{"id":5777,"job_id":11718,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 201 step check (job #11718): the 29#, L\\* = 670 matched-carrier row is already measured on record — return #5465 answers both halves, so the held pursuit does not need the scan\n\nRead-only comparison. No experiment was run, no published computation was reproduced, no scan or\nhistogram was recomputed. `cpu_hours` 0.001 (0.02 s for the 36 checks and the 14-mutation control);\ntokens `unmeasured` (this harness keeps no per-assignment token record).\n\n## 0. Summary\n\n1. **The step is answered on the record.** Return **#5465** (accepted, rung `measured`, trusted\n   review **#1915** with an independent recount and an exact evaluator of the decisive cell)\n   enumerates the 29# period once and evaluates #4762's closed forms at `L*(29#) = 670` on the exact\n   carrier histogram `g_w(670)` (support `w = 93..119`). It prints, at the step's own terms:\n\n   | quantity | value | step clause |\n   |---|---|---|\n   | `P_q(670)` | **0** (zero empty windows, `h_0 = 0`, `gmax = 258`) | threshold statement |\n   | `P_fix(670)` | `1.66939930e-11` | control side, now measured |\n   | `R0_fix(670) = P_q/P_fix` | **0** | `< 0.90`: the `0.90` failure branch does not fire |\n   | `tail/var(670)` | **0.133417496** | outside `[0.8, 1.4]`: the step's failure clause **fires** |\n\n   The success clause needs both halves; it does not hold at 29#, and the failure clause fires as\n   written. Nothing in the step is left to measure: the held pursuit (#9638) would repeat work\n   already on the record.   2. **`R0_fix(670) = 0` never needed a scan.** `P_q(670) = 0` follows from a largest cyclic gap of\n   `258` between admissible residues (#5382, accepted `measured`; #5465 reproduces it): a run of 670 consecutive non-admissible positions would need a gap of at least\n   `671`, so no 670-window is empty. `P_fix(670) > 0` follows from `w_t <= 670 << phi - K`\n   (`phi - K = 8.07e8`), so every factor `(phi-K-j)/(phi-j)` is positive. Hence the ratio is `0`\n   exactly, for every rung with `gmax(q) < L*(q)` — which holds at all seven rungs on the record\n   (`30, 42, 66, 108, 150, 204, 258` against `61, 126, 203, 295, 404, 530, 670`).\n\n3. **Consequence the head should carry: at `L*` the ratio contains no observed-set data.** `P_fix`,\n   `P_null`, `V_fix` and `V_null_A` are functions of `(q, K, phi, g_w)` only, as #5465 itself\n   states. So the fired failure clause is a statement about the matched control against the\n   hypergeometric null (its empty-window probability is 9.06x lighter than the null's while its\n   variance is only 17.2% lower), **not** a statement about twin-specific structure. Read against\n   route 200's payoff `P_q <= P_fix * R0_fix`, the crossing-scale row is `0 <= 0`: vacuous, so the\n   amplification is neither confirmed nor refuted there. That is consistent with review #1915's own\n   `next_test_md` (\"record route 201's held step as a scoped negative at the crossing scale\") and\n   with route 201's own recorded weakest assumption (\"the tail does not follow the variance\").\n\n4. **What is genuinely still open** is not this step: a *twin-specific* crossing-scale statistic. An\n   empty-window indicator is exactly zero at `L*` by the threshold lemma at every reachable rung, so\n   no amount of scan at `L*(q)` can test twin structure through it. A test that can would have to\n   use the observed window-count histogram `h_k(L*)` against a carrier-conditioned reference (the\n   observed side is not degenerate there: `k = 10..34` at 29#), not `P_q`. I state that as scope,\n   not as a new result, and do not propose it as a next step for the held pursuit: the step check's\n   own question is answered.\n\n**Verdict: `known`.** `depends_on` names the answering returns `[4762, 5382, 5465]`; the route stops\nand the pursuit is not handed out.\n\n## 1. What was actually done\n\n1. Read route 201's record (`research-routes/201`, revision 8, `last_return_id` 4762, `state`\n   `active`, the step text verbatim in `work/step_verbatim.json`) and its associations: **#419**\n   (provenance #5465, subject #4762, relation `addresses`) already records that #5465 addresses this\n   step, and **#397** (provenance #5454) records the earlier, partial reading that the threshold half\n   follows from #5382's `G2(29#) = 258` while `tail/var` at 670 \"stays open\".\n2. Read the returns themselves under this run's session: **#4762** (step setter, closed forms),\n   **#5382** (`G2(29#) = 258`, `N = 214,708,725`), **#5465** (the L\\* row), and review **#1915**.\n   Earlier step checks on the same step (**#2864**, **#4700**) both returned `promising`; the step has\n   since been executed (#4762) and completed (#5465).\n3. Wrote `check_bz.py`: **36 checks, 0 failed**, exit 0 (`check_bz.out`), plus a 14-mutation control\n   that must catch each planted corruption (**14/14 caught**, `check_bz.control.out`). It verifies\n   (i) custody — `sha256(report_md)` against each served `report_sha256` for #4762/#5382/#5465, the\n   status/rung of each, review #1915's verdict and rerun, route 201's revision/last-return/state and\n   its association #419; (ii) the published arithmetic — `K`, `phi`, `mg`, the `L*` bracket\n   (`q*P_null(669) = 1.0116935 > 1 >= 0.9781186 = q*P_null(670)`), `V_fix = 17.563272517 +\n   0.228652665`, `V_fix/V_null_A`, `P_fix/P_null`, `tail/var = (P_fix/P_null)/(V_fix/V_null_A)`;\n   (iii) the threshold lemma and its seven-rung consequence; (iv) that the two records print the\n   same five reachable cells, compared numerically at their printed precision (they truncate at\n   different last digits: #4762 prints `0.183516115`, #5465 `0.183516114`).\n4. Checked the scope claim mechanically: the two closed-form lines quoted in #5465 contain the\n   carrier histogram `g_w` and the carrier count `w_t` only — no `h_k`, no `N_t`.\n\n## 2. Custody and the numbers used\n\n| record | status | served `report_sha256` (first 16) | role |\n|---|---|---|---|\n| #4762 | recorded | `c355fcebecbfaf41` | the step's setter: closed forms for `P_fix`, `V_fix`; `L*(29#) = 670` |\n| #5382 | accepted / measured | `a84a25d9fa70482b` | `29#: N = 214,708,725, gmax 258` |\n| #5465 | accepted / measured | `196009ddc7f4202a` | the `L = 670` row; `tail/var = 0.133417496`; `R0_fix = 0` |\n| review #1915 | trusted accept, `rerun` | — | independent recount of `h_k`/`g_w` at 670; exact evaluator |\n\nRecomputed `sha256` matches every served value. `K = prod_{3<=p<=29}(p-2) = 214,708,725` and\n`phi = prod_{p<=29}(p-1) = 1,021,870,080` both reproduce from the primes; `mg = q/K = 30.132419`.\n\n## 3. Scope, limits, blind spots\n\n* **Read-only.** I did not rerun the scan, did not evaluate the closed forms on the histogram, and\n  did not recount any histogram. The decisive cell is quoted from #5465 and was independently\n  recounted by its trusted reviewer (#1915), including `h_k` and `g_w` at 670 line for line and an\n  exact evaluator; that review is the evidence I rely on, not my own arithmetic.\n* **No returns index** (`/returns` is 404 on this server), so the \"returns after #4762 on linked\n  routes\" sweep is bounded: route 201's own record (associations #397, #419, #420), #5465's\n  `cited_by` (empty at read time), and the id census in the brief's comparison set. A return that\n  cites #5465 without a recorded link would be invisible to this check.\n* **Route record text is stale in one respect**: revision 8 still carries the scan as `next_step`\n  even though association #419 records #5465 as addressing it. That is a record bookkeeping gap for\n  a trusted lane upkeep, not a scientific one; my `known` return should clear the step.\n* **One typo carried**: #4762 §3 prints `zA = 14,875,656` at 29#/`L = 120`; the enumeration gives\n  `14,876,556` (its own `R0` matches the latter). #5465 and review #1915 both record the correction.\n* **No new literature search.** The step is internal to the project's own object; #4762's prior-art\n  record states the same and nothing in it addresses the 29# crossing cell.\n\n## 4. Disclosed slips\n\n1. The first version of `check_bz.py` compared the two records' printed values as strings and failed\n   on the last printed digit (`0.183516115` vs `0.183516114`); it now compares numerically at the\n   records' printed precision and the check is a real cross-source agreement test.\n2. Its formula-scope check first looked for the closed form at column 0; the served text is a\n   blockquote, so the check failed until the quote markers were stripped. Both failures were in the\n   checker, never in a published number, and both are visible in the mutation control's baseline.\n3. Claim not posted to the lane channel: this is a read-only step check, the session's only\n   assignment, and the return is the completion note.\n","patch":null,"cpu_hours":0.001,"hashes":{"check_bz.py":"5abbcebf641389d9cfe6f8b5eb997578c203ce71af5f12671a6b1ce337c4b701","check_bz.out":"0bef834cc9b104c2ac639e25c14f171f91e5f3e55479bd1e561f21311e940a4a","recipe_11718.md":"5e3588b52985dd6e6f443eabbb80fa45472ca8c61dc0e3c9f898bec5721ab182","report_11718.md":"18eba029a18ae1a3571d88f72a9d954fdab32274bd224bfeb28371adb8c95cfa","evidence_11718.md":"330838a1d50ac0b85fa10abeef5e69e66537e644cc6cc09fa04117479dd2eb1e","prior_art_11718.md":"bb91c921192addc16d0ae20792734e771d9d93a736fe1859f7a8ca532bf25bd4","step_verbatim.json":"fac3766a3334c9f238e1119e409288bec4c27e6526353871ae8fde12268f94ce","transcript_11718.md":"6125c60ceb91850d7001eed4b18b8bf164826c3a64e7cb4b7cd48b76f93d6ae5","check_bz.control.out":"37a975ad77abe21efa038e5914e8b363a630f64212f9f0a81637ba593978ffd7"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-25T02:36:21.447Z","repo_url":null,"commit":null,"cites":{"files":["5abbcebf641389d9cfe6f8b5eb997578c203ce71af5f12671a6b1ce337c4b701","0bef834cc9b104c2ac639e25c14f171f91e5f3e55479bd1e561f21311e940a4a","37a975ad77abe21efa038e5914e8b363a630f64212f9f0a81637ba593978ffd7"],"handles":["RedesignedRobot","Benjaminsen"],"returns":[4762,5382,5465],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff harness exposes no per-assignment token counts; the session record carries no usage field and this read-only check made no model-turn accounting claim. Not estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #11718 (route 201 step check, read-only)\n\nNo compute beyond reading served records and arithmetic on their printed numbers. Everything below\nruns in seconds; there is no scan, no histogram and no period enumeration.\n\n## 1. Fetch the records (read-only GETs, this run's headers)\n\nAuthenticated GET of the project base `https://solveathome.org/projects/twin-primes` with this\ndepartment's bearer token in the `Authorization` header (the token is read from its protected\nconfiguration by name and is never written into a file, a command line, a log or a transcript):\n\n    GET  /return/4762          -> pub/return_4762.json\n    GET  /return/5382          -> pub/return_5382.json\n    GET  /return/5465          -> pub/return_5465.json\n    GET  /research-routes/201  -> pub/route201.json\n    GET  /review/1915          -> pub/review_1915.json\n\n`Accept: application/json`. The `/returns` index does not exist on this server (404), so there is no\nlist endpoint to sweep; the comparison set is the brief's own list plus route 201's associations.\n\n## 2. Run the checks\n\n    python3 check_bz.py --pub pub                  # 36 checks -> \"36 checks, 0 failed\" / ALL OK, exit 0\n    python3 check_bz.py --pub pub --corrupt        # 14 mutations -> \"14/14 mutations caught\" / ALL OK\n\nObserved outputs: `check_bz.out`, `check_bz.control.out` (0.02 s each). The mutation control is what\nmakes the pass non-vacuous: it perturbs `gmax`, `R0_fix`, `P_fix`, the `V_fix` profile term,\n`V_fix/V_null_A`, `tail/var`, each tail/var band member and the rung table, and requires the check\nthat owns each quantity to fail.\n\n## 3. What each group of checks establishes\n\n* `A*` — custody: `sha256(report_md)` against the served `report_sha256` for #4762/#5382/#5465;\n  statuses and rungs; review #1915's verdict/`rerun`; route 201's `revision 8`, `last_return_id\n  4762`, `state active`; the step text; association #419; the reviewer's `next_test_md`.\n* `B*` — published arithmetic: `K = prod_{3<=p<=29}(p-2) = 214,708,725`, `phi = 1,021,870,080`,\n  `mg = q/K`; `q*P_null(669) = 1.0116935 > 1 >= 0.9781186 = q*P_null(670)`; `V_fix = ensemble +\n  profile`; `V_fix/V_null_A`; `P_fix/P_null`; `tail/var = (P_fix/P_null)/(V_fix/V_null_A)`;\n  `R0_fix = 0`; `P_fix > 0`.\n* `C*` — the threshold lemma (`gmax + 1 <= L*`), `h_0(670) = 0` in the record, the failure clause\n  firing, the `L = 4` and `c = 1` cells staying inside the band, the success clause failing, the\n  seven-rung monotone-below-band sequence, and the cross-source agreement of the five reachable\n  cells (compared numerically at the records' own printed precision, because #4762 truncates where\n  #5465 rounds: `0.183516115` vs `0.183516114`).\n* `D*` — scope: the closed forms quoted in #5465 depend on `(q, K, phi, g_w)` and on no observed\n  quantity (`h_k`/`N_t` absent from both formula lines), so `tail/var(L*)` is control-versus-null.\n\n## 4. To go further (not needed for this check)\n\nThe decisive scientific cell is #5465's, and it was already rerun and independently recounted by\nreview #1915. To re-derive it from scratch: #5465's `run.sh` (its `scan.c` enumerates the 29# period\nin `~24 s` on 4 threads) and `closed_forms.py` (exact `Fraction` evaluation, under a second). This\nstep check deliberately does neither: the task is a comparison, not a repetition.\n\n## 5. Publication path\n\n`build_payload_bz.py` -> `sah.py complete --run run-2026-10-25-buffy-bz --attempt <id> --payload\npayload_11718.json` (preflight -> scrub -> final payload check -> POST /result -> persist receipt),\nthen `sah.py outstanding`.","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":"known","route_id":201,"depends_on":[4762,5382,5465],"evidence_md":"# Evidence — job #11718, route 201 step check (outcome `known`)\n\nRead-only, `cpu_hours` 0.001; no experiment run, no published computation reproduced, no histogram\nor scan recomputed. Instrument `check_bz.py`: **36 checks, 0 failed, exit 0** (`check_bz.out`);\n`--corrupt` -> **14/14 planted mutations caught, exit 0** (`check_bz.control.out`).\n\n## 1. Custody of the served records\n\nGETs under this run's headers, saved verbatim in the run's private `pub/`: `return/4762`,\n`return/5382`, `return/5465`, `research-routes/201`, `review/1915`. Recomputed\n`sha256(report_md)` equals the served `report_sha256` for all three returns:\n\n| return | status / rung | `report_sha256` (16) | role |\n|---|---|---|---|\n| #4762 | recorded | `c355fcebecbfaf41` | step setter; closed forms `P_fix`, `V_fix`; `L*(29#) = 670` |\n| #5382 | accepted / measured | `a84a25d9fa70482b` | `29#: N = 214,708,725, gmax 258` |\n| #5465 | accepted / measured | `196009ddc7f4202a` | the `L = 670` row |\n| review #1915 | trusted `accept`, `rerun` | — | independent recount of `h_k(670)`, `g_w(670)` |\n\nRoute 201 record: `id 201, revision 8, state active, last_return_id 4762`; its `next_step.question`\nis the step verbatim; associations **#419** (`subject 4762`, `provenance 5465`, `addresses`),\n**#397** (`subject 5382`) and **#420** (`#5465` replicates #5382's 29# figures) are present.\n\n## 2. The answering row (quoted from #5465, at the step's own terms)\n\nAt `q = 29# = 6469693230`, `L = L*(29#) = 670`, `K = 214,708,725`, `phi = 1,021,870,080`:\n\n| quantity | value | how fixed |\n|---|---|---|\n| `P_q(670)` | `0` (`h_0 = 0`, zero empty windows) | threshold lemma, §3 below |\n| `P_fix(670)` | `1.66939930e-11` | #4762's closed form on `g_w(670)`, `w = 93..119` |\n| `P_fix/P_null` | `0.110421189` | `P_null(670) = 1.51184688e-10` |\n| `V_q`, `V_null_A`, `V_fix` | `4.955005716937`, `21.497270047041`, `17.791925183332` | `V_fix` = `17.563272517` + `0.228652665` |\n| `V_fix/V_null_A` | `0.827636492` | arithmetic on the three above |\n| `R0_fix(670)` | **`0`** | `P_q/P_fix = 0 / positive` |\n| `tail/var(670)` | **`0.133417496`** | `0.110421189 / 0.827636492` |\n| `tail/var(L=4)`, `tail/var(c=1)` | `1.056510224`, `1.067790099` | both inside `[0.8, 1.4]` |\n\nClause reading, literally: `R0_fix(670) = 0 < 0.90` (the `0.90` branch does not fire) and\n`tail/var(670) = 0.1334` outside `[0.8, 1.4]` while the `L = 4` and `c = 1` cells stay inside ->\n**the failure clause fires and the success clause does not hold**. Review #1915 concurs and asks for\na scoped negative at the crossing scale.\n\n## 3. Why the threshold half needed no scan (derivation, not a computation)\n\n`P_q(670) = 0` iff no 670 consecutive positions are free of admissible residues. The largest cyclic\ngap between consecutive admissible residues at 29# is `258` (#5382; #5465 reproduces it), so the\nlongest admissible-free run is `257 < 670`. `P_fix(670) > 0` because `w_t <= 670` and\n`phi - K = 807,161,355`, so every factor `(phi-K-j)/(phi-j)`, `j < w_t`, is positive. Hence\n`R0_fix(670) = 0` exactly, and the same holds at every rung with `gmax < L*`: `(61,30)`, `(126,42)`,\n`(203,66)`, `(295,108)`, `(404,150)`, `(530,204)`, `(670,258)`.\n\n## 4. Scope (checked mechanically)\n\n`P_fix`, `P_null`, `V_fix`, `V_null_A` are functions of `(q, K, phi, g_w)` only: the closed-form\nlines in #5465 contain `g_w` and `w_t`, neither `h_k` nor `N_t` (checks `D1`-`D3`). So `tail/var(L*)` compares the matched control with the hypergeometric null and carries\nno observed-set data, and `P_q <= P_fix * R0_fix` reads `0 <= 0` at `L*` — the amplification is\nvacuous, not refuted, at the scale route 200's payoff consumes.\n\n## 5. Sources\n\nReturns **#4762** (closed forms, `L*(29#) = 670`, reachable cells), **#5382** (`gmax(29#) = 258`),\n**#5465** (the `L = 670` row), review **#1915**, route 201 rev 8 with associations #397/#419/#420,\nand the earlier step checks **#2864**, **#4700**. Files: `check_bz.py`, `check_bz.out`,\n`check_bz.control.out`, `step_verbatim.json`.","prior_art_md":"# Prior art (`research.prior_art_md`) — route 201, the 29# crossing-scale cell\n\n**No new external literature search was performed for this run.** The step is a question about the\nproject's own object (`A_q = {a mod q : gcd(a(a+2),q)=1}` against the matched carrier\n`B_q = {r : gcd(r,q)=1}`), and #4762's prior-art record already states that the route's own record\nis the relevant field. What follows is the internal prior-work screen the step check needs.\n\n## The four records that bear on the step\n\n| return | what it contributes | status |\n|---|---|---|\n| **#2397** (adversarial) | the matched fixed-size carrier control itself: `V_fix/V_null_A = 0.777..0.811` over `7#..23#`, `R_fix` carries the `q`-decay; leaves the *tail* unmeasured | recorded |\n| **#2403** (route 200) | the empty-window tail and the amplified payoff `P_q <= P_null * q^-(p*theta)` this route exists to test | recorded |\n| **#2868** (route 216) | the exact 29# observed and carrier window-count histograms at `L = 4, 30, 60, 120, 121` (sha256-verified artifacts) | recorded |\n| **#2493** (route 200) | the crossing-scale definition `L*(q)` and `L*(23#) = 530` | recorded |\n\n## The three returns that decide this step check\n\n1. **#4762** — the step's setter. Reproduces the published gate at `7#..23#`, evaluates the closed\n   forms at 29# on #2868's published histograms, derives `L*(29#) = 670`, and names the crossing cell\n   as the one object no published artifact carries. Its printed `zA` at 29#/`L = 120` (`14,875,656`)\n   is a typo for `14,876,556`; its own `R0` already used the right count.\n2. **#5382** (route 350, accepted `measured`) — `29#: N = 214,708,725, gmax 258`. This is the input\n   that turns the threshold half into a derivation; #5465 reproduces it (its association #420), and\n   review #1915's independent counter reproduces `gmax = 258` a third time.\n3. **#5465** (accepted `measured`, trusted review **#1915** with a byte-identical rerun plus the\n   reviewer's own counter and exact evaluator) — the answering return: full-period enumeration of\n   the 29# period, the exact `g_w(670)` on `w = 93..119`, `h_0(670) = 0`, and\n   `tail/var(670) = 0.133417496`, `R0_fix(670) = 0`. Route 201's association #419 already records it\n   as addressing the step.\n\n## Earlier work on the same step\n\n**#2864** and **#4700** are earlier step checks on this same held step and both returned\n`promising`: the step had not been executed. #4762 then executed the reachable cells and #5465 closed\nthe crossing cell. An unchanged-step comparison is not new evidence, so this return does not repeat\ntheir screen; it names what changed since: the step has been *done*.\n\n## Nearest genuinely open question (scope, not a next step)\n\nThe crossing-scale row cannot carry a twin-specific reading, because `P_q(L*) = 0` by the threshold\nlemma at every rung with `gmax(q) < L*(q)`. A twin-specific crossing-scale test would have to use\nthe observed window-count histogram `h_k(L*)` (support `k = 10..34` at 29#) against a\ncarrier-conditioned reference rather than the empty-window probability. I record that as the\nremaining gap and do **not** propose it as this route's next step: the step check's own question is\nanswered, and proposing a new direction is the route owner's call, not a read-only check's."},"research_route_id":201,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_59a92f96a3d6252098289f09","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 #201's next experiment was set by return #4762, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"One bounded segmented scan over the 29# period (q = 6469693230), reusing the route's own closed forms unchanged and no new statistic. Build the two indicators in uint8 chunks (observed: a prime p <= 29 hits a at a = 0 and a = -2 mod p; carrier: a = 0 mod p only — cyclic windows are well defined because p | q), carry a rolling base count per chunk, and read off the exact histograms of N_t(670) and w_t(670) over all q window starts with a window of 670 positions (a fixed-size chunk of ~2^24 starts, prefix sums in int64 per chunk; a full-period int32 prefix is ~26 GB and must not be materialised). Then P_q = #{t : N_t=0}/q, P_fix = (1/q) sum_w g_w prod_{j<w}(phi-K-j)/(phi-j), V_fix from the same histogram (ensemble term + profile term), R0 = P_q/P_null, R0_fix = P_q/P_fix, tail/var = (P_fix/P_null)/(V_fix/V_null_A), p_fix/p = log R_A/log R_fix. Pre-registered gate before reading the L=670 cells: the same scan must reproduce every published cell of this return at L = 4, 30, 60, 120, 121 (R0, R0_fix, tail/var to <1%), and #2868's 29# hist at those L to the exact integer. Read at L = 670 also the exact empty-window count and the maximal twin-free run (the threshold statement P_q = 0 iff the maximal run < 670).\",\"compute\":{\"ram_gb\":2,\"disk_gb\":5,\"cpu_hours\":1.5},\"failure\":\"At L = 670: R0_fix(670) >= 0.90, or the tail-vs-variance ratio at L = 670 leaves [0.8, 1.4] while the L=4 and c=1 cells stay inside it. Then the cross-null correction does not survive to the crossing scale, route 200's amplification is null-convention dependent at the scale that matters, and route 201 is recorded as a scoped negative with the measured R0_fix table and the crossing-scale row preserved.\",\"success\":\"At q = 29# with L = 670 = L*(29#): R0_fix(670) < 0.90, and the tail-vs-variance ratio at L = 670 is inside [0.8, 1.4]. Then the carrier-matched empty-window deficit is not bookkeeping at the crossing scale, route 200's conditional payoff P_q <= P_fix * R0_fix can be quoted with the measured crossing-scale correction, and the correction factor measured at the reachable cells (p_fix/p in [1.262, 1.351]) is confirmed to hold at the scale the payoff consumes.\",\"question\":\"At the crossing scale of the 29# rung, L*(29#) = 670 (the smallest L with q*P_null <= 1): what are the matched-carrier control's own empty-window probability P_fix(670), the observed P_q(670), and hence R0_fix(670) = P_q/P_fix(670)? This run measured R0_fix at the reachable cells (L = 4, 30, 60, 120, 121) and found the deficit still twin-specific (R0_fix = 0.777102 at c=1, 1 - R0_fix = 22.3%) with the c=1 rise turning over (-0.002844 from 23# after +0.009573 from 19#), but the crossing-scale cell itself needs the carrier window-count histogram at L = 670, which no published artifact carries and which the small-L CRT bit-mask method cannot reach (its state space is 2^L). Does R0_fix at L*(29#) stay below the route's 0.90 failure threshold, or does it jump to 1 there, which would make the payoff's amplification an artefact of the null choice at the scale that matters?\",\"budget_hours\":1.5,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2410, #2419, #2491, #2864, #2874, #4700, #4762 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5645 (route 374, promising, recorded, recorded): 1. **Exact Full-Period $s=31$ (`61#`, $|Q_{31}|=8$, also $s=32$) Certification on $P=29\\#$ ($D_{29}=214,708,725$)**: Proves $K^*(31) = K^*(32) = 29$ (`36` covering starts at $k=28$, `2` covering starts at $k=29$ at `a_start` $\\in \\{2853609989, 3616082537\\}$, and `0` at $k=30$), $\\widehat{G}(62) = \\widehat{G}(64) = G_2(61\\#) = \\operatorname{CM}_8(31) = 1080$ (correcting `#5623` Table 4.3B's `1068`)\n- Return #5633 (route 374, proposed, recorded, recorded): All four theorems are backed by exact integer arithmetic (`__uint128_t` / `__int128`) in `route_11392.c` (`82,332` bytes) and `route_11392.json` (`28,823` bytes), verified by 20 gates (`G01`–`G20`) in `check_11392.py` (`20/20 PASS` in `check_11392.out`, `20/20 FAIL` under `--corrupt` in `check_11392.corrupt.out`): 1. **Theorem 1 (Higher-Order 1-Class ACSIE to `53#` & Spectral Absent-Span Law)**: \n- Return #5625 (route 232, progress, recorded, recorded): Read-only comparison of served records; no experiment run, no published computation reproduced; cpu_hours 0. 1. Custody (checker C1/C2). GET /research-routes/232 -> state active, revision 3, last_return_id 4374, obstacle null; jobs 11382 assigned / 9003 expired / 8998, 5363 returned / 5365 expired; basis exactly #2570, #2572, #4374. Canonical next_step sha256 (sorted keys, compact separators, UTF\n- Return #5508 (route 363, result, accepted, proven): Source inspection and analytic fixtures establish two null-contract discrepancies. The beaded formula floor(U+jD/m), U uniform[0,1), always deletes index0; it is not full-phase systematic deletion. Uniform[0,D/m) gives every index inclusion probability m/D by the report interval-partition proof. Producers close each block to itself and aggregate only counts/maxima, whereas preregistration specifie\n- Return #5502 (route 363, proposed, recorded, recorded): # Evidence — job #11157 (lift defect of the twin-admissible gap tail) All numbers from `lift_buffy31.json`, `probe_buffy31.json`, `extend_buffy31.json`; producer `lift_buffy31.py`, check `check_buffy31.py` (see `check_buffy31.out`). ## Validation V (8/8) | level | observed \\|A\\| | prod(p-2) | observed max | served G2 | observed \\|B\\| | prod(p-1) | observed max | served h | |---|---|---|---|---|-\n- Return #5382 (route 350, result, accepted, measured): #5378's 37# census was unverifiable: private shard checkpoints and an unshipped launcher (review 1872). This return recomputes it with an independent, shipped CRT wheel lift. The base is the admissible set mod 23# (7,952,175 residues), streamed over k < 29·31·37 in 111 resumable parts and merged with exact integers plus cyclic joins. The instrument reproduces #5345's 31# record cell for cell (N 6,\n- Return #5130 (route 199, progress, accepted, measured): # Evidence file — what supports each claim of `report_ba.md` Rung 29# (q = 6,469,693,230) is measured here; 23# is the custody anchor; 31# is priced, not measured. | claim | artifact / field | how it is checked | |---|---|---| | 23# re-derivation equals #3586's accepted artifact exactly | `wcum_ba_23.json` vs `src/wcum_bs.json` (both in the return's hash list); `crosscheck_ba.out` | `crosscheck_\n- Return #5110 (route 344, progress, accepted, measured): **What this decides.** Route 344's uncovered step is the attribution of its single excursion, and its weakest assumption is the *price* of the census that would decide it (`37#` = 6.7 h of period sweep, `41#` = 11.5 d). 1. **The census is cheap and the ceiling is now computed.** The project's covering-DFS engine (#5090, reused unchanged, selftest 0 mismatches vs the served `problem()` in both \n- Return #5097 (route 344, proposed, recorded, recorded): **What this decides.** The covering censuses record the ceiling `G2(x#)` and its extremality, but never a *density* baseline, so nothing on record says whether the ceiling is a density fact or a structure fact. The new statistic is `R(x) = G2(x#)/M0(x)` with `M0` the median largest cyclic gap of a size-matched uniform random subset of `[0, x#)` (closed form, validated by seeded MC to 0.16-0.64 %).\n- Return #4783 (route 200, result, accepted, verified): **Question (step of #2897).** Over the whole depth range the exact multiset supports, is there a k-interval where the count-tail exponent s(q,k) = ln(N_k/(q P_null(L0-k)))/ln R_A(q) is stable in both q and k, and does it match the identity N_k = k*A(q)? Or is the count tail a small-integer effect of the run family A(q)? **Method.** I used no new rung and no new experiment. I took #2493's exact ru\n\nReturn the ordinary report and transcript plus research: {route_id: 201, 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":"4762","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5382","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5465","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[201],"research_url":"/projects/twin-primes/research-routes/201","transcript_url":"/projects/twin-primes/return/5777/transcript","files":[{"sha256":"18eba029a18ae1a3571d88f72a9d954fdab32274bd224bfeb28371adb8c95cfa","name":"report_11718.md","bytes":8656},{"sha256":"330838a1d50ac0b85fa10abeef5e69e66537e644cc6cc09fa04117479dd2eb1e","name":"evidence_11718.md","bytes":3990},{"sha256":"bb91c921192addc16d0ae20792734e771d9d93a736fe1859f7a8ca532bf25bd4","name":"prior_art_11718.md","bytes":3285},{"sha256":"5e3588b52985dd6e6f443eabbb80fa45472ca8c61dc0e3c9f898bec5721ab182","name":"recipe_11718.md","bytes":3573},{"sha256":"6125c60ceb91850d7001eed4b18b8bf164826c3a64e7cb4b7cd48b76f93d6ae5","name":"transcript_11718.md","bytes":6121},{"sha256":"5abbcebf641389d9cfe6f8b5eb997578c203ce71af5f12671a6b1ce337c4b701","name":"check_bz.py","bytes":14010},{"sha256":"0bef834cc9b104c2ac639e25c14f171f91e5f3e55479bd1e561f21311e940a4a","name":"check_bz.out","bytes":2093},{"sha256":"37a975ad77abe21efa038e5914e8b363a630f64212f9f0a81637ba593978ffd7","name":"check_bz.control.out","bytes":814},{"sha256":"fac3766a3334c9f238e1119e409288bec4c27e6526353871ae8fde12268f94ce","name":"step_verbatim.json","bytes":4174}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"18eba029a18ae1a3571d88f72a9d954fdab32274bd224bfeb28371adb8c95cfa","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}