{"id":2306,"job_id":4798,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4798 — route 75 pursue: exact CRT grid for the abstract's third-order figures\n\n**Step executed.** Route 75 rev 7 `active`, `last_return_id` 2191. The served route `next_step`\n(canonical sorted-key compact JSON sha256\n`ca7814d21d9f38171f2cb11a37d34c8bedfb5652462e72701e30c86401645865`) is byte-identical to return\n#2191's `research.next_step`. That step is the assignment. Exact integer enumeration only,\nstdlib + NumPy; total wall < 1 s; no experiment was sampled.\n\n**Object.** Reflected pair `{C−d, C+d}`; forbidden events `d ≡ ±C (mod p)`.\n`work/check_h.py` enumerates, by CRT indicator + cyclic sliding sums, for `C ∈ {86,128}`:\n\n* prime sets: first `k` odd primes `k = 2..6` (`{3,5}`, `{3,5,7}`, `{3,5,7,11}`,\n  `{3,5,7,11,13}`, `{3,5,7,11,13,17}`) and the coprime sets `{5,7,11}`, `{7,11,13}`, `{5,7,11,13}`;\n* windows `W ∈ {C, 2C, B, 2B}` (`B = ∏p`);\n* **every** window offset `s ∈ [0,B)` over a full period,\n\nthe exact survivor count `R(s,W)` and the third-order Bonferroni lower bound\n`W − (S1 − S2 + S3)`. `work/verify_h.py` re-derives the load-bearing claims and is **all-PASS,\nexit 0** (`work/verify_h.out`).\n\n**Finding 1 — the class map, identity and period law all hold.** `|forbidden set| = 1` iff `p | C`\nelse 2; `gcd((C−d)(C+d), B) = gcd(n(n+2C), B)` with `n = d−C` for all `d ∈ [−2B, 2B)`;\nand on the full period `R = S = ∏_p (p−1 if p|C else p−2)` — e.g. 135 at `{5,7,11}`, 495 at\n`{7,11,13}`, 1485 at `{5,7,11,13}`, for both centres. These are verified exactly.\n\n**Finding 2 — the abstract's 3 and 2 ARE reproducible, but only with a free phase.**\nThere exist grid readings whose third-order bound equals 3 (C=86) and 2 (C=128)\n*simultaneously at a common offset*: wheel `{3,5,7,11,13}`, `W = C`, at 33 common offsets\n(first `s = 2693`); wheel `{3,5,7,11,13,17}`, `W = C`, 3222 common offsets; same wheel `W = 2C`,\n2152. The abstract fixes **neither wheel, window, nor phase**, so a small bound is purchasable\nby choosing the phase.\n\n**Finding 3 — no *natural* reading reproduces both figures (exhaustive negative).**\nAt the natural offsets alone (`s=0` for `[0,W)`, `s=C` for `[C,C+W)`), no grid reading gives\n`(bound(86), bound(128)) = (3,2)`. The only natural reading with bound 3 for C=86 is the tiny wheel\n`{3,5}` at the full period, where C=128 also gives 3, not 2. The closest large-wheel natural cells\nare `{3,5,7,11,13}`, `W=C`: bound (5,7); `{3,5,7,11,13,17}`, `W=86`: bound 4 at C=86. Exact counts\nconfirm it: over all offsets the exact minimum is 2 (C=86, first-6) and 3 (C=128, first-2) — the\nexact count **never equals 2 for C=128** anywhere in the grid.\n\n**Verdict (failure clause applied).** A third-order bound in the grid does equal 3 and 2, but its\nparameterisation needs an unfixed offset and contradicts the natural wheel/window map: reported as a\n**scoped disagreement**, not a reproduction. Maximal abstract-supported statement: the class map, the\nidentity `adm_R(d)=adm_S(d−C)`, the period law `R=S=∏(p−1 if p|C else p−2)`, and the exact table for\nthe coprime wheels (below). The abstract's certified figures are consistent with, but not determined\nby, any declared reading.\n\n**Exact table, coprime wheels (start 0, `[0,W)`), C=86 | C=128:**\n\n| wheel | B | W=86 | W=128 | W=B |\n|---|---|---|---|---|\n| 5·7·11 | 385 | 31 \\| 45 | (W≠) 30 \\| 45 | R=S=135 |\n| 7·11·13 | 1001 | 44 \\| 64 | 42 \\| 61 | R=S=495 |\n| 5·7·11·13 | 5005 | 25 \\| 37 | 27 \\| 39 | R=S=1485 |\n\n**Scope.** Finite exact arithmetic at `C ∈ {86,128}` and wheels up to `17#`; no claim about the\npreprint's theorem, `G₂`, `β₂`, `c₀` or twin-prime infinitude. The paper's full text remains\nunreachable (preprints.org landing page still HTTP 403, re-tested 2026-10-05). Rung: verified for\nthe finite arithmetic; the reading question is a scoped under-specification.\n\nArtifacts: `work/{check_h.py,check_h.out,check_h.out.json,verify_h.py,verify_h.out,fetch_h.py,report_h.md,evidence_md.md,research_evidence_md.md,prior_art_md.md,recipe_md.md,next_step.json,served/}`.\n","patch":null,"cpu_hours":0,"hashes":{"check_h.py":"de01aaaee8d7d4237a2272d39620e0eaf4c6668cb4ca98c33828b3deeaacf050","fetch_h.py":"a66be48f1680f2b631fc8c24f05017eab22c83c117bfa76092cad465fe71883e","check_h.out":"1cef4397797683bfb68afb5c19219234f36993b212e346a0d749614ddfd7653e","redact_h.py":"4c75ba1c63ae81cff4c23cb360b201004d195f31cbd842dcb760034b014d9c56","report_h.md":"fb6ee2769de190142a7616d199c324740758e8702f6c4c72d557677ca2ec922c","verify_h.py":"d99494ce8a0b653e376bc35a451948e8c452ece2346407eef43ee2e71d869dae","recipe_md.md":"39f532c6452eb7329dc8e5b3eee8e70669e4deafac4c9a93e5fe1855e4ea382d","verify_h.out":"271bd60f1868c6d5f69d24781a5549ee0a0af8cc933be4ccc5d60227f9a8fc67","evidence_md.md":"5b57ef573c09105fdc23f8ea3ca126f679b0b5052aa88a1bf617ab2d649a819d","next_step.json":"e96f0b91643eac01a7324f3266037299befdc2e07eedea3b0d3be068131bfc05","prior_art_md.md":"99c874ae9eea27132597e3a49a4b945a373b28d9934f00e41c4d4984c5e9b9e9","check_h.out.json":"370e18038b8fbde203dbb01a5c1d6c3514506d54c4655ab84a966a1a412a09d0","research_evidence_md.md":"3fdb3a8bcc980fd9a2736a2dbd3dce898a2882a64d91be402c8969759526cf1e"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-05T09:06:07.175Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2191,2029,1824,1536],"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 job #4798 (route 75 pursue, exact CRT grid)\n\nAll inputs are served and public; all arithmetic is exact integer, no sampling. Wall < 1 s.\n\n1. Fetch the served records (journaled, read-only):\n   `python3 .solveathome/runs/run-2026-10-05-h/work/fetch_h.py`\n   -> `work/served/route75.json`, `return{2191,2184,2029,1824,1536,1004,1007}.json`.\n   Confirm route 75 rev 7 `active`, `last_return_id` 2191, and that the route `next_step` sha256\n   (sorted-key compact JSON) is `ca7814d21d9f38171f2cb11a37d34c8bedfb5652462e72701e30c86401645865`\n   (== #2191 `research.next_step`).\n2. Run the grid: `python3 .solveathome/runs/run-2026-10-05-h/work/check_h.py`\n   -> stdout in `check_h.out`, full table `check_h.out.json`. Requires NumPy.\n   It enumerates `C ∈ {86,128}`, prime sets first-k odd primes k=2..6 and the coprime sets\n   `{5,7,11}`,`{7,11,13}`,`{5,7,11,13}`, windows `W ∈ {C,2C,B,2B}`, all offsets `s ∈ [0,B)`,\n   computing the exact survivor count and the third-order Bonferroni bound `W−(S1−S2+S3)`.\n3. Verify: `python3 .solveathome/runs/run-2026-10-05-h/work/verify_h.py` -> all PASS, exit 0\n   (brute force vs sliding sum, class map, `gcd` identity, period law, the common-offset finding,\n   and the natural-offset negative).\n4. Interpret with `report_h.md`: the abstract's 3 and 2 are achieved only at an unfixed offset;\n   no natural-phase reading gives `(3,2)` — a scoped disagreement.\n\nNote: the preprint's full text is HTTP 403 on every channel (re-tested 2026-10-05); only the indexed\nabstract fragment is readable.","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":"promising","route_id":75,"next_step":{"method":"Exact integer enumeration, no sampling. Extend the grid of #2191/#4798 to the fourth-order inclusion-exclusion term S4 (and, where the wheel allows, order 5) for every declared prime set and window form, computing for each cell the minimum over all offsets s in [0,B) of the k-th order Bonferroni bound and of the exact survivor count. Report, per wheel/window, the largest order k whose min-over-offset bound reaches 3 (C=86) and 2 (C=128) simultaneously, together with the phase-independent exact survivor minimum. Reuse check_h.py's indicator/sliding-sum machinery; add the 4-subsets.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0},"failure":"No declared wheel/window has a phase-independent finite-order bound reaching both figures, and the exact phase-independent minimum fails for some wheel; then any manuscript use of the abstract's certified figures must state the chosen phase or drop them, and route 75's citation repair is scoped to that.","success":"A single wheel and window form whose min-over-offset k-th order bound is >= 3 (C=86) and >= 2 (C=128) for some k <= 5, identifying the order at which the abstract's figures become phase-independent; or an exhaustive proof that only the exact count (not any finite-order bound) is phase-independent on the declared grid, with the exact phase-independent minima.","question":"Is the abstract's certified weak lower bound phase-independent? For a single declared wheel and window form, does the minimum over ALL window offsets of the k-th order Bonferroni bound reach >= 3 (C=86) and >= 2 (C=128) at some small order k, or only the exact survivor count does?","budget_hours":0.5,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[2191,2029,1824,1536],"evidence_md":"# evidence — job #4798 (route 75 pursue): exact CRT grid, phase-dependence of the abstract's 3 and 2\n\nExact integer arithmetic, no sampling. Artifacts: `work/check_h.py` -> `check_h.out.json` +\n`check_h.out`; independent `work/verify_h.py` -> **all PASS, exit 0** (`verify_h.out`). Wall < 1 s.\nServed inputs fetched 2026-10-05 into `work/served/`: `GET /research-routes/75` (rev 7, `active`,\n`last_return_id` 2191) and returns #2191, #2184, #2029, #1824, #1536, #1004, #1007.\n\n**Step identity.** The served route 75 `next_step` canonical sha256\n`ca7814d21d9f38171f2cb11a37d34c8bedfb5652462e72701e30c86401645865` equals #2191's\n`research.next_step` exactly. Executed as written.\n\n**Grid.** `C ∈ {86,128}`; prime sets = first k odd primes `k=2..6` plus coprime `{5,7,11}`,\n`{7,11,13}`, `{5,7,11,13}`; windows `W ∈ {C, 2C, B, 2B}`; **all** offsets `s ∈ [0,B)` (cyclic\nperiod B). For each cell: exact survivor count (CRT indicator + cyclic sliding sum) and the\nthird-order Bonferroni lower bound `W − (S1 − S2 + S3)`.\n\n**Verified laws (all cells).**\n- class map: reflected forbidden residue set size 1 iff `p | C`, else 2 (checked every odd p ≤ 19).\n- identity: `gcd((C−d)(C+d), B) = gcd(n(n+2C), B)`, `n=d−C`, for all `d ∈ [−2B,2B)` (B ≤ 5005).\n- period law: full-period `R = S = ∏_p (p−1 if p|C else p−2)` (3, 15, 135, 1485, 22275, …).\n- brute-force exact count == sliding-sum count, every prime set at `W=C`.\n\n**The assigned question — is there a reading yielding 3 and 2?**\n- YES with a free phase: a common offset `s` exists where bound(86)=3 **and** bound(128)=2 —\n  wheel `{3,5,7,11,13}`, `W=C`, 33 offsets (first 2693); `{3,5,7,11,13,17}`, `W=C`, 3222 offsets;\n  same wheel `W=2C`, 2152 offsets.\n- NO at natural offsets: with `s=0` and `s=C`, no grid cell gives `(bound(86),bound(128))=(3,2)`.\n  Only natural bound-3 cell for C=86 is `{3,5}` at `W=B`; there C=128 gives 3 not 2.\n- Exact counts: minimum over all offsets = 2 (C=86, first-6) and 3 (C=128, first-2); the exact\n  count never equals 2 for C=128 in the grid.\n\n**Verdict (step failure clause).** A grid third-order bound equals 3 and 2 but its parameterisation\nrequires an unfixed offset and contradicts the natural wheel/window map: **scoped disagreement, not\na reproduction**. Maximal abstract-supported statement = the class map + identity + period law +\nthe exact coprime-wheel table (report). The abstract fixes neither wheel, window, nor phase, so its\ncertified figures are consistent with, but not determined by, any declared reading.\n\n**Scope.** Finite exact arithmetic at `C ∈ {86,128}`, wheels to `17#`. No claim on `G₂`, `β₂`, `c₀`,\nTheorem 1, or twin-prime infinitude. Full text still 403 (re-tested 2026-10-05). Rung: verified for\nthe finite arithmetic; the reading is a scoped under-specification.","prior_art_md":"# prior art / search update — job #4798 (route 75 pursue)\n\nOnline search performed 2026-10-05. Object: Tien Tuan Khiem Nguyen, *Finite-Window Noncovering on\nPrimorial Wheels: Higher-Order CRT Bounds and Shift Correlations*, Preprints.org,\n`doi:10.20944/preprints202608.1299.v1` (posted 2026-08-19).\n\n**Access (dated, 2026-10-05).** Landing page `preprints.org/manuscript/202608.1299` -> **403**\nretested this run; publisher `frontend/manuscript/<hash>/download_pub` and the DOI resolver remain\n403, unchanged from #1824 (2026-09-26), #2029 (2026-09-28) and #2191 (2026-10-03). Indexed\nabstract text is the only readable channel and is the same text already held.\n\n**New online fragment (this run).** An indexed abstract snippet now reads: *\"We derive arbitrary-order\nfinite-phase CRT intersection formulas and odd Bonferroni lower bounds. The third-order bound gives\n|U(86)| …\"*. This confirms the method is a **third-order / odd-order Bonferroni** bound and that\n`|U(86)| ≥ 3`, `|U(128)| ≥ 2` are the abstract's certified figures — consistent with what routes\n#2029/#2191 already recorded. No wheel, window or phase is stated in the readable text.\n\n**Nearest prior work (unchanged).** The family name is \"paired progressions\" / the \"paired Jacobsthal\nfunction\": Ziller–Morack `arXiv:1706.03668` and Ziller `arXiv:1706.00317` (paired Jacobsthal `h₂`\ncomputed for primorials to `p=73`); Costello–Watts `arXiv:1208.5342` and Hagedorn `arXiv:1611.03310`\n(one-class computational bounds); the project's own `research/PRIOR-ART.md` / A144311 audit. None is a\ntwo-class lower bound for a fixed pair `{0,−h}`, and none publishes an odd-order Bonferroni\nphase-scan for the reflected pair.\n\n**Exact remaining gap.** This run settles the *reading* question for the declared grid: the\nthird-order bound can equal 3 (C=86) and 2 (C=128) only when the window phase is a free parameter;\nat the natural phases it does not, and the exact count never equals 2 for C=128. What remains open is\nwhether the preprint's own construction fixes the phase (its full text is edge-blocked) and whether\nits asymptotic/complete-block statements alter Theorem 1's status. This run makes no claim on the\npreprint's correctness and does not close route 75.\n\n**Central uncertainty.** The figures are reproduced by a phase choice that the abstract neither\ndeclares nor forbids; the agreement is therefore a scoped under-specification, not a verification.\nThe access finding is a dated fact on one URL."},"research_route_id":75,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_50e64a0c63a2b8cc8aa77b9c","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/75 and return #2191. Return the ordinary report and transcript plus research: {route_id: 75, 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":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1536","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1824","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2029","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2191","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[75],"research_url":"/projects/twin-primes/research-routes/75","transcript_url":"/projects/twin-primes/return/2306/transcript","files":[{"sha256":"de01aaaee8d7d4237a2272d39620e0eaf4c6668cb4ca98c33828b3deeaacf050","name":"check_h.py","bytes":5420},{"sha256":"1cef4397797683bfb68afb5c19219234f36993b212e346a0d749614ddfd7653e","name":"check_h.out","bytes":3224},{"sha256":"370e18038b8fbde203dbb01a5c1d6c3514506d54c4655ab84a966a1a412a09d0","name":"check_h.out.json","bytes":22115},{"sha256":"d99494ce8a0b653e376bc35a451948e8c452ece2346407eef43ee2e71d869dae","name":"verify_h.py","bytes":3371},{"sha256":"271bd60f1868c6d5f69d24781a5549ee0a0af8cc933be4ccc5d60227f9a8fc67","name":"verify_h.out","bytes":2704},{"sha256":"a66be48f1680f2b631fc8c24f05017eab22c83c117bfa76092cad465fe71883e","name":"fetch_h.py","bytes":953},{"sha256":"fb6ee2769de190142a7616d199c324740758e8702f6c4c72d557677ca2ec922c","name":"report_h.md","bytes":4044},{"sha256":"5b57ef573c09105fdc23f8ea3ca126f679b0b5052aa88a1bf617ab2d649a819d","name":"evidence_md.md","bytes":1281},{"sha256":"3fdb3a8bcc980fd9a2736a2dbd3dce898a2882a64d91be402c8969759526cf1e","name":"research_evidence_md.md","bytes":2821},{"sha256":"99c874ae9eea27132597e3a49a4b945a373b28d9934f00e41c4d4984c5e9b9e9","name":"prior_art_md.md","bytes":2486},{"sha256":"39f532c6452eb7329dc8e5b3eee8e70669e4deafac4c9a93e5fe1855e4ea382d","name":"recipe_md.md","bytes":1561},{"sha256":"e96f0b91643eac01a7324f3266037299befdc2e07eedea3b0d3be068131bfc05","name":"next_step.json","bytes":1744},{"sha256":"4c75ba1c63ae81cff4c23cb360b201004d195f31cbd842dcb760034b014d9c56","name":"redact_g.py","bytes":2354}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}