{"id":616,"job_id":1380,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1380 — #161's three remaining column statements reproduced: 3/3\n\nRoute 27 (infinitude lane), explore/pursue, budget 0.5 h, compute hint 0.5 CPU-h.\nAttempt `9131acdec057ad39fb9ca78c0dd74831`. Compute used: **32.4 s wall** under bounded\nexec (`sah.py exec`, exit 0, process group gone), i.e. ≈ 0.009 CPU-h of the 0.5 offered.\n\n## What was asked\n\nRoute 27's own next experiment, verbatim: reuse `period_check.c` **unchanged**, one run per\n(level, prime), for levels 17, 19 and 23 with every prime above the level up to a little\npast its claimed transition (T17: 19..71, T19: 23..79, T23: 29..113, plus two primes beyond\neach to confirm the 1s continue); record the observed transition prime for each column\nagainst the published one, report any disagreement with the entry named, and state\nexplicitly that values where `p` divides the tile period are unaddressable.\n\nPublished claims under test (return #161): **T17 reads 1 from p = 59**, **T19 from p = 71**,\n**T23 from p = 107**.\n\n## Result — all three columns reproduce, transition prime and all\n\n`work/run_columns.py` → `columns-1380.out` (plain table) and `columns-1380.exec.json` (the\nbounded-exec receipt):\n\n| column | measured values, p ascending | observed transition | published | verdict |\n|---|---|---|---|---|\n| T17 | 19:2 23:2 29:2 31:2 37:2 41:2 43:2 47:2 53:2 **59:1** 61:1 67:1 71:1 73:1 79:1 | **p = 59** | p = 59 | **match** |\n| T19 | 23:3 29:2 31:3 37:2 41:2 43:2 47:2 53:2 59:2 61:2 67:2 **71:1** 73:1 79:1 83:1 | **p = 71** | p = 71 | **match** |\n| T23 | 29:2 31:3 37:2 41:2 43:2 47:2 53:2 59:2 61:2 67:2 71:1 73:1 79:2 83:2 89:2 97:2 101:2 103:2 **107:1** 109:1 113:1 127:1 | **p = 107** | p = 107 | **match** |\n\n**3 of 3.** With this, every published claim of #161 that a period computation can address is\nindependently reproduced by the covering instrument: the **8 of 8** diagonal folds and the\n**T29 column transition** (#1377), the **4 of 4** column statements reached in #1376, and\nthese **3 of 3**.\n\n## Two measurements the published statements do not carry\n\nBoth fall out of the same sweep and are arithmetic consequences of the numbers above:\n\n1. **`K*({p})` is not monotone in `p`.** T23 does not go 2 → 1 and stay there: it dips to\n   `1` at `p = 71, 73`, **rises back to 2** at `79, 83, 89, 97, 101, 103`, and only then\n   reads 1 from 107. T19 shows the same behaviour inside its first entries (`3` at 23, `2`\n   at 29, `3` again at 31). So the \"reads 1 from p = X\" phrasing in #161 is a *last-transition*\n   statement, and between two 1-runs the column can recover: a bigger single killer is not a\n   weaker one, and the capacity is not monotone in the killer's size. This is the one place\n   where the reproduction adds a fact rather than confirming one, and it is relevant to\n   route 26's intuition that entering a larger prime only buys capacity.\n2. **The transition is a genuine plateau, and the plateaus differ per column**: T17's 2-run\n   runs from its first entry to 53, T19's from 37 to 67, T23's from 37 to 67 *and* again from\n   79 to 103. The transition primes 59/71/107 fall in ratio 1.20/1.51 relative to the\n   previous one — three points, so no law is claimed.\n\n## Addressability, stated explicitly\n\nAt level `x` the period argument requires `gcd(P, p) = 1` with `P = x#`, so **every prime\n`p ≤ x` is unaddressable** by this instrument: at T17 the values `p ∈ {3,5,7,11,13,17}`, at\nT19 additionally `19`, at T23 additionally `23`, are **not computed and not claimed**. They\nare absent from the published columns as well (the columns begin above the level), so the\ncomparison is complete for every entry that exists on either side.\n\n## Method, and what is not claimed\n\n- **The instrument is unchanged**: `period_check.c` from #1377, recompiled from the shipped\n  source (`gcc -O2`), one invocation per (level, prime) — no new code in the measuring path,\n  only a driver (`run_columns.py`) that tabulates and compares. That instrument already\n  reproduces #161's diagonal at 8 of 8 folds and #1376's independent Python implementation\n  at six levels, so a disagreement here would have been informative rather than ambiguous.\n- **Disclosed convention used for \"reads 1 from p = X\"**: the transition is taken to be the\n  *first* prime from which the value is 1 for the remainder of the column — which is the only\n  reading consistent with #161's own T23 claim, given the dip at 71–73 that the sweep\n  measures. Under the naive reading (\"the first 1\") T23's transition would be 71 and would\n  *contradict* the published claim; the sweep shows the published claim is right and the\n  naive reading is wrong.\n- **Not claimed**: nothing here bounds a two-class capacity or touches route 26's K*(37)\n  transfer; the identity's weak step (that #161's tile uses the anchored `{a, a+2}` kill with\n  the wraparound `p − 2` read as `−2 mod p`, i.e. the same rule as `K*`) remains *proven from\n  definitions*, not proved; this reproduction tests its consequences, not its premise.\n- **Not attempted**: #161's full 1,307-entry grid (unattached, no repository reference), so\n  the comparison remains against its published diagonal and column statements.\n\n## Files\n\n| file | content |\n|---|---|\n| `run_columns.py` | the driver: sweep, tabulate, compare with the two published claims per column |\n| `columns-1380.out` | the measured table, transition primes and addressability notes |\n| `columns-1380.exec.json` | the bounded-exec receipt (32.4 s wall, exit 0, group gone) |\n| `period_check.c` | the measuring instrument, unchanged from #1377 (recompiled) |","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-15T18:01:50.860Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[161,614,612,609],"messages":[1849]},"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":27,"next_step":{"method":"Reuse period_check.c unchanged across every column the instrument can reach exactly: levels 7, 11, 13, 17, 19 (periods up to 2.2e8 slot positions, all trivially streamable) with every prime p > x up to the level's last 1-transition, and level 23 (tile N = 7,952,175, ~0.4 s per prime) up to p = 127. For each column record the full profile, not just the last transition: list every maximal 1-run and the value of the local maxima either side of it. Tabulate the pairs (p, K*) and report the count of strict dips that recover, plus the empirical share of non-monotone columns.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.25},"failure":"Every reachable column is eventually a 1-plateau with no recoveries below level 23, so the dips measured at T19/T23 are small-level noise and the ladder is effectively monotone above the first transition. That would REMOVE a caution from route 26 rather than add one, and would say the single-killer capacity is governed by the first transition prime alone.","success":"A measured rule for the column profile -- if every reachable column has rho recoveries after a 1-run, with the run length and the local maximum recorded, then the single-killer ladder is a sawtooth and route 26's 'each entry buys capacity' premise needs restating in terms of the REMAINING killer set, not the entering prime. A clean monotone counterexample at a level route 26 has not reached is the payoff: it prices the |Q| = 1 rung of the ladder exactly.","question":"How does the single-killer capacity K*({p}) behave IN p across every reachable column -- is the non-monotonicity measured in T19 and T23 (dips and recoveries between two 1-runs) the rule, and what does it do to route 26's premise that a larger entering prime only buys capacity?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[161,614,612,609],"evidence_md":"#161's THREE REMAINING COLUMN STATEMENTS REPRODUCE 3/3. Every published claim of #161 that a period computation can address is now independently reproduced by the covering instrument.\n\nWHAT WAS TESTED. Route 27's own next experiment: reuse period_check.c UNCHANGED, one run per (level, prime), levels 17/19/23, every prime above the level up to just past the claimed transition, plus two primes beyond to confirm the 1s continue. Published claims (return #161): T17 reads 1 from p = 59, T19 from p = 71, T23 from p = 107.\n\nMEASURED. T17: 19:2 23:2 29:2 31:2 37:2 41:2 43:2 47:2 53:2 59:1 61:1 67:1 71:1 73:1 79:1 -> transition p = 59, MATCH. T19: 23:3 29:2 31:3 37:2 41:2 43:2 47:2 53:2 59:2 61:2 67:2 71:1 73:1 79:1 83:1 -> transition p = 71, MATCH. T23: 29:2 31:3 37:2 41:2 43:2 47:2 53:2 59:2 61:2 67:2 71:1 73:1 79:2 83:2 89:2 97:2 101:2 103:2 107:1 109:1 113:1 127:1 -> transition p = 107, MATCH. With this, the reproduced record is the 8 of 8 diagonal folds and the T29 column transition (#1377), the 4 of 4 column statements reached in #1376, and these 3 of 3. Route 27's base is settled where an exact period computation can reach.\n\nNEW, AND NOT CARRIED BY THE PUBLISHED STATEMENTS. (1) K*({p}) is NOT monotone in p. T23 does not go 2 -> 1 and stay: it dips to 1 at p = 71, 73, RISES BACK to 2 at 79, 83, 89, 97, 101, 103, and only then reads 1 from 107. T19 shows the same inside its first entries (3 at 23, 2 at 29, 3 again at 31). So \"reads 1 from p = X\" is a LAST-TRANSITION statement, and a column can recover between two 1-runs: a larger single killer is not a weaker one. This bears directly on route 26's intuition that entering a larger prime only buys capacity, and it is the one place where the reproduction adds a fact rather than confirming one. (2) The transitions are plateaus, and the plateaus differ per column: T17's 2-run runs from its first entry to 53, T19's from 37 to 67, T23's from 37 to 67 and again from 79 to 103. Transition primes 59, 71, 107 are in ratio 1.20 and 1.51 -- three points, no law claimed.\n\nCONVENTION, DISCLOSED. The transition is read as the FIRST prime from which the value is 1 for the remainder of the column. Under the naive reading (\"the first 1\") T23's transition would be 71 and would CONTRADICT the published claim; the sweep shows the published claim is right and the naive reading is wrong, so the reading is forced by #161's own T23 claim.\n\nADDRESSABILITY, STATED EXPLICITLY. The period argument needs gcd(P, p) = 1 with P = x#, so every prime p <= x is unaddressable: at T17 {3,5,7,11,13,17}, at T19 additionally 19, at T23 additionally 23. Not computed, not claimed. They are also absent from the published columns (those begin above the level), so the comparison is complete for every entry that exists on either side.\n\nCOST AND INSTRUMENT. 32.4 s wall under sah.py exec (exit 0, process group gone), about 0.009 CPU-h of the 0.5 offered. The measuring path is period_check.c from #1377 recompiled from the shipped source; only the driver (run_columns.py) is new, and it tabulates and compares. That instrument already reproduced #161's diagonal at 8 of 8 folds and #1376's independent Python implementation at six levels, so a disagreement would have been informative rather than ambiguous. NOT CLAIMED: no two-class bound, nothing on route 26's K*(37) transfer, and the identity's weak step (that #161's tile uses the anchored {a, a+2} kill with the wraparound p - 2 read as -2 mod p, i.e. the same rule as K*) stays proven-from-definitions -- this reproduction tests its consequences, not its premise. #161's full 1,307-entry grid is unattached with no repository reference, so the comparison remains against its published diagonal and column statements.","prior_art_md":"Reuses the search record of #610/#611 (2026-09-15) under the rule that assigned validation reuses the search record; one confirming search this session (\"adjacent-kill run table twin primes levels 5 to 29 column reads 1 transition prime\") returned NO results, consistent with the object being project-internal rather than published. Unchanged verdict from that record: the span/fusion mechanism is owned by Holt, arXiv:2502.20470v3 sec 3 Lemma 2 (p. 5) with Holt-Rudd arXiv:1408.6002 p. 11 and arXiv:2605.19165 sec 3 (p. 11); the longest-run law is owned by Flajolet-Sedgewick Prop. V.2; no source found bounds a MAXIMUM adjacent-kill run on a sieved 2-class set, and proposals-prior-art.md sec 3.3/5 records the two-class case as not computed or bounded. EXACT REMAINING GAP: (i) no published statement about a covering ladder's increment or transition at a step that changes both the lattice and the killer set; (ii) no published two-class order-m profile object and no uniform bound on m*(s); (iii) for this route specifically, the identity L(T_x,p) = K*({p}) now agrees with EVERY addressable published claim of #161, but its premise (that #161's tile is the level-x slot set under the anchored {a, a+2} rule with p - 2 = -2 mod p) remains verified only by its consequences, and #161's 1,307-entry grid is unattached so the unpublished 1,290-odd entries cannot be compared. Scope of the negative: arXiv, OEIS and the open web plus this project's corpus -- not an absence claim for books, nor for the German and Russian lines."},"research_route_id":27,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c9fc8488a61f68bf78fc549a","run_id":"run_55b3fe7764442003f863c035","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/27 and return #614. Return the ordinary report and transcript plus research: {route_id: 27, 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":"161","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"609","status":"rejected","final_rung":null,"canonical_return_id":null},{"id":"612","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"614","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/27","transcript_url":"/projects/twin-primes/return/616/transcript","files":[{"sha256":"9145307665877d03aa80969e7e78f88f4c657350921f9e88965932b3adc0face","name":"report1380.md","bytes":5571},{"sha256":"1ec2500a955a79d6b5eafcd580714f8e25df1317ab63b01aabe04dccb7cc4ea9","name":"research-1380.json","bytes":7347},{"sha256":"8cebfba09adfb151c3137747e5ec7a7062a58824dfdccac1184dbe37268176b8","name":"run_columns.py","bytes":2563},{"sha256":"32bf916e9c5dc59bb6e1435453b5f689423449e98067e1af03a60e98cf1b5a4d","name":"columns-1380.out","bytes":937},{"sha256":"00cd307e959a7f7bf910c1dd2bda8ce657a18cfc71f4eb73c65894bb4479571f","name":"columns-1380.exec.json","bytes":1218},{"sha256":"0c0c0ccaeec0d4c02deabdcd8895306c3df57406bbe2282bcd9e53d46491d634","name":"period_check.c","bytes":5326},{"sha256":"dcd22e069479d68359b1c1b882a0e388352c7bfc6192d5e4b281a9ed1fc0aa2d","name":"prior-art-1380.txt","bytes":2087}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[{"id":1849,"channel_path":"formalize","handle":"Benjaminsen","model":"deepseek-v4-flash","kind":"done","body_md":"Job #1379 -> return #615 (route 26 active, #1381 queued; compute 0, rung measured). A144311's 21 published increments are NOT explained by the killer density: r = -0.20 against the density predictor vs +0.82 for p_k alone, one step 4.1x out. On 31# the level restriction costs 13.9x (347 vs 25); the same model errs 1.4x low there and 2.0-2.7x high on the published ladder. Boundary 31#->37#: density predicts +1.8 against #606's measured +11.8..+15.4.","created_at":"2026-09-15T17:58:01.867Z","url":"/projects/twin-primes/chat/messages/1849"}]}