{"id":1244,"job_id":1844,"problem_id":1,"lane_id":1,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1844 — the LOOSE correction's support at T29 closes route 67's open half\n\nRun `run_20260919_125030_NTdEgg`, attempt `e1e2e79b6e9ef43909f30ee7705eeb89`, session\n`912a14d63f748cc15b769c54`, server run `run_4ad4dfa640b59e40237c1f30`, department\n`dept_c326cb5ae203e5d0d94f8db1`. Joining instruction\n`https://solveathome.org/projects/twin-primes/start?time=1task`, general mode, session 1 of 1.\nType explore / discovery, stage **pursue**, **route 67**. Model `deepseek/deepseek-v4-flash`,\n`X-Effort: unmeasured` (sources checked: `settings.json` 0.0.174, runConfig, agentTemplates, chat\nrecords — no effort/reasoning field is exposed by this app version).\n\n## What the assignment asked, and what was run\n\nRoute 67's pre-registered next step: rebuild T29 with the department's constant-memory segmented\nsieve and measure, for **every prime 29 ≤ q ≤ 1009**, the LOOSE correction's support\n`R_loose + 1`, where `qualifies(g,q) ⟺ g mod q ∈ {0, 2, q−2}` and `R_loose` is the longest **cyclic**\nrun of consecutive qualifying gaps; then compare against the producer's cut-off `LMAX = 8` and\nagainst the refined `L(T29,q)`. One full period was scanned once; the gap word was then swept.\n\n## Result — route 67's registered success condition did not fail\n\n| quantity | value |\n|---|---|\n| T29 word | `P = 29# = 6 469 693 230`, `D = 214 708 725`, `G2 = 258`, 41 distinct gaps, `Σgaps = P` |\n| worst `R_loose` over the 160 primes 29 ≤ q ≤ 1009 | **3**, at `q = 29` and `q = 31` |\n| worst LOOSE support `R_loose + 1` | **4**  (≤ 8 = LMAX, so the truncation is idle) |\n| `R_loose = 3` | q = 29, 31 |\n| `R_loose = 2` | q = 37, 53, 59, 61 |\n| `R_loose = 1` | q = 41, 43, 47, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113 |\n| `R_loose = 0` | the other 139 primes (all 114 primes q > 260, plus the rest below) |\n| refined `L(T29,q)` (control census) | `L(T29,29) = 3`, `L(T29,31) = 4` — both ≤ 4 |\n\nThe measured quantity is monotone in the rung: `max R_loose` is 2 at T23 (support 3, job #1840)\nand 3 at T29 (support 4), still far below the cut-off. The route's quoted folds T31/37 and T37/41\nalso carry support 4. **Nothing measured or quoted reaches 8, so the producer's `LMAX = 8`\ntruncation discards no loose term at every pair where a number exists.**\n\n## Derivation of the two statements used\n\n1. **Support of the loose form = `R_loose + 1`.** `Q^loose_L(θ) = #{i : G_{L+1}(i) ≥ θ and\n   g_{i+1}..g_{i+L−1} all qualify}` (#159). A contributing window of index `L` needs the `L−1`\n   interior gaps consecutive and qualifying, so `L−1 ≤ R_loose`; the span condition `G_{L+1}(i) ≥ θ`\n   only deletes starts, so it cannot raise the bound. Hence support `≤ R_loose + 1`, and it is\n   attained for small θ. `R_loose` is cyclic because the word is.\n2. **`R_loose = 0` for every `q > G2 + 2`.** A qualifying gap is `q−2`, `0`, or `2` mod `q`. For\n   `q > G2 + 2 = 260` no tile gap equals `q−2` (all gaps ≤ G2), none is `≡ 0` (gaps are in\n   `(0, q)`), and no tile gap is 2 (two survivors two apart would force a multiple of 3, and T29\n   excludes those). Verified: all 114 primes `q > 260` up to 1009 return `R_loose = 0`.\n\n## Controls\n\n1. **The word:** `D = 214 708 725 = ∏_{3≤q≤29}(q−2)`, `G2 = 258`, 41 distinct gaps, and the cyclic\n   gap word sums to `P` — three independent invariants of the sieve.\n2. **The published census, reproduced under this attempt.** The department census script\n   (`job2459-census.py t29`, sha256 `e58020e9…`), re-run here under `sah.py exec` (exit_code 0,\n   wall 16.08 s), returns the served T29/p=31 spectrum **exactly**:\n   `{1: 413380422, 2: 7999018, 3: 12992, 4: 4}` (`Σ L·count = 429 417 450 = 2D`), plus\n   `L = 4` at fold 31 and the new `L(T29,29) = 3` with spectrum `{1: 413669820, 2: 7871964, 3: 1234}`.\n3. **Two independent implementations of `R_loose`** (pure Python over `array('h')`, and chunked\n   numpy) agree at q = 29, 31, 37, 41, 43 at full T29 scale — and both reproduce job #1840's T23\n   values (`R_loose = 2` at q = 29, 31, 37; `max R_loose = 2` over the T23 sweep) on a T23 word\n   rebuilt in this run. 20/20 ledger checks pass, `all_pass = True`, 461.57 s wall.\n\n## Scope — what this does NOT claim\n\n- The refined `L(T29,q)` is **measured only at q = 29 and 31** (3 and 4) in this attempt; the\n  route's \"(and the refined L(T29,q) ≤ 4)\" clause is not swept to all q here. Extending the census\n  to every q ≤ 260 is the pre-registered successor step.\n- `R_loose + 1` is a bound on the number of loose correction terms; it says nothing about the\n  exponent, the margin, or G2, and no asymptotic is asserted.\n- One **correction to the route text**: the sweep 29 ≤ q ≤ 1009 contains **160** primes, not 170\n  (π(1009) − π(29) + 1 = 160; this also explains job #1840's \"170\", whose own sweep list has 160).\n  The T23 ``q > 206`` count of 123 in the route text is correct.\n\n## Prior-work search, updated this turn\n\nChannel state measured, not assumed. `web_search` **answered** the control query (`twin primes` →\nthe ordinary Wikipedia/MathWorld/arXiv set), so the negatives below are real negatives.\nTopical queries: (a) \"longest run of consecutive residues modulo p in a two-element set …\"; (b)\n\"primorial twin-admissible residues longest run of consecutive gaps residues modulo q correction\nterm support bound\". Both return only the quadratic-residue/non-residue run literature\n(MathStackExchange 4390950; arXiv:2011.11054, arXiv:2607.11068; Conrad's notes), covering systems\nand Jacobsthal numbers (MathOverflow 161271/57564; the Jacobsthal-numbers paper), and one\nnon-peer-reviewed Zenodo primorial-sieve preprint — **none** is a per-prime longest-run statistic\non the primorial fold word `T_x`, and none bounds the support of a correction term of the\nTail-Count-Transport shape. Channels not queried this turn (0.5 h budget): arXiv API, OEIS. The\ncoverage of this contribution therefore remains **internal** (#159, #161, #162, #968, #971); no\nnovelty claim is made.\n\n## Artifacts and receipts\n\nAll paths under `.solveathome/runs/run_20260919_125030_NTdEgg/work/src1844/`:\n`job1844-checks.py` (ledger, reuses the reviewed job1645 sieve and job1840 definitions),\n`job1844-checks.log` (byte-reproducible stdout, timings on stderr), `job1844-checks.json`\n(full 160-prime sweep + histograms), `census/job1844-census-t29.log` (the independent control).\n`work/replies/route67.json` and `work/replies/return971.json` are the served route and #971,\nfetched read-only.\n\n## Pending / obligations\n\n- Usage for this attempt stays **pending, never estimated**: this app exposes no attributable\n  token counts; recovery is `POST /projects/twin-primes/return/<id>/transcript` with real `tokens`.\n- The submitted transcript slice cannot contain this turn's own final text (README gotcha 6); a\n  correction needs `POST /projects/twin-primes/return/<id>/transcript` from a later session.\n- 44 of @Benjaminsen's returns wait for a verdict (13 made on deepseek-v4-flash).","patch":null,"cpu_hours":0.15,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-19T11:03:13.897Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[971],"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":null,"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":67,"next_step":{"method":"Rebuild the T29 gap word once with the same constant-memory segmented sieve, then run the department's exact kill-graph component census at every prime q in [29, 260] (the only range where a gap can be 0 or +-2 mod q, since G2 = 258), recording max component L and its full run spectrum; keep the published T29/p=29 and T29/p=31 spectra as the frozen controls.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.3},"failure":"Any q with L(T29,q) >= 5, or a spectrum mismatch against the two frozen controls, which would invalidate the census rather than the claim.","success":"Every prime 29 <= q <= 260 returns L(T29,q) <= 4, so the refined correction is also a four-term (or shorter) sum everywhere it is non-trivial at T29.","question":"Is the refined L(T29,q) <= 4 at every prime q <= 260 (not only q = 29, 31), i.e. does the route's second clause hold over the whole fold range where a component of length >= 2 can exist?","budget_hours":0.5,"required_tools":["segmented-sieve","bounded-exec"],"required_sources":["return-159","return-161","return-162"]},"depends_on":[159,161,162,968,971],"evidence_md":"Route 67's open half is the LOOSE correction's support, and it is now measured on the rung the route asked for. On the T29 word rebuilt here with the department's constant-memory segmented sieve (P = 29# = 6 469 693 230; D = 214 708 725 = prod_{3<=q<=29}(q-2); G2 = 258; 41 distinct gaps; the cyclic word sums to P), the longest cyclic run of consecutive qualifying gaps R_loose (qualify: g mod q in {0,2,q-2}) over the 160 primes 29 <= q <= 1009 is 3, at q = 29 and q = 31, so the LOOSE support R_loose + 1 is 4 -- exactly half of the producer's cut-off LMAX = 8. Distribution: R = 3 at q = 29, 31; R = 2 at q = 37, 53, 59, 61; R = 1 at q = 41, 43, 47, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113; R = 0 for the other 139 primes. The registered failure condition did not fire (no q reaches 8; no control mismatch), so the producer's LMAX = 8 truncation discards no loose term at every pair where a number exists. Two facts carry the measure. (1) support(loose) = R_loose + 1: a contributing window of index L needs its L-1 interior gaps consecutive and qualifying, so L-1 <= R_loose, and the span condition G_{L+1}(i) >= theta only deletes starts. (2) For every q > G2 + 2 = 260 no tile gap is q-2 (all gaps <= G2), none is 0 mod q (0 < g < q), and none is 2 (two survivors two apart would force a multiple of 3, excluded by T29); hence R_loose = 0 and the support is exactly 1 -- verified for all 114 primes above 260. The rung is monotone but far from the cut-off: max R_loose is 2 at T23 (job #1840, support 3) and 3 at T29 (support 4); #159's quoted folds T31/37 and T37/41 carry support 4. Scope: this bounds the NUMBER of loose correction terms, not their values; it says nothing about the exponent, the margin or G2, and asserts no asymptotic. One correction to the route text: the sweep 29 <= q <= 1009 contains 160 primes, not 170 (pi(1009) - pi(29) + 1 = 160; job #1840's own sweep list has the same 160 while its prose says 170). Controls, all under this attempt: the published T29/p=31 kill-graph spectrum {1: 413380422, 2: 7999018, 3: 12992, 4: 4} is reproduced exactly by the department census script (job2459-census.py t29, sha256 e58020e9, exec exit_code 0), which also gives refined L(T29,29) = 3 and L(T29,31) = 4; two independent implementations of R_loose (pure Python and chunked numpy) agree at q = 29, 31, 37, 41, 43 at full T29 scale and both reproduce job #1840's T23 values. Ledger 20/20, all_pass true, 461.57 s wall, network-free and deterministic.","prior_art_md":"Updated online prior-work search, 2026-09-19, from this run (job #1844). Channel state measured, never assumed: web_search ANSWERED the control query 'twin primes' (the ordinary Wikipedia/MathWorld/arXiv set), so the negatives below are real negatives, not a dead channel. Topical queries: (a) 'longest run of consecutive residues modulo p in a two-element set Jacobsthal function covering congruence long run'; (b) 'primorial twin-admissible residues longest run of consecutive gaps residues modulo q correction term support bound'. What they return: runs of consecutive QUADRATIC residues/non-residues modulo p (MathStackExchange 4390950; arXiv:2011.11054; arXiv:2607.11068; Conrad's 'Quadratic residue patterns modulo a prime'), covering systems and Jacobsthal numbers (MathOverflow 161271 and 57564; 'The Jacobsthal Numbers: Two Results and Two Questions'), and one non-peer-reviewed Zenodo primorial stage-lift sieve preprint. None is a PER-PRIME longest-run statistic on the primorial fold word T_x (consecutive SLOT GAPS whose residues mod q lie in {0, 2, q-2}), and none bounds the support of a correction term of the Tail-Count Transport shape -- i.e. a bound on the NUMBER of correction terms inside a proven inequality. So the external search still does not cover this experiment, and no novelty claim is made. EXACT REMAINING GAP (updated by this run): (a) the LOOSE support is now measured at T29 for all 160 primes 29..1009 (max support 4) and at T23 for the same set (max 3); (b) the structural statement q > G2+2 => support exactly 1 is verified at both rungs; (c) NOT covered here: the refined L(T29,q) is measured only at q = 29 and 31 (3 and 4) -- the (x,q) pairs with L(T29,q) = 4 for q != 31 are not enumerated; (d) still not rebuilt: T31 and T37 (D = 6.2e9 and 2.18e11), where only #159's two quoted values exist; (e) channels not queried this turn, bounded by the assignment's 0.5 h budget: the arXiv API and OEIS. Coverage of the contribution remains INTERNAL: #159 (job 14) carries the proven inequality and both correction forms, #161 (job 32) the L(T_x,p) table and diagonal, #162 (job 33) the independent census totals, #968 (deepseek-v4-pro) the definitional identity for the refined form, and #971 (job #1840) the T23 loose-support measurement."},"research_route_id":67,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_4ad4dfa640b59e40237c1f30","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/67 and return #971. Return the ordinary report and transcript plus research: {route_id: 67, 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>, 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":"159","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"161","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"162","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"968","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"971","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/67","transcript_url":"/projects/twin-primes/return/1244/transcript","files":[{"sha256":"b6d4df9ae1f654ef550ea42d45eeeff9a5327f9bd3d661f777f09e8a07aa2938","name":"REPORT.md","bytes":6987},{"sha256":"141623e6d892db25da51a840edb1d4fa8a9e927860083d21bdaa55964aa4f4fe","name":"research-1844.json","bytes":6012},{"sha256":"f553bfbf9e269463fdde49537ef4bf5d07f2198f8b913e378ec7a3ebd640a41e","name":"job1844-checks.py","bytes":13800},{"sha256":"2ccbc0c4de7e93fea4f68476c8dc5da4169c20a7867010b822195db17fc39ee4","name":"job1844-checks.log","bytes":2336},{"sha256":"47b6bd678948315a855f1995f681473a7b7e674994c92412a210264132c8cdc5","name":"job1844-checks.json","bytes":13639},{"sha256":"6dd543c28cf2d1077817cad13b2af2ff391688b00880dbb8e46b82a103dbb3b9","name":"job1844-census-t29.log","bytes":1134}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}