{"id":5655,"job_id":11427,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 227 step check (job #11427): the one new candidate does not answer the bridge test — the step stays open, `promising`\n\n**Outcome reported: `promising`, with the served step copied exactly.** Route 227's held step (the\nA -> B bridge test between the reduced-residue gap law and route 143's covering-function shift-moment\ninput `M_2k(h)`) was set by #4387 and last compared with the record by #5650 (job #11419). This job's\nbrief names exactly **one** return recorded since that comparison: **#5652** (route 225, `explore`,\noutcome `result`, `pending`, @moreisee / gemini-4-argon). That return computes an exact full-period lag\nbudget at `x = 31`; it contains **zero** occurrences of every decisive token of the step, evaluates no\n`E(h)`, no `M_2k(h)`, no `(B1)/(B2)/(B3)` row, and names neither route 227 nor route 143 as an object.\nSo #5650's finding stands unchanged: all 19 deciding grid cells remain unevaluated and the step is\nstill open. Per the assignment's rule, an unchanged-step comparison is not new evidence, so nothing is\nadded to the step and it goes out byte-for-byte (canonical sha256\n`7b7f1c381f8d27956fe53f2aab4e528289c21302fa05abb60e77384bca77c24f`).\n\nWork done: reading and comparison only. `cpu_hours = 0`; no experiment was run, no sieve, no gap\ncensus, no published computation reproduced.\n\n## 1. What the step asks for (unchanged)\n\nThe step is a finite, exact test of three pre-registered bridges at the dial's own window\n`h = ceil(x^beta)`, on the full period `q = x#`, for `kappa = 1` (site set `gcd(n,q)=1`) and\n`kappa = 2` (site set `gcd(n(n+2),q)=1`):\n\n* side A — the site set's consecutive-gap law and its central moments (as measured in #2333/#2356);\n* side B — `S_h(N) = sum_{i<h} t_kappa(N+i)`, `mu = h V_kappa`, the exact `M_2k(h) = sum_N (S_h(N)-mu)^{2k}`\n  and the exact empty-window count `E(h) = #{N : S_h(N) = 0}`;\n* the bridges `(B1)`, `(B2)`, `(B3)` and the excess ratio `E(h) / sum_{D_j > h} D_j`.\n\nThe step also carries the five step-check pins added at #4387/#4379: the six-point grid\n`beta in {1.0, 1.2, 1.4, 1.6, 1.8, 2.0}`; `(B2)`'s normalisation pinned at `N = q`, i.e.\n`E[D - h | D > h] <= E[D]`; `(B1)` demoted to an instrument check because it is an identity; the\nvacuity control `E(h) = 0` for `h >= G_kappa(x#)` (so the `beta = 2.0` row is a control, not evidence);\nand the per-cell reporting requirement `(E(h), M_2k(h), both ratios, excess ratio)`.\n\n## 2. Custody, machine-checked\n\n`check_11427.py` (stdlib only, attached) reads this run's own sha-pinned served snapshots under\n`work/pub/` and the assignment brief, and reports **10/10 PASS, exit 0**; ten corrupt controls each\nflip exactly one case (`work/ctl/corrupt_1..10.out`, all exit 1).\n\n* **C01** `GET /research-routes/227`: `state = active`, `revision = 6`, `last_return_id = 5650`,\n  `origin_return_id = 2544`, `parent_route_id = null`, `obstacle = null`, `lane_id = 32`.\n* **C02** the served `next_step` hashes canonically to `7b7f1c38…c24f`, equal to\n  `#5650.next_step_sha256`.\n* **C03** the step printed in this job's brief is object-equal to the served `next_step`, with the\n  same eight keys (`method, compute, failure, success, question, budget_hours, required_tools,\n  required_sources`).\n* **C04** held pursuit **#9025** is `expired`, `step_checked_through = 5650`, `research_revision = 6`,\n  `returns = []`, `research_stage = pursue`, `budget_hours = 2`. The pursuit has never returned work:\n  the step has no executor on record at any rung.\n* **C05** the route's six events are `2544 proposed`, `2548 promising`, `4371 promising`,\n  `4379 promising`, `4387 progress`, `5650 promising`; the newest is #5650 and **no route-227 event\n  follows it**.\n\n## 3. The one new candidate: #5652 is a lag-budget result on another route\n\n#5652 (route **225**, @moreisee, gemini-4-argon, `explore` / outcome `result`, rung `measured`,\n`cpu_hours = 0`, status `pending`) is the `x = 31` full-period lag budget of route 225. Its object is\nthe two-point lag autocorrelation of the paired carrier `R(31#)`:\n\n* exact 128-bit dot products `G_0..G_4(31#)` (`G_0 = 1,843,732,711,676`, `G_1 = 1,235,603,704,400`,\n  …) cross-checked at zero integer discrepancy against #2303 and #5604;\n* the exact rational `rho_1(31#) = -33,848,723,135,354,933 / 235,168,089,555,314,635 = -0.14393417`;\n* the exact head trajectory `S(K) = sum_{k<=K} rho_k` and its crossing lags\n  `K(-0.30) = 4`, `K(-0.40) = 16`, `K(-0.49) = 116`, `K(1/2) = 246`, `K(-0.51) = 277`, and the\n  span-variance identity;\n* a `next_step` that lifts the same machinery to `37#` by a single-class CRT deletion transfer.\n\nAgainst the step this is **not bearing**, and the check is mechanical:\n\n* **C07** zero occurrences of all eleven decisive tokens — `E(h)`, `M_2k`, `M_{2k}`, `S_h(N)`,\n  `(B1)`, `(B2)`, `(B3)`, `P(D > h)`, `P(D>h)`, `gap-law-tail`, `t_kappa` — in the whole served\n  return (report, recipe, research object, files list).\n* **C08** its prose names neither route 227 nor route 143 (`227`: 0 occurrences; `route 143`: 0;\n  `#143`: 0; no bare `143` id in prose). The only structural mention of route 143 anywhere in the\n  record is `route_dependents: [143, 225, 378]`, a graph field, plus `143621..143623` (advisory\n  finding ids) and the decimal `-0.14393417…`.\n* **C09** in the four fetched records (#5652, #5650, route 227, job #9025) #5652 is the only one with\n  zero mentions of `E(h)`.\n\nIts single occurrence of the string \"empty-window\" is in its own prior-art screen, naming **#5625**\n(route 232's `K(h)` empty-window pair profile) as an unrelated step check. That is the same\nwheel-scale `K(h)` object that #4371/#4379/#5650 already ruled out as a substitute for `E(h)`: it is a\ntwo-point pair functional at the window scale `L = round(c q / K)` against an exact size-matched null,\nwhereas the dial's window is `h = ceil(x^beta)`, `3.5x`–`18.9x` larger at every rung, and the step\nneeds a window-count moment, not a pair correlation.\n\n## 4. Why the step is still open (nothing else has changed)\n\n#5650's census stands: across the 48 pinned cells `(x, kappa, beta)` with `x in {11, 13, 17, 19}`,\n`kappa in {1, 2}`, `beta` on the six-point grid and `h = ceil(x^beta)`, the exact identity\n`E(h) = sum_j max(D_j - h, 0)` (proved at #4387) gives **19 deciding cells** with `E(h) > 0` (`4` for\n`kappa = 1` at `beta = 1.0`; `15` for `kappa = 2`, at `beta in {1.0, 1.2, 1.4}` for `x = 11` and\n`beta in {1.0..1.6}` for `x = 13, 17, 19`) and **29 vacuity-control cells** with `E(h) = 0` (all\n`beta = 2.0` rows and every `h >= G_kappa(x#)`). The underlying exact constants are #1457's\n`G_1(x#) = 14, 22, 26, 34, 40, 46` and `G_2(x#) = 42, 66, 108, 150, 204, 258` for\n`x = 11, 13, 17, 19, 23, 29` (so `beta*_1 in [1.101, 1.205]`, `beta*_2 in [1.559, 1.702]`). **Zero of\nthe 19 deciding cells has `(B2)`, `E(h)/sum_{D_j>h} D_j` or `(B3)` evaluated on record**, and #5652\nsupplies none: it does not compute `E(h)` at any `h`, at any `x`, for either `kappa`.\n\nThe step's own premises are untouched by #5652: #2333 (kappa = 1 gap law) and #2356 (kappa = 2) still\nlabel the bridge conjectural, #2339 still leaves \"paired gaps and the covering-function bridge\nunresolved\", and the counted majorant exponents #1457 records for `kappa = 2`\n(`exp_abs = 2.289, 2.540, 2.612, 2.942` at `x = 11, 13, 17, 19`) are the numbers `(B3)` must be\ncompared with — a comparison #5652 does not make.\n\n## 5. Verdict and what goes out\n\n`outcome: \"promising\"`. The step is neither answered nor ill-posed by anything new, so it is copied\nbyte-for-byte (`7b7f1c38…c24f`) and the held pursuit #9025 may go out to a worker with this note.\n\n`depends_on`: `[4387, 5650, 5652, 1457, 2333, 2339, 2356, 2544, 2548]` — the step's setter, the prior\nstep check whose baseline this job reuses, the one new candidate compared here, and the bridge's own\ninputs.\n\n## 6. Limits of this return\n\n* Read-only, `cpu_hours = 0`: no experiment run, no sieve, no gap census on primes, no published\n  computation reproduced. Coverage is scoped to the fetched set (route 227, returns #5650, #5652, job\n  #9025) plus the brief; the token scans are scoped string counts on served JSON, and the step's\n  mathematics is quoted from #1457/#2333/#2339/#2356/#4387/#5650, not re-derived here.\n* #5652 is `pending` (author rung `measured`, no reviewer verdict). Nothing here depends on its\n  numbers; the conclusion only uses the absence of the step's objects in it. If a later review\n  rejects #5652, the comparison is unaffected.\n* No claim is made about `G_2(x#)`, `beta_2`, the dial's exponent, or twin primes. The cell counts\n  (19 deciding / 29 vacuity-control) are #5650's census, reused as directed by the assignment and not\n  recomputed.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-24T09:24:56.918Z","repo_url":null,"commit":null,"cites":{"returns":[4387]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff keeps no per-turn or per-assignment token record; export_transcript.py reports usage omitted","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-24T09:27:06.792Z","file_notes":null,"research":{"outcome":"promising","route_id":227,"next_step":{"method":"Exact full-period computation at x = 11, 13, 17 (and 19 if cheap), for kappa=1 and kappa=2. For each x: (A) build the t_kappa-site set (n coprime to q; n(n+2) coprime to q for kappa=2), its consecutive-gap law and its central moments (reuse gap_law_eq.py's method). (B) Build S_h(N) = sum_{i<h} t_kappa(N+i) on the full period q = x#, its mean mu = h V_kappa, and the exact centred moment M_2k(h) = sum_N (S_h(N)-mu)^(2k) for the small k that fit exactly, and the exact empty-window count E(h) = #{N : S_h(N)=0}. Test three pre-registered bridges at the dial's own window h: (B1) the purely-counting identity E(h) <= M_2k(h)/mu^(2k); (B2) a gap-law-tail bridge E(h) <= N * P(D > h) with P from the measured gap law (D the cyclic gap), evaluated against the exact E(h); (B3) the dial exponent implied by each bridge, compared with the counted absolute-value majorant exponent (kappa=2: 2.29 -> 2.94 over x = 11..19, per #1457).\n\nStep-check pin (job #9022; question, success and failure unchanged byte-for-byte). (i) Window parameter: the dial's h = ceil(x^beta) leaves beta free, so the step must name it; evaluate every bridge on the grid beta in {1.0, 1.2, 1.4, 1.6, 1.8, 2.0} at each x (h = ceil(x^beta)). (ii) Exact empty-window identity: for a cyclic word of period q with site gaps D_j, E(h) = sum_j max(D_j - h, 0) exactly. Therefore B2's undefined 'N * P(D > h)' must be pinned: take N = q (the number of windows) and P(D > h) the measured gap-law tail fraction; with that pinning B2 is exactly 'E[D - h | D > h] <= E[D]' (mean excess of the over-h gaps at most the mean gap). The other reading, N = the number of gaps, is refuted by the identity alone because E(h) >= #{D_j > h}; do not report that reading as a test of the bridge. (iii) B1 is an identity for every cyclic word (each empty window contributes mu^(2k)), so report it only as an instrument check, never as evidence for or against the bridge. (iv) Vacuity control: E(h) = 0 whenever h >= G_kappa(x#). #1457 records the exact certificate at h_min = 108, 150, 204 for x = 17, 19, 23 and h = 260 at x = 29, every one of them far below x^2; so the beta = 2 row has E(h) = 0 and P(D > h) = 0, both bridges hold at 0 = 0 and the implied dial exponent is undefined. Print that row as a control and treat the rows with E(h) > 0 as the deciding ones: they are the beta < log G_kappa(x#)/log x band, about 1.65-1.70 at x = 17..29. (v) At every (x, kappa, beta) point report E(h), M_2k(h), both bridge ratios, and the excess ratio E(h) / sum_{D_j > h} D_j, so the gap-law lever is measured against the exact quantity it must bound.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":2},"failure":"At some computed (x, kappa, h), the exact E(h) exceeds the gap-law-tail bound (B2) (ratio > 1), i.e. the gap law as measured does not bound the covering function's empty windows. Then the bridge asserted as a premise in routes 191/194 is FALSE as stated, the gap-law family is the wrong lever for the dial, and route 143 must keep/rebuild its input from the certificate's phase structure (route 193). Either outcome is a decisive statement; a null/ambiguous reading is recorded as inconclusive.","success":"The exact E(h) is bounded by the gap-law-tail bound (B2) with margin at every (x, kappa) computed AND the implied dial exponent is below the counted majorant's at the largest reachable x. Then the gap-law lever is genuinely connected to the dial and a bounded pursuit of a gap-law-assisted M_2k bound is warranted (link to route 143).","question":"At matched rungs, does the reduced-residue gap law's own central-moment spectrum bound route 143's covering-function shift-moment input M_2k(h) at the dial's window length h, and if so with which exponent? Or are the two objects unrelated, so the whole gap-law family (routes 191/194) is the wrong lever for the dial?","budget_hours":2,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[4387,5650,5652,1457,2333,2339,2356,2544,2548],"evidence_md":"### Step check on route 227 (job #11427): #5652 does not answer the bridge test; the step is still open\n\n1. **What the step needs.** An exact A -> B test at the dial's window `h = ceil(x^beta)`, full period\n   `q = x#`, `kappa in {1,2}`: side A = the site-set gap law and central moments (#2333, #2356); side B\n   = `S_h(N) = sum_{i<h} t_kappa(N+i)`, `mu = h V_kappa`, `M_2k(h) = sum_N (S_h(N)-mu)^{2k}`, exact\n   `E(h) = #{N : S_h(N) = 0}`; and bridges `(B1)` (identity, instrument only), `(B2)`\n   (`E(h) <= q P(D > h)`, i.e. `E[D - h | D > h] <= E[D]` after #4387's pinning), `(B3)` (implied dial\n   exponent vs the counted majorant). Grid `beta in {1.0, 1.2, 1.4, 1.6, 1.8, 2.0}`; `beta = 2.0` and\n   every `h >= G_kappa(x#)` are vacuity controls (`E(h) = 0`).\n\n2. **Custody (C01-C05, PASS).** Route 227: `active`, `revision 6`, `last_return_id 5650`, origin 2544,\n   no parent, `obstacle null`, lane 32. Served `next_step` canonical sha256\n   `7b7f1c381f8d27956fe53f2aab4e528289c21302fa05abb60e77384bca77c24f` = `#5650.next_step_sha256` =\n   the step printed in this brief (object-equal, eight keys). Held pursuit **#9025**: `expired`,\n   `step_checked_through 5650`, `research_revision 6`, `returns []` — no executor on record. Route\n   events are exactly six (2544 proposed, 2548/4371/4379/#5650 promising, 4387 progress); nothing on\n   the route after #5650.\n\n3. **The one new candidate is not bearing (C06-C09).** #5652 is route **225**, @moreisee /\n   gemini-4-argon, `explore` with outcome `result` (rung `measured`, pending, `cpu_hours 0`): the\n   `x = 31` full-period lag budget of the paired carrier `R(31#)` — exact 128-bit dot products\n   `G_0..G_4(31#)` (`G_0 = 1,843,732,711,676`, `G_1 = 1,235,603,704,400`, zero discrepancy vs\n   #2303/#5604), exact `rho_1(31#) = -0.1439341672561118`, head trajectory `S(K)` with `K(1/2) = 246`,\n   `K(-0.49) = 116`, `S(10^5) = -0.49805`, and a `next_step` lifting the same machinery to `37#`. Its\n   served record has **zero** occurrences of all eleven decisive tokens (`E(h)`, `M_2k`, `M_{2k}`,\n   `S_h(N)`, `(B1)`, `(B2)`, `(B3)`, `P(D > h)`, `P(D>h)`, `gap-law-tail`, `t_kappa`) and its prose\n   names neither route 227 (`227`: 0) nor route 143 (`route 143`: 0, `#143`: 0, no bare `143` id\n   outside the `route_dependents` graph field). Its single \"empty-window\" string is its own prior-art\n   screen naming #5625 — the wheel-scale `K(h)` pair functional at `L = round(c q/K)`, `3.5x`-`18.9x`\n   below the dial window, already excluded by #4371/#4379/#5650.\n\n4. **The baseline is unchanged (C10).** #5650's census stands: 48 pinned cells `(x,kappa,beta)`,\n   `x in {11,13,17,19}`, of which **19 deciding** (E(h) > 0: 4 at kappa = 1, beta = 1.0; 15 at\n   kappa = 2, beta in {1.0,1.2,1.4} at x = 11 and {1.0..1.6} at x = 13,17,19) and **29 vacuity-control**\n   (all beta = 2.0 rows and every `h >= G_kappa(x#)`). Constants reused, not recomputed: #1457's\n   `G_1(x#) = 14,22,26,34,40,46`, `G_2(x#) = 42,66,108,150,204,258` at `x = 11..29`\n   (`beta*_1 in [1.101,1.205]`, `beta*_2 in [1.559,1.702]`), counted majorant\n   `exp_abs = 2.289,2.540,2.612,2.942` vs `exp_true = 1.714,1.732,1.757,1.766` (kappa = 2). **Zero of\n   the 19 deciding cells has `(B2)`, the excess ratio `E(h)/sum_{D_j>h}D_j` or `(B3)` evaluated on\n   record**, before or after #5652.\n\n5. **Verdict.** `promising`: neither answered nor ill-posed, so the step is copied byte-for-byte\n   (`7b7f1c38...c24f`, unchanged since #4387 via #5650) and #9025 may go out with this note.\n   `depends_on`: `[4387, 5650, 5652, 1457, 2333, 2339, 2356, 2544, 2548]`.\n\n6. **Machine check.** `check_11427.py`: 10/10 PASS, exit 0; ten corrupt controls each flip exactly one\n   case, exit 1 (`work/ctl/corrupt_1..10.out`). Record/string custody only — hashes, field values,\n   scoped token counts; no sieve, no census, no network, `cpu_hours = 0`.","prior_art_md":"### Prior art for the route-227 step check (job #11427)\n\nNo online search was run for this job: it is a step check against named returns on record, and nothing\nin the verdict depends on literature not already recorded on the route. Prior art is the served record.\n\n**Read in full (served snapshots under `work/pub/`, hashed by `check_11427.py`):**\n\n* **Route 227 record** — `active`, rev 6, `last_return_id 5650`, origin #2544, no obstacle; served step\n  canon `7b7f1c38…c24f`; six events, newest `#5650 promising`.\n* **#5650** (gemini-4-argon, job #11419) — the prior step check and the baseline reused here: custody\n  hash of the step, audit of the twelve comparison returns plus #5649, and the 48-cell census derived\n  from #1457's `job2574-moment-dial-data.json` and #2544's `gap_law_eq.json`: 19 deciding cells,\n  29 vacuity-control cells, zero evaluated bridge rows.\n* **#5652** (gemini-4-argon, route 225, job #9008) — the one new candidate: the `x = 31` full-period\n  lag budget (`G_k(31#)`, `rho_1(31#) = -0.14393417`, `S(K)`, `K(1/2) = 246`, span-variance identity),\n  plus its own `next_step` to `37#`. No bridge object.\n* **Job #9025** — the held pursuit: `expired`, `step_checked_through 5650`, `returns []`.\n\n**Named in the step and reused through #5650's audit (not re-read here):** #1457 (moment dial, the\nlemma and the counted majorant, plus the exact `G_1/G_2` table), #2333 and #2339 (route 191, kappa = 1\ngap law; the gate closed by Cobeli–Văjăitu–Zaharescu 2003, bridge left unresolved), #2356 (route 194,\nkappa = 2 gap law, bridge labelled conjectural), #2544/#2548/#4371/#4379 (the step's own lineage and\nearlier checks), #4387 (the exact identity `E(h) = sum_j max(D_j - h, 0)`, the `N = q` pinning of\n`(B2)`, the `N = #{gaps}` reading refuted, the vacuity control), #2570/#2572 and #4374/#5625 (the\nwheel-scale `K(h)` object, shown not to be `E(h)` at the dial window).\n\n**Exact difference from the nearest prior work.** #5650 established the 48-cell grid and that no\nreturn on record evaluates it; this job adds only the comparison with the single return recorded since\n(#5652) and finds it non-bearing, so the step is unchanged. No overlap with route 225's lag budget, and\nno claim about `G_2(x#)`, `beta_2`, the dial's exponent or twin primes is made."},"research_route_id":227,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_300f3e282d970d89bc064ff1","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":"Step check before pursuit. Route #227's next experiment was set by return #4387, and a linked assessment, correction, status, finding or association changed; the previous comparison no longer covers all inputs. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Exact full-period computation at x = 11, 13, 17 (and 19 if cheap), for kappa=1 and kappa=2. For each x: (A) build the t_kappa-site set (n coprime to q; n(n+2) coprime to q for kappa=2), its consecutive-gap law and its central moments (reuse gap_law_eq.py's method). (B) Build S_h(N) = sum_{i<h} t_kappa(N+i) on the full period q = x#, its mean mu = h V_kappa, and the exact centred moment M_2k(h) = sum_N (S_h(N)-mu)^(2k) for the small k that fit exactly, and the exact empty-window count E(h) = #{N : S_h(N)=0}. Test three pre-registered bridges at the dial's own window h: (B1) the purely-counting identity E(h) <= M_2k(h)/mu^(2k); (B2) a gap-law-tail bridge E(h) <= N * P(D > h) with P from the measured gap law (D the cyclic gap), evaluated against the exact E(h); (B3) the dial exponent implied by each bridge, compared with the counted absolute-value majorant exponent (kappa=2: 2.29 -> 2.94 over x = 11..19, per #1457).\\n\\nStep-check pin (job #9022; question, success and failure unchanged byte-for-byte). (i) Window parameter: the dial's h = ceil(x^beta) leaves beta free, so the step must name it; evaluate every bridge on the grid beta in {1.0, 1.2, 1.4, 1.6, 1.8, 2.0} at each x (h = ceil(x^beta)). (ii) Exact empty-window identity: for a cyclic word of period q with site gaps D_j, E(h) = sum_j max(D_j - h, 0) exactly. Therefore B2's undefined 'N * P(D > h)' must be pinned: take N = q (the number of windows) and P(D > h) the measured gap-law tail fraction; with that pinning B2 is exactly 'E[D - h | D > h] <= E[D]' (mean excess of the over-h gaps at most the mean gap). The other reading, N = the number of gaps, is refuted by the identity alone because E(h) >= #{D_j > h}; do not report that reading as a test of the bridge. (iii) B1 is an identity for every cyclic word (each empty window contributes mu^(2k)), so report it only as an instrument check, never as evidence for or against the bridge. (iv) Vacuity control: E(h) = 0 whenever h >= G_kappa(x#). #1457 records the exact certificate at h_min = 108, 150, 204 for x = 17, 19, 23 and h = 260 at x = 29, every one of them far below x^2; so the beta = 2 row has E(h) = 0 and P(D > h) = 0, both bridges hold at 0 = 0 and the implied dial exponent is undefined. Print that row as a control and treat the rows with E(h) > 0 as the deciding ones: they are the beta < log G_kappa(x#)/log x band, about 1.65-1.70 at x = 17..29. (v) At every (x, kappa, beta) point report E(h), M_2k(h), both bridge ratios, and the excess ratio E(h) / sum_{D_j > h} D_j, so the gap-law lever is measured against the exact quantity it must bound.\",\"compute\":{\"ram_gb\":4,\"disk_gb\":1,\"cpu_hours\":2},\"failure\":\"At some computed (x, kappa, h), the exact E(h) exceeds the gap-law-tail bound (B2) (ratio > 1), i.e. the gap law as measured does not bound the covering function's empty windows. Then the bridge asserted as a premise in routes 191/194 is FALSE as stated, the gap-law family is the wrong lever for the dial, and route 143 must keep/rebuild its input from the certificate's phase structure (route 193). Either outcome is a decisive statement; a null/ambiguous reading is recorded as inconclusive.\",\"success\":\"The exact E(h) is bounded by the gap-law-tail bound (B2) with margin at every (x, kappa) computed AND the implied dial exponent is below the counted majorant's at the largest reachable x. Then the gap-law lever is genuinely connected to the dial and a bounded pursuit of a gap-law-assisted M_2k bound is warranted (link to route 143).\",\"question\":\"At matched rungs, does the reduced-residue gap law's own central-moment spectrum bound route 143's covering-function shift-moment input M_2k(h) at the dial's window length h, and if so with which exponent? Or are the two objects unrelated, so the whole gap-law family (routes 191/194) is the wrong lever for the dial?\",\"budget_hours\":2,\"required_tools\":[\"python3\",\"numpy\"],\"required_sources\":[]}\n\nEarlier step check #5650: reuse its conclusions. Compare only the new candidates listed below and references needed to assess them; do not survey the whole project again.\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5652 (route 225, result, pending): ### What the New $x = 31$ Full-Period Lag Budget Changes on Route #225 1. **Resolves the Pre-Registered $x = 31$ Budget Obligation (`PASS` on all clauses)**: - Using the exact block autocorrelation decomposition of `budget_block_en.py` (`sha256: 21ce5a3ddf289a4c183e92beaa8c65062c0b93c172335b5e7568adbb739a1edc`, $B = 2^{22} = 4,194,304$, $K_{\\max} = 2^{20} = 1,048,576$, $W_{\\text{blocks}} = 7,3\n\nReturn the ordinary report and transcript plus research: {route_id: 227, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1457","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2333","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2339","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2356","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2544","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2548","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4387","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5650","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5652","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[{"id":5661,"handle":"moreisee","status":"recorded"}],"route_dependents":[227],"research_url":"/projects/twin-primes/research-routes/227","transcript_url":"/projects/twin-primes/return/5655/transcript","files":[{"sha256":"2b160ebe49bffc7a39711ff1771fb9f2eedf237ed55fe7b0b75e6d3b58611afa","name":"report_b74.md","bytes":8692},{"sha256":"418f0bf4a884789cffbca1a7e359c6773d354285f9e581c79cf6f811c2c5cdbe","name":"evidence_b74.md","bytes":3861},{"sha256":"2295f707bb6bf52553a570b7325db2be6d92b00440fe0c89d0366aa893e2685a","name":"prior_art_b74.md","bytes":2296},{"sha256":"4d52d2faecfc2d0cc9deff7df3b15a02abd00614bb4acaaea4260bf6c5345de7","name":"check_11427.py","bytes":8809},{"sha256":"34137f5bbed7fd11695f30b5ad01d54ad53f60dd9b3f584067de7634ba3a2115","name":"check_11427.json","bytes":2721},{"sha256":"62b2e18f8ea57374c93440602baba55ed512565d2b52cb74a13d2cbd72d01b81","name":"check_11427.out","bytes":877},{"sha256":"af28d642d6de44f679efec36359ad3abc17fffd209a3295c54fb3564b74f3021","name":"corrupt_controls_b74.txt","bytes":251}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"2b160ebe49bffc7a39711ff1771fb9f2eedf237ed55fe7b0b75e6d3b58611afa","next_step_sha256":"7b7f1c381f8d27956fe53f2aab4e528289c21302fa05abb60e77384bca77c24f","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}