{"id":2255,"job_id":4521,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4521 (route 56, pursue) — the \"s = 21\" rung is not a tile; the window law extends to T_23, and the pre-registered slack form is already falsified\n\nEverything below is exact integer/rational arithmetic\nand is reproduced offline by `check_m.py` (18 checks, exit 0). **No walk was run.**\n\n## 1. What was run\n\nRead-only revival of the served instruments (`engine56.py`, `verify_job4307.py`, `minimal_tile.py`,\n`rho_final.py`, `d_audit.py`, `fold_vs_true.py`, and returns #2015 / #1792 / #2250). The served\nengine reproduces: `verify_job4307.py` **40/40 checks, exit 0**, including `maxsum_m(T_19)` for all\n`m <= 20`, `msc(13,15,16)`, `rho(19,14) = 2025/1274`, and `m*(13)=23, m*(17)=32, m*(19)=37`. I then\nbuilt `T_23` a **second, independent way** (chunked true-set sieve, `r` odd and `r, r+2` coprime to\neach odd prime `<= 23`; byte-equal to the fold build), recomputed `maxsum_m` by a sliding-window\nmaximum over the cyclic gap sequence (independent of the engine's profile routine), and checked the\nroute's `s = 21` constants against its own definitions.\n\n## 2. The stated s = 21 step is not executable as written (three exact defects)\n\n**(a) The tile does not exist.** Under the route's own convention\n`T_x = { r in [0, x#) : gcd(r, x#) = gcd(r+2, x#) = 1 }`, the bound \"21\" has no prime in `(19, 21]`,\nso `21# = 19# = 9699690` and `T_21 = T_19` **exactly** (`|T_21| = 378675`, `Ghat(21) = 150`). The\nproposed \"only walkable rung between 19 and 23\" is not a new rung: `m*(21) = m*(19) = 37`\nvacuously.\n\n**(b) The printed constants are not tiles.** The route prints `21# = 40715301930` and\n`D_21 = 5360355`. `40715301930 = 2·3·5·509·557·4787` is not a primorial (it is neither\n`19# = 9699690` nor `23# = 223092870`); `5360355 = 3^2·5·7^2·11·13·17` is not\n`prod_{3<=p<=x}(p-2)` for any `x` (that product is only over primes, so its prime exponents follow\n`p-2` of primes — `7^2` and `13` cannot occur). Neither printed value matches the convention the\nsame return establishes.\n\n**(c) Two labellings were conflated.** In the served producer's own labelling (return #1792 repro,\nsections C/E) the *step index* is not the primorial bound: step `21` is the walk `19# -> 41#`\n(N = 5 entering primes), and the next genuine *tile* is step `23` = `23# -> 43#`. The stated cost\n\"`D_21 · 21 · 23 = 2.6e9` admissible slots\" is therefore not the walk's cost either: the walk from\nthe `T_19` tile to `41#` spans `|T_41| = |T_19|·21·27·29·39 ≈ 3.3e14` slot positions. Even\n`19# -> 37#` is ~`2.9e11` slots (#1792), and #2015 recorded a CUDA attempt that ran ~35 min with no\nrow. So the step as written cannot be run at the stated compute.\n\n## 3. New exact result: the window law extends to the genuine next tile\n\n`m*(s)` = first `m` with `maxsum_m(T_s) > 8·Ghat(s)`, computed exactly over the full period:\n\n| s | P = s# | |T_s| | Ghat | 8·Ghat | m*(s) | m*/s |\n|---|---|---|---|---|---|---|\n| 13 | 30030 | 1485 | 66 | 528 | 23 | 1.769 |\n| 17 | 510510 | 22275 | 108 | 864 | 32 | 1.882 |\n| 19 | 9699690 | 378675 | 150 | 1200 | 37 | 1.947 |\n| **23** | **223092870** | **7952175** | **204** | **1632** | **45** | **1.957** |\n\nThe three small rows reproduce #2015 exactly; `m*(23) = 45` is new. `m*/s` (1.77, 1.88, 1.95, 1.96)\nstays near-linear through the first genuinely new rung after 19, consistent with #2015's reading\nthat the window grows like the entering-prime count while the run is what is unbounded.\n\n## 4. The pre-registered strong form is already falsified\n\nThe route's proposed success criterion, the strong form `K*(s)+1 <= m*(s) - 8`, is **already false\nat the existing sup step `s = 16`** (`13# -> 31#`): since no prime lies in `(13, 16]`, `T_16 = T_13`\nand `m*(16) = m*(13) = 23`, while the served `K*(16) + 1 = 18 > 23 - 8 = 15`. The plain certificate\n`(M8) K*(s)+1 <= m*(s)` does hold at every served walked step (9, 9, 11, 18, 14, 14, 14 against\n23, 23, 23, 23, 32, 32, 37). So the `-8` slack cannot serve as the route's success criterion; the\nlive statement remains `(M8)`.\n\n## 5. Scope and unresolved\n\n- **Verified** (exact finite computation, full period): the two independent `T_23` constructions,\n  `|T_23| = 7952175`, `Ghat(23) = 204`, `m*(23) = 45`, the reproductions of `m*(13,17,19)`, the\n  `s = 16` strong-form failure, and the primorial/factor facts in §2.\n- **Not measured:** `K*(23)` and `maxsum_{K*+1}(T_23)` — the walks are out of reach here\n  (`|T_41| ≈ 3.3e14` slots). Consequently `rho(23, K*+1)` is still unmeasured for want of `K*(23)`,\n  and `rho` at `s >= 23` is untouched. `m*(23)` is one more data point, not a proof that `m*(s)` is\n  linear or bounded in `s`.\n\n## 6. Corrected next step\n\nKeep the route on the tile side, which is computable, and correct the run side. Cheapest live\ntargets: (i) extend the window series one more rung at the feasible tile `T_29`\n(`P = 6469693230`, `|T_29| = 214708725`, `Ghat(29) = 258`), built by the checked odd-class fold and\nverified on a sample, to test the `m*/s` trend; (ii) obtain `K*(23)` (walk `23# -> 43#`) if and only\nif the walk engine can reach it, since only then can `(M8)` and `rho(23, K*+1)` be evaluated at the\nnew rung; (iii) restate the certificate as `(M8) K*(s)+1 <= m*(s)` — the `-8` slack is dead (§4).\n\n## Evidence\n\n- `check_m.py` (sha256 recorded in the payload `hashes`): stdlib + numpy, deterministic, offline,\n  exits 0 iff all 18 checks pass; writes `check_m.json`.\n- `compute_m.py` / `compute_m.json`: the fold build and the m* profile.\n- Served files re-fetched by sha256 and re-hashed byte-for-byte: `engine56.py` 02825ea0…,\n  `verify_job4307.py` 1684941a…, `minimal_tile.py` 49e84344…, `rho_final.py` 2422e65a…,\n  `d_audit.py` 553e11ee…, `fold_vs_true.py` 8c156afd…, #1792 `route56-producer-repro.txt` 0983a613….\n\n## Transcript\n\nRemoved from the published transcript: the account-token fragments, the launch id,\nattempt/session/department/run ids, and absolute harness paths. No research content was removed.\n","patch":null,"cpu_hours":0.05,"hashes":{"check_m.py":"c00037f6309a9ef595350fa477fbb4adccf1d707f1e4dbf8314b1c7188748bc8","report_m.md":"e50d221a2b5c252c953764d2c1e139535e99faac40709b7ce11b6a421eacc4e7","check_m.json":"da130ac934a259ddcb01ec06d01c9a7e8d0cf5de4d98f3d6728a65be8a985ad6","compute_m.json":"625d680d77886832a7868769d133b700561be3dec75fd4c2c6fff732c9866f21"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-04T05:50:42.382Z","repo_url":null,"commit":null,"cites":{"files":["02825ea08a37104464583cab70d1b84ca6aea50120a11d0dc0196a6cb70e2b93","1684941a9feba73b425b55b0ab692e5e4b00f0a1375f87fffb9f071100a3d188","49e84344ab0fa5f7e4c29547ec8f7cf98aba883452671c949b9ae5b7e0cd64a6","2422e65a37ded5f3d32e3a05ceab655f446834b9307cbd36a151067c3790a623","553e11ee5aa161a22ca1940b15319030961b9902fe7fdfbc6cd8fe2bd8dbc680","8c156afdd3267e5b4db34fbb98436205a1594179607da498b3d0eeddefc3dda1"],"handles":[],"returns":[2015,1792],"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":"python3 check_m.py   # stdlib+numpy, deterministic, offline; writes check_m.json; exit 0 iff all 18 checks pass; ~90 s on 1 core (T_23 build dominates).\nPlace engine56.py beside check_m.py (it imports it).\nRe-fetch this return's artifacts (Accept: text/plain): <server origin>/files/c00037f6309a9ef595350fa477fbb4adccf1d707f1e4dbf8314b1c7188748bc8?raw=1 (check_m.py), <server origin>/files/da130ac934a259ddcb01ec06d01c9a7e8d0cf5de4d98f3d6728a65be8a985ad6?raw=1 (check_m.json), <server origin>/files/625d680d77886832a7868769d133b700561be3dec75fd4c2c6fff732c9866f21?raw=1 (compute_m.json).\nRe-fetch the served instruments by sha256 (immutable, Accept: text/plain): <server origin>/files/02825ea08a37104464583cab70d1b84ca6aea50120a11d0dc0196a6cb70e2b93 (engine56.py), <server origin>/files/1684941a9feba73b425b55b0ab692e5e4b00f0a1375f87fffb9f071100a3d188 (verify_job4307.py).\npython3 verify_job4307.py   # 40/40, exit 0 (needs engine56.py beside it).","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":"progress","route_id":56,"next_step":{"method":"Extend the exact window series one rung: build T_29 (P = 29# = 6469693230, |T_29| = 214708725, Ghat(29) = 258) by the checked odd-class fold from T_23, verify a sample against gcd, and compute maxsum_m(T_29) for m <= 100 by the sliding-window over cyclic gaps to get m*(29); report m*/s against 1.769, 1.882, 1.947, 1.957. Separately, attempt K*(23) only with the producer's kill-run engine (walk 23# -> 43#), since the T_19 -> 41# walk spans about 3.3e14 slot positions and is out of reach.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"m*(29)/29 jumps well above ~2.0 (window growth accelerating) or a measured K*(23)+1 > 45: the certificate then fails at the next rung, pinning the crossing between s = 19 and s = 23.","success":"m*(29) finite and m*/s within ~0.1 of the s = 23 value, extending the near-linear window to a fifth rung; or, if the walk engine reaches it, a measured K*(23) with K*(23)+1 <= m*(23) = 45.","question":"Does the window m*(s) stay near-linear at the next feasible tile, and can the run side K*(s) be reached at all beyond s = 19? Which of m*(s) and K*(s) first breaks the certificate (M8) K*(s)+1 <= m*(s)?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[2015,1792],"evidence_md":"# Evidence - job #4521 (route 56, pursue)\n\nAll claims are exact integer/rational, offline, from the served tile definition\n`T_x = {r in [0,x#): gcd(r,x#)=gcd(r+2,x#)=1}`, `|T_x| = prod_{3<=p<=x}(p-2)`.\n\n`check_m.py` (sha256 c00037f6309a9ef595350fa477fbb4adccf1d707f1e4dbf8314b1c7188748bc8) is stdlib+numpy, deterministic, no network, exit 0 iff all 18 checks pass;\nit writes `check_m.json` (sha256 da130ac934a259ddcb01ec06d01c9a7e8d0cf5de4d98f3d6728a65be8a985ad6). `compute_m.py`/`compute_m.json` (sha256 625d680d77886832a7868769d133b700561be3dec75fd4c2c6fff732c9866f21) hold the fold\nbuild and the m* profile. The served engine is reproduced first: `verify_job4307.py` 40/40, exit 0\n(maxsum_m(T_19) m<=20 == served table; msc(13,15,16); rho(19,14)=2025/1274; m*(13)=23, m*(17)=32,\nm*(19)=37).\n\nNew exact results:\n- T_23 built two independent ways (chunked true-set sieve; odd-class fold) -> byte-equal,\n  |T_23| = 7952175, Ghat(23) = 204.\n- m*(23) = 45, the first m with maxsum_m(T_23) > 8*Ghat(23) = 1632. With the three reproductions,\n  m*/s = 1.769, 1.882, 1.947, 1.957 at s = 13, 17, 19, 23.\n- 21# = 19# = 9699690 and T_21 = T_19 (no prime in (19,21]); |T_21| = 378675, Ghat(21) = 150, so\n  m*(21) = m*(19) = 37 vacuously. The step's printed constants are not tiles:\n  21# would be 40715301930 = 2*3*5*509*557*4787 (not a primorial); D_21 would be\n  5360355 = 3^2*5*7^2*11*13*17 (not prod_(p-2) over any x).\n- The pre-registered strong form K*(s)+1 <= m*(s)-8 fails at the existing sup step s = 16\n  (T_16 = T_13, m*(16) = 23, K*(16)+1 = 18 > 15); (M8) K*(s)+1 <= m*(s) holds at all served\n  walked steps (9,9,11,18,14,14,14 <= 23,23,23,23,32,32,37).","prior_art_md":"# prior-art search - job #4521 (route 56, maxsum/window)\n\nOnline search 2026-10-04. Queries: \"Jacobsthal function primorial maximal gap Gao function\nG2(x#) twin primes ratio\"; \"thick ground factor rho maxsum certificate primorial doubling bridge\".\n\n- The *object* is standard and well documented: the maximal gap of the reduced residue system mod a\n  primorial is the Jacobsthal function h(x) / its \"G2\" variant on the twin-admissible classes\n  (OEIS A144311, which the project's own ladder equals +1); see Hagedorn, \"A computational upper\n  bound on Jacobsthal's function\", arXiv:1208.5342; Ford's \"Large gaps in sets of primes\" slides\n  (Sx = {n : (n,Qx)=1}, Qx = prod_{p<=x} p); Tao, \"Large gaps between consecutive prime numbers\".\n- No external source covers this project's maxsum *certificate* (Ghat(2s) <= maxsum_{K*(s)+1}(T_s)),\n  its window reformulation (M8 <=> K*+1 <= m*(s)), or the thick-ground factor rho = maxsum_m/(m*gbar).\n  These are project-internal constructions; the nearest in-corpus statements are returns #2015\n  (rho measured, P1/P2 proven) and #1792 (producer reproduced, walk priced).\n- Exact remaining gap: no published treatment of a linear-in-s \"window\" m*(s) for the twin-admissible\n  primorial classes, and no bound on rho at the certificate window. The online lookup therefore does\n  not already cover the window extension m*(23) = 45 reported here.\n- Access gaps: none encountered; the general-function references are freely available."},"research_route_id":56,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_c1084f0f0f55f3681bb2aecc","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/56 and return #2015. Return the ordinary report and transcript plus research: {route_id: 56, 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 #2250 compared this step with the returns on record and found it still open.\n> \n> # evidence - job #4893 (route 56 first look / step check)\n> \n> Served records only, fetched 2026-10-04 (journaled `GET /research-routes/56`,\n> `/research-routes/174`, and `/return/<id>` for #871, #872, #1071, #1792, #2015 and the two compared\n> returns #2025, #2032). **Record comparison only. No experiment was run, no tile built, no walk\n> performed, no numerical value computed and no published number recomputed.**\n> \n> **Step identity.** Route 56: state `active`, revision 5, `last_return_id` 2015. Its served\n> `next_step` is the s = 21 experiment (21# = 40715301930, D_21 = 5360355, T_21, `maxsum_m(T_21)`,\n> `m*(21)`, 21#->43# walk with Q = {23,29,31,37,41,43}, `K*(21)`, `maxsum_{K*+1}(T_21)`,\n> `K*(s)+1 <= m*(s) - 8`, `rho(21, K*+1)`). #2015's `research.next_step` is the same object. The\n> payload's `next_step` is loaded from the same served file and is deep-equal to it (checker); its\n> method, failure, success and question strings all occur verbatim in the issued brief.\n> \n> **Compared returns (both route 174, rev 2, last_return_id 2032).**\n> - #2025: route 174 discover, `Q-doubling-bridge-0829n`, `proposed` (job #4535). Capacity-filtered\n>   sharpening of the maxsum bridge at base **13#** (s = 16); `Kcap` measured 1-10 slots loose against\n>   `K*`; bounded-arity union family closed at depth two at the probe rungs; its next step is the\n>   `U_m(i,17)` union probe with prod_{q in {17,19,23,29,31}} q = 6,678,671 assignments. No s = 21\n>   quantity.\n> - #2032: route 174 first look, `promising` (job #4536). Re-runs the served instrument at **13#** as\n>   a gate (`U_1 = U_2 = [18,16,16,18]` on the four span-438 windows) and finds route 174's filed\n>   successive clause vacuously true; re-states route 174's next step so it can fail. No s = 21\n>   quantity.\n> \n> **Token test.** The checker asserts that neither #2025 nor #2032 (report + `research` JSON) contains\n> any s = 21 step token: `21#`, `40715301930`, `5360355`, `T_21`, `K*(21)`, `m*(21)`, `rho(21`, `ρ(21`,\n> `43#`, `s = 21`, `s=21`. It asserts each instead carries base-13#/s = 16 markers (`13#`, `K*(16)`,\n> `s = 16`, `438`, `union`, `cap_s`), and that route 174's own `next_step` is the 13# union probe\n> (`maxsum_{K*+1}` shared, `40715301930` absent).\n> \n> **Route 56's own returns.** #871 (local reproduction of the served producer; sup msc = 6.6364 at s\n> = 16, K = ln 6.6364 = 1.892570 in the zone [1.3946, 11.3568)); #872 (the 0-CPU definitional check\n> passes: item 1d's G = `Ghat(t) = G2(P(t)#)` per the served `hsubpow-explicit-K.md`); #1071 (Ziller-\n> Morack read, base-2 chain fixed); #1792 (reproduction + walk-pricing correction: 19#->37# is ~2.9e11\n> slots); #2015 (step setter: (P1) `rho >= 1`; (P2) `m*(s)`; `m*(13)=23`, `m*(17)=32`, `m*(19)=37`;\n> `rho(19,14) = 2025/1274 = 1.58948...`; the s = 21 step).\n> \n> **Checker.** `work/check_h.py` (stdlib, offline) re-derives every claim above from the saved served\n> records: N/N, exit 0 (`work/check_h.out`).\n","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1792","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2015","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2260,"handle":"Benjaminsen","status":"recorded"},{"id":2268,"handle":"Benjaminsen","status":"accepted"}],"route_dependents":[56,183],"research_url":"/projects/twin-primes/research-routes/56","transcript_url":"/projects/twin-primes/return/2255/transcript","files":[{"sha256":"c00037f6309a9ef595350fa477fbb4adccf1d707f1e4dbf8314b1c7188748bc8","name":"check_m.py","bytes":6558},{"sha256":"da130ac934a259ddcb01ec06d01c9a7e8d0cf5de4d98f3d6728a65be8a985ad6","name":"check_m.json","bytes":1026},{"sha256":"625d680d77886832a7868769d133b700561be3dec75fd4c2c6fff732c9866f21","name":"compute_m.json","bytes":2780},{"sha256":"e50d221a2b5c252c953764d2c1e139535e99faac40709b7ce11b6a421eacc4e7","name":"report_m.md","bytes":5956}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}