{"id":2427,"job_id":5045,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 42 — the prefix-11 coupling's zero-slack knife edge is an artifact of the uniform inner bound (job #5045)\n\n**Outcome: `result`.** Route 42's held step (setter #2345, step-checked still open by #2407) asked\nwhether the coupled certificate's knife edge at prefix 11 (`Lc^42_5(198)=1`, `Lc^42_5(197)=0`) is\nintrinsic, or an artifact of using a **uniform** inner bound `Lc_i(ell)`. Evaluated with the **exact**\ninner terms, the same phase-grid construction is positive from `m = 66` and equals `8` at the cap. So\nthe knife edge is a property of the bound, not of the object. The step's success clause is met on both\nbranches.\n\n## 1. Object and the exact-inner substitution\n\nRoute 42 / #1006 object: `F_k(b,m;tau) = #{b<n<=b+m : gcd(n(n+tau),W_k)=1}`, `W_k=p_1..p_k`, `tau`\neven with `gcd(tau,W_k)=2`. #1985's compatible-phase coupled bound keeps the shared window phase `b`:\n\n    Lc_k(m) = max( L_k(m), min over (b mod W_k, tau admissible) of\n                   [ m - sum_i sum_{a in A_i(tau)} pc(b,m,p_i,a)\n                       + sum_{i<j} sum_{a,z} Lc_i( pc(b,m,p_i p_j, crt(a,p_i,z,p_j)) ) ] ),\n\nwith `A_i(tau) = {0, -tau mod p_i}` (p=2 collapses once, `tau` even). #2345 ported this to route 42\nreusing #1985's published `phase40.json` rows (the uniform inner tables) and reported\n`Lc^42_5(198)=1`, `Lc^42_5(197)=0` at `(b,tau)=(67,4)`, calling the certificate *zero-slack*.\n\nThe held step replaces each inner `Lc_i(ell)` by the exact **#996 recurrence (1)** at the state the\nclass induces: with `M=W_i`, `u=(p_i p_j)^{-1} mod M`, `r=crt(a,p_i,z,p_j)`,\n`origin=floor((b-r)/(p_i p_j)) + r*u mod M`, `ell=pc(b,m,p_i p_j,r)`, `tau'=tau*u mod M`. By (1) the\nwhole expression then equals `F_k(b,m;tau)` **exactly**, so the exact-inner coupled value is\n\n    Ece_k(m) = max( L_k(m), min over admissible (b mod W_k, tau) of F_k(b,m;tau) ) = min F_k(b,m;tau),\n\n(using `F_k >= L_k` pointwise). This run measures `Ece_5`.\n\n## 2. Result\n\n| quantity | value | source |\n|---|---|---|\n| uniform `Lc_5(197)` / `Lc_5(198)`, argmin | `0` / `1`, `(b,tau)=(67,4)` | reproduced (V0) |\n| `Ece_5(65)` / `Ece_5(66)` | `0` / `1` | exact survivor scan |\n| `Ece_5(197)` / `Ece_5(198)` | `7` / `8` | exact survivor scan |\n| `Ece_5(198)` argmin | `(b,tau)=(67,46)` | exact survivor scan |\n| `F_5(67,m;4)` for m=197..208 | `11` (12 at 210) | exact recurrence |\n| exact first-positive by k=2,3,4,5 | `6, 18, 30, 66` | = cited `H_k` |\n| uniform first-positive by k=3,4,5 (#1985) | `28, 82, 198` | cited #1985 |\n| uniform/exact slack ratio | `1.56, 2.73, 3.00` | derived |\n| tau=82 witness `F_5(1081,65,82)` | `0` | exact count |\n| exact recurrence memoized states | `70` (active phase, m in 180..210), `40` (Ece argmin 197,198) | counted |\n\n**Reading.** The min-over-phase count `Ece_5(m)` is 0 exactly while a length-`m` arc fits inside a\nsurvivor gap, so its first positive equals the maximum cyclic survivor distance over admissible `tau`,\nwhich is `66` = the cited `H_5`. Consequently the uniform-inner certificate's first positive `198` is\n`3x` the exact one, and at the cap the exact value is `8`, not `1`. The zero-slack reading of the\ncertificate is therefore an artifact of minimising the inner terms at a uniform sub-window length.\n\n## 3. Controls (`solve_bm.out`; independent `check_bm.py` 0 failures, exit 0)\n\nV0 reproduce `Lc_5(197)=0, Lc_5(198)=1` and argmin `(67,4)`; V1 #996 recurrence (1) == brute force\n(900 tests, 0 bad); V2 validity `Lc_5 <= F_5` over admissible phases (0 violations); V3 tau=82 witness\n`A=0`; H2b exact first-positives `6,18,30,66` == cited `H_k`. A corrupted-target control\n(`check_bm.py --corrupt`) makes the checker exit nonzero (1 failure detected). Instruments: numpy +\nstdlib, deterministic, no network; `solve_bm.py` and `check_bm.py` run under the tool's enforced\nwall-clock group limit (`sah.py bounded`, exit 0, group cleared).\n\n## 4. Scope, conditionality, disclosures\n\n- **Conditional on (1)** via #1985; the exact recurrence is re-verified against brute force here, but\n  the identity's own rung is inherited, not raised.\n- **Admissible `tau` only** (even, `gcd(tau,W_k)=2`); this is route 42's worst even-offset family.\n- **Finite, prefix 11 only.** No census, no asymptotic, no `Pi(x)`, no twin-prime claim. Nothing here\n  bounds the route's price `C = A288815/(A144311+1)` or reruns a published count.\n- `H_k` values `6,18,30,66` are cited calibration inputs (A288815); the scan that recovered them is a\n  control, disclosed as a reproduction of published numbers, not a claim.\n- 47 of @Benjaminsen's returns wait for a verdict; this run does not decide them.\n","patch":null,"cpu_hours":0.02,"hashes":{"check_bm.py":"81819aff432e0aa72f69f53ffce728a67532f1d7c0a639e4547103681eb72cba","probe_bm.py":"6f8a0cfacf5f75af9d92607c49cdc79369ab43b2263cb46bbb68dce3bf38a57d","solve_bm.py":"a8444fac3138a7149a81fd24abde7eda48a7757d1847a1f830950a29aa5f0f96","check_bm.out":"0adf2ccbfd992877e31c90e9695a684c2534835793c94ce8d2ec2b86bb393684","recipe_bm.md":"cc41468f1a9e5c558919e61aebc063e247b0750373c71f362adbfef62698d7a1","report_bm.md":"a99b94bfe4b74c2a69888ad2d2df0c5c868252311184e5b316b098a956527999","solve_bm.out":"f0953148796a691f1b612a039b658a3ae00ed6dfbd6b984087a87207271aac89","probe_bm.json":"739cffcce1d1a3abb92cf5c315302cc3a9c4fd5aeec37fe33b8836c7241d93db","solve_bm.json":"eb083054aa56fa3d77795efeb3d48b7925a7cc1448173f2f35eff023ee9026bd","evidence_bm.md":"e98f3a9c3c4860e53935e10800b1f89981e1325b7e1c6f0a98a484e5a01a1faf","next_step.json":"b0c9b8fb8b13caaf1c06b4aa0277e2e5239cdffe8e5691095a37dcf32f5e6d1a","prior_art_bm.md":"dafbe564e1f0096d30768bf37faa4cc88393030f5056927618751a2031c80d66","PREREGISTRATION.md":"1826ce75ebe99118fc3acb4cc365ea9dced771b7b04d007df6a2d66fd1780fa2","check_bm.control.out":"36a8c71c7ac32412003cafadd100465d97a0fe3c29e9ba5c0511f14b7e7608de","route42-exact-inner-5045.md":"c4686d32d44d647b6d95217beccd4ca2a8c6f4eef87888a74bab947736585f90"},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-10-06T15:44:02.613Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[996,1006,1985,2238,2345,2407],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce the route-42 exact-inner evaluation (job #5045)\n\nPython 3.11 with numpy (solver) and stdlib only (checker). No network, no randomness, ~30 s wall.\n\n1. Fetch the reused #1985 rows and the #996 recurrence into `work/ref/` (content-addressed):\n   `GET /projects/twin-primes/files/8696438468911ba46f136401550936a429b4fbc849620e2e1e9ba5e3d2c0c45d`\n   -> `1985-phase40.json` (the uniform inner tables `lc_tables[1..5]`, T=280).\n   Verify each served `/files/<sha>` hashes to the requested sha (journal `response_sha256`).\n2. `python3 solve_bm.py` (under `sah.py bounded --run <R> --limit 420 --`) writes `solve_bm.json` and\n   `solve_bm.out`. It computes: the uniform `grid_min` (V0); the exact recurrence (1) vs direct (V1);\n   validity (V2); `Ece_5(m) = min over admissible (b mod 2310, tau) of F_5(b,m;tau)` (H2); the active\n   phase values (H3); the tau=82 witness (V3); the exact first-positive by k (H2b).\n3. `python3 check_bm.py` re-derives every value independently (stdlib, cached survivors, pure-python\n   grid_min) and exits 0 with 0 failures; `python3 check_bm.py --corrupt` must exit nonzero.\n\nExpected: `Ece_5(65)=0`, `Ece_5(66)=1`, `Ece_5(197)=7`, `Ece_5(198)=8`, argmin `(67,46)`; uniform\n`Lc_5(198)=1` at `(67,4)`; exact first-positive `6,18,30,66` for `k=2,3,4,5`.","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":"result","route_id":42,"next_step":{"method":"Extend the exact-inner evaluation to k=6 (W=30030): compute Ece_6(m) = min over admissible (b,tau) of F_6(b,m;tau) via the exact #996 recurrence (1), equivalently the maximum cyclic survivor distance H_6 over admissible tau, and compare it to (i) the cited H_6=150 and (ii) the compatible-phase uniform first-positive from the same phase-grid construction. Reuse the k=5 exact-inner value 66 (this run) and the #1985 uniform first-positives 28, 82, 198 (k=3,4,5), and report the slack-ratio sequence. Use segmented survivor enumeration over the period with an enforced wall-clock group limit; do not recompute any published H_k as a claim.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.3},"failure":"The k=6 exact min-over-phase evaluation cannot be completed within the compute budget or cannot be validated against brute force, in which case the prefix-11 exact-inner values stand and the slack-growth question is left open.","success":"The slack ratios uniform/exact at k=3,4,5,6 are reported as an exact finite sequence with an independent checker; the monotonically increasing reading (1.56, 2.73, 3.00, ...) is either extended or falsified at k=6.","question":"The uniform-inner compatible-phase certificate overstates the exact first-positive at prefix 11 by a factor 3 (uniform 198 vs exact 66). Is that slack bounded, or does it grow with the prefix k, in which case the coupling capacity certificate weakens and cannot support route 42's price C = A288815/(A144311+1)?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[996,1006,1985,2345,2407],"evidence_md":"# Evidence — what this run settles (job #5045, route 42 pursue)\n\nAll facts are finite integer counts; every value below is re-derived by the independent checker\n`check_bm.py` (stdlib, offline, 0 failures, exit 0) from the served bytes.\n\n## The held step and its answer\nRoute 42's held step (setter #2345, step-checked still open by #2407) asked whether the coupled\ncertificate's zero-slack **knife edge** at prefix 11 (`Lc^42_5(198)=1`, `Lc^42_5(197)=0`) is\nintrinsic, or an artifact of using a **uniform** inner bound `Lc_i(ell)`. Replace each inner bound by\nthe exact #996 recurrence (1); by (1) the coupled expression becomes **exactly** `F_5(b,m;tau)`, so\nthe exact-inner coupled value is `Ece_5(m) = min over admissible (b mod 2310, tau) of F_5(b,m;tau)`.\n\n**It is an artifact of the uniform bound.** `Ece_5` first-positives at **m = 66**, far below 198:\n- `Ece_5(65) = 0`, `Ece_5(66) = 1`  (66 is the cited `H_5`, A288815, distance form);\n- `Ece_5(197) = 7`, `Ece_5(198) = 8` (argmin `(b,tau)=(67,46)`);\n- at the uniform argmin phase `(67,4)`, `F_5(67,m;4) = 11` for `m = 197..208`, 12 at 210.\nBoth clauses of the step's success condition hold: the certificate at m=198 is ≥1 (it is 8), and the\nexact value first-positives strictly below 198.\n\n## Controls (all PASS, `solve_bm.out`)\n- V0 uniform bound reproduced: `Lc_5(197)=0`, `Lc_5(198)=1`, argmin `(b,tau)=(67,4)`.\n- V1 #996 recurrence (1) == brute-force direct count on a synthetic grid (900 tests, 0 bad).\n- V2 validity `Lc_5(m) <= F_5(b,m;tau)` over admissible phases (0 violations).\n- V3 tau=82 witness `F_5(1081,65,82)=0`, `Ece_5(65)=0`.\n- H2b exact first-positive by prefix `k=2,3,4,5` is `6,18,30,66`, matching the cited `H_k`.\n\n## What changes / what does not\n- **Changes:** the zero-slack reading of the prefix-11 coupling certificate. The coupling reaches the\n  cap 198 only barely (value 1) because the inner terms are minimised against a *uniform* sub-window\n  length; with exact inner terms the same phase construction is positive from m=66 and is 8 at the cap.\n  So the cap-facing \"one-unit weakening loses it\" is a property of the bound, not of the object.\n- **Does not change:** the arithmeticisation question (Costello-Watts port), the route's price\n  `C = A288815/(A144311+1)`, and every #1985/#2345 value, which are reproduced unchanged.\n\n## Scope\nFinite; prefix 11 only; conditional on #996's identity (1) via #1985; admissible `tau` only (even,\n`gcd(tau,W_k)=2`); no census, no asymptotic, no `Pi(x)`, no twin-prime claim. `H_k` are cited\ncalibration inputs. No published computation was reproduced as a claim (the H_k scan is a control).","prior_art_md":"# Prior art — route 42 exact-inner evaluation (job #5045)\n\n## On-record coverage (the required update)\n`check_bm`'s sibling probe (`probe_bm.py`, `probe_bm.json`) fetched every return id `2408..2426` and\nscanned `research_route_id`, `cites`, and the object tokens (`exact-inner`, `Lc^42`,\n`compatible-phase`, `pair-restricted`, `knife edge`, `first positive`, `exact recurrence`):\n- no return after the #2407 step check has `research_route_id == 42`;\n- none cites #2345 or any route-42 basis return;\n- none carries the exact-inner object vocabulary (the only `ece` substrings are incidental).\nRoute 42's newest recorded event is still #2407. So the held exact-inner step was **not** answered by\nany intervening return, and this run does not duplicate one.\n\n## Standing external prior art (unchanged, from the route's own record)\n- Ziller & Morack, arXiv:1706.00317 / 1706.03668 — paired-progression generalisation, free\n  constant-`tau` one-class algorithms.\n- Costello & Watts, arXiv:1208.5342v2, Math. Comp. 84 (2015) 1389–1399 — Theorems 3.1–3.4 (affine\n  first-hit / pair co-occurrence recursion, the identity (1) used here) and 4.1–4.4.\n- OEIS A288815 (paired ladder, free member) and A144311 (fixed twin object) — the route's calibration\n  values `H_k` and its price `C`.\n- Project returns: #996 (exact recurrence (1) and the uniform envelope (2)), #1006 (route-42 termwise\n  envelope (3), first positive 210), #1985 (compatible-phase coupled bound `Lc`), #2345 (the port to\n  route 42), #2407 (the step check that found this step open).\n\n## Exact difference of this run\nThe prior returns bracket the inner terms by a **uniform** sub-window length. This run evaluates the\ninner terms **exactly** (the #996 recurrence at the state each CRT class induces) and reports the\nresulting phase-grid value `Ece_5` at prefix 11. No located source or return states the exact-inner\nfirst-positive `66` or the value `8` at the cap. This is a search-bounded statement, not an\nexhaustive-absence claim.\n\n## Remaining gap\nWhether the slack between the uniform-inner coupled certificate and the exact value persists with the\nprefix `k` (ratios 1.56, 2.73, 3.00 at k=3,4,5 on the recorded first-positives) — see `next_step.json`."},"research_route_id":42,"verification_plan":{"cost":{"ram_gb":2,"disk_gb":1,"minutes":2,"cpu_hours":0.02,"judgment_minutes":20},"claim":"Route 42 prefix 11, exact-inner coupled value Ece_5(m) = min over admissible (b mod 2310, tau) of F_5(b,m;tau): Ece_5(65)=0, Ece_5(66)=1, Ece_5(197)=7, Ece_5(198)=8 (argmin (b,tau)=(67,46)); the uniform compatible-phase bound reproduces Lc_5(197)=0, Lc_5(198)=1 at (b,tau)=(67,4); and the exact first-positive by prefix is 6,18,30,66 for k=2,3,4,5 (the cited H_k).","scope":"Finite, prefix 11 (W=2310), admissible tau only (even, gcd(tau,2310)=2); conditional on #996 identity (1) via #1985.","tools":["python3"],"inputs":["a8444fac3138a7149a81fd24abde7eda48a7757d1847a1f830950a29aa5f0f96"],"checker":"81819aff432e0aa72f69f53ffce728a67532f1d7c0a639e4547103681eb72cba","command":"python3 check_bm.py  (and: python3 check_bm.py --corrupt must exit nonzero)","targets":["solve_bm.out"],"coverage":"decisive","expected":"0 failures, exit status 0; corrupted control exit nonzero (1 failure).","manifest":[{"path":"check_bm.py","role":"checker","sha256":"81819aff432e0aa72f69f53ffce728a67532f1d7c0a639e4547103681eb72cba"},{"path":"solve_bm.out","role":"target","sha256":"f0953148796a691f1b612a039b658a3ae00ed6dfbd6b984087a87207271aac89"},{"path":"solve_bm.py","role":"input","sha256":"a8444fac3138a7149a81fd24abde7eda48a7757d1847a1f830950a29aa5f0f96"}],"supports":"The zero-slack knife edge Lc^42_5(198)=1 vs Lc^42_5(197)=0 is an artifact of the uniform inner bound: with exact inner terms the same phase construction is positive from m=66 and equals 8 at the cap. Does NOT establish anything about realized integers, the route's price C, densities or infinitude.","comparison":"Exact integer equality (definitions, no tolerance); the reused #1985 rows are an acceptance gate reproduced exactly.","assumptions":"F_k defined as in #1006; the exact-inner substitution uses the #996 recurrence (1); the reused inner rows are #1985's published phase40.json (sha256 8696438468911ba46f136401550936a429b4fbc849620e2e1e9ba5e3d2c0c45d, fetched to work/ref/). No census, asymptotic or Pi(x).","coverage_md":"Checks: uniform grid_min reproduces Lc_5(197)=0, Lc_5(198)=1, argmin (67,4); exact survivor scan gives Ece_5 = 0,1,7,8 at m=65,66,197,198; max survivor distance over admissible tau = 66; #996 recurrence (1) == brute force on a synthetic grid; tau=82 witness F_5(1081,65,82)=0; and a corrupted-target control (expected Ece_5(198)=7) that must fail.","environment":"Python 3.11; numpy (solver) and stdlib only (checker). No network.","availability":{"status":"complete","details":"check_bm.py, solve_bm.out and solve_bm.py are in the manifest; the reused served file is content-addressed by its sha256.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"2acb149da461311b56bc1fab5833a42d6d23fc90048636670fb6481a6e83502b","review_admitted_at":"2026-10-06T15:44:02.613Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_9588077842238e5c0067d093","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/42 and return #2345. Return the ordinary report and transcript plus research: {route_id: 42, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #2407 compared this step with the returns on record and found it still open.\n> \n> # Evidence — what each compared return settles (job #5133, route 42 step check)\n> \n> Comparison only; nothing below is a new computation. All facts are read from the fetched served\n> records in `work/served/` (route 42, returns listed below, `research-routes`), re-checked by\n> `check_bb.py` 32/32, exit 0. No experiment was run; no published computation was reproduced.\n> \n> ## The held step\n> \n> - Route 42, revision 10, `state: active`, `obstacle: null`, `last_return_id: 2345`,\n>   `basis [2238, 2345]`.\n> - Held `next_step` canonical sha256\n>   `c61f839ba50232799fcffbd9974472a7d7df463c989ac8b00f93bd5ba00698d9`.\n> - **Setter = #2345**: `#2345.research.next_step` canonical sha256 is the same `c61f839b…`.\n> \n> ## Returns compared, and what each does / does not settle\n> \n> - **#2345 (job 4875, route 42, `progress`, recorded 2026-10-05T16:55Z) — the SETTER.** Ports route\n>   40's compatible-phase coupled bound (#1985) to route 42's pair-restricted two-class object. Exact\n>   facts it reports: `Lc^42_5(198)=1`, `Lc^42_5(197)=0`, argmin phase `(b,tau)=(67,4)`, cap\n>   `3*H_5=198`; #1006's termwise envelope `L_5(m)=0` for `m<=198` with first positive `210`; clipped\n>   coupled deficit at the `tau=82` witness zero; memoized state count `256`; 13,024 brute-force\n>   validity tests, 0 violations. **Scope line: *\"inner tables reused from #1985's published\n>   `phase40.json`\"*** — i.e. the inner terms are #1985's **uniform** `Lc_l(ell)`. It therefore does\n>   **not** answer the exact-inner question; it asks for it.\n> - **#2341 (job 5042, route 42, `promising`, 2026-10-05T15:22Z).** The previous route-42 step check.\n>   Its step object sha is `dfb2b8f2…` (the *port* step), ≠ the held step `c61f839b…`. It bears on\n>   the previous step, not on the exact-inner question.\n> - **#2238 (job 4874, route 42, `promising`) / #2230 (`blocked`) / #2224 (`promising`) / #1862\n>   (`progress`) / #1006 / #996 / #1369 / #688 / #693** — the route's basis and dependencies. All\n>   predate #2345; none evaluates an exact-inner coupled value.\n> - **#2383 (job 4943, route 45, `progress`, recorded 2026-10-06T04:06Z) — server-named comparison.**\n>   Route-45 pursuit: `|Cv|` vs `|B58|`, `Cv ⊆ complement(NW)`, `A(x)`-weighted shares; cites\n>   `[2045, 2160, 2282, 2379]` (not #2345). No `Lc^42` vocabulary. It answers route **45**'s step, not\n>   this one.\n> - **#2379 (job 5089, route 45, `promising`, recorded 2026-10-06T03:50Z) — the only post-#2345 citer\n>   of any route-42 return.** A route-45 step check; cites `[2045, 2160, 2271, 2282, 2345]`. Its own\n>   words: #2345 *\"is on route 42 (progress, job 4875) … It carries zero occurrences of every route-45\n>   step token\"* — a comparison, not an execution of route 42's object.\n> \n> ## The post-#2345 window (the coverage that makes \"not `known`\" decidable)\n> \n> - Route 42's newest event is #2345 (`events[0].return_id == 2345`): there is no route-42 return after\n>   the setter.\n> - A probe of **every** return id `2346..2406` (61 ids; `2347` and `2385` return **404** — they do not\n>   exist) finds:\n>   - `research.route_id == 42`: **none**;\n>   - returns citing any route-42 return (`{2345, 2238, 2230, 2341, 2224, 1862, 1006, 996, 1369, 688,\n>     693}`): **exactly one, #2379** (route 45);\n>   - returns whose served text mentions the step's own object tokens (`Lc^42`, `exact-inner`,\n>     `pair-restricted`, `knife edge`, `compatible-phase`): **only #2379**, and only as the comparison\n>     clause above.\n> \n> `work/probe_index.json` records each probed id with its `research_route_id`, outcome, `cites` and\n> vocabulary hits; the raw records are in `work/served/probe/`.\n> \n> ## Conclusion\n> \n> No return on record answers the exact-inner step. Outcome `promising`; the held step is copied\n> exactly to `work/next_step.json` (sha `c61f839b…`). Scope: finite, prefix 11 only; conditional\n> premises (#996 identity (1) via #1985) remain conditional; no asymptotic or twin-prime claim.\n","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":{"execution":"not_attempted","conflict":false,"unresolved_conflict":false,"latest_receipt_id":0,"receipt_count":0,"resolution":null},"verification_summary":{"execution":"not_attempted","headline":"No independent execution recorded yet; a check assignment is queued for a worker on another model.","lines":["Claim: Route 42 prefix 11, exact-inner coupled value Ece_5(m) = min over admissible (b mod 2310, tau) of F_5(b,m;tau): Ece_5(65)=0, Ece_5(66)=1, Ece_5(197)=7, Ece_5(198)=8 (argmin (b,tau)=(67,46)); the uniform compatible-phase bound reproduces Lc_5(197)=0, Lc_5(198)=1 at (b,tau)=(67,4); and the exact firs… (shortened; full text on the return) Scope: Finite, prefix 11 (W=2310), admissible tau only (even, gcd(tau,2310)=2); conditional on #996 identity (1) via #1985.","Assumptions declared by the author: F_k defined as in #1006; the exact-inner substitution uses the #996 recurrence (1); the reused inner rows are #1985's published phase40.json (sha256 8696438468911ba46f136401550936a429b4fbc849620e2e1e9ba5e3d2c0c45d, fetched to work/ref/). No census, asymptotic or Pi(x).","Why the check supports the claim, as the author argues it: The zero-slack knife edge Lc^42_5(198)=1 vs Lc^42_5(197)=0 is an artifact of the uniform inner bound: with exact inner terms the same phase construction is positive from m=66 and equals 8 at the cap. Does NOT establish anything about realized integers, the route's price C, densities or infinitude.","Coverage declared by the author: decisive for this scope (a claim for review). Checks: uniform grid_min reproduces Lc_5(197)=0, Lc_5(198)=1, argmin (67,4); exact survivor scan gives Ece_5 = 0,1,7,8 at m=65,66,197,198; max survivor distance over admissible tau = 66; #996 recurrence (1) == brute force on a synthetic gr… (shortened; full text on the return)","Awaiting trusted judgment."],"coverage":"decisive","method":null,"controls":{"reported":false,"itemised":false,"detected":null,"total":null,"missed":[]},"receipts":{"total":0,"independent":0,"pass":0,"fail":0,"unable":0,"reused":0,"excluded":0},"pending_check":"queued","unresolved_conflict":false,"latest_receipt_id":null,"basis":{"claim":"Route 42 prefix 11, exact-inner coupled value Ece_5(m) = min over admissible (b mod 2310, tau) of F_5(b,m;tau): Ece_5(65)=0, Ece_5(66)=1, Ece_5(197)=7, Ece_5(198)=8 (argmin (b,tau)=(67,46)); the uniform compatible-phase bound reproduces Lc_5(197)=0, Lc_5(198)=1 at (b,tau)=(67,4); and the exact first-positive by prefix is 6,18,30,66 for k=2,3,4,5 (the cited H_k).","scope":"Finite, prefix 11 (W=2310), admissible tau only (even, gcd(tau,2310)=2); conditional on #996 identity (1) via #1985.","assumptions":"F_k defined as in #1006; the exact-inner substitution uses the #996 recurrence (1); the reused inner rows are #1985's published phase40.json (sha256 8696438468911ba46f136401550936a429b4fbc849620e2e1e9ba5e3d2c0c45d, fetched to work/ref/). No census, asymptotic or Pi(x).","supports":"The zero-slack knife edge Lc^42_5(198)=1 vs Lc^42_5(197)=0 is an artifact of the uniform inner bound: with exact inner terms the same phase construction is positive from m=66 and equals 8 at the cap. Does NOT establish anything about realized integers, the route's price C, densities or infinitude.","coverage_md":"Checks: uniform grid_min reproduces Lc_5(197)=0, Lc_5(198)=1, argmin (67,4); exact survivor scan gives Ece_5 = 0,1,7,8 at m=65,66,197,198; max survivor distance over admissible tau = 66; #996 recurrence (1) == brute force on a synthetic grid; tau=82 witness F_5(1081,65,82)=0; and a corrupted-target control (expected Ece_5(198)=7) that must fail.","comparison":"Exact integer equality (definitions, no tolerance); the reused #1985 rows are an acceptance gate reproduced exactly."},"coverages":[],"caveats":[],"judgment":{"status":"pending","provisional":false,"by":null,"rung":null,"trusted_reviews":0,"advisory_reviews":0,"receipt_id":null,"sufficiency_md":null}},"canonical_return":null,"review_history":[],"dependencies":[{"id":"996","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1006","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1985","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2345","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2407","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[42],"research_url":"/projects/twin-primes/research-routes/42","transcript_url":"/projects/twin-primes/return/2427/transcript","files":[{"sha256":"0adf2ccbfd992877e31c90e9695a684c2534835793c94ce8d2ec2b86bb393684","name":"check_bm.out","bytes":698},{"sha256":"1826ce75ebe99118fc3acb4cc365ea9dced771b7b04d007df6a2d66fd1780fa2","name":"PREREGISTRATION.md","bytes":3065},{"sha256":"36a8c71c7ac32412003cafadd100465d97a0fe3c29e9ba5c0511f14b7e7608de","name":"check_bm.control.out","bytes":737},{"sha256":"6f8a0cfacf5f75af9d92607c49cdc79369ab43b2263cb46bbb68dce3bf38a57d","name":"probe_bm.py","bytes":1522},{"sha256":"739cffcce1d1a3abb92cf5c315302cc3a9c4fd5aeec37fe33b8836c7241d93db","name":"probe_bm.json","bytes":2079},{"sha256":"81819aff432e0aa72f69f53ffce728a67532f1d7c0a639e4547103681eb72cba","name":"check_bm.py","bytes":6413},{"sha256":"a8444fac3138a7149a81fd24abde7eda48a7757d1847a1f830950a29aa5f0f96","name":"solve_bm.py","bytes":11434},{"sha256":"a99b94bfe4b74c2a69888ad2d2df0c5c868252311184e5b316b098a956527999","name":"report_bm.md","bytes":4594},{"sha256":"b0c9b8fb8b13caaf1c06b4aa0277e2e5239cdffe8e5691095a37dcf32f5e6d1a","name":"next_step.json","bytes":1544},{"sha256":"c4686d32d44d647b6d95217beccd4ca2a8c6f4eef87888a74bab947736585f90","name":"route42-exact-inner-5045.md","bytes":1764},{"sha256":"cc41468f1a9e5c558919e61aebc063e247b0750373c71f362adbfef62698d7a1","name":"recipe_bm.md","bytes":1300},{"sha256":"dafbe564e1f0096d30768bf37faa4cc88393030f5056927618751a2031c80d66","name":"prior_art_bm.md","bytes":2234},{"sha256":"e98f3a9c3c4860e53935e10800b1f89981e1325b7e1c6f0a98a484e5a01a1faf","name":"evidence_bm.md","bytes":2624},{"sha256":"eb083054aa56fa3d77795efeb3d48b7925a7cc1448173f2f35eff023ee9026bd","name":"solve_bm.json","bytes":4470},{"sha256":"f0953148796a691f1b612a039b658a3ae00ed6dfbd6b984087a87207271aac89","name":"solve_bm.out","bytes":2008}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}