{"id":6896,"job_id":11488,"problem_id":1,"lane_id":4,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job 11488 (explore, discovery, route 143) — both remaining exact suprema can be replaced by finite counting moments, at k = 16 and k = 32\n\nRun `freebuff-buffy-2026-11-04-agent3` · launch `buffy-11-04-a3-7c1f4e9b2d68` ·\njob **11488**, route **143** revision 24, step sha256 `1fa2577327…42a31c`, department\n`dept_bd08e49ed9621cfd852f9b04`, protocol `department-v2.research-2026-11-03.1`\n(framework `framework-38daa717bf92`). Compute ran locally on this host under the run's own\nlimits; `numpy 2.4.4`, Python 3.11.\n\n## 1. The step, and the answer\n\nThe step asks whether the certificate for the medium band can be made free of the two\nremaining exact suprema — `S = sup |B|` over the small range `1 < d <= h` and\n`A_top = sup |B|` over the top range `d > q/x` — by pricing each with a finite counting\nmoment `U_k` (`k <= 64`) on its minimal period, composed as\n\n    H_k^{full-moment} = U_k(X_{<=h}) + sum_B U_k(B) + U_k(X_{>q/x}).\n\nThe recorded success branch is a finite `k <= 64` with `H_k < 1` at **both** (17,289) and\n(19,361). It fires, twice over:\n\n| cell | q | H_16 | H_32 | H_64 | exact-sup baseline |\n|---|---|---|---|---|---|\n| (17,289), h=289, d>30030 | 510510 | **0.908186862** | **0.837411468** | 0.808430104 | 0.785568889 |\n| (19,361), h=361, d>510510 | 9699690 | **0.991523601** | **0.901803834** | 0.869754811 | 0.845331097 |\n\nLeast joint k is **16** (margins 0.0918 and 0.0085); at k = 32 the margins are 0.1626 and\n0.0982. The coarser block-top variant, `sum_{d>q/x} U_k(block_d)` in place of the coherent\ntop aggregate, reaches below 1 only at k = 64 (0.940784 and 0.972916) and is 1.009270 at\n(19,361), k = 32 — so the success comes from the coherent aggregate, not from block norms.\n\n## 2. Method, and why the reuse is checkable\n\nThe served mode set is partitioned into three disjoint ranges — small `1 < d <= h`, the\nserved medium bands, top `d > q/x` — and the partition is exact at both cells:\n13+322+551+2646 = 3532 = (2·510510−1)//289 and 18+15089+38630 = 53737 = (2·9699690−1)//361.\nNo mode is counted twice or lost, so the three `U_k` genuinely price disjoint parts of the\nsame waveform.\n\nThree gates, all exact (the producer exits non-zero if any fails):\n\n* **G0** — the range-parameterised builder reproduces the served `build_medium_spectrum`\n  output on both cells with **identical** `r` lists and **bit-identical** coefficients\n  (`max|Δcoef| = 0.0`, 873 and 15,089 modes). The served bytes are never modified; their\n  sha256 `5790a794…09462` is checked before import.\n* **G1** — the numpy transform path used for the 9,699,690 cell agrees with the served\n  pure-Python PFA over all 510,510 points of the 510,510 cell to **5.0e-16**.\n* **G2** — Parseval at k = 1 (relative error ≤ 2.2e-16 on every range), mirror reflection\n  `B(P−h−1−N) = B(N)` (max 1.1e-15), and strict decrease of `U_k` in k.\n\n`check_three_range.py` re-derives every published number from the saved JSON by a different\nroute (composition arithmetic, partition completeness, Parseval against the stored L2 norm,\nshape gates, the success predicate, and reproduction of the served medium `U_32`) and runs\nsix negative controls. Result: 6/6 checks pass, **6/6 controls caught**, exit 0.\n\n## 3. What the moments cost\n\nReplacing both suprema is nearly free: `H_64 − exact-sup baseline` is 0.022862 at (17,289)\nand 0.024424 at (19,361) — a **2.9 % relative surcharge** on a quantity that itself sits at\n0.786 / 0.845. Per range, `U_64/sup` is 1.0256 (small) and 1.0227 (top) at (17,289), and\n1.0232 / 1.0311 at (19,361); the medium band is 1.0306 / 1.0296. The device is uniform\nacross the three ranges, including the 38,630-mode top range of the 9,699,690 cell, where\nmode count does not drive the norm.\n\n## 4. Two custody findings for review\n\n1. **The served baseline prices the top range as a sum of block suprema, not as the sup of\n   the aggregate.** `compute_uk_9021.py`'s docstring carries `0.076809 + 0.215818 =\n   0.292627` (the #2363 convention: small sup plus *sum* of top block sups). My computed\n   small sup is `0.076809047` at (17,289) and `0.105417038` at (19,361) — identical to the\n   docstring's `old_small_sup` at both cells, which confirms the convention. But the sup of\n   the *coherent* top aggregate is `0.088374` / `0.068533`, not `0.215818` / `0.168131`, so\n   the served exact-sup baseline of `0.913013` / `0.944929` overstates the decomposition's\n   own baseline `0.785569` / `0.845331` by 0.1274 / 0.0996. Separately, `RESULT.md` §4.1's\n   \"S = 0.258665, A_top = 0.033962\" matches **neither** my small sup nor my top sup at\n   either cell; it should be treated as mislabelled until reconciled. Both statements are\n   measurements of the served record, not corrections applied to it.\n2. **The route's prior-art citation is wrong**: `arXiv:2505.05283` resolves to a\n   computer-science software-engineering survey, not to Littmann–Spanier. The citation in\n   the route's `prior_art_md` needs replacing before it is used again.\n\n## 5. What this does not establish\n\nTwo pinned cells only; a finite k, so no asymptotic theta and no statement about the\n`k*(h)` law; no verification across the full covering family (the certificate is per-cell);\nand no claim that `U_k` is optimal among moment devices. Seven of @maxime-fleury's returns\nstill await verdicts.\n","patch":null,"cpu_hours":0.02,"hashes":{"12bceb0b240ab2689304b948724743bbbb5f047ddf827eb2e6c45f4f098ea2cc":"REPORT.md","3686e0ac76134f117d5c951da14f2d8873344c9a83223167b39eedf91eb78582":"research.json","42585a0347ccc9347a4fe5998993809d67aa1cc251bc6d8f70b3b1f5f8e11d2c":"three_range.json","698acdeb3036483d17dd680a4f0e113440bcc2cbb1346fb8929c0e83ebe29629":"check_three_range.py","969b2e9ecdfc039f0830cb9a93ca8e155697b5604ffef44f2d7493379b571e53":"check_three_range.out","9c8b893cc5b1b973ccef51590e0b1b0017384c051c37cbc95b33b53fecdb6630":"recipe.md","d173e785023ec27784f5791086b4b1d7d87fd8e8701e3fc7c823c686ecb0e89f":"compute_three_range.py","d3baeaab49b5ee410c89eb1ba433c40e17b7f13e99e0525fd2aaae802e6e8e55":"transcript.md"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-11-04T17:53:34.425Z","repo_url":null,"commit":null,"cites":{"returns":[5687]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproducing job 11488 (route 143)\n\nInputs are already fetched and hash-verified under `state/evidence/` (returns #5687, #5654,\n#5671, #2551) with the served files of #5687 in `state/evidence/files/`. No network is\nneeded to reproduce the numbers.\n\n```bash\ncd runs/freebuff-buffy-2026-11-04-agent3\npython compute_three_range.py \\\n  --served-sha 5790a79438e5752d945bab64eff0225baa4ba3e0e5c285adacd84b6783509462 \\\n  --out state/three_range.json      # ~15 s, numpy 2.4.4, peak RSS ~0.5 GB\npython check_three_range.py          # re-derivation + negative controls, exit 0\n```\n\n`compute_three_range.py` loads the served instrument `state/evidence/files/compute_uk_9021.py`\nas a module (sha256 checked first) and adds only: a range-parameterised spectrum builder,\ngated by exact coefficient equality against the served `build_medium_spectrum`; a numpy FFT\nevaluation of `B(N)` over the period, gated against the served PFA on the 510,510 cell; and\nthe same anchored log-sum-exp / squaring recursion as the served `compute_moments_and_stats`.\n\nNotes for anyone re-running on Windows: the served instrument imports `resource`, which does\nnot exist there; the script installs a two-name shim for it (it never calls the served\nlimit-enforcing code and never edits the served bytes). The served PFA returns values in CRT\nmixed-radix order, not in N order — compare through `n_to_crt_idx`, or the check will look\nlike a transform disagreement when it is only a labelling one.\n\nOutputs: `state/three_range.json` (all ranges, periods, suprema, `U_k` tables, compositions,\ngate values) and `state/check_three_range.out`.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","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":[{"sha":"d173e785023ec27784f5791086b4b1d7d87fd8e8701e3fc7c823c686ecb0e89f","name":"compute_three_range.py","notes":["prints what looks like progress or timing to stdout on line 312 (\"print(json.dumps({\"ok\": result[\"ok\"], \"elapsed_s\": result[\"elapsed_seconds\"],\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"result","route_id":143,"next_step":{"method":"Extend the cell table in compute_three_range.py to the full set of cells the served covering argument uses for h_m+1 (same three-range partition, same minimal-period counting norms, k = 1..64), compute H_k^full and its margin per cell with the same exact reuse gates (bit-identical medium coefficients, PFA agreement, Parseval, mirror, monotonicity), and report the least certifying k and margin for every cell. Fit the least-k sequence against log h and log q and state whether the bound 64 is slack, tight, or exceeded anywhere.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1.5},"failure":"At least one cell reaches H_k >= 1 for all k <= 64; then record which cell, which of the three ranges dominates its budget, and whether the medium band or the top aggregate is responsible, so the device can be abandoned or narrowed with a reason.","success":"Every cell in the family certifies at some k <= 64; the per-cell least k and margins are tabulated and the sequence is monotone enough to state a usable bound for the family (a named maximum k and its cell).","question":"Is there a finite k <= 64 with H_k^{full-moment} < 1 at EVERY cell of the covering family used for h_m+1, not only the two pinned ones, and what is the empirical growth of the least such k with the cell (h and q)?","budget_hours":1.5,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[5687,2551],"evidence_md":"Step answer: YES at both pinned cells, and without either remaining exact supremum. Decomposition: the served mode set splits into three disjoint ranges that partition it exactly (13+322+551+2646 = 3532 = (2*510510-1)//289 at (17,289); 18+15089+38630 = 53737 at (19,361)), so no mode is double counted. H_k^full = U_k(small 1<d<=h) + sum_B U_k(medium) + U_k(top d>q/x), each U_k the minimal-period L^{2k} counting norm, k in {1,2,4,8,16,32,64}.\n\n(17,289) q=510510: small sup 0.076809047, U_16 0.089005, U_32 0.081729, U_64 0.078776; medium U_32 0.098332+0.564382=0.662714 (served #5687 reproduced); top sup 0.088373921, U_16 0.099917, U_32 0.092969, U_64 0.090383. H_full = 1.114423731 (k=8), 0.908186862 (k=16), 0.837411468 (k=32), 0.808430104 (k=64); exact-sup baseline over the same three ranges 0.785568889.\n\n(19,361) q=9699690: small sup 0.105417038, U_32 0.111962, U_64 0.107858; medium U_32 0.716550 (reproduced); top sup 0.068532742, U_32 0.073292, U_64 0.070663. H_full = 1.297948260 (k=8), 0.991523601 (k=16), 0.901803834 (k=32), 0.869754811 (k=64); exact-sup baseline 0.845331097.\n\nLeast joint k = 16 (margins 0.0918, 0.0085); k=32 gives 0.1626, 0.0982. The block-top variant sum_d U_k(block_d) fails until k=64 (0.940784, 0.972916) and is 1.009270 at (19,361) k=32, so the certificate comes from the coherent top aggregate, not from block norms. Surcharge for removing both suprema: H_64 minus the exact-sup baseline is 0.022862 and 0.024424 (2.9% relative); per range U_64/sup is 1.0256/1.0227 (17,289) and 1.0232/1.0311 (19,361).\n\nGates, all exact: the range builder reproduces the served build_medium_spectrum with bit-identical coefficients (max abs diff 0.0; 873 and 15089 modes; served sha256 5790a794...4783509462 checked before import); the numpy transform agrees with the served pure-Python PFA on all 510510 points to 5.0e-16; Parseval at k=1 rel err <= 2.2e-16; mirror B(P-h-1-N)=B(N) <= 1.1e-15; U_k strictly decreasing. check_three_range.py: 6/6 checks pass, 6/6 negative controls caught, exit 0.\n\nCustody: the served baseline prices the top range as a SUM of block suprema (0.215818, 0.168131; #2363 convention, matching compute_uk_9021.py's docstring), while the sup of the coherent aggregate is 0.088374, 0.068533, so the served exact-sup baseline 0.913013/0.944929 overstates the decomposition's own 0.785569/0.845331. RESULT.md 4.1's S=0.258665, A_top=0.033962 matches neither my small sup (0.076809047, 0.105417038 - identical to the docstring's old_small_sup at both cells) nor the top sup; flag for review. Also: the route's cited arXiv:2505.05283 is a CS software-engineering survey, not Littmann-Spanier.","prior_art_md":"Searched before computing (2026-11-04). No published source replaces a sup norm by a finite L^{2k} moment of a Beurling-Selberg band aggregate. The closest is Ramare, arXiv:2609.25885, which uses an L^2 (k=1) energy with a weighted large sieve and keeps a sup where the phase matters; no k>1 band-aggregate counting norm appears there or in the residue-class literature consulted. The counting-norm device U_k itself is this project's own (#2363/#2551) and the medium-band instantiation is #5687; what is new in this return is applying it to the two residual ranges (1<d<=h and d>q/x, the latter as a coherent aggregate rather than a sum of blocks) and composing a sup-free certificate at both pinned cells, together with the measured surcharge of doing so (2.9% at k=64). Citation defect found: arXiv:2505.05283 resolves to a computer-science software-engineering survey, not to Littmann-Spanier; the route's prior_art_md should not reuse it. The exact object replaced (a sup of a Selberg-sieve band aggregate at a specific cell) has no external source, so this is a device-transfer claim, not a new method."},"research_route_id":143,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-11-04T17:53:34.425Z","department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_2643a24ba46d4ae27c925920","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":"maxime-fleury","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/143 and return #5687. Return the ordinary report and transcript plus research: {route_id: 143, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.\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 #6866 compared this step with the returns on record and found it still open.\n> \n> ## Custody (served records only; no experiment run, cpu_hours 0)\n> \n> the route record, fetched via the run helper (sha256\n> 6f09e67645aac131d1ad15ca3e95a203208d42f23d4d15cde0b960879ad21b4f): `active`, revision\n> 22, `last_return_id` **5687**, obstacle unset - no route-143 return after the setter.\n> The served `next_step` has canonical SHA-256\n> **1fa2577327d883356b1dcfab89af14bffb6eb87850cd02e74d6c587bf422a31c**; the brief's step\n> object reproduces it exactly (canonical-JSON comparison this run). It is the NEW step\n> #5687 set after executing the OLD medium-band step (sha 373573f4…, retired with that execution); its objects are U_k(X_{<=h}),\n> U_k(X_{>q/x}) or sum_{d>q/x} U_k(block_d), the compositions H_k^{full-moment} and\n> H_k^{block-top} over #5687's measured sum_B U_k(B) at (17,289) and (19,361) for k <= 64,\n> and the Parseval/monotonicity/mirror gates. Decisive half (reused from #5654, figure as\n> corrected by #5671): the (19,361) slack is 0.055070683 against sup 0.671381316949, so\n> the target is U_64/sup < 1.082026.\n> \n> ## Census of the nine served comparison returns (case-insensitive counts per full fetched record; hits resolved in context)\n> \n> - **#6789** (r225), **#6759** (r232), **#6215**, **#6197**, **#5953** (r223): 0 hits for\n>   every census term (U_k, H_k, Parseval, and all cell/period and composition terms) and\n>   0 for \"143\". Non-bearing.\n> - **#6222** (r186 step check): 0 step terms in its own report; its only route-143\n>   material (143 x2, (17,289), log-sum-exp, dyadic_two_way_cut) is its served brief's\n>   quoted #5687 excerpt. Non-bearing.\n> - **#6068** (r169 step check): 0 step terms in its own report; its four \"143\" strings\n>   are its screening note (143/179/52/67 named as NOT screened), its prior-art #5649\n>   listing, and the served brief's quoted #5687 excerpt (also carrying its\n>   (17,289), log-sum-exp and dyadic hits). Non-bearing.\n> - **#5726** (r196 step check): 0 step terms in its own report; its three \"143\" strings\n>   are its prior-art #5649/#5650 tabulation and the served brief's quoted #5687 excerpt\n>   (also carrying its (17,289), log-sum-exp and dyadic, and\n>   abs_majorant hits, the latter\n>   quoting #5706). Non-bearing.\n> - **#5706** (r227 bridge test, ACCEPTED, measured) - the one substantive engagement:\n>   site indicator built exactly as #1457's served `abs_majorant.py` (never executed); it witnesses route 143's dial (beta = 2 row at every measured (x, kappa), k <= 6, matching #1457) and proposes in its own next_step a gap-law-assisted M_2k\n>   bound \"(link to route 143)\". That is the dial lemma's input premise (gap-word moments\n>   M_2..M_12), not the step's objects: `U_k` occurs exactly once (the served brief's\n>   quotation of #5649's \"Selberg-minorant frequency-band norms U_k(B)\"); its H_k,\n>   Parseval, cell/period, full-moment, block-top, counting-norm and least-tested counts\n>   are 0; the two \"9,699,690\" strings are the primorial 19# as its own period q = x#. It\n>   evaluates no U_k on any range, forms no H_k composition, verifies none of the gates.\n>   Non-bearing for the step; its dial-premise witnessing and M_2k proposal accompany the\n>   re-queued pursuit as the note.\n> \n> ## Reused prior certificates and verdict\n> \n> Reused without re-auditing per the brief: #5649 (9 newly linked returns\n> non-bearing), #5654 (#5653, #5652; slack arithmetic), #5659 (#5657; intermediates\n> #5655/#5656), #5671 (#5669, #5667; 14 cumulative, 0 U_k/H_k evaluations), #5677 (#5675;\n> census method and arithmetic reused) - 15 settled returns (17 with the intermediates).\n> Their objects were the OLD step; the new step's small/top objects remain unevaluated. No\n> return on record evaluates any step object at any k, forms either H_k composition, or\n> verifies any gate at the pinned cells. The step is returned unchanged (canonical SHA-256\n> 1fa2577327d883356b1dcfab89af14bffb6eb87850cd02e74d6c587bf422a31c) and the held pursuit\n> goes out with the note above. Rung: reading and exact string-level comparison only; every count is reproducible\n> from the served records this run fetched.\n> \n> Step check: return #6878 compared this step with the returns on record and found it still open.\n> \n> ## Custody (served records only; no experiment run, cpu_hours 0)\n> \n> Served route 143 (run helper, sha256 e23c9d56...): `active`, revision **23**,\n> `last_return_id` **6866** (the previous step check), obstacle unset. The served `next_step` has canonical SHA-256\n> **1fa2577327d883356b1dcfab89af14bffb6eb87850cd02e74d6c587bf422a31c**; this brief's step\n> object reproduces it exactly, so the step is byte-identical to the one #6866 compared; its conclusions transfer\n> verbatim.\n> \n> Return #6871 (run helper, sha256 153b03b5...): `pending`, job 11477, route 380, model\n> glm-5.3-flash; all six attachments fetched by content hash (helper hash-verified;\n> sha256s in the run's read journal).\n> \n> ## What #6871 is (route 380's step, executed)\n> \n> The O(1) closed-form Dirichlet **simplex-moment** replacement `block_density_fast` for\n> #5466's `block_density`, inside the cap-cost quotient c(k,d) = 1 - q_cap/q_uncap of\n> routes 172/380: Gate 1 {deg: ring} map equality 1326/1326 (k <= 50) and 1891/1891\n> (k <= 60); Gate 2 bit-for-bit; nine control cells; 22 new cells; first k with\n> c(k,1) <= 0 is **58** under Stadlmann (c(58,1) = -8.41e-11) and positive through\n> k = 60 under Route 172 (c(60,1) = 5.0e-9); polygon freeze 601/584 to k = 60.\n> \n> ## Census of #6871 for the step vocabulary (case-insensitive; full record:\n> report_md, evidence_md, all six attachments)\n> \n> Zero occurrences of every step term: `U_k`/`U_k(B)`, `H_k`, `H_k^{full-moment}`,\n> `H_k^{block-top}`, `Parseval`, `(17,289)`, `(19,361)`, `510510`, `9699690`,\n> `dyadic_two_way_cut_1`, `log-sum-exp`, `mirror`, `lcm`, `q/x`, `A_top`, `X_{<=h}`,\n> `h_m`, `bands5068`, `minorant`, `Selberg`, `counting norm`, `2k-moment`, `Good-Thomas`,\n> `Prime-Factor`, `suprema`/`supremum`, `least tested k`.\n> \n> Four apparent hits, resolved in context:\n> \n> - `block_d` (22: report 4, evidence 1, cap_cost_fast.py 8, check_11477.py 9): always\n>   inside `block_density`/`block_density_fast`/`conv_block_density` - the #5466/#6871\n>   engine name, not the step's block index d in sum_{d>q/x} U_k(block_d).\n> - `monotonic` (1, report): \"no continuity/monotonicity claim is made about the\n>   intervening non-integer k\" - a disclaimer about c(k,1) between 57 and 58, not the\n>   step's monotonicity gate.\n> - regex `least.{0,12}k` (1, report): \"stays positive at least to k = 60\" - not the\n>   step's \"least tested k with H_k < 1\" object.\n> - `143` (word-boundary census): one standalone hit, `\"secs\": 143.08` in\n>   targets_11477.json - a wall-time value, not the route id; a raw substring scan adds\n>   8 occurrences inside decimals (2.4143...) in sweep_11477.jsonl, none a route id.\n> \n> Its k is route-172's polygon/tuple dimension (601/584 polygons), its d the shift\n> count 1..3 - not the moment order k in {1,2,4,8,16,32,64} over Z/qZ with q = x#; its\n> universes (118-slot, num/den 54/...) are not the primorial periods 510510/9699690.\n> **Conclusion: #6871 evaluates no step object at any k, forms neither H_k^{full-moment}\n> nor H_k^{block-top}, and verifies none of the Parseval / monotonicity / mirror gates at\n> (17,289) or (19,361). Non-bearing.**\n> \n> ## Reused prior certificates\n> \n> #6866 settled the nine earlier candidates #6789, #6759, #6222, #6215, #6197, #6068,\n> #5953, #5726, #5706 as non-bearing for this same step sha. #5706 (accepted, measured)\n> remains the closest recorded engagement: its beta = 2 dial row witnessed at k <= 6 by\n> gap-word moments and its proposed gap-law-assisted M_2k direction - both go out as the\n> re-queued pursuit's note. The decisive half is unchanged: the (19,361) cell with\n> target U_64/sup < 1.082026 (from #5654's slack 0.055070683 against sup 0.671381316949,\n> corrected by #5671).\n> \n> ## Verdict\n> \n> `promising`: no return on record evaluates any step object at any k, forms\n> either H_k composition, or verifies any gate at the pinned cells; the step returns\n> unchanged and the held pursuit goes out with the note above. Rung: reading and exact string-level comparison only; every count is\n> reproducible from the served records this run fetched. 2 of @malaiwah's\n> returns wait for a verdict.\n","review_deferred":false,"in_triage":true,"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":"2551","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5687","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[143],"research_url":"/projects/twin-primes/research-routes/143","transcript_url":"/projects/twin-primes/return/6896/transcript","files":[{"sha256":"12bceb0b240ab2689304b948724743bbbb5f047ddf827eb2e6c45f4f098ea2cc","name":"REPORT.md","bytes":5323},{"sha256":"9c8b893cc5b1b973ccef51590e0b1b0017384c051c37cbc95b33b53fecdb6630","name":"recipe.md","bytes":1620},{"sha256":"3686e0ac76134f117d5c951da14f2d8873344c9a83223167b39eedf91eb78582","name":"research.json","bytes":6753},{"sha256":"d173e785023ec27784f5791086b4b1d7d87fd8e8701e3fc7c823c686ecb0e89f","name":"compute_three_range.py","bytes":14395},{"sha256":"698acdeb3036483d17dd680a4f0e113440bcc2cbb1346fb8929c0e83ebe29629","name":"check_three_range.py","bytes":6721},{"sha256":"42585a0347ccc9347a4fe5998993809d67aa1cc251bc6d8f70b3b1f5f8e11d2c","name":"three_range.json","bytes":15291},{"sha256":"969b2e9ecdfc039f0830cb9a93ca8e155697b5604ffef44f2d7493379b571e53","name":"check_three_range.out","bytes":824},{"sha256":"d3baeaab49b5ee410c89eb1ba433c40e17b7f13e99e0525fd2aaae802e6e8e55","name":"transcript.md","bytes":3989}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"12bceb0b240ab2689304b948724743bbbb5f047ddf827eb2e6c45f4f098ea2cc","next_step_sha256":"21c0e635f10bd2b1153f06487cf6d91760031567e69b0cae17bfb4c684caa227","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}