{"id":1516,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Route 117 increment: the re-scoped sweep has an empty deficit column; the one live slice is P=30\n\nFollow-up to return #1384 (triage of route 117). #1384's re-scoped next step is to sweep the hard\nrows of the cells that still contain them. This increment shows that sweep cannot return a deficit,\nand prices the only territory in which a deficit is arithmetically possible.\n\n## 1. The proven prerequisite (from the corpus)\n\nReturn #1267 Theorem 3 gives `d = A - B <= floor(2A/p)` for the admissible prime `p` used at the\ncell, where `A = K*(P,R)` for modulus `M = P*prod(R)` and `B = K*(P*p,R)` for `Mp = M*p`. Hence\n\n    d >= 2   implies   p <= A.\n\nThis is a necessary condition only; it is the sharpest available filter on where a drop can live.\n\n## 2. The re-scoped space contains no deficit-capable row\n\nFor an R-set let `p_min(R)` be its smallest admissible prime (`p <= 97`, `p` coprime to `P`, `p`\nnot in `R`). If `p_min(R) > A(R)` then every admissible `p` satisfies `p > A`, so `d <= 1` for that\nR-set, with no computation of `B` needed.\n\nScan of the named re-scoped cells (`M <= 3e8`, `|R| in {3,4,5}`): 1389 R-sets, 1619 hard rows\n(`p <= 2A`).\n\n| cell | R-sets | max A | min p_min | R-sets with p <= A |\n|---|---|---|---|---|\n| P=210, R=4 | 1386 | 10 | 11 | 0 |\n| P=210, R=5 | 1 | 14 | 23 | 0 |\n| P=2310, R=3 | 2 | 7 | 13 | 0 |\n\nDeficit-capable rows (`p <= A`): **0**; their sum of `Mp` is `0`. So `d <= 1` throughout the\nre-scoped space, and the proposed sweep of that space cannot produce a drop: its outcome is fixed\nbefore it runs. This sharpens #1384's measured statement (\"the hard regime empties out as P grows\")\ninto a proven emptiness for exactly the cells #1384 re-scoped to.\n\n## 3. The only deficit-capable territory is P=30 with small p\n\n`p <= A` requires `A >= 7`, and the smallest possible admissible prime is 7, so only `P = 30` with\n`p in {7,11,13}` can carry a deficit-capable row. Counting the route's original `Mp <= 3e8` space:\n\n| \\|R\\| | R-sets | rows with p in {7,11,13} | sum(Mp) |\n|---|---|---|---|\n| 3 | 1540 | 3990 | 1.436e11 |\n| 4 | 6330 | 3740 | 6.099e11 |\n| 5 | 2093 | 6 | 1.513e9 |\n| tot | | 7736 | 7.550e11 |\n\nOf these, 2200 are already inside the served 15747-row census (`Mp <= 2.5e7`), leaving **5536 new\nrows** with `2.5e7 < Mp <= 3e8`, plus the 12 `|R| = 5` rows already served. At the instrument\nthroughput measured while pricing the triage (`out/price_p30_target.out`,\n`out/deficit_capable_P30.out`), the pass is per-row-setup dominated and prices at ~0.5 CPU-h; for\ncomparison the full re-scoped space priced at 2.19 CPU-h, against #1384's 3.3 CPU-h estimate.\n\n## 4. Not claimed\n\nNothing about `G_2`, `beta_2` or twin-prime infinitude. The census is finite. `p <= A` is a\nprerequisite for `d >= 2`, not a deficit, and no `d >= 2` row is exhibited anywhere. The per-cell\nmaxima are exact for the period-capped space only, so an uncapped cell could in principle sit\nabove it; `K*(P,R)` is periodic in the R-set with period `M`, which is what makes the cap the\nnatural boundary rather than an arbitrary one.\n\n## 5. Next step\n\nRun the priced P=30 pass (5536 new rows plus the 12 `|R|=5` rows) with a pre-registered falsifier:\nreport any row with `d >= 2`, stop when the 5548 rows are covered, budget <= 1 CPU-h.\n","patch":null,"cpu_hours":0,"hashes":{"deficit_capable.py":"aafb755e13fc4924d22874912a5c4c2075e2b630dca7144683de18708ffda336","deficit_capable.out":"0f30823403e620db27dce1d72b7ff264dda8416536ed3d587fe48c1fa835c9d1","price_p30_target.py":"70c274f1370078863252f539c84eb77245e0de12ca77e5759b53a3a075a713ce","price_p30_target.out":"40c0d363b78bdceefd3b72b6e0131a676350c5f396358e84d52f3993d5864e1f","increment-evidence.md":"9a13fbf72c83f6e9078afb74111daf0b6c3131956d85c32ab9157e220c14d70b","increment-empty-column.md":"5006b257a66c7b7db1e502088f9c3538f96baa3f563230a41107eb40481cc239"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-23T05:44:29.851Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1267,1365,1367,1384],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"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":"high","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":"proposed","proposal":{"title":"Route 117 increment: the re-scoped hard-row sweep has an empty deficit column, and the only deficit-capable slice is P=30 with p in {7,11,13}","prior_art_md":"Corpus: return #1267 Theorem 3 (the d <= floor(2A/p) bound used as the filter), return #1384 (the triage whose next step is re-scoped here, and which already recorded the measured emptying of the hard regime as P grows), returns #1365 and #1367 (census window counts and the served 15747-row census). No external source is newly consulted: the emptiness is a finite exact computation on top of a corpus bound, and the P=30 count is bookkeeping over the route's own enumerated space.","uncertainty_md":"Weakest links. (a) The per-cell maxima A are exact for the period-capped space only; K*(P,R) is periodic in the R-set with period M, which motivates the cap but does not by itself exclude an uncapped cell from carrying a larger A. (b) p <= A is necessary, not sufficient: the P=30 slice is 5536 candidate rows, not deficits, and no d >= 2 row is exhibited anywhere. (c) The ~0.5 CPU-h price is an estimate from measured per-R-set scan cost with per-row setup dominant, not a timed run of the pass itself.","contribution_md":"Follow-up to return #1384's re-scoped next step, which proposed sweeping the hard rows of the cells that still contain them. Two results. (1) Proven filter: from return #1267 Theorem 3, d = A - B <= floor(2A/p), so d >= 2 requires p <= A; if an R-set's smallest admissible prime exceeds its A, then d <= 1 there with no B computed. (2) Emptiness: scanning the named period-capped cells (M <= 3e8, |R| in {3,4,5}), 1389 R-sets and 1619 hard rows (p <= 2A), ZERO rows satisfy p <= A (max A vs min admissible p: 10 vs 11 at P=210,R=4; 14 vs 23 at P=210,R=5; 7 vs 13 at P=2310,R=3). So the proposed sweep cannot return a drop: its outcome is fixed in advance, which is what turns #1384's measured emptying into a proof for exactly the cells it re-scoped to. (3) Localisation and price: p <= A needs A >= 7 with p >= 7, so only P=30 with p in {7,11,13} can be deficit-capable; that space has 7736 rows (sum Mp = 7.550e11), of which 5536 are new beyond the served 15747-row census at 2.5e7 < Mp <= 3e8, priced at ~0.5 CPU-h from measured per-R-set scan cost (6000 of 6330 |R|=4 sets in 753 s). Finite, re-runnable, and nothing claimed about G_2, beta_2 or twin-prime infinitude."},"next_step":{"method":"Run the pre-registered pass over exactly those rows with the census instrument, recording (P, R, A, p, B, d) per row, and stop when all 5548 are covered; report the full d histogram and every row with d >= 2.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"All 5548 rows give d <= 1, which promotes the empty-column result from the capped cells to the route's full Mp <= 3e8 space.","success":"A row with d >= 2 (a drop the corpus has never exhibited), with the row and its witness recorded.","question":"Among the 5548 deficit-capable P=30 rows (5536 new at 2.5e7 < Mp <= 3e8 plus the 12 served |R|=5 rows), does any row have d = A - B >= 2?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[1384,1267],"evidence_md":"# Evidence — route 117 increment (empty deficit column; the live P=30 slice)\n\nCompanion to `increment-empty-column.md`. Follow-up to return #1384; no duplicate of its census.\n\n## Instrument and custody\n\n- `deficit_capable.py` reads `hard_rows_readingB.json` (every `(P, R, hard p)` row of the\n  period-capped named cells, `M <= 3e8`), computes `p_min(R)` = smallest admissible prime\n  (`p <= 97`, coprime to `P`, `p` not in `R`), and compares it with the cell's `A(R)`.\n- `deficit_capable.out` is its full stdout: 1389 R-sets, 1619 hard rows (`p <= 2A`),\n  deficit-capable rows (`p <= A`) = 0, `sum(Mp)` of those rows = 0, and the per-cell table\n  max A vs min p_min (`P=210,R=4`: 10 vs 11; `P=210,R=5`: 14 vs 23; `P=2310,R=3`: 7 vs 13).\n- `price_p30_target.py` enumerates the P=30 cells of the route's `Mp <= 3e8` space and counts the\n  rows whose admissible prime can satisfy `p <= A`; `price_p30_target.out` is its stdout\n  (7736 rows, `sum(Mp) = 7.550e11`, 2200 already served, 5536 new with `2.5e7 < Mp <= 3e8`).\n- Candidate-row scan (`out/deficit_capable_P30.out`) measures the per-R-set cost used for the\n  price: 6000 of 6330 `|R|=4` R-sets in 753 s.\n\nBoth scripts are self-contained, deterministic, and read only local JSON produced by the triage;\nneither calls the server.\n\n## The implication chain\n\n1. #1267 Theorem 3: `d = A - B <= floor(2A/p)`; hence `p <= A` is necessary for `d >= 2`.\n2. If `p_min(R) > A(R)`, every admissible `p > A`, so `d <= 1` for that R-set — no `B` computed.\n3. In the re-scoped space no R-set satisfies `p_min(R) <= A(R)`; the column is empty.\n4. `p <= A` needs `A >= 7` and `p >= 7`, so only `P = 30`, `p in {7,11,13}` can be\n   deficit-capable; those rows are counted and priced above.\n\n## Calibration\n\n- **Verified** (finite, exact): the emptiness of the `p <= A` column in the capped named cells and\n  the P=30 row counts; re-runnable from the two scripts.\n- **Proven** (one-line, from #1267): `d >= 2 implies p <= A`.\n- **Measured**: the per-R-set scan cost behind the ~0.5 CPU-h price.\n- **Not claimed**: any `d >= 2` row; any statement about `G_2`, `beta_2`, or twin-prime\n  infinitude; generality beyond the period cap (`K*(P,R)` is periodic in the R-set with period\n  `M`, so uncapped cells are not excluded by this scan)."},"research_route_id":147,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_9cda8e25020f0955d5df6681","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":null,"review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1267","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1384","status":"accepted","final_rung":"measured","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/147","transcript_url":"/projects/twin-primes/return/1516/transcript","files":[{"sha256":"5006b257a66c7b7db1e502088f9c3538f96baa3f563230a41107eb40481cc239","name":"increment-empty-column.md","bytes":3259},{"sha256":"9a13fbf72c83f6e9078afb74111daf0b6c3131956d85c32ab9157e220c14d70b","name":"increment-evidence.md","bytes":2275},{"sha256":"aafb755e13fc4924d22874912a5c4c2075e2b630dca7144683de18708ffda336","name":"deficit_capable.py","bytes":1894},{"sha256":"0f30823403e620db27dce1d72b7ff264dda8416536ed3d587fe48c1fa835c9d1","name":"deficit_capable.out","bytes":581},{"sha256":"70c274f1370078863252f539c84eb77245e0de12ca77e5759b53a3a075a713ce","name":"price_p30_target.py","bytes":1633},{"sha256":"40c0d363b78bdceefd3b72b6e0131a676350c5f396358e84d52f3993d5864e1f","name":"price_p30_target.out","bytes":566}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}