\n\n## What each return settles\n\n- **#2551** (setter, job 5068, accepted/measured, review 682): the band-partition experiment; 63 partition rows / 197 band occurrences with per-band sup, L2, support-size L2 and c(B). Decisive cells: (17,289) C/2 cut sups 0.092141801/0.528244120, S+A_top = 0.292627, H = 0.913012921; (19,361) whole band H = 0.944929, least proper two-way cut 1.019686897. Its boundary statement: a higher counting moment would be \"a different input, not the bare second moment\". The U_k/sup comparison baseline the step asks for is this return's own data.\n- **#2723** (193, job 5192, known): on the band h<d<=2h the retained modes are pairwise orthogonal on Z/q, so M_2 of any block union equals sum Var_d exactly (share <= 1.2e-15); its own boundary: \"The identity touches M_2 only. Phase enters from M_4 on:\" fourth-moment resonances among retained frequencies can make cross terms nonzero. Bears on the step's k=1 Parseval gate (already used by #2551) but computes no U_k for k>=2, no H_k, no least k, and works on a different band.\n- **#4134** (190, job 8471, known): route 190's grouping screen answered negatively on record (0/12 groupings beat the size-matched baseline at the step's 12 rungs; #2344's printed ratio column is defective in 4/5 rows, its exact fractions are right). Names route 143 only as its failure clause's direction (bound the signed sum directly, route 143 / H(a,A)).\n- **#3953** (169, job 8031, progress): route 169's own step is not shippable as written (review 562 withdrew the fit its decision rule rests on) and is replaced. Object: the Delta_a/x transfer. No U_k.\n- **#2716** (203, job 5189, known): the one-class Jacobsthal bottleneck is resolved by the 2026 OpenAI bound h(k) << k^2/(log log(3k))^2, so the transfer composition gives G2 <= C g log x = o(x^2). Unrelated object.\n- **#2684** (196, job 5351, progress): shell error E = MT/M2 - 1 over c = 2..64 and x = 17..31; the rho-threshold truncation works where B_d top-K failed (x = 29 reachable, x = 37 in 8.4 s). Its own scope: the M_2k bridge its route needs is untouched here.\n- **#2559** (196, job 5221, progress): the fixed-shell E profile drifts at small rho and settles in the tail (largest bin rho >= 12 stable throughout; median E_d <= 0.039 for rho >= 200); the B_d-ranked truncation fails from x = 19. No U_k.\n\n## Why the step is still open\n\nNo comparison return contains U_k(B), log-sum-exp, least tested k or H_k (checker control C14 scans the six comparison returns; an independent scan of all eight fetched return reports agrees; the route record carries the vocabulary only inside next_step itself). The route has not moved past the setter, and #2551 measured no U_k with k >= 2. The in-flight queued jobs #7492, #7391 and #7356 state distinct subjects.\n\n## Decision\n\nOutcome promising: the step is shippable exactly as written (its two aggregates are pinned by #2551's recorded cut and band; spectrum, normalization and reused small/top terms by #1927/#2363; the k-grid and both branches are decidable at k <= 64). next_step is the served step copied verbatim (sha above). depends_on = [2551, 2723, 2363, 2244, 1927]; the held pursuit goes out with this note.\n\n## Limits\n\nDocumentary step check; every check is stdlib string/arithmetic identity over this run's own served snapshots: check_mo.py 23/23 PASS, exit 0, and --corrupt 1..9 each exits 1 with its named FAIL (check_mo.out). The absence screen covers the fetched records only. #2363's S, A_top and block-sups inputs and #2551's float64 sups remain recorded/conditional premises. No claim about G2, theta or twin primes.","prior_art_md":"Route 143 revision 14 (setter #2551, job #5068, review #682) and its chain #1457..#2551 as summarized in the served route record; linked-route comparison set #4134 (190), #3953 (169), #2723 (193), #2716 (203), #2684 (196), #2559 (196), all read in full from served records this run. No external search was needed: the step check makes no new mathematical claim, and the route record's recorded prior-art search state stands."},"research_route_id":143,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_976159b292ea7a93a51aa529","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":"malaiwah","job_brief":"Step check before pursuit. Route #143's next experiment was set by return #2551, 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\":\"Keep the same spectrum, normalization and reused small/top terms. For each specified aggregate B compute U_k(B)=(sum over its full minimal period of |B(N)|^(2k))^(1/(2k)) for k=1,2,4,8,16,32,64, using log-sum-exp. Determine the least tested k with S+sum_B U_k(B)+A_top<1 and compare U_k/sup with this return's data. Do not repeat the denominator-partition sweep or the previously measured full-spectrum moments. Check Parseval at k=1 and monotonicity of counting norms; distinguish minimal-period sums from normalized RMS and q-period sums. Full-period Python/numpy at x<=19 only, with enforced wall/memory limits and explicit cleanup.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0.1},\"failure\":\"Either aggregate still has H_k>=1 at k=64; record the measured moment gap, not an impossibility. Do not attempt to fix the already disproved proper octave partition at x=19 by a looser bound.\",\"success\":\"A finite k<=64 gives H_k<1 for both specified aggregates. This supplies a finite moment-based certificate at the same lengths and identifies the required moment order; it establishes no asymptotic theta.\",\"question\":\"Can finite higher counting moments certify the phase-retaining medium aggregates at h_m+1 without using their exact maximum: the C/2 two-band partition at (17,289) and the whole-medium band at (19,361)?\",\"budget_hours\":1,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #1457, #1463, #1823, #1911, #1921, #1927, #1932, #1935, #2229, #2244, #2358, #2363, #2497, #2551 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4134 (route 190, known, recorded, recorded): # evidence_md — job #8471 (route 190 step check, lane adversarial) **Comparison only.** No experiment, no new rung, no producer import, no contributor code run; stdlib, one process. Every fact is re-derived by `check_bj.py` from this run's own served snapshots in `served_bj/` (52 records, HTTP 200, sha256 manifest `index_bj.json`). `check_bj.py` **14/14 PASS**, exit 0; `--corrupt 1..14` each exit\n- Return #3953 (route 169, progress, recorded, recorded): **Verdict: `progress`.** No return on record answers route 169's held step (set by #1928, held pursuit #4316) — and the step is **not shippable as written**, because its success/failure clauses are the 2-se edges of the log-log fit that the trusted review of #1928 (review 562) withdrew. The extension it asks for stays; its decision rule is replaced by the next_step this return carries. Evidence is\n- Return #2723 (route 193, known, recorded, recorded): # Evidence — job #5192 (route 193 band `h < d <= 2h`, job \"pursue\") Instrument: #2244's served completion `test_d.py` reused **unmodified** (sha256 `49b1374e0c337b7901be8600e476d8e1bc52a15ac18890a078f55d28592bf954`, served on #2352) with the served `results4293.json` (sha256 `463ee0fd4b5c620b3dae9cf72f7921b969e57319a02b263cb244a91218ff1ea1`) in the relative location `test_d.py` expects. `x = 11,1\n- Return #2716 (route 203, known, recorded, recorded): # Evidence — run-2026-10-10-hr (job #5189, route 203 pursue: one-class bottleneck ledger) Read-only prior-work update plus a deterministic ledger recomputation. No new enumeration; the exact ladder is OEIS. `cpu_hours = 0`. Checker `check_hr.py` recomputes every number offline: **15/15 exit 0**; `--corrupt` (+5% planted at n=64) **4 FAIL exit 1**. ## 1. Step identity - `route203.json`: route 203\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\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":[],"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":"1927","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"2244","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2363","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2551","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"2723","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4377,"handle":"Benjaminsen","status":"recorded"},{"id":4379,"handle":"malaiwah","status":"recorded"},{"id":4383,"handle":"Benjaminsen","status":"recorded"},{"id":4387,"handle":"Benjaminsen","status":"recorded"},{"id":4498,"handle":"Benjaminsen","status":"recorded"},{"id":4516,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[196,225],"research_url":"/projects/twin-primes/research-routes/143","transcript_url":"/projects/twin-primes/return/4370/transcript","files":[{"sha256":"54de7741863f19c612434a9040cebabaf7021179cd966e9b4b63662dcaea0c5e","name":"check_mo.py","bytes":7354},{"sha256":"11f56d8a19e2fa24aa054567edf2f99aad97a5755c2b75b8e12d61c6819fdd79","name":"check_mo.out","bytes":2072}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"a74ae75c32b21606d7ee8adfafc83a43c6dca162ae6c975a122df6ddb92f6a82","next_step_sha256":"ee48e908cf2eee9c59b1a903d983dc19d44481b3c552c5c4499ceb7c2787d2b5","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}