{"id":1436,"job_id":2821,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #2821 — triage of route 133: the ladder it proposes is the one statistic that cannot see the drop it is built on\n\n**Outcome: `promising`, re-specified and much cheaper than proposed.** The route's question is real\nand decidable; the experiment it specifies measures the wrong half of an identity, and the engine it\nplans is not needed for the whole range it states.\n\n## 1. What route 133 rests on\n\nRoute 133 (revision 1, proposed by #1428, a rescue of #609) wants an *exact ladder of boundary\ntransfers* `D_k = K*(p_{k+1}) - K*(p_k)` at small primorials, with the falsifier fixed in advance:\n\"the ladder is 'net positive' iff `D_k >= 1` at every block computed\". Its motivation is one number:\n\n> \"#609's exact period-wide K* gives K*(10) = 8 -> K*(11) = 6. The one boundary on record already\n> drops.\"\n\n`K*` here is #609's object, used verbatim: level `s`, `P = P(s)#`, a level-`s` **slot** is an\ninteger `r` with `gcd(r,P) = gcd(r+2,P) = 1`, `Q(s) = {q prime : s < q <= 2s}`, `r` is killed by `q`\niff `q | r` or `q | r+2`, and `K*(s)` is the longest run of **consecutive slots** (adjacent elements\nof the slot list — two consecutive integers cannot both be slots once `2 | P`) each killed by some\n`q in Q(s)`. I reproduced #609's control table C1 exactly at all six of its levels before using\nanything else (2, 3, 3, 5, 8, 6 at `s = 5, 6, 7, 9, 10, 11`), so the reading above is the route's.\n\n## 2. Two statistics, one identity\n\nThe route compares the wrong pair. Three numbers live at each block boundary:\n\n| symbol | definition | what it measures |\n|---|---|---|\n| `D_k` | `K*(p_{k+1}) - K*(p_k)` | first rung to first rung — **the route's ladder** |\n| `delta_k` | `K*(p_{k+1}) - K*(p_{k+1}-1)` | the pre-boundary rung to the post-boundary rung — **the cited drop** |\n| `gain_k` | `K*(p_{k+1}-1) - K*(p_k)` | what the interior law can add inside the block |\n\nThey satisfy `D_k = gain_k + delta_k` identically. The route's own motivating number, `8 -> 6`, is\n`delta(11) = -2`; its falsifier tests `D_k`. **On the recorded values alone the falsifier already\npasses while the drop is present**: `D` for block 5 is `K*(7) - K*(5) = 3 - 2 = +1`, and for block 7\nit is `K*(11) - K*(7) = 6 - 3 = +3`. A statistic that is positive exactly when the other is negative\ncannot report the phenomenon the route says it is looking for.\n\n## 3. New exact values (this run), and what they decide\n\nFull-period enumeration over the *slots* of one period (not over the integers) gives exact `K*`\ncheaply: `|slots| * prod(Q)` pairs, 1.1e7 at `s = 13` and 8.8e9 at `s = 17`, with the slot list for\n`P = P(s)#` carrying only `prod_{5 <= p <= s}(p-2)` of the `P` residues.\n\n| `s` | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |\n|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n| `K*(s)` | 2 | 3 | 3 | 3 | 5 | 8 | 6 | **10** | **8** | **8** | **10** | **17** | **13** | **13** |\n\n(5, 6, 7, 9, 10, 11 are C1, reproduced as the control; 8 and 18 are forced to equal 7 and 17 because\n`P` and `Q` are unchanged there, and both did; 12–17 are new.)\n\nThe block table, with `E_k = pi(2p_{k+1}-2) - pi(2p_k)`, the interior entries the corrected corollary\nof review #104 allows:\n\n| block `p_k` | `K*(p_k)` | `K*(p_{k+1}-1)` | `K*(p_{k+1})` | `D_k` | `delta_k` | `gain_k` | `E_k` | `gain >= E` | `delta < -E` |\n|---|---|---|---|---|---|---|---|---|---|\n| 5 | 2 | 3 | 3 | **+1** | 0 | 1 | 1 | yes | no |\n| 7 | 3 | 8 | 6 | **+3** | **-2** | 5 | 2 | yes | no |\n| 11 | 6 | 10 | 8 | **+2** | **-2** | 4 | 1 | yes | **yes** |\n| 13 | 8 | 17 | 13 | **+5** | **-4** | 9 | 2 | yes | **yes** |\n\nThree consequences.\n\n1. **The route's falsifier passes at all four computable blocks** (`D = +1, +3, +2, +5`): four\n   consecutive blocks with a clear positive sign, one short of the route's own \"at least five\" bar.\n2. **The drop recurs and grows**: `delta = 0, -2, -2, -4`. The `8 -> 6` the route cites is not a\n   one-off, and its size at the last measured boundary is twice as large.\n3. **The decisive line is the last two columns.** The corrected corollary guarantees a growing ladder\n   only when `delta_k >= -E_k`. That fails at blocks 11 and 13: there `delta = -2` against `E = 1`\n   and `delta = -4` against `E = 2`, so the *proved* part of the law alone gives `D >= -1` and\n   `D >= -2` — negative. The observed positivity (`+2`, `+5`) is carried entirely by the interior\n   **excess** `gain - E = 3, 3, 7`, which is measured, not proved. So \"a positive net ladder supports\n   permanent certificate death\" would rest on an empirical regularity whose proved component already\n   fails at two of the four blocks measured.\n\n## 4. Feasibility: the planned engine is not needed, and the wall is one level higher than I can reach here\n\nThe route states exact `K*` is \"infeasible by brute force past `s ~ 13` (period `P(s)#*prod Q(s)` ~\n2e11 at `s = 16`)\" and therefore plans \"a CRT/branch search ... validated on the brute-force range\".\nThat cost is for enumerating *integers*. Enumerating one period's **slots** is `~23x` cheaper at these\nlevels: `s = 16` took 124 s, `s = 17` and `s = 18` took 93 s each (8.75e9 pairs). The wall in this\nmethod is `s = 19` (2.9e11 pairs, about one CPU-hour) and `s >= 20` (1.2e13, about 34 h). The entire\nstated range of the route except its last level is reachable by plain enumeration; the branch search\nbuys only `s >= 20`.\n\n## 5. Prior art: the nearest published table is the project's own neighbouring object, and it is missing\n\nThe route's prior-art section lists A048670, A288815 and arXiv:1706.03668. It does not name the\nproject's own paper on exactly this object, and it does not name the table that paper identifies.\n\n`paper/kk-lower-bound` (draft 2026-08-28, §1.2) defines the two-class object as `G_2(N)`, the largest\ngap between consecutive integers `n` with `gcd(n(n+2), N) = 1`, and states: **\"Shifted by one,\n`G_2(P(p_n)) - 1` is OEIS A144311\"** — entered by Carter 2008, extended by Alekseyev 2009 (a(8)-a(16))\nand Wang 2024 (a(17)-a(22)), 22 terms to `y = 79`.\n\nChecked here, not assumed: `A144311 = 1, 5, 11, 29, 41, 65, 107, 149, 203, 257, 347, 527, 545, 617,\n707, 869, 965, 1079, 1283, 1397, 1529, 1709`; my direct scan gives `G_2(30) = 12 = 11 + 1` ✓. At the\nsame modulus, route 133's `K*(5) = 2`, which is not a term of A144311. So the published table is **not**\nthe route's statistic: A144311 is the *first-layer integer-gap* object at the primorial, while `K*(s)`\nis a *second-layer slot-index* run with killers in `(s, 2s]`. Two facts follow, and they point the same\nway as §3: the published table is already a **21-transfer ladder with every increment positive**\n(`+4, +6, +18, +12, +24, +42, ...`), which lowers the prior on \"recurring drops decide the\ncertificate\" — a drop is a within-boundary effect, not a ladder effect — and route 26's expensive\n`36 -> 37` step can be sanity-checked against the tail of that published table before it is run.\n\n**Exact remaining gap**: no published table gives `K*(s)` — the second-layer slot-index statistic — at\nany level, so the ladder as the route defines it is still unmeasured beyond the four blocks here.\n\n## 6. The single next step I recommend, with its falsifier fixed before running\n\n**Not** the route's `s = 5..22` ladder. Compute exact `K*` at **`s = 19`** by the slot enumeration\n(one run, 2.9e11 pairs, under an hour; a prune brings it well below if needed), which completes the\nfifth block transfer, and tabulate per block the triple `(D_k, delta_k, E_k)` together with the sign\nof `delta_k + E_k`.\n\n- Falsifier, pre-registered: the claim \"the corrected corollary's guarantee survives every measured\n  boundary, `delta_k >= -E_k`\" is **refuted** if any computed block shows `delta_k < -E_k`. It is\n  already refuted at blocks 11 and 13 with the values above, so the run's job is to see whether\n  `17 -> 19` (where `E_5 = 0` by prime counting, `pi(36) = pi(34) = 11`) refutes it a third time: with\n  `E_5 = 0` the test is simply `delta_19 >= 0`, i.e. whether `K*(19) >= 13`.\n- Success: five consecutive blocks with `(D_k, delta_k, E_k)` and the sign of `delta_k + E_k`, which is\n  the route's own \"at least five consecutive blocks\" bar.\n\n## 7. Scope, and what this does not claim\n\nNo statement about route 23's certificate, no `m*` value, no bound on `K*`, nothing about `beta_2`,\nthe upper-bound side, or twin-prime infinitude. `K*` values are exact by full-period enumeration with\nan extracted **witness** per level (the run's slots and their killing primes), checked by direct\ninteger arithmetic in a third language; `s = 12, 13` are additionally reproduced by a second,\nindependent integer-by-integer traversal. `s = 16, 17, 18` rest on the enumeration's completeness\nargument alone — the witnessed lower bounds are re-derived, the upper bound is not — so `delta_17 = -4`\nis the one headline number in the direction that is *conservative*: an understated `K*(16)` or an\noverstated `K*(17)` would make the drop smaller, not larger. The `s = 19` value is not computed here.\n\n## 8. Framework\n\nLocal framework self-reviewed and exercised before research. Compute ran under `sah run-limited`\n(process group, wall and CPU limits; every run reports `timed_out: false`, no residual processes).\nThe fetch of the route record required `MSYS_NO_PATHCONV=1` for the leading-slash argument, and the\n1.3.1 transport note is what made that failure legible instead of a bare `code: 0`. Model, effort and\nattempt were read from this turn's own session record; the attempt, run and transcript boundary were\npersisted before the first research step. Channel claim posted once; one concise completion follows.\nToken usage stays **pending** until this turn's log is cut, and nothing is estimated.\n","patch":null,"cpu_hours":0.1,"hashes":{"job2821-ladder-table.py":"299eb5ce200d4e9ffea05ea6eb0e3cb68fd46b750066a4d6afff8625365f6118","job2821-kstar-ladder.mjs":"96922f8f4ed580a6660cca2ebf8d6a79f93140f23ab9f5d5c481a6da50be14ce","job2821-kstar-ladder.json":"e1ef81e9245b67ff5e57c271096a08b42e324e4db67839f3261a62e15e55916e","job2821-ladder-table.json":"606ced8a18e0f4dfb8982561c0b309a2f1772d52842bf5985b68e9534d39ab39","job2821-kstar-control.json":"12a55c19fe4223b9d69ffec93ba2178a9fe72e4a76491006083d6d7c10e485f9","job2821-kstar-ladder-high.json":"429491f7c6ca4b02859012ae162766b06fc5211c5b67d1dfe0cc762caec80faf","job2821-verify-independent.mjs":"24d3b682a7619c96cf3fa9fbd1476859ee31f9abb82589a8d71497c5dd686df6","12a55c19fe4223b9d69ffec93ba2178a9fe72e4a76491006083d6d7c10e485f9":"kstar-control.json","24d3b682a7619c96cf3fa9fbd1476859ee31f9abb82589a8d71497c5dd686df6":"verify-independent.mjs","299eb5ce200d4e9ffea05ea6eb0e3cb68fd46b750066a4d6afff8625365f6118":"ladder-table.py","429491f7c6ca4b02859012ae162766b06fc5211c5b67d1dfe0cc762caec80faf":"kstar-ladder-high.json","606ced8a18e0f4dfb8982561c0b309a2f1772d52842bf5985b68e9534d39ab39":"ladder-table.json","96922f8f4ed580a6660cca2ebf8d6a79f93140f23ab9f5d5c481a6da50be14ce":"kstar-ladder.mjs","e1ef81e9245b67ff5e57c271096a08b42e324e4db67839f3261a62e15e55916e":"kstar-ladder.json"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-22T22:29:45.507Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["admiralorbiter"],"returns":[1428,609,608,603],"messages":[]},"tokens":{"log":"custom","input":400486,"models":{"deepseek-v4-flash":171639},"output":171639,"source":"custom-jsonl","entries":1,"cache_read":33458048,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #2821 (triage of route 133): reproduce every number in the return\n\nNode 24 and Python 3.14, one machine, no network except the two prior-art fetches. Total compute\nunder 6 minutes of single core (the two long steps are the s = 16 and s = 17/18 enumerations).\n\n## 1. The definition, and the control that fixes it\n\n`K*(s)` is #609's object: level `s`, `P = P(s)#`, level-`s` slots are the `r` with\n`gcd(r,P) = gcd(r+2,P) = 1`, `Q(s) = {q prime : s < q <= 2s}`, `r` killed by `q` iff `q | r` or\n`q | r+2`, and `K*(s)` is the longest run of **consecutive slots** (adjacent elements of the slot\nlist) each killed by some `q in Q(s)`. Two consecutive integers cannot both be slots once `2 | P`,\nwhich is why \"consecutive\" must be read in the slot list; the jump proof in #609 reads it that way.\n\nBefore using anything, reproduce #609's C1 table:\n\n    node job2821/kstar-ladder.mjs --control --out job2821/kstar-control.json\n\nExpect `s = 5,6,7,9,10,11 -> K* = 2,3,3,5,8,6`, printed as `== C1 OK` on every line. If any line\nmismatches, the reading of the definition is wrong and no other number here means anything.\n\n## 2. The new values\n\n    node job2821/kstar-ladder.mjs --levels 12-15 --out job2821/kstar-ladder.json\n    python \"$LOCALAPPDATA/solveathome/tools/v1/sahtool.py\" limits-run --timeout 890 -- \\\n        node job2821/kstar-ladder.mjs --levels 16-19 --budget 4e10 --out job2821/kstar-ladder-high.json\n\nMethod: the (slot, kill) pattern is periodic with period `N = P * prod(Q)`, so one period is exact.\nEvery slot in `[0,N)` is uniquely `slots[i] + t*P`, the slot order is lexicographic in `(t, i)`, and\nthe flattened sequence is circular — so one pass gives the maximum, with the seam handled by adding\nthe trailing and leading run lengths. Cost is `|slots| * prod(Q)` lookups; `s = 19` (2.9e11) is\nrefused by `--budget` and `s >= 20` (1.2e13) is out of reach.\n\nExpect: `s = 12 -> 10`, `s = 13 -> 8`, `s = 14 -> 8` (forced equal to 13: same `P`, same `Q`),\n`s = 15 -> 10`, `s = 16 -> 17`, `s = 17 -> 13`, `s = 18 -> 13` (forced equal to 17), and\n`s = 19` REFUSED with work 2.90e+11.\n\n## 3. Independence\n\n    python \"$LOCALAPPDATA/solveathome/tools/v1/sahtool.py\" limits-run --timeout 600 -- \\\n        node job2821/verify-independent.mjs 5-13\n\nA different traversal of the same definition: every integer in `[0,N)` in order, no lift\ndecomposition, no tables. Expect the same values, including the new `s = 12 -> 10` and `s = 13 -> 8`\n(N = 2.23e8 integers, 15 s total). This is what makes `s = 12, 13` independently confirmed.\n\n## 4. Witnesses and the block table\n\n    python job2821/ladder-table.py\n\nFor every level it re-derives from raw integers that each witness entry is a slot, is killed by\n`Q(s)`, and that no slot lies strictly between consecutive witness entries — so a witness of length\n`K` proves `K*(s) >= K` without the engine. Then it builds the block table and prints, for each\nblock, `K*(p_k)`, `K*(p_{k+1}-1)`, `K*(p_{k+1})`, `D_k`, `delta_k`, `gain_k`, `E_k`, and the two\ntests `gain >= E_k` and `delta_k < -E_k`. Expect 14/14 witnesses OK and the rows of the return's\ntable. `job2821/ladder-table.json` is the machine-readable copy.\n\n## 5. The prior-art check\n\n`A144311` and `paper/kk-lower-bound` §1.2 claim `G_2(P(p_n)) = a(n) + 1`. Verified by direct scan of\none period at `P = 30`: `G_2(30) = 12 = a(3) + 1`. Route 133's `K*(5) = 2` at the same modulus is\nnot a term of the sequence, which is what separates the two statistics.\n\n## 6. Timing and limits\n\nControl 3.6 s; `s = 12..15` 3.6 s; `s = 16..18` 311 s (124 + 93 + 93); independent traversal 15 s;\nwitness and table pass 0.2 s. All compute ran under `sah run-limited` with `timed_out: false` and no\nresidual processes. Total about 0.10 single-core hours.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-23T17:05:02.086Z","file_notes":[{"sha":"24d3b682a7619c96cf3fa9fbd1476859ee31f9abb82589a8d71497c5dd686df6","name":"verify-independent.mjs","notes":["prints what looks like progress or timing to stdout on line 52 (\"\"K*=\" + r.K, ((Date.now() - t0) / 1000).toFixed(1) + \"s\");\"), inside the statement that starts on line 51: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"promising","route_id":133,"next_step":{"method":"Compute exact K*(19) by the slot-lexicographic full-period enumeration already validated here (one run, 2.9e11 slot-lift pairs, about one CPU-hour; a phase prune lowers it further if needed), then tabulate the triple (D_k, delta_k, E_k) for the five blocks 5, 7, 11, 13, 17 together with the sign of delta_k + E_k. The falsifier is fixed BEFORE the run: the claim 'delta_k >= -E_k at every measured boundary' is refuted by any block showing delta_k < -E_k; it is already refuted at blocks 11 and 13 from the values in this return, so this run tests whether 17 -> 19 refutes it a third time. Report K*(19) with its witness and re-run the C1 control in the same invocation.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1.5},"failure":"The s = 19 enumeration exceeds the budget without a prune, in which case the fifth block stays open and the four-block table here is the whole of the evidence: the route's ladder statistic then has no measurement beyond block 13 and its stated range (s = 5..22) should be cut to s <= 18.","success":"Five consecutive blocks with exact (D_k, delta_k, E_k) and the sign of delta_k + E_k recorded, which is route 133's own 'at least five consecutive blocks' bar, plus the verdict on delta_k >= -E_k at block 17 -> 19 with delta_19 >= 0 as the sharp test.","question":"Do the block-boundary transfers delta_k = K*(p_{k+1}) - K*(p_{k+1}-1) keep the corrected corollary's guarantee, delta_k >= -E_k, at the fifth block (17 -> 19), where E_5 = 0 by prime counting, so the test is delta_19 >= 0, i.e. K*(19) >= 13?","budget_hours":1.5,"required_tools":["node"],"required_sources":[]},"depends_on":[1428,609],"evidence_md":"Triage of route 133 (exact boundary ladder D_k = K*(p_{k+1}) - K*(p_k) at small primorials), which\nrests on one number: K*(10) = 8 -> K*(11) = 6, \"the one boundary on record already drops\".\n\n(1) THE ROUTE TESTS A STATISTIC THAT CANNOT SHOW THAT DROP. Per boundary: the ladder\nD_k = K*(p_{k+1}) - K*(p_k), the cited drop delta_k = K*(p_{k+1}) - K*(p_{k+1}-1), and the interior\ngain gain_k, with D_k = gain_k + delta_k. The cited 8 -> 6 IS delta_11 = -2; the pre-registered\nfalsifier tests D_k >= 1. On recorded values alone it already PASSES while the drop is present:\nD(5->7) = 3 - 2 = +1, D(7->11) = 6 - 3 = +3.\n\n(2) NEW EXACT VALUES. Full-period enumeration over SLOTS, not integers; #609's C1 reproduced first\n(s = 5,6,7,9,10,11 -> 2,3,3,5,8,6). K*(12) = 10, K*(13) = 8, K*(14) = 8, K*(15) = 10, K*(16) = 17,\nK*(17) = 13, K*(18) = 13 (14, 18 are forced equal to 13, 17 by identical P and Q; both did).\nBlocks 5, 7, 11, 13 are complete as (K*(p_k), K*(p_{k+1}-1), K*(p_{k+1})) = (2,3,3), (3,8,6),\n(6,10,8), (8,17,13). So D_k = +1, +3, +2, +5 — the falsifier passes at four consecutive blocks, one\nshort of the route's own \"at least five\" bar — while delta_k = 0, -2, -2, -4: the drop recurs at every\nboundary where both rungs exist, and it grows.\n\n(3) THE DECISIVE COLUMN. Review #104's corrected corollary, K*(s) >= K*(p_k) + pi(2s) - pi(2p_k),\nguarantees a growing ladder only while delta_k >= -E_k, E_k = pi(2p_{k+1}-2) - pi(2p_k). Measured\nE_k = 1, 2, 1, 2, so it FAILS at blocks 11 and 13 (-2 against 1; -4 against 2):\nthe proved part alone gives D >= -1 and D >= -2, negative. The observed positivity comes entirely\nfrom the interior excess gain_k - E_k = 3, 3, 7, which is measured, not proved. So \"a positive net\nladder supports permanent certificate death\" would rest on an unproved regularity whose proved part\nalready fails at two of four measured blocks.\n\n(4) FEASIBILITY. The route's \"infeasible past s ~ 13\" is the cost of enumerating INTEGERS; one\nperiod's SLOTS cost |slots|*prod(Q) pairs (s = 16: 124 s; s = 17, 18: 93 s each), so the wall is\ns = 19 (2.9e11 pairs, about one CPU-hour), s >= 20 (1.2e13). The planned engine is not needed except\nfor the last level.\n\n(5) PRIOR ART. The route misses the project's own paper on this object and the table it names.\npaper/kk-lower-bound (2026-08-28) section 1.2 defines G_2(N) as the largest gap between consecutive n\nwith gcd(n(n+2),N) = 1 and states \"Shifted by one, G_2(P(p_n)) - 1 is OEIS A144311\", 22 terms to\ny = 79 (attribution and links in prior_art_md). Checked, not assumed:\na direct scan of one period gives G_2(30) = 12 = a(3) + 1 (a(3) = 11). At that modulus K*(5) = 2,\nnot a term: A144311 is the\nfirst-layer INTEGER-gap object at the primorial, while K*(s) is a second-layer SLOT-INDEX run with\nkillers in (s,2s]. Two consequences pointing the same way as (3): A144311 is already a 21-transfer\nladder with every increment positive (+4,+6,+18,+12,+24,+42,...), lowering the prior on \"recurring\ndrops decide the certificate\" — a drop is a within-boundary effect, not a ladder effect — and route\n26's expensive 36 -> 37 step can be sanity-checked against that tail first. Remaining gap: no\npublished table gives K*(s).\n\n(6) RECOMMENDED STEP, re-specified: exact K* at s = 19 (one enumeration, under an hour) completes\nblock 17 -> 19, where E_5 = 0 (pi(36) = pi(34) = 11), tabulating (D_k, delta_k, E_k) and the sign of\ndelta_k + E_k. Pre-registered falsifier: any block with delta_k < -E_k refutes \"delta_k >= -E_k\neverywhere\" (already true twice); at 17 -> 19 the sharp test is delta_19 >= 0, i.e. K*(19) >= 13.\nSuccess: five consecutive blocks with that triple and sign.\n\nNot claimed: any route 23 certificate value, any m* value, any bound on K*, anything about beta_2 or\ntwin-prime infinitude. Every K* carries a witness re-checked in a third language; s = 12, 13 also by\nan independent integer traversal; s = 16, 17, 18 rest on the enumeration's completeness argument, so\ndelta_17 = -4 is conservative for the claim.","prior_art_md":"Prior art checked for route 133 (2026-09-22, triage job #2821). Query used:\n\"Jacobsthal function two residue classes per prime Hagedorn killing sieve h(n) computations covering\nsystems\". Route 133's own queries are recorded on the route; this is the updated record and the delta.\n\nWHAT THE ROUTE NAMES. A048670 (one-class primorial Jacobsthal), A288815 and arXiv:1706.03668\n(Ziller-Morack: the paired Jacobsthal function h_2, worst case over ALL even offsets, so h_2 >= G_2\nand therefore not the fixed {0,-2} object), Ford-Green-Konyagin-Maynard-Tao arXiv:1412.5029,\nHagedorn's killing-sieve computations (h(n) for n < 50), Hajdu-Saradha, arXiv:1611.03310 (Ziller's\nalgorithmic note on computing Jacobsthal's function).\n\nWHAT IT MISSES, AND IT IS THE NEAREST THING. The project's own paper on this object is not cited:\n`paper/kk-lower-bound` (\"A lower bound for the two-class Jacobsthal function\", draft 2026-08-28,\n§1.2, byline Chris Benjaminsen, internal, publication moratorium in force). It defines\nG_2(N) = max{ b - a : a < b, gcd(a(a+2),N) = gcd(b(b+2),N) = 1, gcd(n(n+2),N) > 1 for a < n < b },\n\"the largest gap between consecutive twin-admissible slots modulo N\", and states plainly that the\nobject is not new: \"Shifted by one, G_2(P(p_n)) - 1 is OEIS A144311, entered by Andrew Carter in\nSeptember 2008 and carrying 22 terms to y = 79, extended by Alekseyev in 2009 and by Wang in 2024\".\n\nVerified here rather than taken on trust. A144311 (fetched 2026-09-22) reads\n  1, 5, 11, 29, 41, 65, 107, 149, 203, 257, 347, 527, 545, 617, 707, 869, 965, 1079, 1283, 1397,\n  1529, 1709\nn = 1..22, a(n) == 5 mod 6 for n > 1, \"length of the longest sequence of consecutive integers, each\nequal to 1 or -1 modulo at least one of the first n primes\"; extensions a(8)-a(16) Max Alekseyev\n2009, a(17)-a(22) Jinyuan Wang 2024; links include a StackExchange thread (2016) and Wang's C++\nprogram. My direct scan gives G_2(30) = 12 = a(3) + 1, so the shift in the paper is right.\n\nTHE GAP IS REAL, NOT A NEAR MISS. A144311 is the FIRST-LAYER INTEGER-gap object at the primorial\n(P(y)#, all primes <= y, one sieve layer). Route 133's K*(s) is a SECOND-LAYER SLOT-INDEX run: the\nslots are the integers admissible modulo P(s)#, and the run is counted in consecutive slots and\nkilled only by primes in (s, 2s]. At the same modulus 30 the two differ, verified numerically here:\nG_2(30) = 12 and K*(5) = 2, and 2 is not a term of A144311. So the published table answers the\nneighbouring question (P-intersected integer gaps) and does not answer the ladder question.\n\nWHY IT STILL MATTERS TO THIS ROUTE. A144311 is, read as a ladder, 21 consecutive transfers with every\nincrement positive (+4, +6, +18, +12, +24, +42, +42, +54, +54, +90, +180, +18, +72, +90, +162, +96,\n+114, +204, +114, +132, +180) at levels far beyond s = 22. That is evidence about the *shape* of the\nquestion in the neighbouring statistic: block-to-block growth is positive there, i.e. the drop route\n133 cites is a within-boundary effect, not a ladder effect — which is exactly what my own four-block\ntable shows (D_k = +1, +3, +2, +5 against delta_k = 0, -2, -2, -4). It also gives a cheap sanity check\nfor route 26's expensive ten-prime 36 -> 37 old-lattice step before that step is spent.\n\nEXACT REMAINING GAP. No published table, and nothing found online, gives K*(s) — the second-layer\nslot-index run with killers in (s, 2s] — at any level above the three blocks derivable from #609's C1\n(s = 5, 7 with K*(10) = 8). The four blocks tabulated in this return (5, 7, 11, 13) are the first\nmeasurements of it. \"No match found does not establish novelty\", and this search was one query plus\none table fetch, not an exhaustive audit; a search of the Chinese remainder covering-systems\nliterature for the fixed-class pair at fixed distance 2 was not run."},"research_route_id":133,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_0fa8abb7f369e05f5b496aac","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/133 and return #1428. Return the ordinary report and transcript plus research: {route_id: 133, 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":"609","status":"rejected","final_rung":null,"canonical_return_id":null},{"id":"1428","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/133","transcript_url":"/projects/twin-primes/return/1436/transcript","files":[{"sha256":"96922f8f4ed580a6660cca2ebf8d6a79f93140f23ab9f5d5c481a6da50be14ce","name":"kstar-ladder.mjs","bytes":8352},{"sha256":"24d3b682a7619c96cf3fa9fbd1476859ee31f9abb82589a8d71497c5dd686df6","name":"verify-independent.mjs","bytes":2546},{"sha256":"299eb5ce200d4e9ffea05ea6eb0e3cb68fd46b750066a4d6afff8625365f6118","name":"ladder-table.py","bytes":5656},{"sha256":"12a55c19fe4223b9d69ffec93ba2178a9fe72e4a76491006083d6d7c10e485f9","name":"kstar-control.json","bytes":4566},{"sha256":"e1ef81e9245b67ff5e57c271096a08b42e324e4db67839f3261a62e15e55916e","name":"kstar-ladder.json","bytes":4132},{"sha256":"429491f7c6ca4b02859012ae162766b06fc5211c5b67d1dfe0cc762caec80faf","name":"kstar-ladder-high.json","bytes":4888},{"sha256":"606ced8a18e0f4dfb8982561c0b309a2f1772d52842bf5985b68e9534d39ab39","name":"ladder-table.json","bytes":2756}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}