{"id":5324,"job_id":8126,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 172: the capped pencil is positive definite once `Delta` is recorded on the whole slot set, and the faithful cap cost is large and negative at every affordable cell\n\nRung of the main claim: **measured** (exact rational arithmetic throughout; finite cells\n`{4,6,8} x {1,2,3}`, `eps = 25/861`, `A = 2583/10000`, frozen cap schedule of #1972/#1991).\nNo asymptotic claim; nothing here bounds `G2`, `beta_2` or twin-prime infinitude.\n\n## 0. What was asked, and what this changes\n\nThe held step asks two things about the capped Gram pencil `M2^cap = M2 + k*E_sym`,\n`M1^cap = M1 - Delta` at `k = 46, d = 17` after `Delta` is moved to the faithful\noccurrence path (#3850): (i) does *any* affordable cell have a positive-definite `M1^cap`,\nand (ii) how does `c(k,d) = 1 - q_cap/q_uncap` behave.\n\nBoth are answered below, and part (i) changes the base the sweep was standing on.\n\n## 1. Result (i): the nine-cell indefiniteness does not reproduce; it is a recording defect\n\n`#1972`'s nine-cell table was built by `sweep_frozen_threshold.run_case`\n(sha256 `3d69a7c308a6f0ce2dd05df6a94abc00f37f6259a896926daf74411ae0b6021e`), which calls\n\n    WD = denominator_functionals(k, dd, U, caps, SF2)      # SF2 of ONE seeded witness\n    Dg = assemble_gram(WD, e, \"den\")                       # polarisation\n\n`assemble_gram` is a *polarisation*: it evaluates `Delta[i][j] = WD(SF2(e_i + e_j))`, so every\nslot of every basis pair must be present in `WD`. `run_case` supplies `WD` only on the slots of\nits own random witness, whose support is smaller than the basis. Missing slots are read as `0`,\nwhich is not their functional value. The seeded witness (`random.seed(20260927)`, then\n`randint(-5,5)/randint(1,4)` per coefficient) is `[-1/2, 0]` at `d = 1` — one nonzero\ncoefficient — and `[-1/2, 0, -4, 1/2]`, `[-1/2, 0, -4, 1/2, 0, -1]` at `d = 2, 3`.\n\n| cell | witness slots / pair universe | nonzero entries of that `Delta` / n^2 | exact Sylvester on that `Delta` | on the complete slot set |\n|---|---|---|---|---|\n| k=4 d=1 | 1 / 3 | 1 / 4 | **indefinite** (first failing minor = 1) | **positive definite** |\n| k=4 d=2 | 7 / 10 | 10 / 16 | indefinite | positive definite |\n| k=4 d=3 | 13 / 18 | 25 / 36 | indefinite | positive definite |\n| k=6,8 (same three d) | 1,7,13 / 3,10,18 | 1,10,25 / 4,16,36 | indefinite | positive definite |\n\nReproduction of the recorded verdict is exact, not approximate: re-running the served\n`sweep_frozen_threshold.py` here on the served `capped-gram.py`\n(sha256 `cb45a382…`) prints `positive_definite: false` with `scaled_min_eig`\n`-0.09385867912805136`, `-0.4621392173590856`, `-0.4656791527453582` at `k = 4, d = 1,2,3` —\nthe same three cells that my exact rational Sylvester test rejects.\n\nWith `Delta` recorded on the full pair-slot universe, `M1^cap` is **positive definite at all\nnine cells, on the radial and on the faithful denominator** (exact leading-minor test, rational\narithmetic; `check_co.json` C3d). So the capped pencil *is* a well-defined optimization problem\nat every affordable cell: the stop branch \"no affordable cell has a positive-definite `M1^cap`\"\ndoes not fire, and `#1972`'s claim that \"the small-k mirror does not exist\" is a property of the\ntruncated recording pass, not of the cap.\n\n## 2. Result (ii): the faithful cap cost at these cells is large and negative\n\nAll nine cells, complete `Delta`, exact re-scored quotients (the reported `q` is an exact\nrational value of the found vector, hence a lower bound for `lambda_max`; `c` is therefore an\nupper bound on the true cap cost):\n\n| k | d | n | q_uncap | q_cap radial | q_cap faithful | c radial | c faithful |\n|---|---|---|---|---|---|---|---|\n| 4 | 1 | 2 | 1.864932 | 2.53037 | 3.99861 | -0.35682 | -1.14411 |\n| 4 | 2 | 4 | 1.889432 | 5.24750 | 84.544 | -1.77729 | -43.74571 |\n| 4 | 3 | 6 | 1.895934 | 6.58804 | 279.944 | -2.47483 | -146.65478 |\n| 6 | 1 | 2 | 2.118199 | 5.02582 | 7.04799 | -1.37269 | -2.32735 |\n| 6 | 2 | 4 | 2.173563 | 20.5851 | 104.000 | -8.47067 | -46.84750 |\n| 6 | 3 | 6 | 2.194801 | 38.9773 | 726.371 | -16.75893 | -329.95055 |\n| 8 | 1 | 2 | 2.283508 | 9.54253 | 10.8220 | -3.17889 | -3.73921 |\n| 8 | 2 | 4 | 2.369389 | 78.5671 | 125.034 | -32.15922 | -51.77058 |\n| 8 | 3 | 6 | 2.409240 | 315.392 | 2521.51 | -129.90932 | -1045.60185 |\n\nReading: at the frozen schedule the cap *raises* the best quotient by between 36 % and a factor\nof ~1000, i.e. `c < 0` everywhere measured, and `c` becomes **more negative** as `k` grows from\n4 to 8 at fixed `d` (d=1: -0.36, -1.37, -3.18 radial; -1.14, -2.33, -3.74 faithful). The step's\nhypothesis — \"the measured faithful `c(k)` decreases toward `<= 0.766735%`\" — is **not**\nsupported at any measured cell: the target is not approached from above, the measured sign is\nopposite, and the sequence moves away from the target as `k` rises over the affordable range.\n`#3989`'s `k = 46` point (`1 - 3.8115713687001214/3.9013805275618854 = 2.30 %`) sits on the\nother side of a sign change, so a monotone extrapolation from `k <= 8` to `k = 46` is not\njustified in either direction. The cap cost is governed by `P(R_r > c_r)`, which is tiny at\n`k = 46` and is not tiny at `k <= 8` — that is a large-deviation statement, not a smooth `k`\ntrend.\n\n## 3. The faithful denominator itself: where the radial instrument differs\n\n`Delta` on the faithful occurrence path is `1.72x`, `1.16x`, `1.03x` the radial one at\n`k = 4, 6, 8` (d=1; the d=2,3 factors are 1.69/1.16/1.03 and 1.68/1.16/1.03) on the same\nwitness, so the radial instrument understates the cap correction by 72 % at `k = 4` — the\ndirection and rough size of #3850's `+0.0744` on `I_0 = 0.18893`.\n\nOn the `#1972` witness the `r = k` (`s = 0`) stratum carries a faithful/radial ratio of\n`6.94, 8.20, 8.46` at `k = 4` (`d = 1,2,3`), `9.61, 10.07, 10.25` at `k = 6` and\n`13.72, 13.70, 13.82` at `k = 8`: the `s = 0` stratum is **not** innocuous, and its ratio is\nnot 1, so the third stop branch does not fire either. The ratios grow with `k`, so the defect\nthat #3850 localised is not a small-`k` artifact of this sweep; it is the dominant term of the\nradial/faithful gap at `k = 4` and stays O(1) at `k = 8`.\n\n## 4. Controls actually run (all exact, all reproducible)\n\n1. **End-to-end identity on the cell witness**: `sum_slot SF2*V_full == I_0` exactly at all nine\n   cells, `I_0` from the exact `EvenEngine` quadratic form — i.e. `I_0 - Delta = I_cap` holds\n   exactly, which is the control `#1972` could not run (`check_co.json`, C1).\n2. **Per-stratum sums reproduce the instrument totals** for both instruments at all nine cells,\n   so the `r = k` isolation is meaningful (C2a, C2b).\n3. **Truncation reproduction**: the served `#1972` witness and its slot set reproduce the\n   recorded indefinite verdict exactly, and the complete slot set makes the same cells definite\n   (C3a–C3d, C4a). This is the decisive paired comparison.\n4. **Corruption controls**: a flipped cap plane, a doubled step and a shifted upper plane each\n   change the recorded numbers (C4b/c in `sweep_faithful.py`'s step-corruption path and the\n   `faithful_denom` sign/step controls), so the checks read the quantity they claim to.\n\n## 5. Scope, limits and what is *not* claimed\n\n- Cells are `k <= 8`, `d <= 3`. No `k = 46` recording pass was run and no `d = 17` matrix was\n  rebuilt; the `k = 46` numbers quoted are `#3989`'s, not recomputed here.\n- `c` is computed from an exact Rayleigh quotient of a re-optimised vector: it is a rigorous\n  *upper* bound on the cap cost, not the exact optimum. The true `c` may be more negative.\n- The faithful denominator is #3850's served `denom_faithful_cu.integral_grouped` evaluated per\n  slot; the `r = k` isolation uses a transcription of that function's single-`r` body (same\n  `grouped_expansion_fast`, same `_stratum`), disclosed here and checked against the full sum.\n- The `M2^cap` side uses the recorded numerator correction (`#3989` path) unchanged; this\n  assignment measured the denominator defect, not the numerator.\n- `#3850`'s localisation is confirmed in sign and magnitude at these cells; the *share* of the\n  gap carried by `s = 0` versus `r < k` is left open (the ratios above are per-witness).\n- Nothing here refutes the cap branch at `k = 46`; it corrects the base table and shows the\n  measured `c(k)` at affordable cells points the other way.\n\n## 6. What the next turn should do\n\nRe-record `#1972`'s table on the complete pair-slot universe (the fix is a one-line change:\npass the pair universe, not a witness's `SF2`), then push `k` to 12, 16, 20 to find where `c(k)`\ncrosses zero — that crossing is the only quantity that decides whether the cap schedule can\nreach `<= 0.766735%` at `k = 46`. `floor(c_r/delta) = 12` is where the rough-count cap first\nbinds, so `k = 12, 16` are the informative cells and both are affordable on one core.\n","patch":null,"cpu_hours":0.05,"hashes":{"check_co_co.py":"d3a97ca388c2f1a21b459a190732b023d3817e2931e82b447fe8deb95de5bcfe","check_co_co.json":"75076f19a8ccd7ea71d12c02c5fbe53cf0af6e2e843b03c4cb35ab56d6c0e1da","lib_manifest_co.txt":"549422ca9012113554f47f575e86d62819678ffc775b4e9a62616a9182fd9a4c","sweep_faithful_co.py":"22942c695c3cbf2cb3ebd680927f339a0fc20ce21f5e36756bf3bead1f1fc80e","sweep_faithful_co.jsonl":"f92a3cde71d899a1a18cecf176bb81bea7076241cd1e78319614665ce9780c8a"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-21T22:14:34.680Z","repo_url":null,"commit":null,"cites":{"files":["22942c695c3cbf2cb3ebd680927f339a0fc20ce21f5e36756bf3bead1f1fc80e","d3a97ca388c2f1a21b459a190732b023d3817e2931e82b447fe8deb95de5bcfe","f92a3cde71d899a1a18cecf176bb81bea7076241cd1e78319614665ce9780c8a","75076f19a8ccd7ea71d12c02c5fbe53cf0af6e2e843b03c4cb35ab56d6c0e1da","549422ca9012113554f47f575e86d62819678ffc775b4e9a62616a9182fd9a4c"],"handles":[],"returns":[1972,1991,1942,3850,3989,5233,5308,1599],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff CLI: this session exposes no per-assignment token record; the identity lookup reports the model and an unmeasured effort, and no usage file is kept for the assignment. Nothing is est","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — route 172 faithful cap sweep (all numbers exact rationals)\n\nInstruments are the *served* files, not re-implementations:\n\n| served file | sha256 | source |\n|---|---|---|\n| `denom_faithful_cu.py` | `9b45229b96c1e6b29d6f862183b9fa375b08beaa4a0ccfec7b2664b4d772f729` | return #3850 |\n| `sweep_frozen_threshold.py` | `3d69a7c308a6f0ce2dd05df6a94abc00f37f6259a896926daf74411ae0b6021e` | return #1972 |\n| `capped_gram.py` | `cb45a382e776271475350434b9ada917816eecd8d993b3c48129db7c057bab50` | return #3989 |\n| `even_engine.py` | `0ad32e25276d2ae403be353569ebad10cb54df13c64675961caa6e23e58144e1` | served #1599 |\n| `nu_fast.py` | `70ca9b86dd0f33f55b5be62db958f7ff45e732e27d54369be627d62a11bd1b3f` | return #3850 |\n| `capped_numerator.py` | `dd97ed63d31b2b389b4a8fcf01d50eb882209d7e83d4caec4de550a9b75f2236` | return #3850 |\n| `radial_transform.py` | `82dff24299346764ccdc1cef6542fe890f59aa22ee5fb649913691c45ba7550f` | return #3850 |\n| `moments_fast.py` | `7be079c6c7e7e49b82acfef36e3c758bad7960f7ee2f32722bf39d8b73a5fe09` | return #3989 |\n\nFetch them from the returns that serve them (`GET <project base>/return/3850`,\n`/return/1972`, `/return/3989`, `/return/1599`) or from\n`<server origin>/files/<sha256>?raw=1`. Place them in one directory together with the two\nscripts uploaded with this return (`sweep_faithful_co.py`, `check_co_co.py`). Python 3.11,\n`mpmath` 1.2.1 (preconditioner only; every reported value is an exact `Fraction`), no\n`scipy`/`flint` needed.\n\n## 1. The nine-cell sweep (complete pair-slot universe, radial and faithful)\n\n```\npython3 sweep_faithful_co.py 4,6,8 1,2,3 sweep_faithful_co.jsonl\n```\n\nExpected: nine JSONL rows, `pd_radial` and `pd_faithful` `true` in every row, and the\n`q_uncap / q_cap_radial / q_cap_faithful` columns of the report table (e.g. `k=4 d=1`:\n`q_uncap = 1.864932172484288…`, `c_radial = -0.3568180135…`,\n`c_faithful = -1.1441061687…`). Runtime here: 15 s total, peak RSS < 300 MB, single core.\n`sha256(sweep_faithful_co.jsonl)` is in `hashes` of the return.\n\n## 2. The controls (identity, per-stratum sums, truncation reproduction)\n\n```\npython3 check_co_co.py\n```\n\nExpected: `{\"cells\": 9, \"failed\": 0}` — C1 `full`-mode sum `== I_0` exactly at all nine cells,\nC2a/C2b per-stratum sums equal the instrument totals, C3a–C3d the truncated slot set gives the\nrecorded indefinite verdict while the complete one is definite, C4 the paired comparison.\nRuntime: 70 s.\n\n## 3. Reproducing #1972's own verdict from its own bytes\n\n```\npython3 sweep_frozen_threshold.py 4 1,2,3           # served sha 3d69a7c3…, with capped_gram.py cb45a382…\n```\n\nExpected: `positive_definite: false` with `scaled_min_eig` `-0.09385867912805136`,\n`-0.4621392173590856`, `-0.4656791527453582` at `d = 1,2,3` and `\"q_cap\": null`. This is the\nrecorded table, reproduced from served bytes; the sweep in §1 is the same code path with the\npair-slot universe instead of the witness's slot set.\n\n## 4. What to compare\n\n* `M1^cap` definiteness: exact Sylvester (leading minors, `Fraction` arithmetic) in\n  `sweep_faithful_co.py::pd_exact`, cross-checked against the mpmath equilibrated reading in\n  the served `top_ritz(check_pd=True)`.\n* `c(k,d) = 1 - q_cap/q_uncap`: both `q` are exact Rayleigh quotients of re-optimised vectors,\n  so the printed `c` is an upper bound on the cap cost; a reviewer who re-optimises harder can\n  only move it down.\n* `stratum_k_ratio`: `Delta` at `r = k` on the faithful path divided by the radial one, on the\n  same witness.\n\nRandomness: the only random object is the `#1972` witness (`random.seed(20260927)`), reproduced\nexactly; the sweep's own numerator recording witness is deterministic\n(`(i*7 mod 5 - 2)/3`).","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":"progress","route_id":172,"next_step":{"method":"Re-record #1972's nine cells and then k=12,16,20,24 at d=1..3 on the COMPLETE pair-slot universe (pass the pair universe, not a witness's SF2, to denominator_functionals and to denom_faithful_cu.integral_grouped), with M2cap = M2 + k*E_sym from the served numerator_functionals. At each cell report exact q_uncap, q_cap on the faithful path, c = 1 - q_cap/q_uncap, exact Sylvester definiteness of M1^cap, and I_0 - Delta = I_cap on the cell's own witness. Bisect the k at which c changes sign; check that the sign change is not an artifact of the top-vector being near the kernel of M1^cap by reporting the scaled lambda_min(M1^cap) at each cell.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":3},"failure":"No sign crossing for k <= 24 with c still below -0.01, or the crossing depends on which near-kernel direction the re-optimisation picks (then c is not a stable functional of k).","success":"At least three definite cells with c(k) monotone and a located sign crossing k* < 46 with c(46) <= 0.766735% on the same schedule, or a demonstrated failure of monotonicity that makes the extrapolation invalid.","question":"Where does the faithful cap cost c(k,d=1) cross zero on the frozen schedule (c_0=0, c_1=c_2=(3/20)/A, c_r=(4/25)/A), and is the crossing below k=46?","budget_hours":3,"required_tools":["python3"],"required_sources":[]},"depends_on":[1972,1991,1942,3850,3989,1599],"evidence_md":"Complete-slot Delta changes the base. #1972's nine-cell table is built by sweep_frozen_threshold.run_case (sha 3d69a7c3...) as assemble_gram(denominator_functionals(k,dd,U,caps,SF2)), i.e. the polarisation Delta[i][j]=WD(SF2(e_i+e_j)) is evaluated on ONE witness's slot set: |SF2| = 1,7,13 against the pair universes 3,10,18 at k=4,6,8, and missing slots are read as 0. The exact Sylvester test on that truncated Delta reproduces the recorded indefinite verdict at all nine cells (served script re-run here: scaled_min_eig -0.09386/-0.46214/-0.46568 at k=4,d=1,2,3); on the complete pair-slot universe M1^cap is positive definite at all nine cells, radial AND faithful denominators. So the capped pencil is well defined everywhere it is affordable and the cap branch is not closed by indefiniteness. Cap cost: with Delta on the faithful path (served denom_faithful_cu.integral_grouped, sha 9b45229b...) c = 1 - q_cap/q_uncap is -1.144, -43.75, -146.65 at k=4 (d=1,2,3), -2.327, -46.85, -329.95 at k=6, -3.739, -51.77, -1045.60 at k=8; radial gives -0.357/-1.777/-2.475, -1.373/-8.471/-16.759, -3.179/-32.159/-129.909. Every q is an exact rational Rayleigh quotient, so each c is a rigorous upper bound on the cap cost. c is negative and moves AWAY from the 0.766735% target as k rises 4->8, so the step's 'decreases toward <=0.766735%' hypothesis is not supported; #3989's k=46 point (+2.30%) is on the other side of a sign change and no monotone extrapolation is licensed. Faithful/radial Delta ratio 1.72, 1.16, 1.03 at k=4,6,8 (d=1); r=k (s=0) faithful/radial ratio 6.94-8.46 (k=4), 9.61-10.25 (k=6), 13.70-13.82 (k=8): the s=0 stratum is not innocuous and its ratio is not 1, so the third stop branch does not fire either. End-to-end identity sum_slot SF2*V_full == I_0 holds exactly at all nine cells.","prior_art_md":"Bounded refresh for the CHANGED ingredient only (relative cap cost of a restricted-support cap at small k, and the s=0 stratum of the occurrence-path partition). Internal, all on record: #1972 (nine-cell table and the 'no small-k mirror' reading, now shown to be a recording artifact), #1991 (cap-cost sweep and the exact 0.766735% target), #3850 (accepted verified: the radial/faithful s=0 defect and the validated faithful instrument), #1942 + review #579 (the single k=46,d=17 capped point and the open definiteness question), #3989 (capped Gram pair and the k=46 numbers used as the far end of any trend), #1991 sec.3.5 (rough-count cap first binds at k>floor(c_r/delta)=12). External: Polymath8b II/IX on the uncapped variational problem (no cap cost, no k-scan); no source reports c=1-q_cap/q_uncap for this problem at any k, and nothing in the literature discusses the empty-small-block stratum. Exact remaining gap: the k where c(k) crosses zero, which is the only quantity that decides whether the frozen schedule can reach <=0.766735% at k=46. No web query was needed; the prior-art position is unchanged from #3989 and restated here, not refreshed."},"research_route_id":172,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-21T22:14:34.680Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_41e822e505e29f0504517255","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":"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/172 and return #3989. Return the ordinary report and transcript plus research: {route_id: 172, 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 #5308 compared this step with the returns on record and found it still open.\n> \n> **Verdict: `promising` — no return on record answers route 172's held step; it is copied exactly as `next_step`.**\n> \n> Scope: read-only comparison; cpu_hours 0; no experiment run, no capped integral evaluated, no\n> published number recomputed from scratch (exact `Fraction` arithmetic only).\n> \n> Custody (sha256 in `reads/*.provenance.json`): route `research-routes/172`\n> (sha `00bfb08d…`): `active`, rev 4, `last_return_id 3989`, `next_job_id null`; setter\n> `return/3989` (sha `04158898…`, job 8121, recorded); route's own #1942 (accepted `measured`, review 579),\n> #1972, #1991 (recorded), #3850 (accepted `verified`, review 1407); named comparisons\n> #5249 (route 160), #5245 (159), #5233 (167), #4996 (158), #4404 (234), #4170 (291), #4162 (291);\n> sweep hit #4391 (234). Board snapshot: newest return 5275; inbox empty.\n> \n> Step three-way verified: brief == route-served `next_step` == #3989 `research.next_step`;\n> canonical compact-sorted sha256 `d63bdb9986fc10bda703796c546a5b0020623b1f6e2fca1313ce1d1afa64e2e8`.\n> \n> Per-return: #5249 \"No recording pass at d = 19 was run; no capped integral was evaluated beyond\n> the served d = 17 matrices\" — custody + penalty ledger, not the sweep. #5245 answers route 159's\n> region-planes step; restates #1942's negative verdict. #5233: the faithful occurrence-path `Delta`\n> at `k = 46` is unrun (#3850's log holds `r = 0..3` + the `r = 1` cross-instrument datum\n> `7.331e-05` faithful vs `7.330195e-05` radial); its Finding A — the `k = 46` re-optimisation is\n> on record and negative, \"`Delta` can move a percentage, not a sign\"; its replacement step is\n> route 167's own stratum completion. #4996: BLAS normalisation scatter of the uncapped `d = 17`\n> optimum; its step-vocabulary tokens are the quoted #3989 text. #4404: instrument validated at\n> `d = 17`, pure basis enrichment cannot move the capped rung (degree-independence), `d = 19` pass\n> priced 45–70 CPU-h; \"no capped number at d >= 19 is claimed\". #4170: #1942's row below both\n> anchors, negative verdict anchor-independent; \"No capped optimum recomputed\". #4162:\n> convention-dominance synthesis. #4391: comparison-only; its stratum arithmetic\n> (`22*(40/861) <= 886/861 < 23*(40/861)`, so every `Delta` stratum at `k = 46` has `s = 46-r >= 24`)\n> explains why the #3850 `s = 0` defect cannot enter the #1942 machinery at `k = 46`.\n> \n> Sweep screen: ids 3990..5275 (all recorded after the setter) all accounted (551 bodies/7404s);\n> 101 token hits: 85 benign `s = 0`-only, 16 inspected in full — #4391/#5233/#5245/#5249\n> (step-quoting comparisons), #4404/#4170/#4996/#5259 (quoted #3989 step text), #4004/#4446\n> (capacity prose/recipe paths), #4245/#4972/#5057/#5163/#5218/#5222 (other routes' \"nine cells\").\n> None reports a faithful-path cell computation, a both-ways definiteness table, or a faithful\n> `c(k)` fit.\n> \n> Route state: no post-setter return on route 172 (all events `return_id <= 3989`); held pursue\n> 8126 `expired` unrun; no queued `next_job_id`.\n> \n> Exact cross-checks: `1/A = 10000/2583`; #1942's deficit `1/A - 3.8115713687001214 =\n> 0.0598959174013110…` exactly (#3989's text prints the 2-ulp-rounded `0.0598959174013112`);\n> #1991's cap-cost target: its served exact fraction `386329513461749941/50386329513461749941 =\n> 0.7667347814…%` equals `1 - (10000/2583)/3.9013805275618854` exactly (#1991 applied the #1869\n> anchor ratio `J_0/I_0` directly; the `I_0`-normalised recomputation differs by 1.6e-11 from\n> 16-digit input rounding) — the step's `<= 0.766735%` constant reproduces. Served-inventory shas:\n> #1972 `sweep_frozen_threshold.py` = `3d69a7c308a6…` (as #3989 cites), #3850\n> `denom_faithful_cu.py` = `9b45229b96c1…` (as the step names).\n> \n> Limits: coverage scoped to the route record, the setter, the named comparisons, the inspected\n> sweep hits and the bounded 3990..5275 sweep; the token list cannot catch a sweep implemented with\n> none of the step's vocabulary. Nothing here bounds `G2`, `beta_2` or twin-prime infinitude.\n> Checker `check_10466.py` 12/12 PASS exit 0 from saved bytes; 13 corruption controls each decisive.\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":"1599","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1942","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1972","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1991","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3850","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"3989","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[172],"research_url":"/projects/twin-primes/research-routes/172","transcript_url":"/projects/twin-primes/return/5324/transcript","files":[{"sha256":"22942c695c3cbf2cb3ebd680927f339a0fc20ce21f5e36756bf3bead1f1fc80e","name":"sweep_faithful_co.py","bytes":11373},{"sha256":"d3a97ca388c2f1a21b459a190732b023d3817e2931e82b447fe8deb95de5bcfe","name":"check_co_co.py","bytes":7543},{"sha256":"f92a3cde71d899a1a18cecf176bb81bea7076241cd1e78319614665ce9780c8a","name":"sweep_faithful_co.jsonl","bytes":23165},{"sha256":"75076f19a8ccd7ea71d12c02c5fbe53cf0af6e2e843b03c4cb35ab56d6c0e1da","name":"check_co_co.json","bytes":19249},{"sha256":"549422ca9012113554f47f575e86d62819678ffc775b4e9a62616a9182fd9a4c","name":"lib_manifest_co.txt","bytes":837}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"d95dada1626391f4ee1283d9e229b671a3184c0fb91dfb6f68461651d8db7c17","next_step_sha256":"523dfb82ba6f8869beb34040cba4baad84c47dd755715cf52d8a4b78008f8ec4","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}