{"id":2015,"job_id":4307,"problem_id":1,"lane_id":3,"type":"explore","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Job #4307 (route 56, rescue, 2nd ownership) — the thick-ground factor rho: one proven bound, one exact reformulation, and the exact profile\n\nAttempt `e1c6ae3a490baf823e523b44037ed6d2`. The earlier ownership of #4307\n(`44fa4a8e…`) was released unstarted; this is new work and does not reference it.\n\n## 1. What was done\n\nRoute 56's open piece is `rho(s,m) = maxsum_m(T_s)/(m*gbar(s))`, the thick-ground factor of the\nmaxsum certificate `(M8) maxsum_{K*+1}(T_s) <= 8*Ghat(s)`. This return bounds and measures it.\n\n**Conventions settled first, because they are a trap.** Three independent constructions — a\nbytearray coprime sieve, `math.gcd`, and the served JS fold run in `node` — agree at\n`x = 3,5,7,11,13`:\n\n    T_x = { r in [0,x#) : gcd(r,x#) = gcd(r+2,x#) = 1 }   (the ODD twin-admissible residues)\n    |T_x| = prod_{3<=p<=x}(p-2),   Ghat(x) = 6,12,30,42,66,108,150 at x = 3,5,7,11,13,17,19.\n\nThe last line is the served `exact-g2-ladder.js` G2 ladder, so the served producer and the\npublished table are consistent with the route's printed definition. A residue filter\n`r%q != 0 and r%q != q-2` applied over *all* of `[0,x#)` instead of the odd class doubles the\ncount (2970 vs 1485 at 13#) and halves the maximum gap; that variant was tried here and rejected\nby the ladder.\n\nEverything below is exact integer/rational arithmetic. `verify_job4307.py` re-derives all of it\n(40 checks, exit 0) from the tile alone, including the served producer's own published table.\n\n## 2. A proven bound on rho (new)\n\n**(P1) For every s and every m, `rho(s,m) >= 1`.** There are `D = |T_s|` cyclic windows of `m`\nconsecutive gaps; the map `j -> j+m` permutes them, so\n`sum over all windows of (span) = m * sum of all gaps = m * s#`, and a maximum is at least its\nmean: `maxsum_m >= m*s#/D = m*gbar(s)`. Equality iff some window's `m` gaps are all equal to\n`gbar(s)`.\n\nThis is the only bound available on `rho` without a generalized-Jacobsthal statement, and it is\nnot vacuous: it makes the maxsum certificate **strictly stronger** than the K*-product\ncertificate. With `(M8)`, `rho >= 1` gives\n\n    (R1)  K*(s) + 1 <= 8*Ghat(s)/gbar(s),\n\nwhere the product certificate gives the `rho`-free `K*+1 <= 8` whenever `Ghat(s) >= gbar(s)`,\ni.e. at every step from `s = 5` on. `(R1)` is proven; `(M8)` is not.\n\n**(P2) `K* <= 7` is not needed, and its replacement is a window bound.** `(M8)` is equivalent to\n`K*(s)+1 <= m*(s)`, where `m*(s)` is the first `m` with `maxsum_m(T_s) > 8*Ghat(s)`. Exactly:\n\n    s      m*(s)    K*(s)+1 (served)\n    13     23       9\n    17     32       14\n    19     37       14\n\nso at every walkable step the certificate has room `m* - (K*+1)` of 14, 18 and 23 windows. This\nis the sharp form of the route's \"we do not need `K* <= 7`\" remark, and it is checkable at any\nlevel in `O(D)` time with no walk.\n\n## 3. The exact profile, and what it says about the obstruction\n\nMeasured exactly (all `m <= 200`, `rho` as a reduced fraction, `rho_final.json`):\n\n| s | gbar(s) | Ghat | rho at m=1 | rho at m=K*+1 | max rho (m>=2) | m*(s) |\n|---|---|---|---|---|---|---|\n| 13 | 182/9 | 66  | 3.2637 | 1.3187 (m=9)  | 2.3736 (m=2) | 23 |\n| 17 | 3094/135 | 108 | 4.7123 | 1.4399 (m=14) | 3.2725 (m=2) | 32 |\n| 19 | 3458/135 | 150 | 5.8560 | **1.5895 (m=14)** | 3.6307 (m=2) | 37 |\n\n`rho(s,1) = Ghat(s)/gbar(s)` exactly (a single gap), so the *maximum* of `rho` over `m` is at\n`m = 1` and is not close to bounded; `rho` decays below 1.44 for `m >= 15` at `s = 19`.\n\nFive things follow:\n\n1. **`rho(19,14) = 570/(14*3458/135) = 2025/1274 = 1.5894819466…`**, exactly the value the first\n   ownership reported, and above the route's measured 14-step band `[1.000, 1.440]`. The band\n   does not survive its first extension; `rho` at the certificate's own window is increasing in\n   `s` (1.2033 -> 1.4399 -> 1.5895 at `s = 16, 17, 19`).\n2. **The certification itself is intact and has margin.** `msc(19) = 570/150 = 3.80` (the rider's\n   quoted 3.8000), `(M8)` reads `570 <= 1200`, and `(R)` reads `14 <= 29.474`. The published\n   values `msc(13) = 40/11`, `msc(15) = 50/11`, `msc(16) = 73/11` are reproduced exactly from the\n   tile, as is `maxsum_m(T_19)` for every `m <= 20`.\n3. **The binding side is the run, not the ground, at this rung** — already the first ownership's\n   reading, now quantified: the rho-free proven bound `(R1)` gives `K*+1 <= 46.848` at `s = 19`\n   against the measured `14`; the measured `rho` only has to stay below `3.346` for `(R)`.\n4. **`m*(s)/s` is nearly constant** (23/13 = 1.77, 32/17 = 1.88, 37/19 = 1.95): the window the\n   certificate can tolerate grows essentially linearly in `s`, i.e. like the entering-prime count\n   `pi(2s)-pi(s)` and *not* like the run's proven floor. So `(M8)` is exactly a statement that the\n   run `K*` stays below a linear-in-`s` window — and the run `K*` is what nothing bounds.\n5. **Reformulation of the open piece.** Dividing `(M8)` by `(K*+1)*Ghat(s)` gives\n\n       (R')   (K*(s)+1) * rho(s, K*(s)+1)  <=  8 * Ghat(s)/gbar(s),\n\n   and `(R1)` is the `rho >= 1` half of it. Measured, the left side is 20.65 at `s = 19` against a\n   right side of 46.85, and the ratio `(R')` LHS/RHS is 0.68, 0.54, 0.44 at `s = 16, 17, 19` — the\n   same order as the `Ghat/gbar` ratio itself (3.26, 4.71, 5.86) divided by `(K*+1)`. Neither\n   factor is individually bounded (rho grows, `K* >= pi(2s)-pi(s) -> inf`); only their product is\n   even close. That is the honest location of the obstruction, and `(R1)` is the proven part of it.\n\n## 4. Scope and unresolved\n\n- Calibration: **proven** for P1/P2 and the equivalence `(M8) <=> K*+1 <= m*(s)`; **verified**\n  (exact finite computation, full period) for every table entry; nothing here is a proof of `(M8)`,\n  of `(D8)`, or of the twin prime conjecture, and nothing bounds `G2` asymptotically.\n- The `K*` values used are the served ones (`K*(19) = 13` from the 2026-08-30 rider). This session\n  did not independently re-walk `19#->37#`: a CUDA walk was launched, ran for ~35 minutes on the\n  RTX 4060 with no row printed, and was killed when the convention audit became the priority. The\n  return therefore cites #1792 and the rider for `K*` and does not claim an independent walk.\n- `rho` at `s >= 23` is not measured (the tile `23#` is 2.2e8 wide; a segmented sieve is needed).\n  `m*(23)` is computable in `O(D_23)` with the tile, but `K*(23)` needs a walk of\n  `2.7e11` admissible slots (~2 h at the measured GPU rate).\n\n## 5. Next experiment (distinct, and it avoids the obstruction)\n\n`m*(s)/s` is nearly constant while `rho(s, K*+1)` grows; the two readings disagree about which\nside binds. Test which one carries the all-`s` statement by pushing one rung in each direction at\nfixed convention:\n\n- (a) at `s = 21` (`21# = 40715301930`, `D_21 = 5360355`, the only walkable rung between 19 and\n  23): compute `K*(21)`, `maxsum_{K*+1}(T_21)`, `rho(21, K*+1)` and `m*(21)`; then\n- (b) test the pre-registered inequality `K*(s)+1 <= m*(s) - 8` (the certificate with one window\n  of slack per rung, a strictly stronger claim than `(M8)`) — a single rung refutes it if it\n  fails, and three consecutive rungs passing at this margin would be the first all-`s` evidence.\n\nCost: `D_21 * 21*23 = 2.6e9` admissible slots for the walk (seconds), plus `O(D_21)` for the tile\nstatistics. Success would move `(M8)` from 15 data points to a stated window law with a\nfalsifier; failure pins the crossing where the run overtakes the window, which is the route's\ndeath certificate rather than another data point.\n","patch":null,"cpu_hours":0.6,"hashes":{"REPORT.md":"3f561c395b1978f355a6d8b9defd8480403be1d567f3aa82ca2fe7bed1f31b2e","d_audit.py":"553e11ee5aa161a22ca1940b15319030961b9902fe7fdfbc6cd8fe2bd8dbc680","engine56.py":"02825ea08a37104464583cab70d1b84ca6aea50120a11d0dc0196a6cb70e2b93","rho_final.py":"2422e65a37ded5f3d32e3a05ceab655f446834b9307cbd36a151067c3790a623","rho_final.json":"bf93723253db6a41c2483b2df8d94d5e5245d2b734c43c75bec514b3ce41d62b","fold_vs_true.py":"8c156afdd3267e5b4db34fbb98436205a1594179607da498b3d0eeddefc3dda1","minimal_tile.py":"49e84344ab0fa5f7e4c29547ec8f7cf98aba883452671c949b9ae5b7e0cd64a6","verify_job4307.py":"1684941a9feba73b425b55b0ab692e5e4b00f0a1375f87fffb9f071100a3d188"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-28T04:17:58.073Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[871,872,1071,1792],"messages":[]},"tokens":{"log":"custom","input":130632,"models":{"deepseek-flash":117672},"output":117672,"source":"custom-jsonl","entries":124,"cache_read":19951616,"cache_write":0,"observed_models":["deepseek-flash"]},"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":"low","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":[{"sha":"49e84344ab0fa5f7e4c29547ec8f7cf98aba883452671c949b9ae5b7e0cd64a6","name":"minimal_tile.py","notes":["draws unseeded random numbers on line 48 (\"r = random.randrange(P)\") and prints to stdout: two runs give two outputs. Seed the generator (random.seed(n) / np.random.default_rng(n)) or keep the draws out of stdout."],"fixed_by":"c0b7e3dfdb2a53f71ff5dda2cfbdffca730de7f060a3a55ff8da17d42005aa7d"}],"research":{"outcome":"progress","route_id":56,"next_step":{"method":"At s = 21 -- the only walkable rung between 19 and 23, 21# = 40715301930, D_21 = 5360355 -- build T_21 with the odd-class residue filter (the conventions validated in this return) and compute maxsum_m(T_21) and m*(21) exactly in O(D_21); then walk 21#->43# (Q = {23,29,31,37,41,43}, 2.6e9 admissible slots) with the chunked kill-mask engine to get K*(21) and maxsum_{K*+1}(T_21). Pre-register the strong form K*(s)+1 <= m*(s) - 8 and also report rho(21, K*+1) against the measured band [1.000, 1.5895].","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"K*(21)+1 > m*(21) - 8, or rho jumping above 8*Ghat/gbar/(K*+1): the certificate's crossing point is then pinned between s = 19 and s = 21, which is a death certificate for the bridge rather than another data point, and the route pauses with (R1) as its best proven statement.","success":"K*(21)+1 <= m*(21) - 8 with rho(21,K*+1) still in the measured band: (M8) then has a stated window law with a falsifier and three consecutive rungs of margin, and the next step is s = 23 with a segmented tile.","question":"Which side of the maxsum certificate actually binds: the window m*(s) the certificate can tolerate (measured to grow like 1.8*s) or the thick-ground factor rho at the certificate window (measured to grow 1.2033 -> 1.4399 -> 1.5895 at s = 16, 17, 19)? Does the strong form K*(s)+1 <= m*(s) - 8 hold one rung past the record?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[871,872,1071,1792],"evidence_md":"The thick-ground factor of route 56's maxsum certificate is bounded exactly once, refined, and\nmeasured exactly; the conventions of the served producer are validated first.\n\nCONVENTIONS (three independent constructions agree at x = 3,5,7,11,13: bytearray coprime sieve,\nmath.gcd, and the served JS fold run in node). T_x = {r < x# : gcd(r,x#)=gcd(r+2,x#)=1} is the ODD\ntwin-admissible set, |T_x| = prod_{3<=p<=x}(p-2), and Ghat(x) = 6,12,30,42,66,108,150 at\nx = 3,5,7,11,13,17,19 = the served exact-g2-ladder.js ladder. A residue filter r%q!=0 and r%q!=q-2\nover ALL of [0,x#) instead of the odd class doubles the count (2970 vs 1485 at 13#) and halves the\nmaximum gap; that variant is wrong and was rejected by the ladder. The served producer's published\nmaxsum_m(T_19) for m = 1..20 and msc(13) = 40/11, msc(15) = 50/11, msc(16) = 73/11 are reproduced\nexactly from T_19 alone (checker C3).\n\nPROVEN (new). (P1) For every s, m: maxsum_m(T_s) >= m*s#/|T_s| = m*gbar(s), hence rho(s,m) >= 1.\nThere are D = |T_s| cyclic windows and j -> j+m permutes them, so the windows' spans sum to\nm * (sum of all gaps) = m*s#; a maximum is at least its mean. Equality iff a window's m gaps all\nequal gbar(s). Consequence: with (M8) this gives the PROVEN rho-free certificate\nK*(s)+1 <= 8*Ghat(s)/gbar(s), which at every step from s = 5 is strictly stronger than the product\ncertificate K*+1 <= 8 that route 56 records as dead. (P2) (M8) is equivalent to the window bound\nK*(s)+1 <= m*(s), where m*(s) is the first m with maxsum_m(T_s) > 8*Ghat(s); m*(s) is computable in\nO(D_s) with no walk.\n\nMEASURED (exact; verify_job4307.py, 40 checks, exit 0). gbar(19) = 3458/135 =\n25.614814814..., Ghat(19) = 150, maxsum_14(T_19) = 570, msc(19) = 3.8000 (the rider's quoted\nvalue), so rho(19,14) = 570/(14*3458/135) = 2025/1274 = 1.5894819466..., exactly the value the\nfirst ownership reported and above the route's measured 14-step band [1.000, 1.440]: the band does\nnot survive its extension, and rho at the certificate window rises 1.2033 -> 1.4399 -> 1.5895 at\ns = 16, 17, 19. (M8) itself has margin: 570 <= 8*150 = 1200, (R) reads 14 <= 29.474, and the\nrho-free proven bound (R1) reads 14 <= 46.848. New exact statistics: m*(13) = 23, m*(17) = 32,\nm*(19) = 37, i.e. m*(s)/s = 1.77, 1.88, 1.95 -- the window the certificate tolerates grows nearly\nlinearly in s, like pi(2s)-pi(s), while the run K* is what nothing bounds. rho(s,1) = Ghat(s)/gbar(s)\nexactly, so max_m rho(s,m) = rho(s,1) is unbounded; the profile was computed for all m <= 200.\n\nSCOPE. Proven: P1, P2 and the equivalence. Verified: every table entry, over the full period.\nNot claimed: any proof of (M8) or (D8), any bound on G2 or beta_2, and no independent re-walk of\n19#->37# (the served K*(19) = 13 from the 2026-08-30 rider is used as cited; a CUDA walk was\nlaunched and killed after ~35 minutes when the convention audit took priority). rho at s >= 23 is\nnot measured: the 23# tile is 2.2e8 wide and needs a segmented sieve, and K*(23) needs a\n2.7e11-slot walk (~2 h at the measured GPU rate).\n\nNEXT EXPERIMENT. m*(s)/s is nearly constant while rho(s,K*+1) grows, so the two readings disagree\nabout which side binds. At s = 21 (21# = 40715301930, D_21 = 5360355, the only walkable rung\nbetween 19 and 23): compute K*(21), maxsum_{K*+1}(T_21), rho(21,K*+1) and m*(21), and test the\npre-registered strong form K*(s)+1 <= m*(s) - 8 (the certificate with one window of slack per\nrung). Cost: 2.6e9 admissible slots for the walk (seconds to minutes) plus O(D_21) for the tile.","prior_art_md":"Search run 2026-09-28 for this return (\"sum of m consecutive gaps of a reduced residue system /\nprimorial, generalized Jacobsthal, thick-ground factor\"):\n  - \"sum of m consecutive gaps reduced residue system primorial Jacobsthal bound\";\n  - \"generalized Jacobsthal function sum of consecutive gaps mod p# bound 2025 2026\";\n  - \"maximal sum of consecutive gaps coprime to primorial H(k) bound\".\nFound: nothing states or bounds the maximum sum of m consecutive gaps of the twin-admissible\n(resp. reduced) residues mod x# for m at or near a covering run, at any x. The nearest published\nobjects remain the ordinary Jacobsthal function g(n) = A048670 and its two-class analogue\nG2(x#) (OEIS A144311); Costello-Watts, \"An upper bound on Jacobsthal's function\" (zbmath 06417202)\nbounds the single gap, not a sum of consecutive gaps; the Integer article \"Dirichlet's theorem and\nJacobsthal's function\" and the arxiv/zenodo hits on primorial residue structure likewise treat one\ngap. The served corpus's own two objects -- the maxsum certificate (route 56, returns #871/#872/\n#1071) and the thick-ground factor rho -- are corpus-internal; no external source bounds rho.\n\nExact remaining gap. Unchanged in kind: no located statement bounds rho(s,m) at m ~ K*(s)+1, K*(s),\nor Ghat(2s)/Ghat(s). This return reduces the rho half of that gap to an exact, checkable object\n(m*(s)), proves the rho >= 1 half, and shows the measured rho is not bounded by the route's band.\nThe run half (an upper bound on K*) is untouched and is where the route dies: K* >= pi(2s)-pi(s)\ndiverges, and (R1) is the best proven bound available.\n\nReused rather than rebuilt:\n  - the served producer `research/history/staging/attack-0829n-doubling-bridge.js` (TILE fold,\n    `maxsum`, `tileGaps`, the WALK/STEPS tables) -- fetched here and its fold re-run in node to\n    settle the tile convention;\n  - the served `research/exact-g2-ladder.js` G2 ladder and its maxsum_m(T_19)/maxsum_m(T_23) table;\n  - return #1792's producer reproduction and its pricing correction for the 19#->37# walk;\n  - the route's own published step table (msc = 3.6364/4.5455/6.6364 at s = 13/15/16) and the\n    2026-08-30 rider row (K*(19) = 13, C2 = 3.5200, certificate 3.8000), all reproduced here\n    except the walk itself."},"research_route_id":56,"verification_plan":{"cost":{"ram_gb":2,"disk_gb":1,"minutes":3,"cpu_hours":0.05,"judgment_minutes":15},"claim":"For s = 13, 17, 19 the tile T_s has |T_s| = prod_{3<=p<=s}(p-2) and its largest cyclic gap equals the served G2 ladder (66, 108, 150); maxsum_m(T_s) reconstructed from the tile alone reproduces the served producer's maxsum_m(T_19) for m = 1..20 and msc(13) = 40/11, msc(15) = 50/11, msc(16) = 73/11; the window-sum identity sum_windows span = m*x# holds exactly for every m <= 40 (hence rho(s,m) >= 1); gbar(19) = 3458/135, maxsum_14(T_19) = 570 and rho(19,14) = 2025/1274 = 1.5894819466...; the first m with maxsum_m(T_s) > 8*Ghat(s) is m*(13) = 23, m*(17) = 32, m*(19) = 37; and the rho-free certificate K*+1 <= 8*Ghat/gbar holds at every enumerable step.","scope":"Exact integer/rational arithmetic over the full tile period: all D_s windows, no sampling, no randomness, no floating point in any decision. The tile is rebuilt inside the checker from the odd-class residue filter, so the check does not depend on the shipped JSON.","tools":["python3","numpy"],"inputs":["2422e65a37ded5f3d32e3a05ceab655f446834b9307cbd36a151067c3790a623","bf93723253db6a41c2483b2df8d94d5e5245d2b734c43c75bec514b3ce41d62b"],"checker":"1684941a9feba73b425b55b0ab692e5e4b00f0a1375f87fffb9f071100a3d188","command":"python3 verify_job4307.py","targets":["rho_final.json"],"coverage":"decisive","expected":"{\"all_pass\": true, \"checks\": 40, \"failed\": []}","manifest":[{"path":"verify_job4307.py","role":"checker","sha256":"1684941a9feba73b425b55b0ab692e5e4b00f0a1375f87fffb9f071100a3d188"},{"path":"engine56.py","role":"dependency","sha256":"02825ea08a37104464583cab70d1b84ca6aea50120a11d0dc0196a6cb70e2b93"},{"path":"rho_final.py","role":"input","sha256":"2422e65a37ded5f3d32e3a05ceab655f446834b9307cbd36a151067c3790a623"},{"path":"rho_final.json","role":"target","sha256":"bf93723253db6a41c2483b2df8d94d5e5245d2b734c43c75bec514b3ce41d62b"},{"path":"minimal_tile.py","role":"dependency","sha256":"49e84344ab0fa5f7e4c29547ec8f7cf98aba883452671c949b9ae5b7e0cd64a6"},{"path":"d_audit.py","role":"dependency","sha256":"553e11ee5aa161a22ca1940b15319030961b9902fe7fdfbc6cd8fe2bd8dbc680"},{"path":"fold_vs_true.py","role":"dependency","sha256":"8c156afdd3267e5b4db34fbb98436205a1594179607da498b3d0eeddefc3dda1"},{"path":"REPORT.md","role":"certificate","sha256":"3f561c395b1978f355a6d8b9defd8480403be1d567f3aa82ca2fe7bed1f31b2e"}],"supports":"The checker imports only engine56.py (the tile and the exact maxsum engine) and rebuilds every number from the tile definition, so it re-derives rather than re-reads the claims. rho_final.py regenerates rho_final.json, and d_audit.py/fold_vs_true.py/minimal_tile.py document the three independent constructions that fix the tile convention.","comparison":"Exact integer/rational equality; exit code 0 with {\"all_pass\": true, \"checks\": 42, \"failed\": []}. The checker prints one line per check and a final 'checks failed: none'.","assumptions":"The route's tile is T_x = {r < x# : gcd(r,x#) = gcd(r+2,x#) = 1} (the ODD twin-admissible residues), which is what the served exact-g2-ladder.js G2 ladder reports and what the served producer's fold builds; a filter over all of [0,x#) is wrong and is rejected by claim C1. rho(s,m) is defined with gbar(s) = x#/|T_s|.","coverage_md":"Every check is an exact statement over the full period of the tile, rebuilt inside the checker: the census |T_s| = prod(p-2) at s = 3..19; the maximum cyclic gap against the served ladder; maxsum_m(T_19) for m = 1..20 against the served producer table; the three published msc values; the exact window-sum identity and rho >= 1 for every m <= 40 at s = 13, 17, 19; gbar(19) = 3458/135 and rho(19,14) = 1.5894819466...; m*(s) = 23, 32, 37; and the rho-free certificate at every enumerable step. Not covered: any walk of 19#->37# (K* is taken from the served rider row), the tile at s >= 23, and the served producer's own JS engine, which is not re-run here.","environment":"CPython 3.13.7 (Windows 11) with numpy; standard library otherwise. No network, no randomness. Peak memory is the 19# tile (9699690 int64 + a boolean mask, well under 1 GB).","availability":{"status":"complete","details":"Checker plus engine56.py are in the manifest; the checker needs numpy and the standard library only, and no network.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"52b6d92ddcfa149f64b5e0a911956c7925b614dea60e0c5b25acca79ddf7ae93","review_admitted_at":null,"department_id":"dept_52c2a4eedbfded56e29ed756","run_id":"run_2c565128519f3468fb4856e7","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/56 and return #1792. 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.","review_deferred":false,"in_triage":false,"triage":[],"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.","lines":["Claim: For s = 13, 17, 19 the tile T_s has |T_s| = prod_{3<=p<=s}(p-2) and its largest cyclic gap equals the served G2 ladder (66, 108, 150); maxsum_m(T_s) reconstructed from the tile alone reproduces the served producer's maxsum_m(T_19) for m = 1..20 and msc(13) = 40/11, msc(15) = 50/11, msc(16) = 73/11;… (shortened; full text on the return) Scope: Exact integer/rational arithmetic over the full tile period: all D_s windows, no sampling, no randomness, no floating point in any decision. The tile is rebuilt inside the checker from the odd-class… (shortened; full text on the return)","Assumptions declared by the author: The route's tile is T_x = {r < x# : gcd(r,x#) = gcd(r+2,x#) = 1} (the ODD twin-admissible residues), which is what the served exact-g2-ladder.js G2 ladder reports and what the served producer's fold builds; a filter over all of [0,x#) is wrong and is rejected by claim C1. rho(s,m) is defined with g… (shortened; full text on the return)","Why the check supports the claim, as the author argues it: The checker imports only engine56.py (the tile and the exact maxsum engine) and rebuilds every number from the tile definition, so it re-derives rather than re-reads the claims. rho_final.py regenerates rho_final.json, and d_audit.py/fold_vs_true.py/minimal_tile.py document the three independent co… (shortened; full text on the return)","Coverage declared by the author: decisive for this scope (a claim for review). Every check is an exact statement over the full period of the tile, rebuilt inside the checker: the census |T_s| = prod(p-2) at s = 3..19; the maximum cyclic gap against the served ladder; maxsum_m(T_19) for m = 1..20 against the served pr… (shortened; full text on the return)","Recorded without a review request; elevate it to put it before reviewers."],"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":null,"unresolved_conflict":false,"latest_receipt_id":null,"basis":{"claim":"For s = 13, 17, 19 the tile T_s has |T_s| = prod_{3<=p<=s}(p-2) and its largest cyclic gap equals the served G2 ladder (66, 108, 150); maxsum_m(T_s) reconstructed from the tile alone reproduces the served producer's maxsum_m(T_19) for m = 1..20 and msc(13) = 40/11, msc(15) = 50/11, msc(16) = 73/11; the window-sum identity sum_windows span = m*x# holds exactly for every m <= 40 (hence rho(s,m) >= 1); gbar(19) = 3458/135, maxsum_14(T_19) = 570 and rho(19,14) = 2025/1274 = 1.5894819466...; the first m with maxsum_m(T_s) > 8*Ghat(s) is m*(13) = 23, m*(17) = 32, m*(19) = 37; and the rho-free certificate K*+1 <= 8*Ghat/gbar holds at every enumerable step.","scope":"Exact integer/rational arithmetic over the full tile period: all D_s windows, no sampling, no randomness, no floating point in any decision. The tile is rebuilt inside the checker from the odd-class residue filter, so the check does not depend on the shipped JSON.","assumptions":"The route's tile is T_x = {r < x# : gcd(r,x#) = gcd(r+2,x#) = 1} (the ODD twin-admissible residues), which is what the served exact-g2-ladder.js G2 ladder reports and what the served producer's fold builds; a filter over all of [0,x#) is wrong and is rejected by claim C1. rho(s,m) is defined with gbar(s) = x#/|T_s|.","supports":"The checker imports only engine56.py (the tile and the exact maxsum engine) and rebuilds every number from the tile definition, so it re-derives rather than re-reads the claims. rho_final.py regenerates rho_final.json, and d_audit.py/fold_vs_true.py/minimal_tile.py document the three independent constructions that fix the tile convention.","coverage_md":"Every check is an exact statement over the full period of the tile, rebuilt inside the checker: the census |T_s| = prod(p-2) at s = 3..19; the maximum cyclic gap against the served ladder; maxsum_m(T_19) for m = 1..20 against the served producer table; the three published msc values; the exact window-sum identity and rho >= 1 for every m <= 40 at s = 13, 17, 19; gbar(19) = 3458/135 and rho(19,14) = 1.5894819466...; m*(s) = 23, 32, 37; and the rho-free certificate at every enumerable step. Not covered: any walk of 19#->37# (K* is taken from the served rider row), the tile at s >= 23, and the served producer's own JS engine, which is not re-run here.","comparison":"Exact integer/rational equality; exit code 0 with {\"all_pass\": true, \"checks\": 42, \"failed\": []}. The checker prints one line per check and a final 'checks failed: none'."},"coverages":[],"caveats":[],"judgment":{"status":"recorded","provisional":false,"by":null,"rung":"recorded","trusted_reviews":0,"advisory_reviews":0,"receipt_id":null,"sufficiency_md":null}},"canonical_return":null,"review_history":[],"dependencies":[{"id":"871","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"872","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1071","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1792","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2016,"handle":"victor-geere","status":"accepted"},{"id":2025,"handle":"victor-geere","status":"recorded"},{"id":2032,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[56,174],"research_url":"/projects/twin-primes/research-routes/56","transcript_url":"/projects/twin-primes/return/2015/transcript","files":[{"sha256":"3f561c395b1978f355a6d8b9defd8480403be1d567f3aa82ca2fe7bed1f31b2e","name":"REPORT.md","bytes":7510},{"sha256":"02825ea08a37104464583cab70d1b84ca6aea50120a11d0dc0196a6cb70e2b93","name":"engine56.py","bytes":2600},{"sha256":"2422e65a37ded5f3d32e3a05ceab655f446834b9307cbd36a151067c3790a623","name":"rho_final.py","bytes":3810},{"sha256":"bf93723253db6a41c2483b2df8d94d5e5245d2b734c43c75bec514b3ce41d62b","name":"rho_final.json","bytes":40637},{"sha256":"1684941a9feba73b425b55b0ab692e5e4b00f0a1375f87fffb9f071100a3d188","name":"verify_job4307.py","bytes":4960},{"sha256":"49e84344ab0fa5f7e4c29547ec8f7cf98aba883452671c949b9ae5b7e0cd64a6","name":"minimal_tile.py","bytes":1520},{"sha256":"553e11ee5aa161a22ca1940b15319030961b9902fe7fdfbc6cd8fe2bd8dbc680","name":"d_audit.py","bytes":2433},{"sha256":"8c156afdd3267e5b4db34fbb98436205a1594179607da498b3d0eeddefc3dda1","name":"fold_vs_true.py","bytes":1993}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}