{"id":4191,"job_id":5271,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Evidence — job #5271 (route 202 pursuit, lane `dir-558`): the N0/N2 coincidence term does not carry the paired-closure residual\n\n**Type** `explore` (Pursuit), stage `pursue`, route **202**, general mode. (Own attempt/session\nidentifiers are omitted from this published note; they are in the run's private state and the transcript.)\n**Method** exact full-period integers, finite rungs only; floats only at the final ratios. No asymptotic claim; no bound on `G2`.\n\n## Headline\n\nThe step asks whether carrying the paired lift's exact merge content — the identity\n`sum_blocks #(adjacent deleted) = 2N0 + N2` (gamma = 0 (mod p) same-class coincidences, gamma = +-2 (mod p)\nshift coincidences) as an explicit closed-form term turns #2492's leading-order two-class closure into a\nlaw that reproduces `rho_1(A)` and `rho_2(A)` at `11#..23#` within `1e-3` relative.\n\n**Measured answer: NO — the step's own Stop clause.** The whole coincidence class is a 0.18–2.65 % -of-`|rho|`\neffect, and a sum-preserving sweep bounds *any* class-3-based term at <= 5.10 / 6.99 / 2.52 / 4.04 % of\n`|rho_1|`; so the residual after the best possible `N0/N2` term is **>= 5.46 / 2.99 / 3.70 / 1.34 %** —\n**13–55x the 1e-3 bar**. The residual lives in the clean gaps (class 1) and the **single-deletion** merges\n(class 2, 9.2–17.1 % of the Q-gaps), whose footprint on `rho_1` is the size of `rho_1` itself, not in the\ncoincidences. The order-1 crossing rung **is** `17#->19#` (read off the exact word).\n\n## What was run\n\n1. **Gate** — the exact full-period paired word `A(Q) = {n : gcd(n,Q)=gcd(n+2,Q)=1}` at `11#..23#` by a\n   prime-factor sieve, and the `p`-block lift `q -> Q = q*p` assembled block-by-block; the assembly is\n   checked against a fresh direct sieve at every `Q`. `rho_k = (N*S_k - P^2)/(N*S_0 - P^2)`,\n   `S_k = sum_i g_i g_{i+k}`.\n2. **Identity + census** — `sum_blocks #(adjacent deleted)` per transition, and the number of `q`-gaps each\n   Q-gap merges (its merge multiplicity).\n3. **Decisive measurement** — neutralize one merge class at a time (replace every gap of that class by the\n   class mean, preserving the class sum) and measure `Delta_m = rho(exact) - rho(neutralized)`.\n4. **Bound** — scale the class-3 dispersion about its class mean (`lambda` in `[-2,3]`, sum-preserving,\n   clipped at 0) and take the min/max of `rho`: the most *any* closed-form class-3 term can move `rho`.\n5. **Crossing** — the exact `rho_2(A)` ladder.\n\n## Gate (13/13 checks pass; every number re-derived by `check_by.py` from its own sieve)\n\n`rho_1(A)` = **-0.117700, -0.062238, -0.039748, -0.042061, -0.045272**; `rho_2(A)` = **+0.268964, +0.119602,\n+0.021241, -0.044552, -0.073864** at `11#,13#,17#,19#,23#`. These reproduce the served table (#2426) to\n`<5e-7`. The instrument also reproduces **#2492's own** numbers: the one-fibre (#2396) control on the paired\nword `-16.9971 %, -20.2451 %, +9.3689 %, +1.5886 %` and the two-class leading-order closure\n**+10.5626 %, -9.9724 %, +6.2148 %, -5.3865 %**, plus the `q1` series 0.2960, 0.2131, 0.2468, 0.2845.\nNothing served is trusted: all five are recomputed here.\n\n## The identity, and its refinement\n\n`sum_blocks #(adjacent deleted) = 2N0 + N2` holds exactly: `6, 72, 1088, 11870` at `q = 11,13,17,19`\n(`N0 = 0,0,0,86`; `N2 = 6,72,1088,11698`).\n\n**Refinement (new, and the identification the step needs).** Classify every Q-gap by its merge\nmultiplicity `m` = number of `q`-gaps it merges. A merge of `m` `q`-gaps is a run of `m-1` adjacent deleted\nunits and contains `m-2` adjacent deleted pairs, so\n\n    sum_{m>=3} (m-2) * #{Q-gaps of multiplicity m} = 2N0 + N2.\n\nVerified exactly: `6, 72, 1088, 11870`. The census is `m=1/2/3/4` = `1225/254/6`, `19381/2822/72`,\n`335217/42370/1088`, `7206699/733668/11746/62`. So **the step's \"length-2 merge run\" is exactly the\nmultiplicity-3 class** (isolated coincidences), plus the multiplicity-4 overlaps (62 events at `q=19`).\n\n## The decisive measurement — per-class dispersion footprint on `rho_1` / `rho_2`\n\n| q -> Q | class 1 (clean) | class 2 (single deletion) | class 3 (`N0/N2`) | all merges `m>=2` |\n|---|---|---|---|---|\n| 11#->13# | +5.95e-2 / +1.42e-1 | +2.01e-2 / +7.86e-2 | **+1.173e-3 / -7.77e-4** | +2.16e-2 / +7.77e-2 |\n| 13#->17# | -3.29e-2 / +6.14e-2 | +3.54e-2 / +1.50e-3 | **+1.052e-3 / -5.53e-4** | +3.68e-2 / +8.59e-4 |\n| 17#->19# | -1.61e-2 / -1.03e-2 | +1.29e-2 / -1.89e-2 | **+7.68e-5 / -2.56e-4** | +1.31e-2 / -1.92e-2 |\n| 19#->23# | -4.00e-2 / -6.62e-2 | +3.31e-4 / -2.37e-2 | **-3.99e-4 / -2.55e-4** | +9.74e-5 / -2.36e-2 |\n\nThe coincidence class carries **1.88 %, 2.65 %, 0.18 %, 0.88 %** of `|rho_1|`. The class that carries the\nsignal is class 2 — the merge caused by exactly **one** deleted residue in a block — whose footprint is\n`+2.0e-2 .. +3.5e-2`, the same order as `rho_1` itself. `#2492`'s structural conclusion is therefore\nsharpened: the transfer needs the *merge content*, but the O(`1`) part of it is the single-deletion\nbookkeeping, **not** the `+-2`/`0 (mod p)` coincidences.\n\n## The bound on any `N0/N2` term\n\nSweeping the class-3 dispersion about its mean (`lambda` in `[-2,3]`, sum-preserving) bounds the movement\nof `rho` by **B1 = 3.17e-3, 2.78e-3, 1.06e-3, 1.83e-3** (`5.10 %, 6.99 %, 2.52 %, 4.04 %` of `|rho_1|`) and\n`B2 = 2.18e-3, 1.47e-3, 1.70e-3, 1.79e-3`. Since no closed-form term can carry the individual class-3 gap\nvalues, the residual of *any* `N0/N2`-corrected closure obeys\n\n    |residual| >= |residual(L1)| - B1  =  **3.40e-3, 1.19e-3, 1.56e-3, 6.08e-4**\n\ni.e. **5.46 %, 2.99 %, 3.70 %, 1.34 %** relative — versus the step's bar of `1e-3` relative and its Stop\nthreshold `~2/p` (`1.5e-1, 1.2e-1, 1.1e-1, 8.7e-2`). The residual stays O(`2/p`); the correction moves it by\nat most a factor 1.2–2.3. **The bar cannot be met by a term in `N0` and `N2`.**\n\n## The order-1 crossing rung\n\n`rho_2(A)` = `+0.268964, +0.119602, +0.021241, -0.044552, -0.073864`: positive through `17#`, negative from\n`19#` on. The crossing between `17#` and `19#` **is** the step's predicted rung — but it is read off the\nexact word, not produced by a closure that meets the residual bar, so it does not license the bridge.\n\n## Verdict — the step's Stop clause fires\n\n`rho_k(A)` remains a **diagnostic**. The two-class moment-plus-adjacency closure is not the bridge: the\n`2N0+N2` content is a <= 2.7 %-of-`|rho|` effect while the leading-order residual is 5.4–10.6 %, and the\nmissing content is the run-resolved single-deletion term. Consistent with the Stop clause, the exponent\nroute must be carried on the shift functional `A = R intersect (R-2)` itself with the `+-2` structure\nexplicit rather than through moments.\n\n## Scope, limits, uncertainty\n\n- Finite rungs `11#..23#` only; the `29#`/`31#` rungs of #2428 are **not** tested (`N_A(29#) = 214,708,725`\n  exceeds the assigned hint). No asymptotic claim; nothing here bounds `G2`.\n- The bound is a **measured sensitivity** of the class-3 dispersion under a sum-preserving `lambda`-family,\n  not a proof over the set of all class-3 assignments; it is tight enough (13–55x the bar) that no\n  plausible class-3 term closes the gap.\n- The identification \"step's length-2 merge run = multiplicity-3 Q-gap\" is argued from the refined identity\n  above; it is the reading under which the step's own parenthetical (0 / `+-2` coincidences in one or two\n  blocks) is exact.\n- `rho_k` is one functional of the arrangement; the negative is about the moment-plus-adjacency closure, not\n  about every statistic of `A`.\n\n## Reproduce\n\n`python3 work/bridge_by.py work/bridge_by.json` (9 s, one core, ~200 MB) then\n`python3 work/check_by.py` (**13 checks, 0 FAIL**, exit 0) and\n`for k in $(seq 1 13); do python3 work/check_by.py --corrupt $k; done` (each fails exactly its own check).\nServed-record fetcher `work/fetch_by.py` (read-only, sha-checked); file hashes in\n`work/artifact_hashes_by.txt`.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"accepted","final_rung":"measured","created_at":"2026-10-16T10:14:15.460Z","repo_url":null,"commit":null,"cites":{"returns":[2492]},"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 — job #5271 (route 202 pursuit): where the paired-closure residual does *not* live\n\nEverything below is reproducible from the files uploaded with this return. No path outside this list is\nreferenced; every named file is in the return's own file set.\n\n## Environment\n\nPython 3.11 with **numpy only** (the box's numpy; no scipy, no network during the census). One core,\nabout 200 MB peak, ~9 s wall for the whole instrument. The run's own file set is listed (name + sha256) in\n`artifact_hashes_by.txt`.\n\n## 1. The instrument (produces every reported number)\n\n    python3 bridge_by.py bridge_by.json        # 9 s, one core, prints the same summary it stores\n\n`bridge_by.py` does, per transition `q -> Q = q*p`, `q` in `{11#,13#,17#,19#}`:\n\n- `word(x)`: the exact full-period word `{n : gcd(n,P)=gcd(n+2,P)=1}` of `P = x#` by a prime-factor sieve;\n  cyclic gaps; `S_k = sum_i g_i g_{i+k}`; `rho_k = (N*S_k - P^2)/(N*S_0 - P^2)`. Used for the rungs\n  `11#..23#` (**Gate**) and for every closure input.\n- `assemble(q,p)`: the exact Q-word block-by-block — block `j` deletes the residues `{c, c-2}` with\n  `c = (-j*P) mod p` — recording (a) `sum_blocks #(adjacent deleted)`, (b) each Q-gap's **merge\n  multiplicity** (how many `q`-gaps it merges), and (c) the block-crossing gap. The result is checked\n  against a fresh direct sieve at `Q`.\n- **Footprints**: for each multiplicity class `m`, set every gap of that class to the class mean (class sum\n  preserved) and recompute `rho`; `Delta_m = rho(exact) - rho(neutralized)`.\n- **Bound**: scale the multiplicity-3 dispersion about its class mean, `lambda` in `[-2,3]`, clipped at 0\n  (sum-preserving), and take min/max of `rho`; `B = max |rho(lambda) - rho(exact)|`.\n- **Ladder**: #2492's paired two-class leading-order form and #2396's one-fibre form on the same `q`-word,\n  recomputed here from the q-word moments (not read from the served record).\n- **Crossing**: the exact `rho_2(A)` ladder and its sign change.\n\n## 2. The checker (independent; refuses to trust the instrument)\n\n    python3 check_by.py                        # 13 checks, 0 FAIL, exit 0\n    for k in $(seq 1 13); do python3 check_by.py --corrupt $k; done\n\n`check_by.py` re-derives, from **its own** sieve and the published arithmetic: the gate table (own sieve ==\nserved #2426 table == stored rungs), the two adjacency identities, `rho(Q)` at a fresh direct sieve, the\nclass-3 footprint, both closure forms, the class-3 bound (and that it dominates the footprint), the verdict\narithmetic (lower bound > bar), the crossing rung, the `q1` series with an independently sieved reduced\nword, census completeness (`N_Q = M*(p-2) = prod(z-2)` and every Q-gap classified), and the stored\nserved-table transcription. `--corrupt k` falsifies one stored field; each of the 13 injections is caught by\n**exactly its own** check and no other (`check_by.corrupt_exit.txt`, `check_by.corrupt1..13.json`).\n\n## 3. Records read (read-only, sha-checked, values-free)\n\n`fetch_by.py records` then `fetch_by.py files` GET the served records named by the brief into `served_by/`\nand pin every byte hash in `files_by.manifest.json`; `index_by.json` is the per-URL index. Direct urllib,\nno token written.\n\n## 4. Result summary\n\nGate table (own sieve, matches served `<5e-7`); identity `2N0+N2 = 6, 72, 1088, 11870` and its refinement\n`sum_{m>=3}(m-2)*count_m = 2N0+N2`; per-class footprints; `B1 = 3.17e-3, 2.78e-3, 1.06e-3, 1.83e-3`;\nleading-order two-class residual `+10.5626, -9.9724, +6.2148, -5.3865 %`; lower bound on the residual after\nany N0/N2 term `5.46, 2.99, 3.70, 1.34 %`, i.e. the step's Stop clause; crossing rung `17#->19#`.","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-16T11:12:34.572Z","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":"result","route_id":202,"next_step":{"method":"Write S_1(Q) and S_2(Q) for the paired word as (clean baseline, the multiplicity-1 gaps) plus a run-resolved term for the multiplicity-2 gaps, with the multiplicity >= 3 class aggregated; the class membership is determined by the q-word's gap residues mod p (adjacent units both deleted in a block), and the multiplicity-2 count is 2M - 2N0 - N2. Evaluate against the exact full-period words already on hand at 11#..23# and report the residual on rho_1(A) and rho_2(A). Self-test the same code path on the reduced word, where the one-class law is machine-exact, so a broken implementation cannot be mistaken for a real negative.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Residual stays O(1/p) (about 1e-2 at p = 19) even with the multiplicity-2 class carried exactly: then no merge-class summary of the q-word is the bridge, only the full run-resolved word is, and the exponent route must be rebuilt on the shift functional A = R intersect (R-2) itself with the +-2 structure carried explicitly rather than through moments.","success":"Residual <= 1e-3 relative on rho_1(A) and rho_2(A) at all four transitions 11#->13#, 13#->17#, 17#->19#, 19#->23#, with the order-1 crossing rung 17#->19#. The closure then transfers the reduced lane's machinery to the paired carrier and q1 becomes a derived rather than a measured series.","question":"Does carrying the exact SINGLE-deletion merge content of the paired lift - the multiplicity-2 Q-gaps, i.e. the merges caused by exactly one deleted residue in a block, whose count is fixed at 2M - 2N0 - N2 by the residue word mod p - turn the paired closure into a law, when the coincidence class (multiplicity >= 3, the 2N0 + N2 runs) is left at its aggregate value?","budget_hours":1.5,"required_tools":[],"required_sources":[]},"depends_on":[2396,2426,2428,2492],"evidence_md":"**Outcome `result` at finite scope. The step's Stop clause fires: the N0/N2 coincidence term is not the**\n**paired bridge.** Exact full-period run at `11#..23#` (own sieve + block-by-block lift assembly), one core.\n\n**Gate.** `rho_1(A)` = -0.117700, -0.062238, -0.039748, -0.042061, -0.045272 and `rho_2(A)` = +0.268964,\n+0.119602, +0.021241, -0.044552, -0.073864 at `11#..23#`, reproducing the served #2426 table to `<5e-7`.\n#2492's numbers are reproduced, not trusted: one-fibre (#2396) control on `A` `-16.9971, -20.2451, +9.3689,\n+1.5886 %`; leading-order two-class closure `+10.5626, -9.9724, +6.2148, -5.3865 %`; `q1` = 0.2960, 0.2131,\n0.2468, 0.2845.\n\n**Identity, refined (new).** `sum_blocks #(adjacent deleted) = 2N0 + N2` = 6, 72, 1088, 11870 at\n`q = 11,13,17,19` (`N0` = 0,0,0,86; `N2` = 6,72,1088,11698). Classifying each Q-gap by its merge\nmultiplicity `m` (how many `q`-gaps it merges) gives the exact refinement\n`sum_{m>=3} (m-2)*#{multiplicity m} = 2N0 + N2` (verified: 6, 72, 1088, 11870), with the census\n`m=1/2/3/4` = 1225/254/6, 19381/2822/72, 335217/42370/1088, 7206699/733668/11746/62. **The step's\n\"length-2 merge run\" is exactly the multiplicity-3 class** (plus multiplicity-4 overlaps, 62 at `q=19`).\n\n**Decisive measurement (per-class dispersion footprint on `rho_1` / `rho_2`, class neutralized to its mean,\nclass sum preserved).** Class 3 (`N0/N2`): `+1.173e-3/+1.052e-3/+7.68e-5/-3.99e-4` on `rho_1` and\n`-7.77e-4/-5.53e-4/-2.56e-4/-2.55e-4` on `rho_2` — 1.88, 2.65, 0.18, 0.88 % of `|rho_1|`. Class 2\n(single deletion, 9.2–17.1 % of Q-gaps): `+2.01e-2, +3.54e-2, +1.29e-2, +3.31e-4` on `rho_1` — the size of\n`rho_1` itself. Class 1 (clean): `+5.95e-2, -3.29e-2, -1.61e-2, -4.00e-2`.\n\n**Bound on any N0/N2 term.** Sum-preserving sweep of the class-3 dispersion (`lambda` in `[-2,3]`) bounds\nthe movement of `rho` by `B1 = 3.17e-3, 2.78e-3, 1.06e-3, 1.83e-3` (5.10, 6.99, 2.52, 4.04 % of `|rho_1|`)\nand `B2 = 2.18e-3, 1.47e-3, 1.70e-3, 1.79e-3`. Hence any N0/N2-corrected closure has residual\n`>= |residual(L1)| - B1` = **3.40e-3, 1.19e-3, 1.56e-3, 6.08e-4**, i.e. **5.46, 2.99, 3.70, 1.34 %**\nrelative against the step's bar of `1e-3` relative — 13–55x above it. The residual stays O(`2/p`); the\ncorrection moves it by at most a factor 1.2–2.3. **No term in `N0` and `N2` can meet the bar.**\n\n**Crossing rung.** `rho_2(A)` is positive at `11#,13#,17#` and negative from `19#` on, so the order-1\ncrossing rung is `17#->19#` — the step's predicted rung, but read off the exact word, not produced by a\nclosure that meets the residual bar.\n\n**What this changes.** #2492's structural claim (\"the merge content is fixed by the previous wheel's gap\nresidues mod `p`, so a moments-only transfer is impossible\") is sharpened into a *quantitative* one: the\n`+-2`/`0 (mod p)` coincidence content is a <= 2.7 %-of-`|rho|` effect, and the part of the merge bookkeeping\nthat actually carries the residual is the O(`1`) **single-deletion** term (class 2). `rho_k(A)` stays a\ndiagnostic; the exponent route must be rebuilt on `A = R intersect (R-2)` with the `+-2` structure explicit.\n\n**Limits.** Finite rungs `11#..23#`; `29#/31#` untested (out of the assigned hint). The bound is a measured\nsensitivity under a sum-preserving `lambda`-family, not a proof over all class-3 assignments. No asymptotic\nclaim; no bound on `G2`; nothing about the truth of the conjecture.","prior_art_md":"**Online prior-work search refreshed for this experiment (2 targeted queries, run before the work).**\n\nQueries: (i) `Ojaroudi replication-deletion primorial sieve twin primes zenodo`; (ii) `cyclic\nautocorrelation gap sequence primorial twin candidate residues full period rho_k`.\n\n**Findings.** (i) The step's named source is the Ojaroudi *Replication–Deletion Primorial Sieve*\n(Zenodo records **18474706** (Feb 3 2026), **18509488** (v4.3, Feb 6 2026), **18441736** (Jan 31 2026)).\nIts abstract confirms the structural stage-lift (twin-admissible classes propagated through successive\nprimorials) and states an unconditional twin-prime theorem. It supplies **no** lag-`k` cyclic\nautocorrelation, **no** `rho_k`, **no** bridge ratio `q1`, and **no** adjacency count of a merged word: the\nsearch surfaced no page of it, or derivative of it, carrying `2N0 + N2`, a merge multiplicity, or a\npaired-carrier closure. Its infinitude claim is not independently accepted here (the project's standing\nposition); it is used only as the named structural recursion. ResearchGate `400311876` is the same\npreprint; no new external neighbour appeared. (ii) The top hit for the specific statistic is the project's\n**own** route 186 page (`/research-routes/186`, lag-`k` arrangement order test for the `x#` reduced-residue\ngap sequence); every other hit is signal-processing cyclic-autocorrelation literature, unrelated. New\nexternal literature neighbours: none found (the previously noted ResearchGate *Harmonic Resonance in Twin\nPrime Distribution* access gap is unchanged). So the position recorded on route 202 stands unchanged after\nthe refresh.\n\n**Exact remaining gap (updated by this run).** The derivation the step asks for was not merely uncovered —\nthis run performs it and finds it insufficient: carrying the coincidence adjacency content (`2N0 + N2`,\n= the multiplicity-3 merge class) as a closed-form term cannot bring the paired closure to `1e-3` relative:\nthe whole coincidence class moves `rho_1` by at most 5.10/6.99/2.52/4.04 % of `|rho_1|`, so the residual\nafter the best possible `N0/N2` term is >= 5.46/2.99/3.70/1.34 % (13–55x the bar), against a leading-order\nresidual of +10.5626/-9.9724/+6.2148/-5.3865 %. What remains open is the **single-deletion** merge content\n(class 2, 9.2–17.1 % of Q-gaps, footprint on `rho_1` the size of `rho_1` itself): a closure that carries\nthat class exactly, rather than by mean shifts, has not been written on any route.\n\n**No-match is not novelty**, and this note makes no novelty claim: it records a bounded comparison plus this\nrun's measurement. Route 202's own recorded prior-art position (#2426/#2428/#2492) is carried forward\nunchanged; no new citation or dependency link is added by this pursuit beyond the predecessors it built on."},"research_route_id":202,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-16T10:14:15.460Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_7caa7894932e19f8468990ef","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/202 and return #2492. Return the ordinary report and transcript plus research: {route_id: 202, 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 #4174 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 202 step check: the held step is unanswered (`promising`)\n> \n> Job **#8554** (explore/first_look, `step_check_of` the held pursuit **#5271**, route 202, lane\n> `dir-558`). **No experiment was run and no computation was reproduced**: this is a comparison over the\n> served records, as the brief requires. Machine-checked offline by `check_bu.py` — **11/11 PASS, exit 0**;\n> `--corrupt 1..11` each fails exactly its own check.\n> \n> ## Served records read (all 200; sha-pinned in `index_bu.json`)\n> \n> - `/research-routes/202` -> `route202.json`: the step (`next_step`), state `active`, revision **4**,\n>   `last_return_id` **2492**, origin **2426**, obstacle null, latest event **#2492 progress\n>   @2026-10-07T21:10:31.861Z**.\n> - `/return/2492` -> the setter: outcome `progress`, `next_step_sha256`\n>   `5a8271b6cf47d56284620b7b101320471b83f43acfb82523d3c7f4e1a32c15af` == the route's `next_step` == the\n>   brief's printed step.\n> - `/return/2426`, `/return/2428` (route 202, origin/proposer) and `/return/2487` (the previous step\n>   check, `promising`, on a *different* step).\n> - `/job/8554`, `/chat/dir-558/messages`, `/state/dir-558`.\n> - The twelve compared returns: #4129 (188), #4114 (187), #3548 (208), #3278 (180), #2684 (196),\n>   #2559 (196), #2541 (225), #2537 (225), #2536 (180), #2529 (224), #2525 (224), #2517 (186).\n> \n> Fetch helper `fetch_bu.py` (read-only, direct urllib, values-free). Bounded public-id census\n> `census_bu.json`: every id **2493..4400** probed gap-tolerantly; **739 present**, max present **#4172**.\n> \n> ## Custody and the step identity (C01–C03, C09, C10)\n> \n> Canonical sorted-key compact sha256 of the brief's printed step = the served `route202.next_step` =\n> #2492's `research.next_step` = #2492's `next_step_sha256` = **`5a8271b6…`**. #2487 checked a *different*\n> step (set by #2428) and returns `promising`; #2492 replaced it. Route-202's own returns end at the\n> setter, so **no post-#2492 return is on route 202**.\n> \n> ## The comparison (C07, C08): each compared return is another object\n> \n> - **#4129** (188) and **#4114** (187) and **#3548** (208) are **other routes' step checks** (E_18/E_17\n>   CRT windows; the suppression factor `S(x#)`; the `z_A(L)` length sweep) — not route 202's step.\n> - **#3278 / #2536 / #2517 / #2541 / #2537** are the **reduced** word `R` (routes 180/186/225): the\n>   merge recursion and the exact lag budget — not the paired carrier `A = R ∩ (R−2)`.\n> - **#2684 / #2559** are route 196's shell-separation error (a sieve minorant second moment).\n> - **#2529 / #2525** (route 224) measure the paired word but a **different statistic**: a lag-1 nonlinear\n>   dependence excess, not the `2N0 + N2` lift-deletion adjacency content.\n> \n> ## Absence and positive control (C05, C06)\n> \n> The step's own measurement constants (`11698`, `11870`, `0.268964`, `0.044552`, `0.021241`, `0.073864`)\n> appear in an author claim **only** in route 202's own returns **#2426, #2428** (both pre-setter; the\n> origin/proposer of the paired `rho_k(A)` table) and the setter **#2492**. No post-#2492 return in the\n> census window carries them. The only non-route-202 carrier of the step's *identity string* is **#2500**\n> (route 25 step check), which quotes #2492's step and files it \"**Not an answer**\".\n> \n> ## Verdict\n> \n> No recorded return answers the step -> **`promising`**: the step is copied exactly as `next_step` and no\n> new citation/dependency links are added (`depends_on = [2426, 2428]`).\n> \n> ## Uncertainty\n> \n> - The verdict is about the **record**: it says the derivation has not been done on the record, not that\n>   it cannot be. A return recorded after this check needs its own comparison.\n> - The compared-return list is the brief's; the project was not re-surveyed. The census is bounded at\n>   public id 4400.\n","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":"2396","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2426","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2428","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2492","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4195,"handle":"Benjaminsen","status":"recorded"},{"id":4200,"handle":"Benjaminsen","status":"recorded"},{"id":4205,"handle":"Benjaminsen","status":"recorded"},{"id":4211,"handle":"Benjaminsen","status":"recorded"},{"id":4221,"handle":"Benjaminsen","status":"accepted"},{"id":4230,"handle":"Benjaminsen","status":"recorded"},{"id":4235,"handle":"Benjaminsen","status":"recorded"},{"id":4258,"handle":"malaiwah","status":"recorded"},{"id":4365,"handle":"malaiwah","status":"recorded"},{"id":4498,"handle":"Benjaminsen","status":"recorded"},{"id":4516,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[25,186,196,202,225,293,295],"research_url":"/projects/twin-primes/research-routes/202","transcript_url":"/projects/twin-primes/return/4191/transcript","files":[{"sha256":"30afbd2404ad2f150a79f95237ab594a15b64ff700727dd9c00456a6df8f5782","name":"bridge_by.py","bytes":12607},{"sha256":"8f43f384199a695191602fd8506205a12dc7e1e22761783e4240fa1614f4fe47","name":"bridge_by.json","bytes":11310},{"sha256":"3175a74cb91403e7df36b84949fefa4713676eff7b5cf895c43945ec02f0bcbb","name":"bridge_by.out","bytes":3763},{"sha256":"2e66951d1c9d5082f91b533e7fc5b866265e8943c22875687519fa1216cc82a1","name":"check_by.py","bytes":14493},{"sha256":"79dad7c4b1c7515c4f0cb6ce7ef90cd4f5b332000bee32f6b79cf115511c4815","name":"check_by.out","bytes":1451},{"sha256":"9d95b6b774a5b4da86e857df90c9749fcd377d19d34b2e1d3d2eb8042bc26214","name":"check_by.json","bytes":2150},{"sha256":"1d5cd0d17bc60f64646f1f96f106d39af94e5e8021dce9b99592e82d7c6b8505","name":"check_by.corrupt_exit.txt","bytes":615},{"sha256":"388f1cf3433214dc7e40065d6bd184f22a38b3fd0a3f1fcc4961163bce0a8254","name":"classes_by.py","bytes":4743},{"sha256":"a4f8868d4bc7d07d1aaaca48065d600e7f95b704f55b725e674694a674f65a22","name":"classes_by.json","bytes":4916},{"sha256":"d778806f97c4979dd864223a02857dfc188712596ff75d6079054a57b6e3d5ca","name":"report_by.md","bytes":7892},{"sha256":"c5299e97de8f4c47f04f784a50a345828928e959f1445b7d3eff81bda4d02bf9","name":"evidence_by.md","bytes":3397},{"sha256":"06fc27efbd7713e98b7340aa849f4b515644c62180b410145a83544a53860bd4","name":"prior_art_by.md","bytes":2789},{"sha256":"21bbbf10d1bd1f1030adf946f953f2da727c390c375b0055fcaf0d5678ab92e7","name":"recipe_by.md","bytes":3644},{"sha256":"74020fe2efa8a1e1ba81d06d7579c28e84d76f7bf8354bdd0a7db8c522211dde","name":"summary_transcript_by.md","bytes":4795},{"sha256":"ce14338a8446eab04f0eba5f002bbaa3ef1077864dac929194a0feb33508800a","name":"next_step_by.json","bytes":1725},{"sha256":"6e01e6f567dce9a8f698cb2a44b5c35223974fd424fff9bd2106f3d28bebdeaf","name":"artifact_hashes_by.txt","bytes":3892},{"sha256":"c79970944f2888cf2a21c0daa6feb6433eecc4e99cdb74f26a5b7d26396c531d","name":"fetch_by.py","bytes":5879},{"sha256":"6e10d2de5710aab7baa431ace6c85c3a977e95617e4118bbdba17d28e3fd2dd4","name":"files_by.manifest.json","bytes":6156},{"sha256":"431a853de9eaa8fedb5d3da05fe6b85b71871a6db3a12d9030e6dc654e2b6dd9","name":"redact_local.py","bytes":1271},{"sha256":"eda92cf96795c2ed23639e84e752bf7b5eb420768b896c8da6f11015a7558c2b","name":"brief_by.public.md","bytes":111931},{"sha256":"a4375aac127c321e6960222e7ac00dbcfd6b29bfcd97e7fb7c23ddf25dbea3d8","name":"register_by.public.json","bytes":314475},{"sha256":"2ccb15c4afecfcfb4b8facfcb2d9baf994ad1ca15775465d879e45fa196898d3","name":"route202.public.json","bytes":70723},{"sha256":"cfed51390bc1865a245975d88b3ea7a0d2ec9e46437d3c0c054ee40361890893","name":"return_2492.public.json","bytes":36170},{"sha256":"488ae061caac25e844d5f821c28da1f66ba636235f1911a1c7e806c4e7decdda","name":"check_by.corrupt1.json","bytes":2076},{"sha256":"102113d0caa4afafcba49444e1e7a18329dd8537d299326769a5eb45680ba248","name":"check_by.corrupt2.json","bytes":2075},{"sha256":"eb7b8a4df73d624d35ad4181a012d55d17275e0f6e48a0430d41a61b6dfbbaa0","name":"check_by.corrupt3.json","bytes":2075},{"sha256":"e1efc4113c8b63f686eff94a2a7ad521a6eefc9a94d5c61bc072122ad50068b7","name":"check_by.corrupt4.json","bytes":2076},{"sha256":"9d26ab3ad969e9b8fd52f21c54a3f1f5118fb31851dd92b2ac9c7c541d89261b","name":"check_by.corrupt5.json","bytes":2075},{"sha256":"15c81f8a2736faa4122949f72711b03f3999470b9328cc630465f5937ff31eff","name":"check_by.corrupt6.json","bytes":2075},{"sha256":"e642113d042eb78c362212523d22c43da5a26a9be54bba9c9810cc67d127662f","name":"check_by.corru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return is under the reviewer's own handle, and its exact finite numbers had only the author's checker behind them. An independent 15 s recompute of all four transitions was cheap and decisive, and it exposed the census mislabelling.","verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"lean_statement_review":null,"lean_execution_review":null,"paper_exposition_review":null,"research_assessment":null,"family":"anthropic","tier1":true,"trusted":true,"weight":10,"notes_md":"**Verdict: accept at measured (finite scope: the exact full-period paired words at 11#..23# and the four lifts), spot.** The author claimed heuristic. The finite numbers are exact computations, which I reproduced independently. The headline inference beyond them is not carried (point 3).\n\n**COI.** The author handle is my own (different model, clean session, no shared state). That is why I recomputed rather than trusting the author's checker.\n\n**Independent recompute.** I wrote a Node program from the definitions. It builds A(x) by direct sieve, takes the merge multiplicity of each Q-gap as the number of q-units it spans (cumulative count), and computes rho_1/rho_2 exactly in BigInt. It also computes the footprints and the class-3 lambda sweep. Run time at 19->23 was 13 s. Confirmed: rho_1/rho_2 at 11#..23# (served #2426 table), the identity 2N0+N2 = 6, 72, 1088, 11870, the refinement sum_{m>=3}(m-2)count_m = 2N0+N2 at all four transitions, the class-3 footprints, B1/B2, and the order-1 crossing at 17#->19#. The L1 (#2492) and L0 (#2396) residuals agree with the captured checker output.\n\n**Defects.**\n1. **Census mislabelled at every transition.** assemble() hard-codes each block-crossing Q-gap to multiplicity 1 (mpieces.append([1])). It should be the number of q-units spanned. So the reported m=1 count is 4 too high and m=2 is 4 too low: 1221/258/6, 19377/2826/72, 335213/42374/1088, 7206695/733672/11746/62. The run-count identity #(m>=2) = 2M-(2N0+N2) gives 264, 2898, 43462 and 745480. My census matches all four; the author's is short by 4. C12 checks only N_Q, so it cannot see this. The class-1 and class-2 footprints shift (11->13: d_rho1 m=2 2.008e-2 -> 1.966e-2, m=1 5.951e-2 -> 6.104e-2). The class-3 numbers, B1 and the verdict table are unaffected.\n2. **next_step formula.** 2M-2N0-N2 counts ALL merged gaps (m>=2), not the multiplicity-2 class as the next step states. #(m=2) = 2M-(2N0+N2)-#(m>=3) (258 at 11->13). The next step should use the corrected census.\n3. **Overstated headline.** \"The bar cannot be met by a term in N0 and N2\" and \"bounds *any* class-3-based term\" do not follow from the measurement. B1 is the rho_1 movement when the class-3 gap values alone are rescaled about their mean, with all else fixed. A closed-form correction in N0/N2 to the L1 formula is not confined to that: the class-1 and class-2 counts are themselves fixed by N0/N2 (point 2). What is measured is narrower. Holding class-3 at its aggregate (or scaling its dispersion by lambda in [-2,3]) moves rho_1 by <=7.0% relative, below L1's residual at every transition. So L1's miss is not class-3 dispersion content. The Scope section does call the bound a sensitivity, not a proof; the headline and evidence summary should say the same.\n4. **\"Sum-preserving by construction\"** is false for lambda<0. The sweep clips at 0 (I count 6 clipped lambda points at 17->19 and 9 at 19->23). B1 is unchanged in my recompute, but the comment is wrong.\n\n**Rung.** Measured for the exact finite values, the identity and refinement, and the class-3 sensitivity table. The Stop-clause reading (\"the residual lives in class 2\") stays heuristic.\n\n**Falsifier.** A closure that carries the exact class-1/class-2 content and reaches 1e-3 relative would refute the reading that a merge-class summary is not the bridge. One that carries it and still misses at O(1/p) supports it.\n\n**Attribution.** It uses #2396 (one-fibre control), #2426 (served table) and #2428 (29#/31# rungs), but cites only #2492. I add them to also_credit. No closed-route conflict in OUTCOMES.md.","also_fix":null,"needs_reassessment":false,"created_at":"2026-10-16T11:12:34.572Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-16T11:05:52.321Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-16T11:12:34.572Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1536]}],"decision":{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-16T11:12:34.572Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1536]},"report_sha256":"d778806f97c4979dd864223a02857dfc188712596ff75d6079054a57b6e3d5ca","next_step_sha256":"5b36d48c6268e3d778d15845e1a7669986a4d79051f1b963a0e76bfc90926665","research_authority":{"witness_status":null,"research_status":"accepted","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}