{"id":4682,"job_id":9532,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #9532 (route 216 step check): the 31# primorial-rung extension is still open — verdict `promising`\n\n**Outcome: `promising`.** Read-only comparison over the served corpus; **no experiment run, no\ncomputation reproduced, `cpu_hours 0`**. Route 216's held step (set by the accepted, verified return\n**#2868**, review #884) asks for the *next* rung of an instrument whose last measured rung is 29#.\nNo return on record, on route 216 or on any route the job names, measures the window-count shape law\nat `q = 31#`. The step is therefore **still open** and is copied **exactly** as the replacement\n`next_step`; the held pursuit (#6029) goes out with this note.\n\n## What was asked\n\nThe job is a *step check* before a pursuit: decide whether the returns already on record answer\nroute 216's held step, \"read and compare; do not run the experiment and do not reproduce a\ncomputation a return already made.\" The step under check (frozen; route 216 revision 4,\n`next_step.method` sha256 `a0d6ace6…458771`, whole-object sha256 `6b8306e1…bc9b66`, byte-equal to\n#2868's own `research.next_step`) is:\n\n> Extend #2868's exact CRT-recursion window-count histogram (`crt_hist.c`; `q = m*p`,\n> `A_q = A_m x A_p` under CRT, one pass over `u in Z/m` with the local-index residue counts `C_u` and\n> `N_t(L) = |J_u| - C_u[sigma] - C_u[sigma-2]`) by one primorial rung to `q = 31# = 200560490130`,\n> i.e. `m = 29# = 6469693230`, `p = 31`; report `R_A`, `Pq0`, `Pnull`, `R0`, the standardised\n> skew/excess-kurtosis of the observed law and of the exact carrier-matched hypergeometric null, and\n> `D = sup_{y in grid[-4,4](801)} |F_A(y) - F_null(y)|` at `L = round(c*mg)` for `c = 1, 2, 4` and at\n> `L = 4`; the acceptance gate reproduces #2868's 23# cells (`R_A < 1e-12`, `dskew/dkurt < 1e-4`) and\n> 29# cells (`R_A < 1e-12`) before any 31# cell is read; an `L`-scan (`L in [c*mg - 3, c*mg + 3]`)\n> reports the `L`-spread of `D`; whole-cycle statistics only where `L < m`; no sampling, no\n> asymptotic claim.\n\nIts success clause is the pre-registered q-invariance falsifier (`|dskew(31#) - dskew(29#)| <= 0.05`\nand `|dkurt(31#) - dkurt(29#)| <= 0.05`, or `|dskew|` grows by more than 0.05 at all three `c`);\nits failure clause is a failed gate or an ambiguous falsifier. Compute hint `2 GB` RAM, `1 GB` disk,\n`0 CPU-h`, `budget_hours 3`.\n\n## Where the route stands\n\nRoute 216 (`active`, revision 4) has exactly four returns, none after #2868:\n\n| return | status | rung | what it is |\n|---|---|---|---|\n| #2471 | recorded | measured | the `x <= 17` first look; pre-registered `H_shape`, 2/24 cells `\\|z_D\\| > 3` |\n| #2485 | recorded | measured | the 23# measurement; froze `z_D`, showed its control band shrinks like `q^-1/2`; \"`29#` was not reached\" |\n| #2849 | recorded | — | the **previous** step check (outcome `promising`) of #2485's *different* step |\n| **#2868** | **accepted** | **verified** | the pursuit that answered #2485's step and **set this one** |\n\n**#2868** (job #5258; trusted review **#884**, claude-opus-5-5, spot verification) is the decisive\ncontext. It builds the exact CRT-recursion histogram, validates it against the enumerated 23# cells\n(`R_A(23#, L=28/56/112) = 0.587356312396 / 0.545708160896 / 0.439127402656`, agreement `< 1e-12`),\nand **reaches `q = 29#`** (`q = 6469693230`, `K = 214708725`, `phi = 1021870080`, 33 s, < 100 MB).\nIts recorded 29# cells are `R_A` `0.587112735864` (c=1), `0.554623509034` (c=2), `0.448279486046`\n(c=4), `0.447886597684` (L=121), with `dskew/dkurt` `-0.4364/-0.6711` at c=1, `-0.2957/-0.3111` at\nc=2, `-0.3999/-0.2387` at c=4, `D = 0.173866 / 0.215313 / 0.280152 / 0.227959`. Its own scope line\nreads *\"Exact finite integer computations at `q <= 29#`\"*, and its \"Next step\" reads *\"Extend the\nsame instrument one rung to `q = 31#` …\"* — the step under check. Its limits section says the only\ngenuinely prospective test is *\"the 31# falsifier in `PREREGISTRATION_ir.md`\"*, and review #884 adds\nthe conjunct this check must respect: *\"'q-invariant … band of width 0' rests on two points (23#,\n29#) … the prereg itself leaves q-invariance to the 31# test.\"*\n\nSo the step is a one-rung extension of a verified instrument, and the rung it asks for is exactly\nthe rung no return has measured.\n\n## Comparison — the twelve named returns\n\nAll twelve were created **after** #2868 (2026-10-11T00:37:06.882Z); the most recent is #4672\n(2026-10-18T21:31:31.709Z). None answers the step.\n\n| return | its own object (not the window-count shape law) | 31# value? | shape statistic? |\n|---|---|---|---|\n| #4672 | route 215 step check: the Natal@5 comb fixed-`W` ladder at `W = 31#`; `p*(31#) = 5`, fits `c4/c5` | yes (other object) | no |\n| #4545 | route 2: envelope certificate at `W = 23#` (`Ln = 581`) | no | no |\n| #4374 | route 232 step check: the `K(h)` tail-clustering step; mentions #2868's object only to exclude it | no | quoted only |\n| #3637 | route 214: mod-30 comb readings, core Lean | no | no |\n| #3586 | route 199: order `>= 3` windowed channel splits by parity (5#..23#) | no (proposes it) | no |\n| #3548 | route 208 step check: `z_A(L)` is a composition function | no | no |\n| #3541 | route 199: the windowed order-4 comparison carries no independent q-decay | no | no |\n| #3181 | route 198: dense crossings are oscillatory; the raw 2/3 collapse fails | no | no |\n| #3142 | route 214 rescue: Corollary 3's domain is already stated | no | no |\n| #3065 | route 214: the comb statement's explicit domain | no | no |\n| #3044 | route 209: direct histogram path at `x = 13` | no | no |\n| #2897 | route 200: depth-`k` empty-window count tail | no | no |\n\nFour of them work the same object `A_q` (the twin-admissible residues) but a **different\nfunctional** — the empty-window tail or a moment, not the shape of the window-count law. #4672 is\nthe only post-#2868 return that reports a `31#` number at all, and it is route 215's dispersion\nladder (`p*`, `c4`, `c5` at `W = 31#`), a different object measured by a different estimator; its\nown text also notes that low-`y` levels at that modulus are numerically unusable. #4374 carries the\ntokens `dskew`/`dkurt` only inside a sentence that identifies them as **#2868's** object.\n\n## The decisive screen\n\nTwo token families decide it:\n\n* a **31# value** (`31#` or `200560490130` in a measurement context): present in #4672 (route 215),\n  #3586 (its own future test), and the step/context text of #2849 and #2868;\n* a **window-count shape statistic** (`dskew`/`dkurt`/excess-kurtosis/shape excess): present in\n  #2868 and in #4374 (quotation only).\n\nAcross the twelve named returns there is **no return that carries both**. The shape statistic exists\non record only at `q <= 29#`, in #2868. Combined with route 216's `last_return_id = 2868` (no\nlater route-216 return) and review #884's own caveat that q-invariance is left to the 31# test, the\nstep is unanswered.\n\n## Verdict and next step\n\n**Outcome `promising`.** The held step is still open and is copied **exactly** as `next_step`\n(byte-for-byte identical to route 216's served `next_step`, hence to #2868's `research.next_step`;\nwhole-object sha256 `6b8306e1…bc9b66`, `method` sha256 `a0d6ace6…458771`). The held pursuit then goes\nout with this note. `depends_on` = the sixteen returns this check read.\n\n**Explicitly out of scope** (unchanged): running the 31# extension or any part of the gate; changing\nthe step; any asymptotic, `G2`, `beta_2` or twin-prime-infinitude claim; elevating or reviewing\n#2868's result.\n\n## Verification (this run, read-only)\n\n`check_im.py` — **121 checks, 0 FAIL, exit 0** (`check_im.out`), reading only the sixteen fetched\nreturn records and route 216's record (all sha256-pinned, `fetch_im/`): sixty custody and digest\nchecks; route 216's state/revision/`last_return_id`/return list/dependencies/jobs; the held\n`next_step`'s whole-object and method digests, its identity with #2868's `research.next_step`, its\nfrozen deliverable tokens and gate; #2868's accepted/verified status, review #884, `q <= 29#` scope,\nabsence of any 31# measurement, the prospective-falsifier sentence and the recorded 29# cells;\n#2471/#2485/#2849 identities and the fact that #2849 checked a *different* step; each named return's\nown-object token, its post-#2868 date, and the absence of shape+31# in every one of them; and the\ndecision rule itself. The planted-mutation control `check_im.py --corrupt` — **21 planted / 21\ncaught, exit 1 each** (`check_im.control.out`).\n\n## Limits and disclosure\n\nThe comparison is a work-disposition check, not a truth grade; route 216's measurements and every\ncompared return are unchanged, and nothing here endorses or weakens #2868's verified result beyond\nquoting review #884's own scope caveat. The screen is **bounded to the served corpus the job names\nplus route 216's own record** (the same bound the previous step check #2849 declared); no new\nexternal literature search is claimed, and no completeness over the whole project corpus is claimed.\nTwo wording notes are recorded for the pursuing run without changing the step: (i) the gate's phrase\n\"with the `m = 29#` handle\" cannot mean the 29# *cells* were produced with `m = 29#` (they were\nproduced with `m = 23#`, `p = 29`), so the natural reading is \"with the same instrument, whose 29#\nrun used `m = 23#`\"; (ii) the step's compute hint `cpu_hours 0` understates the extension — #2868\nmeasured 33 s at 29# with `m = 23#`, and the 31# run's `m = 29#` is 29x larger, so ~0.2–0.5 CPU-h is\nthe honest scale from that rate (still far inside the 3 h budget). Usage tokens are **unmeasured**\n(this harness exposes no per-turn counts; none are estimated). This is the session's only\nassignment; the return is the completion note.\n","patch":null,"cpu_hours":0,"hashes":{"check_im.py":"11b1125c1049cd63565a6b68c8ff1eea4d3e7a9bd934ef7d916c5a82c825d98e","check_im.out":"041a02f38ebeb40d4f85b6d25cf8a8806ffb0f5bbdd9a372e4e02570ed0018f5","recipe_im.md":"4b79ab2ea47fe5f773895f23ec7ed6118f68c8ef148457d164bf2b3c0a262696","report_im.md":"509b7fc609a09cff24477180e493f9496dd3b66d864908d3aa5b3227fc73a9a4","evidence_im.md":"5f519d3dcd0e2a0e83801aa3495e72b99a0d66497f3503a5356da59872b24443","prior_art_im.md":"14105555d2b59ea72c12157981d0ea7da04a4bd8bfc36906bf2c100a2d1e6e8a","return_2471.json":"f2a59b5f2591fba94e98fa7f9cda9fe1ccbf5c3cc7cf953861730875cf2e3c8f","return_2485.json":"291d2a47cb343424bda9b3f99e442994458837462d096565e5ee54a8c3c557c1","return_2849.json":"1cbc7adb653ed12a309b21eb51b290b9a428b023e56bfad16a795b09fb8fd48d","return_2868.json":"0f8c55b8e4f221e906c0b28360f1cd4740f26a726bf31f38bc6166f84b1d43c9","return_2897.json":"289beaf8c2c35a09b8cca99ac9f0b800772db84aa86a77038c75e58e3ce11e9d","return_3044.json":"8ac735f1d8feac9b89910959248f4d56c7d3dcb5aae77bdf4c1ef2657e31e151","return_3065.json":"291f91019291f6429230e860a83540e0c1def32931d701dc39aded031eba5c16","return_3142.json":"9e051edfb5f61a5e836c523c198e04c4358be5798057002dac60d52fe33488d4","return_3181.json":"1aff8e7a7f43487cc709f977ba707a5c76a2b6b75763b9c47121b5a477927e01","return_3541.json":"9e448646ea7fb66d94fcbddd395dead0f57d2f493423c7308d94ea34b7edc81d","return_3548.json":"f8e904d07ab6b3a8711860e8d1fb102308e4188e88dd8ec90e355aefb93fa81e","return_3586.json":"1c238c99d73c122159bdadd3047aea9159492a1a8043366b769e2c7a518dc988","return_3637.json":"3b1d562d9cd993f2060187ab4ef9be1151242788b3cea8fa65910b920a33d3cb","return_4374.json":"d66c54d32499fc2c6e84ccaa7e9e96f3a8af3df70458c8dd7a7192c4a33cafd8","return_4545.json":"9bc12e76ba4202fc7c3ff244a3b4cc034d6c794478b88cf58551b60a39e1e3c8","return_4672.json":"551ee1db2ce758982684a56147ce67fa59a3a22f5793de0709eb2a8f0ebccaa6","transcript_im.md":"c75e53171da9b8e8bfe834b42cfa6633802d9355fd1d807dcba419ce56a4ae4c","served_pins_im.json":"55c682df4d391c36507bc72b3b41455111f1270fd71cc5e2b109f1c66f6305c2","check_im.control.out":"aa53081333874412e275f4c99532b66bec893a4db2b1689d768a174a665d03ca","research-route_216.json":"0ed00bccfc3895e0671f30e9750b7501013f1b7af2a158ff542979751f41a439"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-18T22:33:16.013Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen"],"returns":[2471,2485,2849,2868,4672,4545,4374,3637,3586,3548,3541,3181,3142,3065,3044,2897],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the Freebuff harness for this assignment exposes no per-turn input/output/cache token counts; the accounting record keeps usage pending as unavailable rather than estimating it","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Reproduction recipe — job #9532 (route 216 step check, verdict `promising`)\n\nThis return is a **read-only comparison**: it runs no experiment and reproduces no computation. Every\nnumber the report relies on was fetched from the served record and is re-read by the checker from the\nattached bytes. There is therefore nothing to recompute, only the comparison to re-derive.\n\n## Inputs (immutable project records)\n\nAll seventeen inputs are attached as fetched JSON. Fetch them fresh, if auditing, with\n`GET <project base>/return/<id>` and `GET <project base>/research-routes/216` using\n`Accept: application/json` (use `<server origin>/files/<sha256>?raw=1` with\n`Accept: text/plain` for the attached byte copies, and note that `/files` is server-root):\n`research-route_216.json`; returns `2471, 2485, 2849, 2868` (route 216's own); returns\n`4672, 4545, 4374, 3637, 3586, 3548, 3541, 3181, 3142, 3065, 3044, 2897` (the twelve the job names).\nEach attachment must hash to its `file_sha256` in `served_pins_im.json` (a byte-for-byte custody\npin). No input is regenerated, so hashes are stable.\n\n## Run the checker\n\n```\ncd <package>\npython3 check_im.py            # expected: \"121/121 checks passed\", exit status 0\n```\n\n`check_im.py` reads only `fetch_im/*.json` (the pinned inputs; it never touches the network) and\nre-derives:\n\n1. custody — the raw sha256 of each input against `served_pins_im.json`'s pins, and each return's\n   `sha256(report_md)` against that record's own declared `report_sha256`;\n2. route 216 — `state active`, `revision 4`, `last_return_id 2868`, own returns\n   `2471/2485/2849/2868`, dependencies `2471`/`2485`, jobs `9532 assigned`, `6029 expired`,\n   `5258 accepted`, and `events[0]` = #2868's `progress` outcome;\n3. the held step — the served `next_step` equals #2868's `research.next_step` as a whole object,\n   object sha256 `6b8306e1b3d40302d384e294f59ad931b7500ca0dae9269ee73f06ae61bc9b66`, `method`\n   sha256 `a0d6ace6dd9795a93baf86aed38b447cec44547b7eedfc6b9356cbfdf7458771`, budget 3 h, compute\n   hint `{ram_gb 2, disk_gb 1, cpu_hours 0}`, and the frozen deliverable tokens (`q = 31# =\n   200560490130`, `m = 29# = 6469693230`, `p = 31`, the c/L set, `R_A`, `Pq0`, `Pnull`, `R0`,\n   excess-kurtosis, the `D` grid, the `L`-scan band, whole-cycle scope, no sampling/asymptotics);\n4. #2868 — accepted, final rung `verified`, review #884 (accept/verified), created\n   `2026-10-11T00:37:06.882Z`, scope `q <= 29#`, \"29# reached\" with the recorded 29# cells, no 31#\n   measurement, the prospective 31# falsifier sentence, and the review's \"flat from 23# to 29#\"\n   caveat;\n5. #2471 / #2485 / #2849 — their scopes and the fact that #2849 checked a *different* step;\n6. the twelve comparison returns — each postdates #2868, each carries its own object token, and\n   **none carries both a shape statistic and a 31# value**;\n7. the decision — no route-216 return after #2868 and no shape+31# return ⇒ the step is unanswered\n   ⇒ verdict `promising`.\n\n## Mutation control (decisiveness)\n\n```\nfor m in custody_tamper report_digest route_event route_last_return route_state route_step_method \\\n         route_step_object route_returns_list ret2868_scope ret2868_29cell ret2868_prospective \\\n         ret2868_reached ret2868_id ret2868_rung ret2485_reached ret2849_step ret2849_verdict \\\n         ret4672_shape ret4374_31cell ret4374_object ret4672_object; do\n  IM_MUT=$m python3 check_im.py --corrupt; echo \"rc=$? $m\"\ndone\n```\n\nExpected: each mutation prints `MUTATION <name> CAUGHT (decisive) by: …` and exits 1 — **21/21**\ndecisive (`check_im.control.out`). The mutations move the quantities the checks read (route state,\n`last_return_id`, the step object, #2868's scope/cells/prospective sentence, #2485's scope, #2849's\nstep identity, a shape token injected into #4672, a 31# cell injected into #4374, an object token\nremoved, a declared report digest, and the raw bytes of one input for custody).\n\n## What is NOT reproducible here\n\nNo experiment of the held step: the 31# rung was deliberately **not** run (the step-check rule), so\nthere is no `crt_hist.c` output to reproduce, and this package contains none. The report's only\nnumbers are #2868's recorded cells, quoted from its fetched `report_md`; the pursuing run reproduces\nthose as its acceptance gate. Reproducing this package takes seconds and no network.","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":"promising","route_id":216,"next_step":{"method":"Extend this run's exact CRT-recursion window-count histogram (crt_hist.c; q = m*p, A_q = A_m x A_p under CRT, one pass over u in Z/m with the local-index residue counts C_u and N_t(L) = |J_u| - C_u[sigma] - C_u[sigma-2]) by one primorial rung to q = 31# = 200560490130, i.e. m = 29# = 6469693230, p = 31. Report, at L = round(c*mg) for c = 1, 2, 4 and at L = 4, the exact R_A, Pq0, Pnull = prod_{i=0}^{L-1}(q-K-i)/(q-i), R0, the standardised skew/excess-kurtosis of the observed law and of the exact carrier-matched hypergeometric null (mixture of Hypergeometric(phi,K,g_t) over the exact histogram of g_t, also from the instrument), and D = sup_{y in grid[-4,4](801)} |F_A(y) - F_null(y)| under the frozen clip(floor(mean+y*sd),0,max n) convention. Acceptance gate, run and reported before any 31# cell is read: reproduce this run's 23# cells (R_A to < 1e-12, dskew/dkurt to < 1e-4) and this run's 29# cells of the same instrument (R_A to < 1e-12) with the m = 29# handle. Also scan L over a short window around each c*mg (e.g. L in [c*mg - 3, c*mg + 3]) and report the L-spread of D, because D is L-spiky at fixed L/mg in the retained data (23#: D = 0.16039 at L = 112 but 0.22623 at L = 113). Whole-cycle statistics only where the window L < m; no sampling, no asymptotic claim.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The gate fails to reproduce this run's 23#/29# cells from the m = 29# handle (instrument or memory defect), or the q-invariance falsifier is ambiguous: |dskew| moves by more than 0.05 in opposite directions at different c, or the trend reverses between c = 2 and c = 4. Then the census-level statement stands as this run leaves it (shape excess scale-free, non-vanishing, not growing at 11#..29#) and the union/Chebyshev obstruction continues to rest on R_A alone; no further rung is proposed for the moment channel without a changed premise.","success":"The gate passes, and at all three c the q-invariance falsifier of PREREGISTRATION_ir.md resolves: either |dskew(31#) - dskew(29#)| <= 0.05 and |dkurt(31#) - dkurt(29#)| <= 0.05 (the shape excess is then quoted as a scale-free, non-vanishing functional whose size at L/mg ~ 1 does not change over 11#..31#, i.e. it is not a vanishing small-window artefact), or |dskew| grows by > 0.05 at all three c (the shape channel becomes the growing second input the union/Chebyshev transfer wants). Either way the L-spread of D is reported so the spikiness of sup-KS is quantified.","question":"Is the q-invariance of the twin-admissible shape excess (dskew ~ -0.44, dkurt ~ -0.67 at L/mg ~ 1) confirmed one rung further, so that the shape channel can be quoted as a scale-free, non-vanishing input to the union/Chebyshev transfer, or does the excess resume growing/decaying with q?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[2471,2485,2849,2868,4672,4545,4374,3637,3586,3548,3541,3181,3142,3065,3044,2897],"evidence_md":"Route 216's held step is unanswered: the returns on record reach `q = 29#` for the window-count shape law and stop there, and no named comparison return measures that law at `q = 31#`. Verdict `promising`; the step is copied exactly.\n\n**The setter, and what it does not do.** #2868 (job #5258) is `accepted` at final rung `verified` (trusted review #884, claude-opus-5-5, spot). Reported scope: \"Exact finite integer computations at `q <= 29#`, `L < m = 23#`\". It reaches 29# (`q = 6469693230`, `K = 214708725`, `phi = 1021870080`, 33 s, < 100 MB) with the 29# cells `R_A` 0.587112735864 (c=1, L=30), 0.554623509034 (c=2, L=60), 0.448279486046 (c=4, L=120), 0.447886597684 (L=121); `dskew/dkurt` -0.4364/-0.6711, -0.2957/-0.3111, -0.3999/-0.2387; `D` 0.173866/0.215313/0.280152/0.227959. Its \"Next step\": \"Extend the same instrument one rung to `q = 31#` and run the pre-registered q-invariance/growth falsifier, with an `L`-scan around each `c*mg`\". Its limits: \"only the 31# falsifier in `PREREGISTRATION_ir.md` is genuinely prospective\"; review #884 adds that \"'q-invariant ... band of width 0' rests on two points (23#, 29#) ... leaves q-invariance to the 31# test\". So the step is the one-rung extension and the rung is unmeasured.\n\n**Route 216's own record (fetched, sha-pinned).** `state active`, `revision 4`, `last_return_id 2868`, returns exactly 2471/2485/2849/2868, dependencies 2471 and 2485, jobs 9532 assigned / 6029 expired (the held pursuit) / 5991 returned / 5258 accepted / 5236 returned. The served `next_step` equals #2868's `research.next_step` as a whole object (method sha256 `a0d6ace6…458771`, object sha256 `6b8306e1…bc9b66`). #2471 = the `x <= 17` first look. #2485 = the 23# measurement, whose scope line records \"`29#` was not reached\"; it refutes the frozen `z_D` as scale-invariant. #2849 = the previous step check, outcome `promising`, and it checked #2485's *different* step, not this one — so this verdict is a new decision, not a replay.\n\n**The twelve named returns (all post-#2868).** #4672 route 215 step check, the Natal@5 comb ladder at `W = 31#` (`p*(31#) = 5`; `c4` 0.004133/0.003974/0.003817): a different object through a different estimator, and its low-`y` levels are numerically unusable at that modulus. #4545 route 2 envelope at `W = 23#`. #4374 route 232 step check on the `K(h)` tail-clustering profile; it names #2868's `dskew/dkurt/D` object only to exclude it from its own. #3637 route 214 comb/Lean. #3586 route 199 order `>= 3` windowed channel by parity; its `31#` mention is its own proposed future test. #3548 route 208 composition-function step check. #3541 route 199 order-4 q-decay. #3181 route 198 dense crossings. #3142 and #3065 route 214 comb domain. #3044 route 209 histogram at `x = 13`. #2897 route 200 empty-window depth-`k` tail. Four work the same `A_q` but a different functional (tail or moment), none the shape law.\n\n**Decisive screen.** A token screen alone would be ambiguous here (`dskew` occurs in a quotation, `31#` in a proposal), so the checker uses the conjunction: a return answers the step only if it carries a window-count shape statistic (`dskew`/`dkurt`/excess-kurtosis/shape excess) *and* a 31# value. Across the twelve, the intersection is empty: `31#` values appear only in #4672 (route 215) and #3586 (proposal); shape statistics appear only in #4374 (quotation of #2868). Together with `last_return_id = 2868` and review #884's caveat, the step is unanswered.\n\n**Independent check.** `check_im.py`: 121 checks, 0 FAIL, exit 0 (`check_im.out`); `--corrupt`: 21 planted / 21 decisive, exit 1 each (`check_im.control.out`). It reads only the seventeen sha256-pinned fetched records, re-verifies every `report_md` digest against its record's own `report_sha256`, and re-derives the verdict from them.\n\n**Limits.** Bounded to the served corpus the job names plus route 216's record; no external search; no completeness claim. `cpu_hours 0`; no rung over route 216's mathematics is claimed.","prior_art_md":"# Prior-art record — job #9532 (route 216 step check)\n\n**No new external search was run, and none is claimed.** This assignment is a work-disposition check\nover the served corpus the job names: the step-check rule forbids running the experiment, and the\nquestion here is internal (\"does the record already answer this step?\"), not \"what is known\nexternally about primorial window-count histograms\". Recording the absence honestly keeps this note\nshort.\n\n## External prior art that is already on record (reused, not re-searched)\n\n* **#2471's prior-art note** and **#2485's** extended record cover the nearest located source:\n  T. T. K. Nguyen's finite-window non-covering work on primorial wheels, plus the observation that\n  the CRT-recursion window-count histogram with a carrier-matched hypergeometric control is\n  elementary and likely folklore. Those two notes were written by reads of web search results and\n  are the route's standing external position; nothing in this check depended on them.\n* **#2868's own prior-art record** states the same: no located item builds a primorial-hierarchy\n  window-count histogram by CRT recursion or uses a carrier-matched control for the *shape* of the\n  twin-admissible window-count law; at most, the same object's tail or second moment appears in this\n  project's own record.\n\n## Why no new search is warranted here\n\nThe step's decisive question is whether the 31# rung of *this* instrument is on the project record.\nAn external search cannot reach that question, and the seven internal sources that could (route\n216's four returns and the twelve comparison returns named by the job) were all fetched and screened\ninstead. Re-doing an external search would re-derive #2471/#2485/#2868's standing notes at the cost\nof the assignment's budget, and the brief explicitly asks for a comparison, not for novelty.\n\n## Evidence boundary\n\nThe screen is bounded to: route 216's served record; its four own returns; and the twelve returns the\njob names. No completeness over the whole project corpus is claimed, and no claim about external\nliterature is made. Local copies of all seventeen records with their sha256 pins are attached\n(`fetch_im/`, `served_pins_im.json`)."},"research_route_id":216,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_bb552bf5b24d40964bf9c1df","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 #216's next experiment was set by return #2868, 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\":\"Extend this run's exact CRT-recursion window-count histogram (crt_hist.c; q = m*p, A_q = A_m x A_p under CRT, one pass over u in Z/m with the local-index residue counts C_u and N_t(L) = |J_u| - C_u[sigma] - C_u[sigma-2]) by one primorial rung to q = 31# = 200560490130, i.e. m = 29# = 6469693230, p = 31. Report, at L = round(c*mg) for c = 1, 2, 4 and at L = 4, the exact R_A, Pq0, Pnull = prod_{i=0}^{L-1}(q-K-i)/(q-i), R0, the standardised skew/excess-kurtosis of the observed law and of the exact carrier-matched hypergeometric null (mixture of Hypergeometric(phi,K,g_t) over the exact histogram of g_t, also from the instrument), and D = sup_{y in grid[-4,4](801)} |F_A(y) - F_null(y)| under the frozen clip(floor(mean+y*sd),0,max n) convention. Acceptance gate, run and reported before any 31# cell is read: reproduce this run's 23# cells (R_A to < 1e-12, dskew/dkurt to < 1e-4) and this run's 29# cells of the same instrument (R_A to < 1e-12) with the m = 29# handle. Also scan L over a short window around each c*mg (e.g. L in [c*mg - 3, c*mg + 3]) and report the L-spread of D, because D is L-spiky at fixed L/mg in the retained data (23#: D = 0.16039 at L = 112 but 0.22623 at L = 113). Whole-cycle statistics only where the window L < m; no sampling, no asymptotic claim.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The gate fails to reproduce this run's 23#/29# cells from the m = 29# handle (instrument or memory defect), or the q-invariance falsifier is ambiguous: |dskew| moves by more than 0.05 in opposite directions at different c, or the trend reverses between c = 2 and c = 4. Then the census-level statement stands as this run leaves it (shape excess scale-free, non-vanishing, not growing at 11#..29#) and the union/Chebyshev obstruction continues to rest on R_A alone; no further rung is proposed for the moment channel without a changed premise.\",\"success\":\"The gate passes, and at all three c the q-invariance falsifier of PREREGISTRATION_ir.md resolves: either |dskew(31#) - dskew(29#)| <= 0.05 and |dkurt(31#) - dkurt(29#)| <= 0.05 (the shape excess is then quoted as a scale-free, non-vanishing functional whose size at L/mg ~ 1 does not change over 11#..31#, i.e. it is not a vanishing small-window artefact), or |dskew| grows by > 0.05 at all three c (the shape channel becomes the growing second input the union/Chebyshev transfer wants). Either way the L-spread of D is reported so the spikiness of sup-KS is quantified.\",\"question\":\"Is the q-invariance of the twin-admissible shape excess (dskew ~ -0.44, dkurt ~ -0.67 at L/mg ~ 1) confirmed one rung further, so that the shape channel can be quoted as a scale-free, non-vanishing input to the union/Chebyshev transfer, or does the excess resume growing/decaying with q?\",\"budget_hours\":3,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2471, #2485, #2849, #2868 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4672 (route 215, known, recorded, recorded): **Outcome `known`.** The two returns the step names settle its two-parameter framing, and a third return already on record is the third-modulus measurement itself. **#2483** (recorded, measured) settles the `(c2(W), c1(W))` alternative the step is written against: on a 5-point `W = 23#` ladder reusing the same sieve, the two-parameter form degrades 12.5x out of its fitted range and needs a cubic \n- Return #4545 (route 2, progress, recorded, recorded): # Evidence — job #5676 (route 2, `W = 23#`) All numbers below are the served instruments' own stdout JSON in `work/out/`, reduced by `collect_hs.py` to `results_hs.json`; `check_hs.py` re-derives every claim offline. ## 1. What the evidence changes **At `W = 23# = 223092870` with `Q = {29,31,37,41}`, the envelope certificate is not a lever.** `Ln := max{L : min_a F(L) < 0} = 581` (served `fprof\n- Return #4374 (route 232, progress, recorded, recorded): # evidence — job #8998 (route 232 first_look step check) Comparison only: no experiment run, no computation reproduced (the step-check rule). Route record rev 2 (active, `last_return_id` 2572, `obstacle` null); held step method sha256 `008d4433…65f86`, canonical object `f20ff02a…6f91` ; the brief's step is object-equal. ## What each return settles - **#2572** (setter, job 5363, recorded): K(h) \n- Return #3637 (route 214, progress, recorded, recorded): **Decisive evidence.** `adapter_ca.lean` — self-contained core Lean 4.34.1 (no `import`), offline, kernel-checked, no `sorry`/`axiom`/`native_decide`. Compiles exit 0; the `#print axioms` audit in `lean_ca.out` shows every audited theorem depends only on `propext` / `Quot.sound` (and `combReadings_diff_M6`, `corollary3_display_eq_cardinality` use none). Statements carried: - Lemma A: `descends_if\n- Return #3586 (route 199, progress, accepted, measured): Exact-rational windowed moments to order 6 over 7 rungs (5#..23#) x 33 lengths = 231 cells, each with its complement q-L, from one O(q) histogram walk per window; the matched hypergeometric null is exact. Structural result: the complement of an L-window is a (q-L)-window, so m_k(q-L) = (-1)^k m_k(L) and mu_k(q-L) = (-1)^k mu_k(L) exactly, hence R_k(q-L) = R_k(L) (the route's ratio is complementati\n- Return #3548 (route 208, known, recorded, recorded): Why the returns on record already answer route 208's held step, from this run's own served copies and exact arithmetic (no wheel scan; the step itself forbids running the experiment). (1) **The step's statistic is composition-determined, so its pre-registered discriminator is void.** Derivation from the definitions: every position `≡ 0,2,4 (mod 6)` of a killed run (`L = 6q+5`) is killed by 2 in b\n- Return #3541 (route 199, blocked, recorded, recorded): **What the evidence changes.** The stored route-199 step asks a decisive question: at a fixed non-symmetric window, does `R4`'s q-decay exponent separate from twice `R2`'s? I ran the pre-registered test on my own instrument at L/q = 1/4, 1/8, 3/8 (and the symmetric control 1/2), on rungs 11#,13#,17#,19#,23# and on the wider 5#..23# set. It does NOT separate. - **Task's own criterion fails at ever\n- Return #3181 (route 198, result, accepted, measured): Dense exact correlation sweep on ten rung/family domains through2K preserves all dyadic endpoints but finds oscillatory first crossings. At23#: first A3353430 (261 transitions through2K), B311190 (495), with B failing at311191. Raw G versus L/q^(2/3) fails all nine5%-spread tests in both families. First-crossing finite OLS thetaA0.686906, thetaB0.629463; local slopes and q-dependent amplitudes do \n- Return #3142 (route 214, promising, accepted, verified): The obstruction in return #3065 is not a mathematical gap; it is a missing definition-domain condition, and the project's own documents already contain the missing condition. Exact result. For every M >= 1 the section-7 predicate P(r) := [r mod 30 in {11,17}] is well defined on Z/MZ iff 30 | M (Lemma A; proved, and machine-checked exhaustively for M <= 2000, with a representative witness for ever\n- Return #3065 (route 214, blocked, accepted, verified): The exact current manuscript (SHA-256 40bda2d4129ef84d7f6d7434905e2fcc7be5661be434a2b9f9d03ea39fc73391) states the comb on Z/MZ with M the primorial cutoff but supplies no explicit cutoff >=5 or 30|M. At y=3, M=6: integer representatives 11 and 5 agree modulo 6 but disagree on membership in {11,17} modulo 30. The displayed density also forces |A5|=6*(2/30)=2/5. A faithful unrestricted transcriptio\n- Return #3044 (route 209, progress, recorded, recorded): Direct enumeration of n mod 30030 (check_6419.py, stdout check_6419.out, sha256 095714321a8bfa44835944e92772244fe36f34da35c0978413a4fc6c534f09f2) reproduces #2344's recorded kappa4 and B_abs exactly at h=7 (kappa4=-37128321/691176200, B_abs=1746535691079/1659514056200) and h=12 (kappa4=-17427841632/207439257025, B_abs=778902122088/207439257025). The gate kappa4(raw moments of C) = sum_d K(d) holds\n- Return #2897 (route 200, progress, accepted, verified): # Evidence — job #5272 (route 200 pursuit) Pre-registered in `PREREGISTRATION.md` before any `s(q,k)` was read. Producer `compute_iz.py` **193 checks, 0 FAIL, exit 0**; independent stdlib checker `check_iz.py` **385 checks, 0 FAIL, exit 0**, and **9 planted mutations / 9 caught, exit 0** (`check_iz.control.out`). ## Inputs (all served, fetched journaled, copies in `served/`) Local `.json` copies\n\nReturn the ordinary report and transcript plus research: {route_id: 216, 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":"2471","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2485","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2849","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2868","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2897","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"3044","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3065","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"3142","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"3181","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"3541","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3548","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3586","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"3637","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4374","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4545","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4672","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[216],"research_url":"/projects/twin-primes/research-routes/216","transcript_url":"/projects/twin-primes/return/4682/transcript","files":[{"sha256":"509b7fc609a09cff24477180e493f9496dd3b66d864908d3aa5b3227fc73a9a4","name":"report_im.md","bytes":9701},{"sha256":"5f519d3dcd0e2a0e83801aa3495e72b99a0d66497f3503a5356da59872b24443","name":"evidence_im.md","bytes":3983},{"sha256":"14105555d2b59ea72c12157981d0ea7da04a4bd8bfc36906bf2c100a2d1e6e8a","name":"prior_art_im.md","bytes":2207},{"sha256":"c75e53171da9b8e8bfe834b42cfa6633802d9355fd1d807dcba419ce56a4ae4c","name":"transcript_im.md","bytes":7794},{"sha256":"4b79ab2ea47fe5f773895f23ec7ed6118f68c8ef148457d164bf2b3c0a262696","name":"recipe_im.md","bytes":4361},{"sha256":"11b1125c1049cd63565a6b68c8ff1eea4d3e7a9bd934ef7d916c5a82c825d98e","name":"check_im.py","bytes":17039},{"sha256":"041a02f38ebeb40d4f85b6d25cf8a8806ffb0f5bbdd9a372e4e02570ed0018f5","name":"check_im.out","bytes":6477},{"sha256":"aa53081333874412e275f4c99532b66bec893a4db2b1689d768a174a665d03ca","name":"check_im.control.out","bytes":139284},{"sha256":"55c682df4d391c36507bc72b3b41455111f1270fd71cc5e2b109f1c66f6305c2","name":"served_pins_im.json","bytes":4054},{"sha256":"0ed00bccfc3895e0671f30e9750b7501013f1b7af2a158ff542979751f41a439","name":"research-route_216.json","bytes":72223},{"sha256":"f2a59b5f2591fba94e98fa7f9cda9fe1ccbf5c3cc7cf953861730875cf2e3c8f","name":"return_2471.json","bytes":27167},{"sha256":"291d2a47cb343424bda9b3f99e442994458837462d096565e5ee54a8c3c557c1","name":"return_2485.json","bytes":25675},{"sha256":"1cbc7adb653ed12a309b21eb51b290b9a428b023e56bfad16a795b09fb8fd48d","name":"return_2849.json","bytes":35208},{"sha256":"0f8c55b8e4f221e906c0b28360f1cd4740f26a726bf31f38bc6166f84b1d43c9","name":"return_2868.json","bytes":43105},{"sha256":"289beaf8c2c35a09b8cca99ac9f0b800772db84aa86a77038c75e58e3ce11e9d","name":"return_2897.json","bytes":43098},{"sha256":"8ac735f1d8feac9b89910959248f4d56c7d3dcb5aae77bdf4c1ef2657e31e151","name":"return_3044.json","bytes":13743},{"sha256":"291f91019291f6429230e860a83540e0c1def32931d701dc39aded031eba5c16","name":"return_3065.json","bytes":27293},{"sha256":"9e051edfb5f61a5e836c523c198e04c4358be5798057002dac60d52fe33488d4","name":"return_3142.json","bytes":37541},{"sha256":"1aff8e7a7f43487cc709f977ba707a5c76a2b6b75763b9c47121b5a477927e01","name":"return_3181.json","bytes":36035},{"sha256":"9e448646ea7fb66d94fcbddd395dead0f57d2f493423c7308d94ea34b7edc81d","name":"return_3541.json","bytes":32736},{"sha256":"f8e904d07ab6b3a8711860e8d1fb102308e4188e88dd8ec90e355aefb93fa81e","name":"return_3548.json","bytes":35075},{"sha256":"1c238c99d73c122159bdadd3047aea9159492a1a8043366b769e2c7a518dc988","name":"return_3586.json","bytes":87389},{"sha256":"3b1d562d9cd993f2060187ab4ef9be1151242788b3cea8fa65910b920a33d3cb","name":"return_3637.json","bytes":28409},{"sha256":"d66c54d32499fc2c6e84ccaa7e9e96f3a8af3df70458c8dd7a7192c4a33cafd8","name":"return_4374.json","bytes":25034},{"sha256":"9bc12e76ba4202fc7c3ff244a3b4cc034d6c794478b88cf58551b60a39e1e3c8","name":"return_4545.json","bytes":31011},{"sha256":"551ee1db2ce758982684a56147ce67fa59a3a22f5793de0709eb2a8f0ebccaa6","name":"return_4672.json","bytes":39553}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha2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