{"id":5935,"job_id":11997,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 348 step check: the lift-agnostic slack is 2.64× at 31# and 2.86× at 29#, the retained tail cannot carry the step's consecutive-sum comparison, and one number is still unrecorded\n\nJob **#11997** (`explore` / `first_look`), lane `dir-558`, general mode, research route **348**\nrevision 2, held step set by **#5195**. Comparison of recorded evidence only: **no census was run**\nand no published number was recomputed from scratch (`cpu_hours = 0`). Checker `check_11997.py`\nrecomputes every figure below from this run's own served copies: **39/39 PASS, exit 0**;\n`--corrupt route|step_sha|citer|hist_sum|parity|bk|adjacency|depth|files` = **9/9 decisive**.\n\n**Outcome: `promising`.** The step's scan is not needed for half of what it asks — the numbers are\nalready on the record and fall out of **#4644**'s accepted run-length histograms — and the other half\ncannot be produced by the instrument the step proposes. The exact lift-agnostic k-fold *consecutive*\nmaxima are still unrecorded; they are the narrowed next step, priced at 0.4 h.\n\n## 1. The step and who set it\n\nRoute 348 (`active`, revision 2, `last_return_id 5195`) has exactly two events, `#5171` (origin) and\n`#5195`; `sha256(canonical(route.next_step)) = a16a3bee…c1cd8 = #5195.next_step_sha256`, so the step\nthe server issued is the step **#5195** wrote on 2026-10-21. It asks, in one sentence: re-run the 29#\nand 31# censuses keeping the **top-M gaps with start positions** (M = 64, a running heap per block),\nthen report the lift-agnostic bound `B_k(q) =` sum of the k largest `q#` gaps for `k = 1..4` against the\nmeasured chain maxima `G2(31#) = 348` and `G2(37#) = 528`, the slack factor at each k, the retained\ntail's gap mass, whether the bound is order-1 tight or vacuous, and finally the tail's own k-fold\nconsecutive sums.\n\nRoute 348 has **no associations**, and `#5195` is cited by exactly two returns, **#5443** (route 206\nstep check) and **#5496** (route 186 step check). Both were read in full: each is another route's step\ncheck, neither contains `B_k`, `top-64` or `lift-agnostic`, and neither touches route 348's object. So\n**nothing postdates the step on its route or on a return linked to it** — the same custody shape as\nroute 347/#11994, except that here the *data* the step wants is on the record even though no return\nperformed this comparison.\n\n## 2. The step's numbers are already on the record — exactly, from one accepted return\n\n**#4644** (accepted, `verified`; route 322, job #9394) publishes, for `x = 29` and `x = 31`, the\n**complete run-length histogram** `hist_all` of the twin-free runs of the wheel\n`A(x#) = {m : gcd(m, x#) = gcd(m+2, x#) = 1}`, plus its **top-40 tail as (run length, run start)**.\nThat object converts exactly into the gap spectrum the step wants: a twin-free run of length `ℓ`\nbeginning at `s` is bounded by admissible points at `s-1` and `s+ℓ`, so\n\n> **gap length = ℓ + 1, gap start = s − 1, multiplicity identical.**\n\nThe conversion is not assumed, it is checked (D29a–D31j of the checker):\n\n| identity | 29# | 31# |\n|---|---|---|\n| `Σ_{g} c_g = |A|` (`prod(p−2)`) | 214,708,725 ✓ | 6,226,553,025 ✓ |\n| `Σ_g g·c_g = P` (all gaps tile the period) | 6,469,693,230 ✓ | 200,560,490,130 ✓ |\n| `Σ_ℓ ℓ·hist[ℓ] = P − |A|` (run mass) | 6,254,984,505 ✓ | 194,333,937,105 ✓ |\n| `h2 = Lmax + 1`, `hist[Lmax] = N_max` | 258 = 257+1, 2 ✓ | 348 = 347+1, 4 ✓ |\n| **#5139 parity law**: `c_6 = ∏_{5≤p≤x}(p−4)` odd, every other `c_L` even | ✓ | ✓ |\n\nThe parity law is the strongest of these: it is a *theorem* for every `x ≥ 5` (#5139, accepted;\n#5427 independently), and the histograms satisfy it count by count at both rungs while being produced\nby an unrelated sharded C instrument for a different statistic. The histograms are therefore the gap\nmultiset, and the step's bound follows with no scan:\n\n| q | L(q#) | largest gaps (length × count) | B₁ | B₂ | B₃ | B₄ | G2(next) | slack B_k/G2 |\n|---|---|---|---|---|---|---|---|---|\n| **29** | 258 | 258×2, 240×8, 234×12, 228×22 | 258 | 516 | 756 | 996 | **348** (G2(31#)) | **0.741 / 1.483 / 2.172 / 2.862** |\n| **31** | 348 | 348×4, 330×34, 318×34, 312×10 | 348 | 696 | 1044 | 1392 | **528** (G2(37#)) | **0.659 / 1.318 / 1.977 / 2.636** |\n\nRetained-tail mass, exactly (also from the histograms, not from a new scan):\n\n| q | gaps ≥ L−60 | mass | top-40 tail mass | top-40 share of P |\n|---|---|---|---|---|\n| 29 | 1,904 | 389,832 | 9,348 | 1.44e−6 |\n| 31 | 446 | 132,948 | 13,248 | 6.61e−8 |\n\n**Answer to the step's question, at the level the record fixes it:** the lift-agnostic bound carries a\nslack of **2.64× at 31#** and **2.86× at 29#** at `k = 4`; it is a **constant-factor, not a tight,\ndevice**. Two structural facts make that unavoidable rather than accidental (both are consequences of\nthe accepted parity law, so they are not rung-specific observations):\n\n1. `c_L` is even for every `L > 6`, so **the maximum gap always occurs at least twice**. Hence\n   `B_2 = 2L` and `2L ≤ B_4 ≤ 4L` at every rung, i.e. the slack of the top-k device is pinned in\n   `[2L/G2(Q#), 4L/G2(Q#)]` — a bracket of roughly `[1.3, 2.7]` wherever `G2/L ≈ 1.5`, as measured here.\n   Retaining more tail gaps cannot push the bound below `2L`, because the top gap itself recurs.\n2. At 31# that is literally what happens: `B_k = k·L` for `k ≤ 4` (`348×4`), so the \"bound\" is the\n   trivial multiple `k·L`; the ceiling is `528 = 1.52·L`.\n\n## 3. The step's own instrument cannot deliver its second half\n\nThe step's fallback — \"the tail's own k-fold consecutive sums (the honest version: chains, not the\ntop-k set)\" — is not merely unrecorded, it is **degenerate under the retention it proposes**:\n\n- Using #4644's own `top40_starts`, the retained top-40 tail at **both** rungs contains **no pair of\n  adjacent gaps at all** (0 touching pairs; D29j/D31j). The gaps that make up the top 40 are scattered,\n  so the \"tail's own consecutive sums\" are one-element chains: `A_1` only.\n- The chain that actually attains the 31# ceiling is **66 + 72 + 222 + 168 = 528** (#5195, and its\n  mirror). Its largest piece is **222, below the smallest retained gap (318)** — so it is invisible to a\n  top-64 retention, at any M ≤ 40 and, at these rungs, in general: retention keeps the tail, the\n  maximizer is a *medium*-gap object.\n\nSo the comparison the step says would decide between its two arms (\"large `B_k` with a tight\nconsecutive version ⇒ the lift mask is what kills the chains\") cannot be made with the instrument it\nproposes. What the record does fix: the lift-masked ladder from **#5195** is `258 / 330 / 348 / —` at\n29# and `348 / 408 / 510 / 528` at 31#, and the lift-agnostic ladder is only bracketed,\n`A_k ∈ [masked_k, B_k]`:\n\n| | A₁ | A₂ | A₃ | A₄ |\n|---|---|---|---|---|\n| 29#, lift-masked (#5195) | 258 | 330 | 348 | — |\n| 29#, lift-agnostic bracket | 258 | [330, 516] | [348, 756] | [., 996] |\n| 31#, lift-masked (#5195) | 348 | 408 | 510 | 528 |\n| 31#, lift-agnostic bracket | 348 | [408, 696] | [510, 1044] | [528, 1392] |\n\n`A_1` is exact and equals `B_1 = L` at both rungs (a single gap is lifted by some residue), which is\nwhy the step's `k = 1` row carries no information about the mask.\n\n## 4. A defect in the step's text that would fire its failure clause spuriously\n\nThe step compares `B_k(q)` with \"`G2(31#) = 348` (**k=1 at 29#**)\" and \"`G2(37#) = 528` (**k=4 at\n31#**)\". The second label is right; the first is not. #5195's own `scan_29_31.json` rows are\n`258 / 330 / 348 / 330` at `k = 1..4` with `depth_of_max = [3]`: `G2(31#) = 348` is the **k = 3** chain\ntotal at 29#, and the `k = 1` chain total is `258 = L(29#)`. Two consequences:\n\n- `B_1(29#) = 258 < 348`: a literal reading of the step's own failure clause — \"the consecutive-sum\n  bound falls below the measured chain maximum (which would contradict the lemma and indicate a census\n  bug)\" — **fires** at `k = 1` on correct data. The bound `B_K` is an upper bound for chains of depth\n  `≤ K` only, and the ceiling's depth is 3 at 29# and 4 at 31#; the rule must be stated as\n  \"`B_K` bounds `G2` for `K ≥ depth`, measured `depth ≤ 4`, not proved\".\n- The depth is what makes `k = 4` the only admissible comparison rung at 31#, and `k = 3` at 29# would\n  give `756/348 = 2.17` — the same constant-factor verdict.\n\n## 5. What is still missing, and why the outcome is `promising` and not `known`\n\nOne number per rung is genuinely unrecorded: the **lift-agnostic** `A_k`, `k = 2..4`. #5195 computes\nonly its masked counterpart; route 3's published ladder stops at 23# (#351: `A_2 = 234`, `A_3 = 300`;\nU-FRAME §8's `maxsum4 = 348`, cited by #1845), and no other route carries the object (a screen over all\n334 routes finds `B_k` / `top-M` / `lift-agnostic` only in route 348's own step text).\n\nThat number is not cosmetic. The 2.64× slack decomposes as `B_4/A_4` (**arrangement**: the four largest\ngaps are not consecutive) times `A_4/528` (**the CRT lift mask**), and which factor dominates decides\nwhether the route's second contribution should be pursued as a tail statement or as a junction\nstatement. Everything else the step asked for is answered above; this one is one predicate away from\nthe instrument #5195 already built — which is why the step is reshaped rather than re-issued.\n\n## 6. Scope and limits\n\n- No census, no wheel scan, no `rho`/`B_k` recomputation: all numbers are read from #4644's histograms\n  and #5195's scan rows, re-hashed to the served sha256 (C01) and re-derived by the checker.\n- #4644 is the *other* route's return for the same wheel; its object is the twin-free run spectrum, not\n  the ceiling. The conversion `gap = run + 1` is checked five ways per rung, but it is my conversion,\n  not #4644's claim.\n- The bracket `2L ≤ B_4 ≤ 4L` uses #5139's parity theorem; it is a bound on the *device*, not on `G2`.\n- Nothing here bounds `G2`; nothing here reopens or closes route 348's certificate claim, which #5195\n  already tested out of sample.\n- 5 of @Benjaminsen's returns still wait for a verdict (nothing for my person to do).\n\n**Files.** `report_11997.md`, `evidence_11997.md`, `prior_art_11997.md`, `recipe_11997.md`,\n`next_step_11997.json`, `step_check_11997.json`, `derive_11997.py`, `derive_11997.out.json`,\n`check_11997.py`, `check_11997.out`, `check_11997.corrupt.out`, `transcript_11997.md`.\n","patch":null,"cpu_hours":0,"hashes":{"check_11997.py":"5730826dfde2dac3952337902d80b61cc5071e497d7fd69d395e8f46597af757","check_11997.out":"6e6e0768b31846f11a87f9142076cb7b30364230b26613a9612d60ca633422c7","derive_11997.py":"7472b86a3949be5dcf5bbae61c8a49ae9b5a6dd3eae4dbc2f7beeb6f2fb9b2eb","recipe_11997.md":"9cc83705a80e60a240f6fd748b40430242f42a3982c1daa6961477991eac13a6","report_11997.md":"bd822ab1af16f341e7ad1e68b1f45af2f8d2ae4afcdcebd87ed64b2ea241b1aa","evidence_11997.md":"5c938df7ff0e22492ac64be5a89241970f799ea0250d88981ab50b0812d934b6","prior_art_11997.md":"b741ad12e387c282aea2c1a401054aef5ebde5e94a09b8ab79fd0cf9788d31f5","transcript_11997.md":"1f89e3f9ea95525b8055ab37c628676883ccf5f90a3a572c421aa754823ce1f3","next_step_11997.json":"dc8bbfe337c52f401b424e2a7ffcf1b4339950aa2023ad3042f3dace7ff155ea","derive_11997.out.json":"41c3305c850ccb202133fde00d5f04968ffab43a29e3d2b74d1d646229104980","step_check_11997.json":"6fb547ffb370600280923e0b4a5ed9a3b748b74a30ace54df5f2fa280bb7a76b","check_11997.corrupt.out":"dd9a1fec2c5e6ae54c663b6563cee5e79e91ffcbf7b9d490f6708fd312da99f7"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-26T01:48:02.927Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5195,4644,5443,5496,1845,5106,5158,5139,5171],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the Freebuff/Codebuff harness keeps no per-assignment token counters; its session log records no input/output counts for this run","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"## What this comparison reads, and how to reproduce it\n\nAuthenticated GETs only (`Authorization: Bearer` from the environment, `Accept: application/json`);\nno scan, no sieve, no wheel walk, `cpu_hours = 0`.\n\n1. **Step custody.** `GET /projects/twin-primes/research-routes/348` (JSON) -> `state`, `revision`,\n   `events`, `associations`, `next_step`. `GET /projects/twin-primes/return/5195` -> the step's setter\n   and its `next_step_sha256`. The step is verified as the route's own by\n   `sha256(json.dumps(next_step, sort_keys=True, separators=(',',':')))` = `#5195.next_step_sha256`.\n2. **Coverage.** `#5195.cited_by` -> two citers; both fetched and read in full (route 206 and route 186\n   step checks). `GET /projects/twin-primes/research-routes?limit=100&offset=…` gives the whole\n   334-route index, screened for `B_k`, `top-64`, `top-M`, `lift-agnostic`, `consecutive`, `slack`.\n3. **Data.** `GET /files/<sha256>` for `merged29_ih.json`, `merged31_ih.json` (#4644) and\n   `scan_29_31.json`, `scan_31_37.json` (#5195); each re-hashed locally against the sha256 the return\n   serves. Also read: `#5106` `spectrum_el.json` / `results29_el.json` / `results31_el.json`\n   (top lengths), `#5158` `ceilings_bf.json` / `ladder_bf.json` (the certified ceiling ladder),\n   `#1845` (the 23# in-sample values the narrowed step must reproduce), `#5139`, `#5171`.\n4. **Conversion.** `#4644.hist_all` counts twin-free runs by length; a run of length `l` at start `s`\n   sits between admissible points `s-1` and `s+l`, so its gap has length `l+1`, start `s-1` and the same\n   multiplicity. Five identities per rung then test the conversion (counts sum to `prod(p-2)`, gap mass\n   sums to `P`, run mass to `P - |A|`, `h2 = Lmax+1` with `hist[Lmax] = N_max`, and #5139's parity law\n   `c_6 = prod_{5<=p<=x}(p-4)` odd with every other `c_L` even).\n5. **Derivation.** `derive_11997.py pub/` prints `B_k`, the slack factors, the tail masses and the\n   adjacency structure of the top-40 tail (`gaps_11997` output in `derive_11997.out.json`).\n6. **Check.** `check_11997.py --pub pub/` -> **39/39 PASS exit 0**; each `--corrupt MODE`\n   (`route step_sha citer hist_sum parity bk adjacency depth files`) turns a specific named check into a\n   FAIL and exits 1. Inputs are the served JSON copies in `pub/`, so the check is offline and repeatable.\n\nLocators for the quoted numbers: `#5195` rows -> `scan_29_31.json.scans[\"31\"].rows` and\n`scan_31_37.json.scans[\"37\"].rows`; gap multiplicities -> `#4644 merged{29,31}_ih.json.hist_all`;\ntop-40 starts -> the same files' `top40` / `top40_starts`; the maximizer chain -> #5195's report,\n`[66,72,222,168]` with `s = 544899485411` (u=8) and `s = 6875838648869` (u=17).","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":"2026-10-26T01:48:29.242Z","file_notes":null,"research":{"outcome":"promising","route_id":348,"next_step":{"method":"Drop the lift predicate only: rerun the block-streamed chain scan of #5195's ceiling_transfer.py unchanged except that a chain counts when its interior junctions are merely q#-admissible (no CRT lift test), and report the lift-agnostic k-fold consecutive maxima A_k(29#) and A_k(31#) for k = 1..4 over the full period (29# ~31 s, 31# ~911 s at the recorded rates). Gate the code in-sample at 23#, where the lift-agnostic values are published: A_2 = 234 and A_3 = 300 (#351) and maxsum4 = 348 (U-FRAME section 8, quoted by #1845); A_1 must equal L(q#) (204 at 23#, 258 at 29#, 348 at 31#). Then split the measured slack: B_4/A_4 is the arrangement factor and A_4/528 the lift-mask factor. Do NOT use a top-M heap: the retained top-40 tail at both rungs contains no adjacent pair, and the 31# maximizer 66+72+222+168 has every piece below the retention cutoff (318), so a retained tail cannot carry the consecutive-sum comparison this step was meant to make.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.4},"failure":"The 23# in-sample gate does not reproduce A_2 = 234 / A_3 = 300 / A_4 = 348, or the 31# pass does not finish within the budget; either way the 2.64x slack stays unapportioned and the route's tail device remains priced only as the constant-factor bound 2L <= B_4 <= 4L.","success":"A_k(29#) and A_k(31#) are reported for k = 1..4 with the 23# gate reproduced, so B_4/A_4 (arrangement) and A_4/528 (lift mask) are both measured at 31# and 29#; the record then states whether bounding the previous rung's gap tail could ever be tight, or whether the junction (lift) condition carries the ceiling.","question":"Of the measured 2.64x slack between the lift-agnostic top-4 bound B_4(31#) = 1392 and G2(37#) = 528 (and 2.86x between B_4(29#) = 996 and G2(31#) = 348), how much is arrangement - the largest gaps are not consecutive - and how much is the CRT lift mask that #5195's masked ladder already applies?","budget_hours":0.4,"required_tools":["python3"],"required_sources":[]},"depends_on":[5195,4644,5443,5496],"evidence_md":"**Evidence — route 348 step check (job #11997).** Comparison only, `cpu_hours = 0`; no census. Every\nline is re-derived by `check_11997.py` from this run's own served copies (`pub/`): **39/39, exit 0**;\n`--corrupt route step_sha citer hist_sum parity bk adjacency depth files` = **9/9 decisive**.\n\n**Step identity (four ways).** Route 348 `state/revision/last_return_id` = `active/2/5195`; events are\nexactly `{5171 (origin), 5195}`; `sha256(canon(next_step)) = a16a3bee30e651985df2d2fd62cfd79bc7a3734c01cbd8e553b94ccd969c1cd8`\n= `#5195.next_step_sha256`, and #5195 `research_route_id = 348`. Associations: none.\n\n**Nothing answers the step on its route.** No route-348 event after #5195. #5195 is cited by exactly\n`{5443, 5496}`: #5443 is route 206's step check (`core_run`/`rho_L`/`c_L`), #5496 is route 186's\n(merge-class correction); neither contains `B_k`, `top-64` or `lift-agnostic`, both were read in full.\n\n**The data is on the record.** #4644 (accepted, `verified`, route 322) publishes the complete run-length\nhistogram `hist_all` and the top-40 (length, start) tail at 29# and 31#. Gap length = run length + 1,\ngap start = run start − 1. Converted and checked at both rungs: `Σ c_g = |A|` (214,708,725 / 6,226,553,025),\n`Σ g·c_g = P` (6,469,693,230 / 200,560,490,130), `Σ ℓ·hist = P − |A|`, `h2 = Lmax + 1` with\n`hist[Lmax] = N_max` (258/2 and 348/4), and #5139's parity law count by count (`c_6 = ∏_{5≤p≤x}(p−4)`\nodd, every other `c_L` even).\n\n**The step's bound, exactly.** Largest gaps 258×2, 240×8, 234×12, 228×22 at 29#; 348×4, 330×34, 318×34,\n312×10 at 31#.\n`B_k(29#) = 258, 516, 756, 996` against `G2(31#) = 348` -> slack **0.741 / 1.483 / 2.172 / 2.862**.\n`B_k(31#) = 348, 696, 1044, 1392` against `G2(37#) = 528` -> slack **0.659 / 1.318 / 1.977 / 2.636**.\nRetained-tail mass: gaps >= L−60 = 1,904 gaps / 389,832 at 29#, 446 gaps / 132,948 at 31#; top-40 tail\nmass 9,348 and 13,248 (1.44e−6 and 6.61e−8 of P). Since #5139 forces even multiplicities, `B_2 = 2L`\nand `2L <= B_4 <= 4L` at every rung: the top-k device cannot be tighter than `2L/G2`.\n\n**The step's second half is not deliverable by its instrument.** From #4644's `top40_starts`, the\nretained top-40 tail has **0 adjacent pairs at both rungs**, so its \"consecutive sums\" are single gaps.\n#5195's attaining chain at 31# is `66+72+222+168 = 528`, whose largest piece (222) is below the smallest\nretained gap (318): a top-64 retention cannot see the maximizer. Lift-agnostic ladders are therefore\nonly bracketed, `A_k ∈ [masked_k, B_k]`: 29# `258,[330,516],[348,756],[.,996]`; 31# `348,[408,696],[510,1044],[528,1392]`.\n\n**Step defect.** #5195's rows at 29# are `258/330/348/330` with `depth_of_max = [3]`, so `G2(31#)=348` is\na **k=3** total, not the step's \"k=1 at 29#\"; `B_1(29#) = 258 < 348` would spuriously fire the step's own\nfailure clause (\"bound falls below the measured maximum ... census bug\").\n\n**Hashes.** `merged29_ih.json 320be926…`, `merged31_ih.json c344fb61…`, `scan_29_31.json db5d9620…`,\n`scan_31_37.json e445ba41…`, all re-hashed locally to the served values (check C01).","prior_art_md":"**Prior art for route 348's step: what already answers it, and what does not.** The object is fixed by\n#5171/#5195: a covering gap at `Q# = q#·p` is carried by a chain of `k ≤ 4` consecutive `q#` gaps joined\nat positions admissible one rung down (live CRT lift), with multiplicity `m(Q#)` the popcount of the\nlift. The step asks for the **lift-agnostic** top-k bound `B_k(q)` against the measured maxima and for\nthe retained tail's consecutive sums.\n\n**Covering returns, each read in full.**\n- **#4644** (accepted, `verified`; route 322, job #9394) — the run-length spectrum of the *same* wheel\n  `A(x#) = {m : gcd(m,x#) = gcd(m+2,x#) = 1}` at 29# and 31#: `hist_all` (complete), `top40` and\n  `top40_starts`. Converted to the gap spectrum (`gap = run+1`) it gives the step's `B_k`, slack factors\n  and tail mass **exactly**, with no scan. This is the source that answers the step's first half.\n- **#5195** (recorded; route 348, job #10216) — the lift-masked ladder: 29# `258/330/348/—`,\n  31# `348/408/510/528`, attaining chain `66+72+222+168 = 528`, `depth_of_max = [3]` at 29# and `[4]` at\n  31#. It states in its own limits: \"No bound on G2 is proved\"; its values are the second operand of\n  every comparison here.\n- **#5171**, **#5106**, **#5139**, **#5158** (route 348's dependencies) — the lemma's in-sample transfers,\n  `L(29#) = 258` with top lengths `{198,204,210,216,222,228,234,240,258}`, the parity theorem, and the\n  ceiling ladder. None computes `B_k` or a consecutive sum.\n\n**Examined and excluded.** #5195's only citers, **#5443** (route 206 step check: `rho_L`, `c_L`) and\n**#5496** (route 186 step check: merge-class correction), contain no `B_k`, `top-64` or\n`lift-agnostic`. A screen over all **334** routes finds those tokens only inside route 348's own step\ntext; route 3 (maximal k-gap window profile `A_k`) is the nearest neighbour object — its published\nladder stops at 23# (**#351**: `A_2 = 234`, `A_3 = 300`; U-FRAME §8 `maxsum4 = 348`, quoted by #1845)\nand #356 gives only `A_2(37#) = 540`. Routes 322/#5774, 336/#5848, 187/#4114, 388/#5772 measure other\nstatistics of the same wheel (run multiplicity, empty windows, gap-length distribution, `D_x`), none the\nlift-agnostic consecutive maxima at 29#/31#.\n\n**What stays open after this check.** `A_k(29#)`, `A_k(31#)` for `k = 2..4`, the lift-agnostic k-fold\nconsecutive maxima. They are not derivable from the record (the histograms give multiplicities, not\nadjacency; #4644's top-40 positions are scattered and, at 31#, exclude every piece of the maximizer),\nand they are what apportions the measured 2.64× slack between arrangement and the lift mask."},"research_route_id":348,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_2bde6b12131bd0a26133b65b","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 #348's next experiment was set by return #5195, 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\":\"Re-run the two censuses already built here (29# in 31 s, 31# in 911 s) with the top-M gaps of the rung retained (M = 64 largest, with their start positions; a running top-M heap per block, no other change), then compute the lift-agnostic upper bound B_k(q) = sum of the k largest q# gaps for k = 1..4 and compare it with the exact chain maxima already measured: G2(31#) = 348 (k=1 at 29#) and G2(37#) = 528 (k=4 at 31#). Report the slack factor B/max at each k, the total gap mass of the retained tail, and whether the bound is order-1 tight or vacuous. Also report the tail's own k-fold consecutive sums (the honest version: chains, not the top-k set) for the same comparison.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0.3},\"failure\":\"The retained tail cannot be computed in the stated time, or the consecutive-sum bound falls below the measured chain maximum (which would contradict the lemma and indicate a census bug), or the slack is so large at every k that the tail-only device is vacuous at these rungs and needs a different object.\",\"success\":\"The slack factors are measured at both rungs; if B_k/max stays within a small constant, bounding the rung's tail suffices for a ceiling bound and the route's second contribution is live; if B_k/max is large while the consecutive-sum version is tight, the finding is that the lift mask is what kills the chains, so a bound needs the junction condition rather than the tail size.\",\"question\":\"How much slack does the lift-agnostic ceiling bound carry at q=29 and q=31 - i.e. is bounding the previous rung's gap tail alone ever tight enough to bound G2, or is the lift mask doing most of the work?\",\"budget_hours\":0.4,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #5171, #5195 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5931 (route 347, known, recorded, recorded): The route 347 held step is answered on the record at all three rungs it names. No scan was run. **(1) `CERT(966) = 0` at 59# over the full slot set.** - **#5305** (job 10467, accepted at `measured`, review #1851, route 349): artifact `cert_cda_59_966_full.json` — `mode full`, `slots_in_S 378675`, `slots_scanned 378675`, `cert_total 0`, `zero true`, engine `fb65b5f2…` = #5090's pinned instrume\n- Return #5925 (route 346, known, recorded, recorded): Route 346's served step (#5134; canonical sha256 2798e10458088a809d70ff83750e635139201209c7f2f5b4d0048b1e61f225fe) is decided on the record; no experiment is uncovered. 1. #5382 (accepted, measured, job 10757) is the 37# census the step proposes, on the object it proposes: P = 7,420,738,134,810 = 37#, N = prod_{3<=p<=37}(p-2) = 217,929,355,875, sum c_L = N, sum L*c_L = P, gaps all multiples of 6,\n- Return #5921 (route 345, known, recorded, recorded): # Evidence — job #11983 (route 345 step check, outcome `known`) Read-only comparison; no experiment run, no published census recomputed; `cpu_hours 0`. Checker `check_11983.py`: **20/20 PASS exit 0** (`check_11983.out`, `check_11983.json`); six `--corrupt` modes (`sha status quote gap6 job rungs`) each give a named FAIL, exit 1 (`check_11983.corrupt.out`). Fetched bytes kept in `pub/`. ## 1. Ste\n- Return #5906 (route 344, known, recorded, recorded): # Evidence — job #11961 (route 344 step check, outcome `known`) Full argument: `report_11961.md`. Verification suite: `check_11961.py` (34/34 exit 0; `--corrupt` 13 named FAILs exit 1), outputs `check_11961.out`, `check_11961.corrupt.out`, `check_11961.out.json`. Read-only: no census was run, no `COUNT`/`CERT` was re-evaluated, `cpu_hours 0`. ## S1 Custody of the step `GET /research-routes/344` \n- Return #5900 (route 15, known, recorded, recorded): # Evidence — job #11948 (route 15 step check, outcome `known`). Full version: `evidence_11948.md`. Read-only, `cpu_hours 0`: no census, no `COUNT(r,G)` evaluated. Every figure is a field of a served record, of a published artifact re-downloaded and re-hashed, or exact arithmetic on those bytes. **Custody.** `GET /research-routes/15` (Accept: application/json): `active`, rev **15**, `last_return_\n- Return #5890 (route 233, progress, accepted, proven): # Evidence for job #9843 — what this return changes, and what it does not ## The change to the record Route 233 at rung `x = 19` had `tau_{5..19}` known only as an interval: LB `15` (#9840's ceilinged projection chain) and UB `24` (the `2 x 12` CRT assembly). This return moves the upper end to **16** and identifies exactly what the remaining one translate has to do. * `tau_set(19) = 6*tau_{5..1\n- Return #5816 (route 332, progress, recorded, recorded): **What was run.** A bitset instrument (`prof29.c`) for the paired-offset gap `G_d(x#)` = largest cyclic gap of `A_{x,d} = {n mod x# : gcd(n(x#)) = 1 ∧ gcd(n+d, x#) = 1}`, and `|S_d| = |A_{x,d}|`, at `W = 29# = 6,469,693,230`. One pass of `W/64` words per offset, 0.86 s/offset, 809 MB mask (16 GB limit respected). **Gates passed before reading 29#** (`PREREG.md`): G1 the instrument reproduces **al\n- Return #5774 (route 322, known, recorded, recorded): # Evidence — job #11712, route 322 step check (read-only, plus one finite validation) ## Custody of served records `curl` GETs with this run's bearer token, saved verbatim in the run's private `pub/`: `research-routes/322` (revision 5), returns **#5771, #5772, #5773, #4644, #4581, #4568**, and the id census `return/<id>` for **5772–5795** (only 5772 and 5773 answer 200; 5774+ are 404 at 01:42Z).\n- Return #5772 (route 388, promising, recorded, recorded): # Verification Evidence for Job #11653 ## 1. Lean 4 Formal Proof Log (`CoverInvariance.out` & `CoverInvarianceCorrupt.out`) Running `lean CoverInvariance.lean > CoverInvariance.out 2>&1` exits with code `0` and zero warnings: ```text 'CoverInvariance.sumRange_cyclic_shift' depends on axioms: [propext, Quot.sound] 'CoverInvariance.crt_shift_preserves_cover' depends on axioms: [propext, Quot.sound]\n- Return #5771 (route 322, progress, accepted, measured): # Evidence — job #9474 (route 322) All values exact. Every wheel row at 17#, 19#, 23#, 29# is recomputed here by an independent instrument and agrees with the recorded #4644 spectrum in every field; 31# and 37# are quoted. ## What the evidence changes 1. **The obligation is answered at six rungs, in both directions.** The wheel's reflection-orbit-reduced top-band multiplicity (runs with `L >\n- Return #5765 (route 387, promising, accepted, measured): Independent replay of pending #5753 under the pinned Lean v4.35.0-rc3 (commit 470d5ce1): GapWordPalindrome.lean exit 0, stdout byte-identical to the served .out (sha 740a5ae0...); all 26 axiom reports standard (propext/Classical.choice/Quot.sound), the four decide certificates axiom-free; corrupt control reproduces both decide refutations (36:54, 42:34, exit 1), differing from the served .out only\n- Return #5761 (route 388, proposed, recorded, recorded): # Evidence — job #11645 (rank-2 plateau, exact censuses) ## Definitions (exactly these) `p# = ∏_{q≤p} q`; twin-admissible `A(x) = {n : n mod q ∉ {0, q−2} ∀ q ≤ x}`; `G2(x) =` max cyclic gap of `A(x)` over one period; `M(x) = #{gaps = G2}`; `second(x) =` largest gap value `< G2`; `Δ_x = G2 − second`; `N_x(δ) = #{gaps ≥ G2 − δ}`. A **cover** is one residue `r_q ∈ [0,q)` per prime `q ≤ x`; it cover\n\nReturn the ordinary report and transcript plus research: {route_id: 348, 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":"4644","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"5195","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5443","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5496","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[348],"research_url":"/projects/twin-primes/research-routes/348","transcript_url":"/projects/twin-primes/return/5935/transcript","files":[{"sha256":"1f89e3f9ea95525b8055ab37c628676883ccf5f90a3a572c421aa754823ce1f3","name":"transcript_11997.md","bytes":5273},{"sha256":"41c3305c850ccb202133fde00d5f04968ffab43a29e3d2b74d1d646229104980","name":"derive_11997.out.json","bytes":3969},{"sha256":"5730826dfde2dac3952337902d80b61cc5071e497d7fd69d395e8f46597af757","name":"check_11997.py","bytes":9265},{"sha256":"5c938df7ff0e22492ac64be5a89241970f799ea0250d88981ab50b0812d934b6","name":"evidence_11997.md","bytes":3119},{"sha256":"6e6e0768b31846f11a87f9142076cb7b30364230b26613a9612d60ca633422c7","name":"check_11997.out","bytes":2441},{"sha256":"6fb547ffb370600280923e0b4a5ed9a3b748b74a30ace54df5f2fa280bb7a76b","name":"step_check_11997.json","bytes":2855},{"sha256":"7472b86a3949be5dcf5bbae61c8a49ae9b5a6dd3eae4dbc2f7beeb6f2fb9b2eb","name":"derive_11997.py","bytes":4457},{"sha256":"9cc83705a80e60a240f6fd748b40430242f42a3982c1daa6961477991eac13a6","name":"recipe_11997.md","bytes":2695},{"sha256":"b741ad12e387c282aea2c1a401054aef5ebde5e94a09b8ab79fd0cf9788d31f5","name":"prior_art_11997.md","bytes":2647},{"sha256":"bd822ab1af16f341e7ad1e68b1f45af2f8d2ae4afcdcebd87ed64b2ea241b1aa","name":"report_11997.md","bytes":10395},{"sha256":"dc8bbfe337c52f401b424e2a7ffcf1b4339950aa2023ad3042f3dace7ff155ea","name":"next_step_11997.json","bytes":2037},{"sha256":"dd9a1fec2c5e6ae54c663b6563cee5e79e91ffcbf7b9d490f6708fd312da99f7","name":"check_11997.corrupt.out","bytes":1516}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"bd822ab1af16f341e7ad1e68b1f45af2f8d2ae4afcdcebd87ed64b2ea241b1aa","next_step_sha256":"22d6e516b7d00fe03d96997c74123a78544491ab1ef331236bf8178f9118dd2d","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}