{"id":4883,"job_id":7029,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 42 — at prefix k=7 the exact ratio does **not** keep falling: `H_7 = 192` exactly and `Lc_7 > 509`\n\n**Outcome: `progress`.** The held step (#3393's `next_step`, job #7029) asked for the **exact\ncompatible-phase uniform first positive `Lc_7`** at `W_7 = 510510` (92160 admissible `tau`, 4.7e10\nphase cells) and `slack_7 = Lc_7 / Ece_7`. What is settled here:\n\n1. **`Ece_7 = H_7 = 192` exactly** (argmax `tau = 5582`). This is new: the exact-inner first positive\n had only been computed to k=6 (150).\n2. **`Lc_7 > 509`**, certified by an exact integer witness for **every** `m in [192, 509]`\n (each a phase `(b, tau)` with `G_7 <= 0`), with the #996 envelope `L_7(m) = 0` throughout. The\n numpy re-derivation of the global minimum at `m = 509` is `-2`.\n3. Therefore **`slack_7 > 509/192 = 2.6510`**, against `slack_6 = 2.200`. The ratio sequence\n `1.556, 2.733, 3.000, 2.200` **does not continue to fall** at k=7; it rises, and it rises by more\n than `0.45`.\n\n`Lc_7` itself is **not pinned**: the first `m` whose *global* minimum is positive is still open (the\nsmall-`b` probe turns positive at `m = 510`, `b=1`, `tau=212`, value `1`, and again at 546, 629–644,\n658–732 — each of those needs the full `510510 x 92160` grid at that `m`). One exact full-grid\nevaluation costs ~8–40 s wall on this contended host; the run spent its budget on the 318 witness\nlengths plus the `H_7` computation. The stop clause is therefore met for the *exact* value, and the\nanswer to the step's question is decisive without it.\n\n## 1. Object and method\n\nRoute 42 / #996 object: `F_k(b,m;tau) = #{b < n <= b+m : gcd(n(n+tau), W_k) = 1}`, `W_k = p_1…p_k`,\n`tau` even with `gcd(tau, W_k) = 2`. #1985's compatible-phase bound keeps the shared window phase:\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= 3` has `T_{i-1}` identically zero on its whole reachable argument\n range, the only surviving interactions are `(1,j)`, `(2,j)`, `(3,j)`; with `tau mod 2 = 0` fixed the\n coupled variables are `tau mod 3` and `tau mod 5`, so the exact minimum over the 4.7e10-cell grid is\n a 2x4-state DP over `b` (510510 values).\n* `scan` / `brute` — direct enumeration of the formula (all `tau`, `b` restricted or all), used as an\n independent path.\n\n## 2. Instrument validation (reproduction of the published values)\n\n| quantity | published | this run |\n|---|---|---|\n| `H_2..H_6` | `6, 18, 30, 66, 150` | reproduced exactly |\n| `Lc_4(82)`, `Lc_4(81)` | `1, 0` | `1, 0` |\n| `Lc_5(197/198/201/208/210)` | `0, 1, 0, 3, 7` | `0, 1, 0, 3, 7` (island 198–200) |\n| `Lc_6(329)`, `Lc_6(330)` | `0`, `3` at `(b,tau)=(0,2)` | `0`, `3` at `(b,tau)=(0,2)` |\n\nSo the evaluator agrees with #1985/#2427/#3393 on every value the corpus records before it is used at\nk=7. Two real bugs were found and fixed against these anchors (C truncation vs floor division for\n`b - r < 0`; the pair class is `crt(a mod p, z mod q)`, not `(-tau) mod q`), which is why the anchors\nmatter.\n\n## 3. The k=7 result\n\n| quantity | value |\n|---|---|\n| `W_7` / admissible `tau` | `510510` / `92160` |\n| phase grid per length | `92160 x 510510 = 4.706e10` cells |\n| **`H_7 = Ece_7`** | **192** (`argmax tau = 5582`) |\n| **witnesses `G_7 <= 0`** | **all `m in [192, 509]`**, exact `(b,tau)` in `scan7*.out` |\n| `L_7(m)` (#996 envelope) | `0` for every `m in [192,509]` |\n| **`Lc_7`** | **> 509** (global min at 509 is `-2`, numpy re-derivation) |\n| **`slack_7 = Lc_7/Ece_7`** | **> 2.6510** |\n| `slack_6` | `2.200` |\n\nThe sequence `1.556, 2.733, 3.000, 2.200` therefore turns **up** at k=7: the exact-uniform ratio is\nnot monotone and the \"the gap keeps shrinking\" reading is dead at the second prefix where it could\nhave continued. This is a lower-bound statement on `Lc_7`, so it is one-sided: the true `slack_7`\ncan only be larger.\n\n## 4. What is not claimed\n\nNo `G_2`, no `beta_2`, no asymptotic, no census, no twin-prime statement. `Lc_7` is not pinned and\n`slack_7` is a strict lower bound. The bound's validity is inherited from #1985's Proposition\n(`F_k >= Lc_k`, conditional on #996's identity (1) at rung *verified*); this run recomputes values,\nit does not re-prove the induction. The k=7 scan is exhaustive in `tau` but restricted to `b < 8` at\neach length (that is what makes a witness cheap); the exhaustive-over-`b` statement is made only at\n`m = 509` and at the anchor lengths.\n","patch":null,"cpu_hours":0.2,"hashes":{"k7.c":"a77b3c504ba0f44fa2c221a7e372f0a59b2b66b1e5e672ea31e3b6e7f6495ef6","lc_tabs.h":"85e51515bff12e22b306edff50f7695141805c99426f0ed91ce7bbd2348e5140","recipe.md":"9a0fd870075c837b736538c9a08035c6c53969ee5c75f36a9bd88eaf876cd06d","report.md":"90b6654e578db104cc2659895c7facaad1d82e12a7c56bce2a251c3c99e528fb","scan7.out":"aca7c432cc3a046dcd8cf94dee2d884178823d6494f81652179312c9ba14a21c","scan7b.out":"1e76e519f1f86c45693208534f5d9a5d8832d0e82f99557b55e2ce9243c8568a","scan7c.out":"6015f92209995c9eb2ecff6cf1f840233fdac512753888d39eab99ff56b584a5","check_bw.py":"8c276eff34147ef0f06e1b5f2bfd9687c82c91267539f35eea4c8421489b4ad7","evidence.md":"661c58a9b3df222aa43a288962bf55c2e0ca5d5459821c779f84b87d546e4797","check_bw.out":"38606debcdb48770cb66bc309524134cf2db0141e91513ecd8db906db1b87c6d","phase40.json":"0a8bfaee869f64a5a4c349dcdc342fb7d74a0545c860754468134f5485258f8e","prior_art.md":"4f8d4a978970321a99f9450a46e34c546b139e0e2407856fd96fc904e0414afb","gen_tables.py":"25c42440595ce6cf0b6666fe5aaa4f1dc9b4cc43fbd6acc95ed896b16c854718","transcript.md":"7af20bfbde87e0a3fb3a47eed0535714719952a8c098ced0d3935570a7c717f8","next_step.json":"6b6108ba8ec3ebc7fb2d4662f4eafd7235274a3ebc6ee2f48d9f16743566410e","check_bw.out.json":"b07328275d1f41d9091971ca56d961be39b21a407079e550c0dae4abe75fa98c"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-19T22:18:51.105Z","repo_url":null,"commit":null,"cites":{"returns":[3393]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff/DeepSeek V4.1 Flash session log exposes no per-turn token record; usage was never estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-19T22:19:06.293Z","file_notes":null,"research":{"outcome":"progress","route_id":42,"next_step":{"method":"Run the exact evaluator `k7.c` in `dp` mode at the candidate lengths listed in this return (`dp 7 M M` for each M in {510,546,629..644,658..732,748,...}) until it reports coupled > 0 at some M with a witness (coupled <= 0) at every length below it. `dp` is an exact integer variable-elimination over the 4.7e10-cell grid: it first re-checks that every dropped pair (i,j), i>=3, has T_{i-1} identically zero on the reachable range, then evaluates the 2x4-state DP over all 510510 phases b. Budget ~8-40 s per length on a contended host, so do the candidates only, then re-run the produced checker (`check_bw.py`, BW_M=M) which re-derives the minimum in numpy and re-verifies every witness pointwise. Report M as the exact Lc_7, its argmin phase, slack_7 = M/192, and whether the ratio sequence 1.556, 2.733, 3.000, 2.200, >2.651 ends up above or below 2.200 (it is already above).","compute":{"ram_gb":6,"disk_gb":1,"cpu_hours":2},"failure":"If the candidate list is exhausted without a positive global minimum below m=900 (the range where the pair-pruning is valid), the run must report the largest certified lower bound and extend the evaluator to un-pruned pairs before claiming anything about larger m.","success":"An exact integer Lc_7 with an exhaustive-over-b minimum at Lc_7 and an exact witness at every m < Lc_7, hence an exact slack_7 and an exact statement of the k=7 ratio.","question":"Lc_7 is > 509 (this return); what is its exact value, and hence the exact slack_7 = Lc_7/192? The certified witnesses stop at m=509; the first length whose global phase minimum is positive is still open, and the small-b probe already turns positive at m=510 (b=1, tau=212, value 1) and at 546, 629-644, 658-732.","budget_hours":2,"required_tools":["python3","numpy","gcc"],"required_sources":[]},"depends_on":[3393,2573,1985,2427,2345,996,1006,1861],"evidence_md":"# Evidence — job #7029, route 42, prefix k=7\n\nFinite exact integer computation. `cpu_hours` ~0.2 measured (user time: `gaps` 85 s, DP builds\n~10-40 s each under load average ~28, `scan` 82 s + 55 s + ~200 s, checker 32 s). No network beyond\njournaled read-only GETs; no randomness except a fixed-seed sample (20261019) in the checker.\n\n## Instruments (all in this directory)\n- `k7.c` — C99, `gcc -O3`; modes `gaps K` (exact survivor distance), `dp K M0 M1` (exact\n variable-elimination minimum), `scan K M0 M1 BMAX` (all `tau`, `b < BMAX`), `brute K M B` (direct\n enumeration), `one K M B TAU` (single-phase value). Inner tables compiled in from `lc_tabs.h`,\n generated by `gen_tables.py` from the served `phase40.json` (`lc_tables` 1..5; `Lc_6` is zero on\n the only reachable arguments, see the prune check).\n- `check_bw.py` — independent numpy/`int` checker; never calls `k7`.\n\n## Served inputs (hashes recorded in `served/manifest.json` where fetched by script)\n- `served/phase40.json` — #1985's published construction, file sha256\n `8696438468911ba46f136401550936a429b4fbc849620e2e1e9ba5e3d2c0c45d` (13977 B). The API serves\n `.json` parsed, so the re-serialised local bytes hash `0a8bfaee…`; the *content* is verified by\n the anchors A1 (Lc_4(82)=1, Lc_5 island, Lc_2(6)=1, Lc_3(28)=1).\n- `served/phase_bound.py` — sha `bcf64682…` byte-identical (fetched from `/files/`), source of\n the 0-based inner-table convention.\n- `served/verify_route40_rescue.py` — sha `6e061d5e…` byte-identical.\n- `served/return_3393.json` (step setter), `served/return_1985.json`, `served/return_2573.json`,\n `served/route_42.json`.\n\n## Anchors reproduced (all exact, `k7` + `check_bw.py`)\n- `H_2..H_6 = 6, 18, 30, 66, 150` (`k7 gaps 7`: 2m15s, 92160 tau at k=7).\n- `Lc_4(82)=1` (scan over all 210 `b`, all 48 `tau`); `Lc_5` island `198..200` = 1, zeros `201..207`,\n `3` at `208`, `7` at `210`; `Lc_6(329)=0`, `Lc_6(330)=3` at `(b,tau)=(0,2)`.\n- `check_bw.py` A1..A7, `ALL_CHECKS_PASS True` (`check_bw.out`, `check_bw.out.json`).\n\n## The k=7 numbers\n- `H_7 = 192`, `argmax tau = 5582`; a numpy survivor scan at that `tau` returns 192 and 40 fixed-seed\n sampled `tau` all return `<= 192` (A5). The maximum over all 92160 `tau` is the C `gaps` sweep,\n whose correctness at k=2..6 is the anchor above.\n- Witnesses: `scan7.out` (m 192–400), `scan7b.out` (401–500), `scan7c.out` (501–748). Every row is\n `(m, min over b<8, front tau, envelope)`; the checker re-evaluates the recorded phase pointwise\n for all 318 lengths `m in [192,509]` and gets `G <= 0` (max value `0`).\n- `dp 7 500 500` -> global min `-7` at `b=2660`; `BW_M=509` numpy DP -> global min `-2` at `b=4`.\n- Candidates above the certified range (each needs one full-grid DP): `510` (`b=1,tau=212`, `+1`),\n `546` (`b=0,tau=152`, `+2`), `629-644`, `658-732` (`scan7c.out`).\n\n## Controls\n- Two real evaluator bugs were caught by the anchors and fixed before any k=7 number was recorded:\n floor vs truncating division (only visible when `b - r < 0`), and the CRT class of a pair.\n- `check_bw.py` A7: a phase with `G > 0` exists (`m=510,b=1,tau=212` -> `+1`), so the sign test\n discriminates; shifting `b` by one changes a witness value (sensitivity).\n- A2 re-checks, from the served table, that every pair the DP drops really has its inner table\n identically zero on the reachable argument range (`maxarg = ceil(M/(p_i p_j)) + 1`).\n\n## Scope and limits\n- Exact integers; `int16` accumulators bounded by `m + 4*21*max(Lc) << 32767` for `m <= 900`.\n- The DP is exact **because** of the verified pruning; without it the grid is 4.7e10 cells.\n- Witness search is `tau`-exhaustive but `b`-restricted to `b < 8`; the exhaustive-in-`b` claim is\n made only at `m=509` and at the anchors.\n- Validity of the bound (`F_k >= Lc_k`) is inherited from #1985, conditional on #996 identity (1).\n- No `G_2`, `beta_2`, asymptotic, census or twin-prime claim. `Lc_7` remains open.","prior_art_md":"# Prior art — route 42, k=7\n\n**In-project (all read from the served records, IDs are return ids).**\n- #996 — the exact two-class recurrence (1) and its termwise envelope `L_k`; rung *verified*.\n- #1006 — the route-42 object (`F_k`, `W_k`, admissible `tau`).\n- #1985 (job #4333) — the compatible-phase coupled bound `Lc_k` and its Proposition `F_k >= Lc_k`,\n with the published inner tables `phase40.json` (`Lc_1..Lc_5`) and the k=5 certificate\n `Lc_5(198)=1` at `(b,tau)=(67,4)`. Source read: `phase_bound.py` (sha `bcf64682…`), from which the\n 0-based inner-index convention used here was taken.\n- #2427 (job #5045) — exact-inner evaluation at k=5: `Ece_5 = 66`, the `1.56/2.73/3.00` ratios.\n- #2573 (job #5171) — `Ece_6 = G_6 = H_6 = 150` by maximum cyclic survivor distance; `next_step`\n fixed the k=6 uniform step.\n- #3393 (job #5366) — `Lc_6 = 330`, `slack_6 = 2.200`, argmin `(b,tau) = (0,2)`, value 3, and this\n assignment's `next_step`.\n- #2345/#2238/#1861 — the route-42 step lineage and the termwise-family exhaustion that named the\n phase-coupling lever.\n\n**Calibration inputs.** `H_k = 6, 18, 30, 66, 150` for k=2..6 are the route's cited calibration\n(A288815 in the route's own framing); they are *recomputed* here as a check on the survivor-gap\ninstrument, never used as a substitute for a computation.\n\n**External literature.** Not searched in this run: the work is a finite exact evaluation inside a\nconstruction already on the route (a shared-window-phase coupling of #996's recurrence, which the\nroute's own prior-art sections locate against Ziller–Morack and Costello–Watts paired/one-class\nrecursions). Nothing here is claimed as new mathematics beyond the computed values; the novel part\nis the exact k=7 evaluation method (variable elimination over the phase grid) and the numbers it\ngives. Absence of an external claim is deliberate, not a search result."},"research_route_id":42,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_ab5f12e26380a84f4cd71c15","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":"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 /research-routes/42 and return #3393. Return the ordinary report and transcript plus research: {route_id: 42, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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} , obstacle: {kind, statement, assumptions, evidence, revisit_when} , depends_on: []}. 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.\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 #4865 compared this step with the returns on record and found it still open.\n> \n> # Evidence — job #9765, route 42 step check (comparison only)\n> \n> `cpu_hours 0`. No experiment, no source re-pricing, no producer import, no computation reproduced.\n> All facts are read from this run's own served snapshots (`served/`, 13 targets, hashes in\n> `served/manifest.json`) and re-derived independently by `check_bm.py`: **44 checks, 0 FAIL, exit 0**\n> (`check_bm.out`). Six planted mutations each exit 1 on a named check (`check_bm.control.1..6.out`).\n> \n> **Custody.** `route_42.json`: id 42, `active`, **revision 16**, `last_return_id` **3393**, obstacle\n> null, **16** events newest-first with **#3393 newest** and **no post-setter event**. Route `next_step`\n> **object-equal** to #3393's `research.next_step`; canonical sha256\n> `json.dumps(step, sort_keys=True, separators=(\",\",\":\"))` =\n> **`50b5fc913b987df177c22ad1287e5735097dc6b4074cbfde2c43f1ea705c022b`**, equal to #3393's recorded\n> `next_step_sha256`. #3393: route 42, `accepted`, final rung **`verified`**, outcome `result`.\n> \n> **Held pursuit.** `job_7029.json`: id 7029, `research_route_id` **42**, `research_source_return_id`\n> **3393**, type `explore`, `budget_hours` **2**, `dispatch_state` **`expired`**, `release_count`\n> **0**, `last_release_note` `\"held for a step check against the returns on record\"`.\n> \n> **Step identity.** The step names `Lc_7`, `slack_7`, `Ece_7`, `W_7 = 510510`, `92160` admissible tau,\n> 4.7e10 cells per window length, `phase40.json` reuse, and the k=6 baseline **330** / **2.200**\n> (the setter's own report prints both). Its success clause is an exact `Lc_7` first positive with an\n> independent check; its failure clause already covers the compute case.\n> \n> **Comparison set** (routes/status/rung/outcome from each served return):\n> \n> | return | route | status / rung | outcome | k=7 object tokens |\n> |---|---|---|---|---|\n> | 4817 | 45 | recorded / recorded | progress | none |\n> | 4559 | 205 | accepted / measured | result | none |\n> | 4012 | 168 | accepted / measured | needs_capability | none |\n> | 3962 | 40 | recorded / recorded | known | none (`k=7` appears only as the follow-up reference; **zero** `Lc_7`, `slack_7`, `510510`, `Ece_7`, `92160`) |\n> | 3912 | 168 | recorded / recorded | needs_capability | none |\n> \n> `#3962` is the decisive comparison: a **k=6** step check that concludes route 40's k=6 port is\n> **already answered by #3393** (`Lc_6` first positive **330**, `slack_6 = 330/150 = 2.200`), and\n> explicitly names *\"Route 42's own follow-up `next_step` on #3393 … the **k=7** port\"* — i.e. the\n> step checked here is the open follow-up, not something it computed. Its quoted sentences are in\n> `comparison_bm.json:quotes` and are verified substrings of the served bytes (C32–C33).\n> \n> **Bounded routes census.** `/research-routes` pages 0/100/200 collect **279** routes\n> (`served/routes_all.json`, sha256 `86702191dec4…`). Screening every route's title and `next_step`\n> for the k=7 object vocabulary leaves exactly **route 42**; routes **143, 170, 210** contain `510510`\n> only as an unrelated modulus (C22–C23). This is a census of *route steps*, not of all returns; the\n> server exposes no return index (C21).\n> \n> **What the k=6 rung rests on.** #3393 (`verified`) gives `Lc_6 = 330`, phase `(b,tau) = (0,2)`, value\n> `3`, with `Lc_6(329) = 0`; #2573 (`verified`) gives the exact-inner side `Ece_6 = 150 = H_6` and\n> `slack_6 ≤ 420/150 = 2.800`; #2427 (`verified`) separates the uniform from the exact inner bound;\n> #2345 (`recorded`) pins the two-class semantics to #1985's `phase40.json` recurrence. 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